Quantitative Modeling and Analysis with FMC-QE

Transcription

1 Hasso Plattner Insttut für Softwaresystemtechnk an der Unverstät Potsdam Quanttatve Modelng and Analyss wth FMC-QE Dssertaton zur Erlangung des akademschen Grades Doktor der Ingeneurwssenschaften (Dr.-Ing. n der Wssenschaftsdszpln "Software Engneerng" engerecht an der Mathematsch Naturwssenschaftlchen Fakultät der Unverstät Potsdam von Stephan Kluth Wolfsburg, den

2 Ths work s lcensed under a Creatve Commons Lcense: Attrbuton Noncommercal Share Alke 3.0 Germany To vew a copy of ths lcense vst nc sa/3.0/de/ Publshed onlne at the Insttutonal Repostory of the Unversty of Potsdam: URL URN urn:nbn:de:kobv:517 opus resolvng.de/urn:nbn:de:kobv:517 opus 52987

3 Gutachter: Prof. Dr.-Ing. Werner Zorn Prof. Dr. Dr. h.c. Otto Spanol Prof. Dr. Paul Müller

4

5 Abstract The modelng and evaluaton calculus FMC-QE, the Fundamental Modelng Concepts for Quanttatve Evaluaton [143], extends the Fundamental Modelng Concepts (FMC for performance modelng and predcton. In ths new methodology, the herarchcal servce requests are n the man focus, because they are the orgn of every servce provsonng process. Smlar to physcs, these servce requests are a tuple of value and unt, whch enables herarchcal servce request transformatons at the herarchcal borders and therefore the herarchcal modelng. Through reducng the model complexty of the models by decomposng the system n dfferent herarchcal vews, the dstncton between operatonal and control states and the calculaton of the performance values on the assumpton of the steady state, FMC-QE has a scalable applcablty on complex systems. Accordng to FMC, the system s modeled n a 3-dmensonal herarchcal representaton space, where system performance parameters are descrbed n three arbtrarly fne-graned herarchcal bpartte dagrams. The herarchcal servce request structures are modeled n Entty Relatonshp Dagrams. The statc server structures, dvded nto logcal and real servers, are descrbed as Block Dagrams. The dynamc behavor and the control structures are specfed as Petr Nets, more precsely Colored Tme Augmented Petr Nets. From the structures and parameters of the performance model, a herarchcal set of equatons s derved. The calculaton of the performance values s done on the assumpton of statonary processes and s based on fundamental laws of the performance analyss: Lttle s Law and the Forced Traffc Flow Law. Lttle s Law s used wthn the dfferent herarchcal levels (horzontal and the Forced Traffc Flow Law s the key to the dependences among the herarchcal levels (vertcal. Ths calculaton s sutable for complex models and allows a fast (re-calculaton of dfferent performance scenaros n order to support development and confguraton decsons. Wthn the Research Group Zorn at the Hasso Plattner Insttute, the work s embedded n a broader research n the development of FMC-QE. Whle ths work s concentrated on the theoretcal background, descrpton and defnton of the methodology as well as the extenson and valdaton of the applcablty, other topcs are n the development of an FMC-QE modelng and evaluaton tool and the usage of FMC-QE n the desgn of an adaptve transport layer n order to fulfll Qualty of Servce and Servce Level Agreements n volatle servce based envronments. Especally the close collaboraton the development of the FMC-QE Tool promotes the valdaton of FMC-QE. Ths thess contans a state-of-the-art, the descrpton of FMC-QE as well as extensons of FMC- QE n representatve general models and case studes. In the state-of-the-art part of the thess n chapter 2, an overvew on exstng Queueng Theory and Tme Augmented Petr Net models and other quanttatve modelng and evaluaton languages and methodologes s gven. Also other herarchcal quanttatve modelng frameworks wll be consdered. The descrpton of FMC-QE n chapter 3 conssts of a summary of the foundatons of FMC-QE, basc defntons, the graphcal notatons, the FMC-QE Calculus and the modelng of open queueng networks

6 ABSTRACT as an ntroductory example. The extensons of FMC-QE n chapter 4 consst of the ntegraton of the summaton method n order to support the handlng of closed networks and the modelng of multclass and semaphore scenaros. Furthermore, FMC-QE s compared to other performance modelng and evaluaton approaches. In the case study part n chapter 5, proofof-concept examples, lke the modelng of a servce based search portal, a servce based SAP NetWeaver applcaton and the Axs2 Web servce framework wll be provded. Fnally, conclusons are gven by a summary of contrbutons and an outlook on future work n chapter 6.

7 Zusammenfassung FMC-QE (Fundamental Modelng Concepts for Quanttatve Evaluaton [143] st ene auf FMC, den Fundamental Modelng Concepts, baserende Methodk zur Modellerung des Lestungsverhaltens von Systemen mt enem dazugehörenden Kalkül zur Erstellung von Lestungsvorhersagen we Antwortzeten und Durchsatz. In deser neuen Methodk steht de Modellerung der herarchschen Bedenanforderungen m Mttelpunkt, da se der Ursprung aller denstbaserenden Systeme snd. We n der Physk snd n FMC-QE de Bedenanforderungen Tupel aus Wert und Enhet, um Auftragstransformatonen an Herarchegrenzen zu ermöglchen. Da de Komplextät durch ene Dekomposton n mehreren Schten und n verschedene herarchsche Schchten, de Unterschedung von Operatons- und Kontrollzuständen, sowe dazugehörge Berechungen unter Annahme der Statonartät reduzert wrd, skalert de Anwendbarket von FMC-QE auf komplexe Systeme. Gemäß FMC wrd das zu modellerende System n enem 3-dmensonalen herarchschen Beschrebungsraum dargestellt. De quanttatven Kenngrößen der Systeme werden n dre belebg fre-granularen herarchschen b-partten Graphen beschreben. De herarchsche Struktur der Bedenanforderungen wrd n Entty Relatonshp Dagrammen beschreben. De statschen Bedenerstrukturen, untertelt n logsche und reale Bedener, snd n Aufbaudagrammen erläutert. Außerdem werden Petr Netze, genauer Farbge Zet-behaftete Petr Netze, dazu verwendet, de dynamschen Abläufe, sowe de Kontrollflüsse m System zu beschreben. Anschleßend wrd ene Menge von herarchschen Glechungen von der Struktur und den Parametern des Modells abgeletet. Dese Glechungen, de auf dem statonären Zustand des Systems beruhen, baseren auf den beden Fundamental Gesetzen der Lestungsanalyse, dem Gesetz von Lttle und dem Verkehrsflussgesetz. Das Gesetz von Lttle defnert herbe Bezehungen nnerhalb ener herarchschen Schcht (horzontal und das Verkehrsflussgesetz wederum Bezehungen zwschen herarchschen Schchten (vertkal. De Berechungen erlauben Lestungsvorhersagen für komplexe Systeme durch ene effzente (Neu-Berechnung von Lestungsgrößen für ene große Auswahl von System- und Lastkonfguratonen. Innerhalb der Forschungsgruppe von Prof. Dr.-Ing Werner Zorn am Hasso Plattner Insttut an der Unverstät Potsdam st de vorlegende Arbet n enen größeren Forschungskontext m Berech FMC-QE engebettet. Während her en Fokus auf dem theoretschen Hntergrund, der Beschrebung und der Defnton der Methodk als auch der Anwendbarket und Erweterung gelegt wurde, snd andere Arbeten auf dem Gebet der Entwcklung ener Anwendung zur Modellerung und Evaluerung von Systemen mt FMC-QE bzw. der Verwendung von FMC- QE zur Entwcklung ener adaptven Transportschcht zur Enhaltung von Denstgüten (Qualty of Servce und Denstverenbarungen (Servce Level Agreements n volatlen denstbaserten Systemen behematet. Spezell de Kooperaton m Berech der Anwendungsentwcklung führte de Entwcklung von FMC-QE n deser Arbet m Berech Valderung voran. Dese Arbet umfasst enen Enblck n den Stand der Technk, de Beschrebung von FMC-QE sowe de Weterentwcklung von FMC-QE n repräsentatven allgemenen Modellen und Fall

8 ZUSAMMENFASSUNG studen. Das Kaptel 2: Stand der Technk gbt enen Überblck über de Warteschlangentheore, Zet-behaftete Petr Netze, wetere Lestungsbeschrebungs- und Lestungsvorhersagungstechnken sowe de Verwendung von Herarchen n Lestungsbeschrebungstechnken. De Beschrebung von FMC-QE n Kaptel 3 enthält de Erläuterung der Grundlagen von FMC-QE, de Beschrebung enger Grundannahmen, der graphschen Notaton, dem mathematschen Modell und enem erläuternden Bespel. In Kaptel 4: Erweterungen von FMC-QE wrd de Behandlung weterer allgemener Modelle, we de Modellklasse von geschlossenen Netzen, Synchronserung und Mehrklassen-Modelle beschreben. Außerdem wrd FMC-QE mt dem Stand der Technk verglchen. In Kaptel 5 werden Machbarketsstuden, we de Modellerung enes denstbaserten Suchportals, ener denst-baserten SAP-NetWeaver Anwendung und des Axs2 Web Servce Frameworks, beschreben. Schleßlch werden n Kaptel 6 ene Zusammenfassung und en Ausblck gegeben. v

9 Acknowledgements Frst of all, I would lke to gratefully thank Prof. Dr.-Ing. Werner Zorn for the opportunty to work n hs research group. Through hs deas of FMC-QE [ ] and a lot of frutful dscussons and support, ths thess was possble. I would also lke to thank the other revewers for the tme and effort they spent to supervse ths thess. Addtonally, I would lke to thank the HPI Research School for the support and the constant pressure especally through the bannually retreats wth t s demandng reports and presentatons. A specal thank n ths area goes to Prof. Dr. Andreas Polze, the head of the research school. I would also thank the other members of the Research School for the cooperatons and dscussons. Especally I would lke to thank Mchael Schöbel for dscussons on mathematcal questons. Another thank goes to my colleagues Tomasz Porzucek, Flavus Copacu, Ram-Habb Ed- Sabbagh, Domnc Wst, Dr.-Ing. Ralf Wollowsk, Sebastan Kuhle, Nanjun L and Raveendra Babu Dars n the research group for ther support as well as the FMC research group especally Dr.-Ing. Peter Tabelng and the Operatng Systems and Mddleware research group especally Dr. Martn von Löws. I would also lke to thank the co-authors of my papers for ther cooperaton. Here I especally would lke to thank Marcel Seelg and Mathas Frtzsche. Another thank goes to the admnstraton of the Hasso Plattner Insttute and Hasso Plattner hmself for provdng the envronment for the research. Here a specal thank goes to Prof. Dr. Chrstoph Menel, Annett Sedler, Sabne Wagner, Ilona Pampern, Ralf Gruner and Jens Luef. Last but not least I would lke to thank my parents, famly and frends for ther support. Especally I would lke to thank Brtta Knüppel for the careful proofreadng of my thess. v

10

11 Contents Gutachter Abstract Zusammenfassung Acknowledgements Contents Lst of Fgures Lst of Tables Revewer v v x xv 1 Introducton 1 2 State of the Art Queueng Theory Quanttatve Measures Fundamental Laws Server Performance Values Open Queueng Networks - Jackson s Theorem Closed Queueng Networks - Gordon-Newell Theorem Mxed Open and Closed Queueng Networks - BCMP Theorem An Algorthm for Product Form Networks - Mean Value Analyss (MVA Tme Augmented Petr Nets Classfcaton Contnuous Tme Stochastc Petr Nets (SPN Generalzed Stochastc Petr Nets (GSPN Product Form Petr Nets v

12 CONTENTS Queueng Petr Nets (QPN Quanttatve Herarchcal Modelng Decomposablty Norton s Theorem Formal Herarches and Combnaton of Models Herarches n Tme Augmented Petr Nets Forced Traffc Flow Law Layered Queueng Networks (LQN Summary FMC-QE Fundamentals Foundatons Fundamental Modelng Concepts (FMC FMC-eCS Basc Defntons Servce Request Herarchcal Servce Requests Quanttatve Measures n FMC-QE Graphcal Representaton Servce Request Structures Statc Structures Dynamc Structures Calculus Fundamental Laws Expermental Parameters Servce Request Secton Server Secton Dynamc Evaluaton Secton Multplexer Secton Computaton Algorthm / Complexty Analyss FMC-QE Example - Open Queueng Network Orgnal Model and Calculaton Transformaton Servce Request Structure and Statc Structure Summary FMC-QE Tool v

13 CONTENTS 4 FMC-QE Extensons Closed Queueng Networks General Dscusson Closed Tandem Network Central Server Network Summary Handlng of Multclass Scenaros Semaphore Synchronzaton GSPN Model FMC-QE Model Summary Comparsons Queueng Theory Tme Augmented Petr Nets Layered Queueng Networks (LQN Performance Smulatons FMC-QE Case Studes HPI Search Portal - a Servce based Case Study Archtecture FMC-QE Model Summary Modelng of a Servce based System: ERMF Introducton Servce Request Structure and Dynamc Behavor Measurements Analyss Smulaton Summary Modelng of nteractng herarchcal Protocol Stacks - Axs Axs2 Web Servces Framework Axs2 Model Testbed Descrpton FMC-QE Tableau Summary x

14 CONTENTS 6 Conclusons 185 Publcatons 189 Bblography 193 Glossary 205 Index 209 A Server Performance Values 213 B Tables 225 C Fgures 239 x

15 Lst of Fgures 2.1 Sample Queueng Net Performance Parameters and Values Lttle s Law Sample Open Network - Jackson s Theorem Sample Closed Network - Gordon Newell Sample Open Network - BCMP Sample Closed Network - MVA Range of Steady State Varables SPN Example SPN Example - Reachablty Graph State Transton Rate Dagram of the SPN Example GSPN Example GSPN Example - Reachablty Graph Queueng Place and Queueng Place Shorthand Notaton QPN Example Herarches Norton s Theorem Norton s Theorem for Queueng Networks - Example Communcatng Tme Petr Nets Example - Modules Communcatng Tme Petr Nets Example - Composton Herarchcally combned Queueng Petr Nets (HQPN - Example Memory Constraned System LQN Example - Fle Server Applcaton - LQN LQN Actvty Graph Example - Quorum Consensus LQN Example - Fle Server Applcaton - Sequence Dagram FMC Block Dagram - Example x

16 LIST OF FIGURES 3.2 FMC Petr Net - Example FMC Entty Relatonshp Dagram - Example Crtcal Secton Crtcal Actonfeld Types of Crtcal Sectons Jont Acton - Clent/Server Jont Acton - Producer/Consumer Jont Acton - Short Notatons Unrelable Servce Checked Servce Transactonal Servce Servce Request Entty Entty Relatonshp Dagram Tree Metastructure Traffc Flow Coeffcent v External Servce Request Generator Barbershop - Servce Request Structure Basc Server Staton Queueng Staton Infnte Server Herarchcal Server Staton Herarchcal Server Staton - Short Notaton Multplexer Server Mappng between Logcal and Multplexer Servers Barbershop - Statc Structures Controlled Operatonal Transton Dynamc Behavor of a Queueng Staton Parallel Server - Actvty Refned Infnte Queues Infnte Server Herarchcal Transton Branch Parallel Actvtes Seral Actvtes Most Smple FMC-QE Petr Net Parallelsm on Logcal Server Level - Threads x

17 LIST OF FIGURES 3.37 Barbershop - Dynamc Behavor Basc FMC-QE Model Steady State Systems - Notaton Steady State Systems - Paternoster Steady State Systems - Soup Ktchen Steady State Systems - Roller coaster Chart External Servce Tme Multplexer/Demultplexer Multplex Example Multplex - Servce Requestor s Vew Parallel Server Infnte Server Herarchcal Actvtes Seral Actvtes Parallel Actvtes Branch Orgnal Whle Loop Whle Loop Transformaton (Seralzaton Whle Loop Transformaton (Combnaton Feed Backward Loop Feed Forward Open Queueng Example - Orgnal Model Open Queueng Example - Intal Petr Net Open Queueng Example - Load Generaton Open Queueng Example - Persst Data Branch Open Queueng Example - Unrelable Executon Open Queueng Example - Feed Forward - Feed Backward Open Queueng Example - Transformed Petr net Open Queueng Example - Servce Request Structures Open Queueng Example - Server Structures Open Queueng Example - Chart: Response Tme - Arrval Rate Open Queueng Example - Chart: External Servce Tme - Populaton Open Queueng Example - Chart: Utlzaton, Response Tme - Arrval Rate FMC-QE Tool - Screenshot Closed Tandem Network - Orgnal Model x

18 LIST OF FIGURES 4.2 Closed Tandem Network - State Transton Dagram Closed Tandem Network - Servce Request Structure Closed Tandem Network - Server Structure Closed Tandem Network - Dynamc Behavor Closed Tandem Network - M/M/1 - Chart: Adjustment External Servce Tme - f Closed Tandem Network - Summaton Method Chart: n System - f Central Server - Orgnal Model Central Server - Orgnal Model Reengneered Central Server - FMC-QE Model (Servce Request Structure Central Server - FMC-QE Model (Server Structure Central Server - FMC-QE Model (Dynamc Behavor Central Server - Chart: Throughput - Populaton Multclass Example - Class A - Servce Request Structures Multclass Example - Class A - Dynamc Behavor Multclass Example - Class B - Servce Request Structures Multclass Example - Class B - Dynamc Behavor Multclass Example - Statc Structures Multclass Example - Chart: Response Tmes A, B - Arrval Rate A Multclass Example - Chart: Response Tmes A, B - Servce Tme Req. A.2 Srv Semaphore Synchronzaton - GSPN Model Semaphore Synchronzaton - Reachablty Graph Semaphore Sync. wth Inter-Server Control Flows (Statc Structures Semaphore Sync. wth Inter-Server Control Flows (Dynamc Structures Semaphore Sync. wthout Inter-Server Control Flows (Statc Structures Semaphore Sync. wthout Inter-Server Control Flows (Dynamc Structures Semaphore Synchronzaton - Chart: Throughput - Crtcal Acton 2 Servce Tme Range of Steady State Varables and Methods Fork-Jon Queueng Model SM/M/1 Queue HPI Search Portal - Archtecture HPI Search Portal - Servce Request Structure HPI Search Portal - Behavor HPI Search Portal - Chart: Overall Response Tme - Arrval Rate HPI Search Portal - Chart: External Servce Tme - Arrval Rate xv

19 LIST OF FIGURES 5.6 HPI Search Portal - Chart: Response Tme - Number of Parallel Man Processors ERMF - Statc Structure ERMF - Servce Request Structure ERMF - Dynamc Behavor ERMF - Chart: Traffc Servce - Response Tme ERMF - Chart: Weather Servce - Response Tme ERMF - Chart: Result Comparson Axs2 Based System - Block Dagram Axs2 Dynamc Behavor - Petr Net Axs2 Herarchcal Servce Request Structure - Entty Relatonshp Dagram Axs2 Input Flow - Petr Net Axs2 Output Flow - Petr Net Axs2 - Optonal Handlers Axs2 - Chart: Response Tme - Arrval Rate Axs2 - Chart: External Servce Tme - Populaton Axs2 - Chart: Response Tme - Number of CPUs Axs2 - Dynamc - All C.1 HPI Search Portal - Archtecture C.2 HPI Search Portal - Servce Request Structure C.3 HPI Search Portal - Behavor C.4 Axs2 - Dynamc - All xv

20

21 Lst of Tables 2.1 SPN Example - Parameters GSPN Example - Parameters Operatonal vs. Control Varables Tableau Example Tableau Example - Expermental Parameters Tableau Example - Servce Request Secton Tableau Example - Server Secton Tableau Example - Dynamc Evaluaton Secton Tableau Example - Multplexer Secton Open Queueng Example - Tableau Closed Tandem Network - M/M/1 Tableau Closed Tandem Network - M/M/1/K Tableau Closed Tandem Network - Summaton Method Tableau Central Server - Orgnal Parameters Central Server - Tableau Multclass Example - Tableau Semaphore Synchronzaton - Parameters Semaphore Synchronzaton - Tableau Comparson of Modelng Aspects Comparson of LQNS to FMC-QE and other Layered Queueng Systems HPI Search Portal - Tableau ERMF - Tableau Axs2 - Tableau A.1 D/D/ A.2 D/D/m xv

22 LIST OF TABLES A.3 M/M/ A.4 M/M/m A.5 M/M/ A.6 M/M/1/K A.7 M/M/m/K A.8 M/M/m/K/M A.9 M/M/m/m A.10 Server Performance Values - Overvew B.1 Tableau Example B.2 Open Queueng Example - Tableau B.3 Closed Tandem Network - M/M/1 Tableau B.4 Closed Tandem Network - M/M/1/K Tableau B.5 Closed Tandem Network - Summaton Method Tableau B.6 Closed Tandem Network - Tableaux - Comparson B.7 Closed Central Server Example - Tableau B.8 Semaphore Synchronzaton - Tableau B.9 Multclass Example - Tableau B.10 ERMF - Tableau B.11 Axs2 - Tableau B.12 HPI Search Portal - Tableau xv

23 Chapter 1 Introducton In the area of mathematcal-analytc performance modelng and analyss there are essentally two methodcal approaches: Queueng Theory and Tme Augmented Petr Nets, whereas both approaches have ther assets and drawbacks. The advantages of the Queueng Theory are the matured analyss technques and the good computablty. A dsadvantage s lmted power n modelng of complex systems, as for example systems are only modeled from the perspectve of the server structures and therefore control flows are often neglected. Tme Augmented Petr Nets are more powerful n concerns of modelng concurrent processes and control flows. But here, wth rsng complexty, the problem of state space exploson arses, whch could set a border for the practcal applcablty. As n Queueng Theory, n Tme Augmented Petr Nets only one vew of the system, here the dynamc vew, ncludng the control flows, s modeled. The server structures behnd are often neglected, whch could cause problems, when shared resources or specal queueng or schedulng strateges should be consdered. The modelng and evaluaton calculus FMC-QE, the Fundamental Modelng Concepts for Quanttatve Evaluaton, extends the Fundamental Modelng Concepts (FMC by aspects of performance modelng and performance predcton. In ths new methodology the herarchcal servce requests are n the man focus, because they are the orgn of every servce provsonng process. Smlar to physcs, the servce requests are a tuple of value and unt, n order to provde a servce request transformaton at the borders of the dfferent herarchcal levels. Through reducng the complexty of the models by decomposng the system n dfferent herarchcal vews, the dstncton between operatonal and control states and the calculaton of the performance values on the assumpton of the steady state, FMC-QE has a scalable applcablty on complex systems. Accordng to FMC, the modeled system s represented n a 3-dmensonal herarchcal representaton space. The system performance parameters are descrbed n three arbtrarly fne-graned herarchcal bpartte dagrams. The herarchcal servce request structures are modeled n Entty Relatonshp Dagrams. The statc server structures, dvded nto logcal and multplexer servers, are descrbed n Block Dagrams. The dynamc behavor and the control structures are specfed n Petr Nets, more precsely Colored Tme Augmented Petr Nets. From the structures and parameters of the performance model, a herarchcal set of equatons s derved. The calculaton of the performance values n ths herarchcal set of equatons s done on the assumpton of statonary processes and s based on fundamental laws of the performance analyss: Lttle s Law and the Forced Traffc Flow Law. Lttle s Law s used wthn the dfferent herarchcal levels (horzontal and the Forced Traffc Flow Law s the key to the connectons among the herarchcal levels (vertcal. Ths calculaton s sutable for complex models and 1

24 CHAPTER 1. INTRODUCTION allows a fast calculaton of dfferent performance scenaros n order to support system development and confguraton decsons. In ths thess FMC-QE s descrbed, extended and appled to dfferent representatve examples as well as compared to other exstng performance modelng and evaluaton frameworks. The thess s structured as follows: In chapter 2 the State of the Art of performance modelng and evaluaton from the vewpont of FMC-QE s explaned. Ths ncludes an overvew on the Queueng Theory and an explanaton of quanttatve measurements, fundamental laws and formulas as well as algorthms for the predcton of performance values for queueng servers and queueng networks n the Queueng Theory. Ths chapter gves also a resume on Tme Augmented Petr Nets, descrbes some classfcatons and outlnes mportant types of Tme Augmented Petr Nets. Whle herarches and the herarchcal modelng are the key to complexty, other herarchcal quanttatve modelng and aggregaton methods are also dscussed. Ths ncludes decomposablty, the adapton of Norton s Theorem to Queueng Theory, formal herarches, aggregaton and herarches n Tme Augmented Petr Nets and the Forced Traffc Flow Law. In addton to ths, Layered Queueng Networks (LQN are summarzed, as a specal quanttatve herarchcal modelng methodology. The man concepts of FMC-QE are descrbed n chapter 3. Ths starts wth a descrpton of the bass of FMC-QE, the Fundamental Modelng Concepts (FMC and the Fundamental Modelng Concepts extended for Communcaton Systems (FMC-eCS. After that, basc defntons of the servce request, the herarchcal modelng as well as performance parameters and values are gven. Then, the 3-dmensonal graphcal representatons dvded n servce request structures, statc (server structures as well as dynamc behavor and control flow are descrbed. The mathematcal calculus and the computaton of the performance predctons n the Tableau are also specfed. An Open Queueng Network s modeled as an example for the modelng and transformatons as well as the calculatons of FMC-QE. The development of an FMC-QE Tool for the quanttatve modelng and evaluaton s n the focus of another PhD. student (Tomasz Porzucek n the research group and s shortly referenced n ths chapter. In chapter 4 FMC-QE s used to model and evaluate selected types of problems, to provde extensons to the core methodology n order to to solve these problems. In the frst part sample Closed Queueng Networks are modeled and the correspondng performance values are derved. Ths ncludes an ntegraton of the summaton method [17] nto FMC-QE n order to extend the range of applcatons to closed models through the summaton approxmaton method. The handlng of multclass scenaros s also descrbed n ths chapter. Then, the semaphore synchronzaton, a classcal Tme Augmented Petr Net problem, s addressed. Ths part descrbes and compares an approxmaton for the performance predcton of semaphore synchronzaton problems. Furthermore, FMC-QE s compared to other performance modelng and predcton approaches. Chapter 5 provdes case studes, modeled and evaluated. The HPI Search Portal llustrates the applcablty to larger systems. The second example, the modelng of a servce based system nsde an SAP NetWeaver / SAP WebAS envronment 1, focuses on the comparson of the performance predctons of a performance model, a smulaton and the correspondng measured values n the real system. The thrd example, the modelng of nteractng herarchcal Proto SAP AG, SAP NetWeaver, Webste: August 2

25 col Stacks n Apache Axs2 2, s a Proof-of-Concept for the herarchcal modelng wth the key aspects of multplex and synchronzaton n the calculatons. Chapter 6 summarzes the man contrbutons and the author proposes possble future work. 2 Apache Software Foundaton, Apache Axs2 Archtecture Gude, Webste: Axs2ArchtectureGude.html, August

26

27 Chapter 2 State of the Art The followng chapter descrbes the State of the Art of performance modelng and performance predctons from the vewpont of FMC-QE. Ths chapter s gven to provde an overvew of other quanttatve modelng and evaluaton approaches and to form the bass for further chapters. The chapter s structured as follows: Secton 2.1 provdes an overvew of the man mathematcal background of FMC-QE, the Queueng Theory, especally the fundamental laws Lttle s Law and the Forced Traffc Flow Law, the calculaton of the dfferent servers and algorthms and theorems for the calculaton of the performance values of Queueng Networks. In addton to the Petr Nets used n FMC to model the dynamc behavor of systems, the Tme Augmented Petr Nets nfluenced the development of the FMC-QE Petr Nets and are therefore descrbed n secton 2.2. The herarchcal modelng s the key to complex systems and therefore of central concern n FMC-QE. Secton 2.3 gves an overvew of other quanttatve herarchcal and aggregaton modelng technques. As a specal herarchcal quanttatve modelng technque, Layered Queueng Networks are also brefly explaned n ths secton. 5

28 CHAPTER 2. STATE OF THE ART 2.1 Queueng Theory The Queueng Theory forms the man mathematcal bass for FMC-QE. Therefore ths secton provdes an overvew of the most mportant measures, laws, theorems and algorthms. Important quanttatve measures are defned n subsecton After that, two fundamental laws of the Queueng Theory, Lttle s Law [98] and the Forced Traffc Flow Law [43, 95] are ntroduced. In order to defne and calculate the quanttatve measures of sngle statons or servers n a Queueng Network, the Kendall Notaton for server statons [83] and equlbrums for general systems are descrbed n Ths s further enrched by formulas for dfferent specal types of queueng statons n the appendx A. In the sectons to 2.1.7, mportant theorems and algorthms for the estmaton of performance values for whole Queueng Networks are explaned. Ths ncludes Jackson s Theorem for Open Queueng Networks [77, 78], the Theorem of Gordon and Newell [64] for Closed Queueng Networks, the networks of Baskett, Chandy, Muntz and Palacos (BCMP-Networks [8] and the Mean Value Analyss (MVA [118]. 1 p 1,1 =0,1 CPU 2 3 Dsk1 p 1,2 =0,4 p 1,3=0,5 Dsk2 Fgure 2.1: Sample Queueng Net [68] Fgure 2.1 provdes a small Queueng network example n order to gve an mpresson of the way of modelng n Queueng Theory as a graph of servers, queues and lnks wth routng probabltes. Ths example shows a Closed Queueng Network wth three servers (CPU, Dsk1 and Dsk2 and queues n front of the servers. The servce requests are processed at the CPU, then routed to Dsk1 wth a probablty of 40%, routed to Dsk2 wth a probablty of 50% or recalculated wth a probablty of 10%. After the servce requests have passed one of the dsks they are routed back to the CPU. Ths example wll be later calculated usng the Theorem of Gordon and Newell n secton and the Mean Value Analyss n secton as well as n FMC-QE n secton Quanttatve Measures In the Queueng Theory a number of quanttatve measures are defned. The most mportant ones are llustrated n fgure 2.2 and descrbed as: Arrval Rate λ The arrval rate denotes the mean rate of servce requests arrvng at a servce staton. Servce Tme X The servce tme s the tme a server needs to process a servce request. Multplcty m The parameter multplcty defnes the number of parallel servers n a server staton. It s possble, that m =. In ths case every servce request has a dedcated server. An example could be the user n a clent server scenaro, where every user-nput request has a dedcated user. 6

29 2.1. QUEUEING THEORY Servce Rate µ The servce rate s a parameter for the mean maxmal number of servce requests a server could process n a gven tme perod. Departure Rate D The departure rate (throughput denotes the rate, fulflled servce requests (servce responses leave the server. Number of Servce Requests n The mean number of servce requests n a servce staton s denoted by n. It s the sum of the servce requests n servce n s and the queued servce requests n q. Watng Tme W The watng tme (queueng tme denotes the mean tme a servce request s queued n a servce staton. Utlzaton ρ The utlzaton denotes the fracton of tme the server s processng servce requests (busy. Response Tme R The response tme s defned as the tme nterval between the arrval of the servce request and the correspondng departure of the servce response at a servce staton. Queue Sze K The queue sze defnes the maxmum number of servce requests nsde the servce staton (queued + n servce. Number of SRqs n the Staton n Arrval Rate λ Number of Queued SRqs n q Queue Sze K Watng Tme W Number of SRqs n Servce n s & & & 1 2 Multplcty m Servce Tme X Servce Rate % Throughput D Utlzaton ρ Response Tme R Fgure 2.2: Performance Parameters and Values In general usage, the term performance s also often used when quanttatve measures are consdered. In a more specfc usage, performance would be a measure lke throughput n terms of servce requests per tme unt or response tme and not measures lke the servce tme or the multplcty. But as sad, n general use servce tme could also be defned as a performance parameter whch then s also the case n ths thess. Furthermore, there s a dstncton between performance parameters and performance values, whle performance parameters are an nput n a the calculaton and performance values are the results. In the further parts the relatons between the dfferent measures are descrbed. 7

30 CHAPTER 2. STATE OF THE ART Fundamental Laws Lttle s Law and the Forced Traffc Flow Law are two fundamental Queueng Theory laws. Lttle s Law Lttle s Law [98] s one of the fundamental laws n the Queueng Theory. As descrbed n [125], the law defnes the dependence between the mean number of jobs n a system n, the mean arrval rate λ and the mean response tme R. n = λ R (2.1 Lttle s Law s llustrated n fgure 2.3. Lttle s Law s a black box law whch ncludes measures of the servers and the queues nsde the black box. Number of SRqs n the Staton n Arrval Rate λ Response Tme R Fgure 2.3: Lttle s Law Forced Traffc Flow Law The Forced Traffc Flow Law [43, 68, 79, 95] defnes the transformaton of a global arrval rate λ nto a specfc arrval rate λ usng a traffc flow coeffcent v as a law, defnng relatons for the network. λ = v λ (2.2 A more detaled dscusson on the Forced Traffc Flow Law can be found n secton and and s therefore omtted here Server Performance Values The Queueng Theory dstngushes between a broad range of dfferent server types wth dfferent queueng strateges, servce tme dstrbutons and other parameters. The Kendall-Notaton was establshed n order to defne and classfy these dfferent types. Ths notaton s descrbed n ths subsecton. Also some general formulas on the server level are descrbed. There are a lot of equlbrums for dfferent Kendall-Servers n lterature. In the dfferent sources, the dfferent varable dentfers are slghtly dfferent (sometmes conflctng and nformaton for one type of server s sometmes scattered over several sources. Therefore, the tables A.1 to A.9 n the appendx A consoldate and recaptulate these equlbrums usng one consstent nomenclature, wthout repeatng the dervatons and proofs. 8

31 2.1. QUEUEING THEORY Kendall-Notaton The Kendall-Notaton s the de facto standard for the descrpton of elementary queueng systems. The bascs of ths notaton are defned by Davd George Kendall n 1953 [83]. The notaton defnes a Queueng Staton and the correspondng arrval process by a seres of symbols ( A x /B/m/K/C, whch are defned as [67]: A Interarrval-tme dstrbuton, where x defnes a group arrval process [125] B Servce-tme dstrbuton m Number of parallel servers K Capacty restrctons C Queueng dscplne A and B are further specfed by the followng dstrbuton notatons [18, 67, 125]: M Markov - exponental dstrbuton (Posson Process D determnstc dstrbuton (constant nterarrval or servce tme GI General Independent, General dstrbuton wth ndependent nterarrval or servce tmes E k H k Erlang-k-dstrbuton Hyperexponentonal dstrbuton wth k phases The number of servers m denotes the number of parallel servers at the staton and s defned n the range from 1 to. The parameter K defnes the queue sze ncludng the places for the servce requests n servce (m. Often ths parameter s omtted, f K = 1 or K = The standard queueng dscplne C s FIFO (frst n, frst out resp. FCFS frst come, frst served and s therefore often omtted. But other possble dscplnes, then defned, are amongst others [67]: LIFO Last In, Frst Out resp. LCFS Last Come, Frst Served RSS Random Selecton for Servce PR Prorty Based GD General Dscplne In specal cases an addtonal /M s defned, whch defnes the overall fxed populaton of servce requests (or customers each wth an "arrvng" parameter λ [88]. The correspondng formulas for ths case are provded n table A.8. 9

32 CHAPTER 2. STATE OF THE ART General Systems Whle n the tables A.1 to A.9 n appendx A dfferent formulas for the calculaton of server performance values are defned for the dfferent server types, some formulas are defned for general systems and are therefore true for all of the systems. The utlzaton ρ s defned as the fracton of tme, a server s busy and could be calculated as the quotent of the arrval rate λ and the servce rate µ (where the servce rate µ s multplcatve nverse of the servce tme X: µ = 1 X [88]: ρ = λ µ (G/G/1 (2.3 For queueng statons wth multple servers m, ths defnton s the same, extended by the work capacty of the system [88]: ρ = λ mµ (G/G/m (2.4 For systems, where the the maxmum servce rate s not ndependent of the system state, ρ s defned as traffc ntensty [88]. The response tme R s defned as the sum of watng tme W and the servce tme X [88]: R = W + X (2.5 Through Lttle s Law [98] the mean number of servce requests n the system are defned as [88]: n = λr, (2.6 where n systems, where the the maxmum servce rate s not ndependent of the system state, the arrval rate λ s substtuted by the effectve arrval rate λ e f f. The mean number of queued servce requests s also defned through Lttle s law [98] as [88]: n q = λw (2.7 or as the dfference from the mean number of servce requests n the system n and the mean number of servce requests n servce n s (where n s = mρ, f the maxmum servce rate s ndependent of the system state [88]: n q = n n s. (2.8 Therefore, Lttle s Law s vald for the whole staton or system (n = λr as well as the server (n s = mρ = λx and the queue (n q = λw. 10

33 2.1. QUEUEING THEORY Open Queueng Networks - Jackson s Theorem The results of Jackson [77, 78] have been consdered as a break through n the analyss of Queueng Networks [18], because he developed Product Form Solutons for open networks. Through Product Form Solutons, the complexty of the computaton of the overall steady-state probabltes and the global performance values could be reduced, as the steady-state probablty of the network can be computed as the product of the steady-state probabltes of every node. Jackson s Theorem [77]: If n an open network the ergodcty (λ < µ m holds for every node ( = 1,..N, then the steady-state probablty of the network can be computed as the product of the steady-state probabltes of every node: p (k 1,.., k N = p 1 (k 1.. p N (k N (2.9 The Jackson Networks have some constrants, whch are lsted n the followng [18]: There exsts only one sngle class of servce requests n the network. The overall number of servce requests s not lmted. Each of the N nodes n the network can be connected to an external Posson source and a servce response can leave the network at each node. All servce tmes are exponentally dstrbuted. The queueng dscplne at each node s FIFO (FCFS. A node consst of m 1 dentcal servers wth the servce rate µ wth = 1,.., N. The arrval rates λ 0 and the servce rates µ could depend on the number of k servce requests n the node. In ths case the servce and arrval rates would be load dependent. 1 An algorthm for the calculaton of open networks, based on ths theorem, s then manly composed of three steps [18, 132]: Step 1: Calculaton of the arrval rates λ for every node = 1,.., N, based on the setup and solvng of the equaton system of the traffc equatons. Step 2: Ergodcty check and computaton of the state probabltes p (k for every staton. Whle every node could be consdered as a M/M/1 resp. M/M/m staton, the performance values of the nodes could be computed usng the formulas n table A.3 and A.4. Step 3: Fnally the overall steady-state probabltes and the global performance values can be computed. The followng network from [18] n fgure 2.4 wll exemplfy ths algorthm. Ths example wll also be used n secton 3.5 for the explanaton of the FMC-QE transformatons and the comparson of classc open network computaton and the FMC-QE computatons. 1 A Queueng Staton wth more than one server and constant arrval rates s equvalent to a servce staton wth one server and a load dependent servce rate [18]: µ = { k µ m µ k m k m 11

34 CHAPTER 2. STATE OF THE ART p 1,2 = 0,5 Prnter 2 p 2,1=1 Source 4 1 I/ODevce CPU p 1,3 = 0,5 Dsk 3 p 3,5=0,4 Snk p 3,1 =0,6 Fgure 2.4: Sample Open Network - Jackson s Theorem [18] In ths example all nodes are sngle server FCFS statons. The servce rates are exponentally dstrbuted and defned as [18]: µ 1 = 25 [Jobs] ; µ 2 = 33 Jobs [s] 3 [1] [s] ; µ 3 = The exponental dstrbuted arrval rate s defned as [18]: λ = λ 0,4 = 4 [Jobs], [s] and the routng probabltes are defned as [18]: [Jobs] ; µ [s] 4 = 20 [Jobs] [s] p 12 = p 13 = 0, 5; p 41 = p 21 = 1, 0; p 31 = 0, 6; p 30 = 0, 4. In ths example the performance values: utlzatons ρ, mean number of jobs at a noden, mean response tmesr, mean watng tmes W, mean queue lengths n q, and mean overall response tme R as well as the steady-state probablty of state (k 1, k 2, k 3, k 4 = (3, 2, 4, 1 are sought. In step 1 the traffc equatons are set up [18]: λ 1 = λ 2 p 2,1 + λ 3 p 3,1 + λ 4 p 4,1 = 20 [Jobs] [s] λ 3 = λ 1 p 1,3 = 10 [Jobs] [s] ; λ 2 = λ 1 p 1,2 = 10 [Jobs] ; [s] ; λ 4 = λ 0,4 = 4 [Jobs]. [s] In step 2 the performance values and state probabltes for the dfferent nodes are calculated as [18]: Utlzatons ρ as ρ = λ µ [18]: ρ 1 = λ 1 µ 1 = 0, 8; ρ 2 = λ 2 µ 2 = 0, 3; ρ 3 = λ 3 µ 3 = 0, 6; ρ 4 = λ 4 µ 4 = 0, 2. Mean number of jobs at a node (M/M/1 n = ρ 1 ρ [18]: n 1 = 4 [Jobs]; n 2 = 0, 429 [Jobs]; n 3 = 1, 5 [Jobs]; n 4 = 0, 25 [Jobs]. 12

35 2.1. QUEUEING THEORY Mean response tmes of the servers R = n λ [18]: R 1 = 0, 2 [s]; R 2 = 0, 043 [s]; R 3 = 0, 15 [s]; R 4 = 0, 0625 [s]. Mean watng tmes W = ρ µ λ [18]: W 1 = 0, 16 [s]; W 2 = 0, 013 [s]; W 3 = 0, 09 [s]; W 4 = 0, 0125 [s]. Mean queue lengths n,q = ρ2 1 ρ [18]: n 1,q = 3, 2 [Jobs]; n 2,q = 0, 129 [Jobs]; n 3,q = 0, 9 [Jobs]; n 4,q = 0, 05 [Jobs]. Margnal probabltes p (k = (1 ρ ρ k for (k 1, k 2, k 3, k 4 = (3, 2, 4, 1[18]: p 1 (3 = 0, 1024; p 2 (2 = 0, 063; p 3 (4 = 0, 0518; p 4 (1 = 0, 16. Fnally n step 3 the mean overall response tme s calculated by aggregatng the mean number of jobs at every node and Lttle s Law [18]: R = n λ = 1 λ 4 n = 1, 545 [s] =1 and the state probablty for state (k 1, k 2, k 3, k 4 = (3, 2, 4, 1 s [18]: p(3, 2, 4, 1 = p 1 (3 p 2 (2 p 3 (4 p 4 (1 = 0, 0, Closed Queueng Networks - Gordon-Newell Theorem Whle n open networks wth servce request sources and snks, the arrval rate s a free parameter of the source and the overall number of servce requests n the system s dependent upon ths arrval rate, n closed networks the overall number of servce requests n the system s the free parameter and the arrval rate, more precsely the throughput, s a dependent value. Furthermore, n closed networks the overall number of servce requests n the system s an nteger n comparson to the resultng real value n an open network. In closed networks not a servce request source s consdered, but a global balance of the network wth ths constant number of servce requests. Therefore a normalzaton constant s defned n the calculaton of the steady-state probabltes. Ths secton wll gve a general ntroducton nto the handlng of closed (Product Form networks n the Queueng Theory. Further general dscussons on the model of closed networks beyond ths ntroducton could also be found later n secton 4.1.1, where questons lke: Who nserted the servce requests n the system? (ntalzaton of a closed network are rased. 13

36 CHAPTER 2. STATE OF THE ART Gordon and Newell [64] transfered and refned the results of Jackson to closed networks (whle Jackson also already consdered closed systems n [78]. Gordon Newell Networks have the same assumptons than Jackson Networks except, that no servce request can enter or leave the system and therefore the number of servce requests n the system s constant [18]: K = N k (2.10 =1 Accordng to the Gordon-Newell Theorem, each network state n a Gordon Newell Network could be calculated usng the followng product-form expresson [18, 64]: p (k 1,.., k N = 1 G(K N =1 The normalzaton constant G(K s then defned as [18]: F (k (2.11 wth F (k defned as [18]: G(K = F (k = Where the vst ratos are defned as [18]: e = and the functon β (k s defned as [18]: N =1 N k =K =1 ( e µ F (k, ( β (k. (2.13 N e j p j f or = 1,.., N. (2.14 j=1 k! k m, β (k = m!m k m k m, 1 m = 1. (2.15 Another defnton of the functon F (k for load dependent servce rates s gven as [18]: wth A (k as [18]: F (k = ek A (k, (2.16 A(k = { k j=1 µ (j k > 0, 1 k = 0. (2.17 Where a constant servce rate s a specal case n ths generalzed functon. 14

37 2.1. QUEUEING THEORY An algorthm for the computaton of Gordon Newell Networks s separated nto four steps [18]: Step 1: Calculaton of the vst ratos e for all nodes. Step 2: Computaton of the functons F (k for all nodes. Step 3: Computaton of the normalzaton constant G(K. Step 4: Computaton of the state probabltes of the network and the requred performance values. Fgure 2.5 llustrates an exemplary Gordon Newell Network (example from [68], exemplary calculatons from [18]. 1 p 1,1 =0,1 CPU 2 3 Dsk1 p 1,2 =0,4 p 1,3=0,5 Dsk2 Fgure 2.5: Sample Closed Network - Gordon Newell [68] In ths example the queueng dscplne s FCFS at every node and the routng probabltes are defned as [68]: p 11 = 0, 1; p 12 0, 4; p 13 = 0, 5; p 21 = 1, 0; p 22 0, 0; p 23 = 0, 0; p 31 = 1, 0; p 32 0, 0; p 33 = 0, 0. The exponentally dstrbuted servce tmes are: There are 3 servce requests n the system: µ 1 = 0, 5; µ 2 = 0, 40; µ 3 = 0, 25. K = 3. Ths network conssts of 10 states: (3, 0, 0; (2, 1, 0; (2, 0, 1; (1, 2, 0; (1, 1, 1; (1, 0, 2; (0, 3, 0; (0, 2, 1; (0, 1, 2; (0, 0, 3. In the frst step the vst ratos e are determned as: e 1 = e 1 p 11 + e 2 p 21 + e 3 p 31 = 1; e 2 = e 1 p 12 + e 2 p 22 + e 3 p 32 = 0, 4; e 3 = e 1 p 13 + e 2 p 23 + e 3 p 33 = 0, 5. 15

38 CHAPTER 2. STATE OF THE ART In the second step the functons F (k wth F (k = F 1 (0 = 1; F 1 (1 = 2; F 1 (2 = 4; F 1 (3 = 8; F 2 (0 = 1; F 2 (1 = 1; F 2 (2 = 1; F 2 (3 = 1; F 3 (0 = 1; F 3 (1 = 2; F 3 (2 = 4; F 3 (3 = 8. In the thrd step the normalzaton constant G(3 s determned: ( e µ k for = 1, 2, 3 are computed: G(3 = F 1 (3F 2 (0F 3 (0 + F 1 (2F 2 (1F 3 (0 + F 1 (2F 2 (0F 3 (1+ F 1 (1F 2 (2F 3 (0 + F 1 (1F 2 (1F 3 (1 + F 1 (1F 2 (0F 3 (2+ F 1 (0F 2 (3F 3 (0 + F 1 (0F 2 (2F 3 (1 + F 1 (0F 2 (1F 3 (2+ F 1 (0F 2 (0F 3 (3 = 49. In the fourth step the dfferent state probabltes p(k 1, k 2, k 3 = 1 N G(3 F (k are calculated: =1 p(3, 0, 0 = 8 49 ; p(2, 1, 0 = 4 49 ; p(2, 0, 1 = 8 49 ; p(1, 2, 0 = 2 49 ; p(1, 1, 1 = 4 49 ; p(1, 0, 2 = 8 49 ; p(0, 3, 0 = 1 49 ; p(0, 2, 1 = 2 49 ; p(0, 1, 2 = 4 49 ; p(0, 0, 3 = Ths leads to the followng margnal state probabltes: p 1 (0 = p(0, 3, 0 + p(0, 2, 1 + p(0, 1, 2 + p(0, 0, 3 = ; p 1 (1 = p(1, 2, 0 + p(1, 1, 1 + p(1, 0, 2 = ; p 1 (2 = p(2, 1, 0 + p(2, 0, 1 = ; p 1 (3 = p(3, 0, 0 = 8 49 ; p 2 (0 = p(2, 0, 1 + p(1, 0, 2 + p(0, 0, 3 + p(3, 0, 0 = ; p 2 (1 = p(2, 1, 0 + p(1, 1, 1 + p(0, 1, 2 = ; p 2 (2 = p(1, 2, 0 + p(0, 2, 1 = 4 49 ; p 2 (3 = p(0, 3, 0 = 1 49 ; p 3 (0 = p(3, 0, 0 + p(2, 1, 0 + p(1, 2, 0 + p(0, 3, 0 = ; p 3 (1 = p(2, 0, 1 + p(1, 1, 1 + p(0, 2, 1 = ; p 3 (2 = p(1, 0, 2 + p(0, 1, 2 = ; p 3 (3 = p(0, 0, 3 =

39 2.1. QUEUEING THEORY Fnally, the performance values are derved, lke the utlzaton ρ = 1 p (0: ρ 1 = 0, 69; ρ 2 = 0, 35; ρ 3 = 0, 69. The mean number of servce requests at a staton n = 3 k=1 kp (k: The throughput at every node D = m ρ µ : n 1 = 1, 27; n 2 = 0, 47; n 3 = 1, 27. and the mean response tmes R = n D : D 1 = 0, 35; D 2 = 0, 14; D 3 = 0, 17, R 1 = 3, 65; R 2 = 3, 38; R 3 = 7, 27. Ths leads to the overall performance values of: D = 0, 35; R = 8, Mxed Open and Closed Queueng Networks - BCMP Theorem Baskett, Chandy, Muntz and Palacos (BCMP [8] extended the results of Jackson and Gordon and Newell to open, closed and mxed networks wth several job classes, dfferent knds of servers and state dependent arrval processes. A BCMP network can consst of an arbtrary but fnte number of queueng statons (servce centers of the followng type [8]: Type-1: -/M/m - FCFS, Type-2: -/G/1 - PS, Type-3: -/G/, Type-4: -/G/1 - LCFS PR. Where the dfferent types of servers are defned as: Type-1: -/M/m - FCFS [8]: frst-come-frst-served (FCFS servce dscplne, all servce requests get the same servce tme, the servce tme dstrbuton s negatve exponental, 17

40 CHAPTER 2. STATE OF THE ART a state dependent servce rate s possble (then: m > 1 2. Type-2: -/G/1 - PS [8]: sngle server, no queue, servce dscplne: processor sharng, each class of servce request may have dstnct servce dstrbutons, the servce tme dstrbutons have Laplace transforms. Type-3: -/G/ (IS [8]: number of servers n the staton maxmum number of servce requests, each class of servce request may have dstnct servce dstrbutons, the servce tme dstrbutons have Laplace transforms. Type-4: -/G/1 - LCFS PR [8]: sngle server, preemptve-resume last-come-frst-served (LCFS PR servce dscplne, each class of servce request may have dstnct servce dstrbutons, the servce tme dstrbutons have Laplace transforms. For open networks two knds of arrval processes can be dstngushed from each other [18]: In the frst case, the arrval process s Posson and all servce requests arrve n the network from one source wth an overall arrval rate λ, whch can depend on the number of jobs n the network. The probablty, that an arrvng servce request s then routed from the source 0 to the servce staton n class r s defned as p 0,r, where [18]: N R =1 r=1 p 0,r = 1, (2.18 where N s the overall number of servce statons and R s the number of classes n the network. In the second case, the arrval process conssts of U ndependent Posson arrval streams, where each source s assgned to a dfferent chan. An arrval rate λ U could be load dependent. A servce request arrves at servce staton wth the probablty p 0,r, where [18]: 2 A queueng staton wth more than one server and constant arrval rates s equvalent to a servce staton wth one server and a load dependent servce rate [18]: µ = { k µ m µ k m k m 18

41 2.1. QUEUEING THEORY For ths knds of networks the BCMP Theorem states [8]: N p 0,r = 1 u = 1,..U. (2.19 r C u ; =1 For a network of servce statons whch s open, closed or mxed n whch each servce center s of type 1,2,3 or 4, the steady-state state probabltes are gven by: p(s 1,.., S N = 1 N G (K d (S f (S, (2.20 =1 where G(K s the normalzaton constant, d(s s a functon of the number of servce requests n the system and each f (S s a functon that depends on the type of the servce center. The functon d(s s defned as [8, 18]: K(S 1 λ( open network wth arrval process 1, =0 d(s = U K(S 1 λ u ( open network wth arrval process 2, u=1 =0 1 closed network. (2.21 Where K(S s the overall number of servce requests n the network and K u (S s the overall number servce requests n from the uth source. The functon f (S s defned as [8, 18]: k! f (S = ( 1 µ k k R r=1 k j=1 R r=1 u r l=1 j=1 u r l=1 1 k rl! e sj Type 1, ( krl er A rl µ rl Type 2, ( krl er A rl µ rl Type 3, 1 k rl! e rj A rj m j µ rj m j, (2.22 wth [18]: s j Class of the servce request n the j th poston of the FCFS queue. µ rl The mean servce rate of the node n the class r at the lth phase (l = 1,.., u r n a Cox dstrbuton. u r The maxmum number of exponental phases of the class r n node. A rl a rl A rl = l 1 a rj s the probablty, that a class-r servce request n node reaches the lth stage j=0 (A r1 = 1 because of a r0 = 1. The probablty, that a class-r servce request at the node moves to the (j + 1th phase. 19

42 CHAPTER 2. STATE OF THE ART k rl Number of class-r servce requests n phase l. k k = R k r overall number of servce requests of all classes n node. r=1 In ther paper [8] Baskett, Chandy, Muntz and Palacos provded also two smplfcatons of the equlbrum state probabltes for closed networks and open network, n whch the arrval rate does not depend on the state of the model. For a closed system the steady-state state probabltes are gven by [8, 18]: where the normalzaton constant G (K s defned as [18]: p(s 1,.., S N = 1 N G (K F (S, (2.23 =1 G (K = and the functon F (S s defned as [8, 18]: N =1 N S =K =1 F (S (2.24 f (S = wth [18]: and [18]: k! 1 β (k ( 1 µ k R k j=1 k! k! R r=1 µ (j R r=1 1 k r! R r=1 1 k r! r=1 1 k r! (e r k r 1 k r! (e r k r Type 1, ( er µ r kr Type 2, 4, ( er µ r kr Type 3. Type 1 (m = 1, load dependent servce rate µ (j, k = (2.25 R k r, (2.26 r=1 k! k m, β (k = m!m k m k m, 1 m = 1. e r = (2.27 N e js p js,r. (2.28 s C q ; j=1 An example for a closed BCMP Queueng network s omtted, whle the calculaton of the normalzaton constant nvolves the calculaton of all states of the network and could therefore 20

43 2.1. QUEUEING THEORY be very large. An optmzed algorthm for the calculaton of these networks s for example provded through the Mean Value Analyss, explaned n the next subsecton, where also a descrptve example s gven. For open BCMP networks wth load-ndependent arrval and servce rates the steady-state state probablty smplfes to [8, 18]: p(k 1,.., k N = N p (k, (2.29 =1 where the ndvdual steady-state state probabltes are defned as [8, 18]: (1 ρ ρ k Type 1, 2, 4 (m = 1, p (k = e ρ ρk k! Type 3. (2.30 wth [18]: k = R k r (2.31 =1 and [18]: ρ = R ρ r, (2.32 =1 ρ r = { e λr r µ Type 1 (m = 1, e λ r r µ r Type 2, 3, 4., (2.33 where the ergodcty condtons (ρ < 1 holds for every server. The mean number of servce requests per class n every server s then provded as [18]: n r = ρ r 1 ρ r. (2.34 For the calculaton of the steady-state state probabltes for Type-1 nodes wth more than one server (m > 1, the formulas from table A.4 are used. An algorthm for the calculaton of the performance values of an open BCMP network wth load-ndependent arrval and servce rates could then be defned as follows [18]: Step 1: Compute the vst ratos e r for = 1,.., N and r = 1,.., R. Step 2: Compute the utlzatons ρ for each node. Step 3: Compute the other performance values. Step 4: Compute the steady-state state probabltes for every node. Step 5: Compute the overall steady-state state probabltes. 21

44 CHAPTER 2. STATE OF THE ART An llustratve example, also used n [18], s defned as follows: p 11,21=0,4 p 12,22 =0,3 2 Type4 /G/1LCFS PR p 21,11 =0,6 p 22,12 =0,7 p 21,31 =0,4 p 22,32 =0,3 Source p 0,11 =1 p 0,12 =1 Type2 /G/1PS 1 p 11,0 =0,3 p 12,0 =0,1 Snk p 11,31=0,3 p 12,32 =0,6 3 Type4 /G/1LCFS PR p 31,21 =0,5 p 32,22 =0,6 p 31,11 =0,5 p 32,12 =0,4 Fgure 2.6: Sample Open Network - BCMP [18] The network s an open network of three nodes (N = 3 and two servce request classes (R = 2, shown n fgure 2.6. The servce tmes are exponentally dstrbuted and the servce rates are defned as [18]: µ 11 = 1 8s ; µ 21 = 1 12s ; µ 31 = 1 8s ; µ 12 = 1 24s ; µ 22 = 1 32s ; µ 32 = 1 36s. The arrval rates are also exponentally dstrbuted as [18]: λ 1 = λ 2 = 1 8 SRq. s In the example the demanded values are the mean number of servce requests at every node n r and the probablty for the state (k 1, k 2, k 3 = (3, 2, 1 In the frst step the vst ratos e r as e r = p 0,r + = 1,.., N [18]: N s C q, j=1 e js p js,r are computed for r = 1,.., R and e 11 = p 0,11 + e 11 p 11,11 + e 21 p 21,11 + e 31 p 31,11 = 3, 333; e 21 = p 0,21 + e 11 p 11,21 + e 21 p 21,21 + e 31 p 31,31 = 2, 292; e 31 = p 0,31 + e 11 p 11,31 + e 21 p 21,31 + e 31 p 31,21 = 1, 917; e 12 = p 0,12 + e 12 p 12,12 + e 22 p 22,12 + e 32 p 32,12 = 10; e 22 = p 0,22 + e 12 p 12,22 + e 22 p 22,22 + e 32 p 32,32 = 8, 049; e 32 = p 0,32 + e 12 p 12,32 + e 22 p 22,32 + e 32 p 32,22 = 8,

45 2.1. QUEUEING THEORY In the second step the utlzatons ρ are computed [18]: ρ 1 = λ 1 e 11 µ 11 + λ 2 e 12 µ 12 = ρ11 + ρ 12 = 0, 833; ρ 2 = λ 1 e 21 µ 21 + λ 2 e 22 µ 22 = ρ21 + ρ 22 = 0, 442; ρ 3 = λ 1 e 31 µ 31 + λ 2 e 32 µ 32 = ρ31 + ρ 32 = 0, 354. In the thrd step the mean number of servce requests at every node (n r = [18]: ρ r 1 ρ are computed n 11 = 2, 500; n 21 = 0, 342; n 31 = 0, 186; n 12 = 2, 500; n 22 = 0, 500; n 32 = 0, 362. In the fourth step the margnal probabltes (p (k = (1 ρ ρ k - Type 1,2,4 (m = 1 are computed [18]: p 1 (3 = 0, 0965; p 2 (2 = 0, 1093; p 3 (1 = 0, Fnally n the ffth step the steady-state state probablty for the state (k 1, k 2, k 3 = (3, 2, 1 s computed [18]: p(3, 2, 1 = p 1 (3p 2 (2p 3 (1 = 0, An Algorthm for Product Form Networks - Mean Value Analyss (MVA The Mean Value Analyss [118] s an algorthm for the calculaton of the performance values of closed queueng networks wth product form solutons. It s based on Lttle s Law [98] and the Theorem of the dstrbuton at arrval tme (Arrval Theorem, Reser/Lavenberg-Theorem [93, 121], whch states, that n a closed Product Form Queueng Network, the probablty mass functon of the number of jobs seen at the tme of the arrval at node when there are k servce requests n the network s equal to the probablty mass functon of the number of jobs at ths node wth one less job n the network [18]. Ths teratve algorthm s then defned as follows [18]: In step 1 the performance values are ntalzed: For = 1,.., N and j = 1,.., (m 1: n = 0; p (0 0 = 1; p (j 0 = 0. (

46 CHAPTER 2. STATE OF THE ART Step 2 s an teraton over the number of servce requests k = 1,.., K wth the correspondng sub steps: Step 2.1 Computaton of the mean response tmes of a servce request at each node: 1 µ (1 + n (k 1 Type 1, 2, 4 (m = 1, ( 1 R (k = µ m 1 + n (k 1 + m 2 j=0 ((m j 1 p (j k 1 Type 1 (m > 1, 1 µ Type 3, (2.36 wth the condtonal probabltes of: ( p (0 k = 1 1 m e D (k 1 + m µ j=1 p (j k = D (k µ α (j p (j 1 k 1. ((m j p (j k, (2.37 and the functon α (j defned as: α (j = { j f j m, m else. (2.38 In Step 2.2 the overall throughput D(k: D(k = k N =1 e R (k (2.39 and the node throughput D (k s computed: D (k = e D(k. (2.40 The mean number of servce requests at every node s computed n Step 2.3: n (k = e D(kR (k. (2.41 The other performance values are computed usng the known formulas. The Mean Value Analyss s furthermore extended to mult class networks, mxed networks and networks wth load dependent servce rates. More nformaton on ths extensons could be found n [18]. 24

47 2.1. QUEUEING THEORY An example of the closed Gordon Newell Networks, llustrated n fgure 2.7, s used for exemplfcaton. CPU 1 p 1,1 =0,1 2 3 Dsk1 Dsk2 p 1,2 =0,4 p 1,3=0,5 Fgure 2.7: Sample Closed Network - MVA [68] The performance parameters are: µ 1 = 0, 5; µ 2 = 0, 40; µ 3 = 0, 25; K = 3; e 1 = 1; e 2 = 0, 4; e 3 = 0, 5. In step 1, the n values are ntalzed. (The condtonal probabltes p (0 0 and p (j 0 are not ntalzed, because there are no Type-1 m > 1 servers n the example: n 1 = 0; n 2 = 0; n 3 = 0. The frst teraton (j = 1 of step 2 leads to the followng results: Second teraton (j = 2: R 1 = 2[s]; R 2 = 2, 5[s]; R 3 = 4[s]; R = 5[s]; D 1 = 0, 2 1 [s] ; D 2 = 0, 08 1 [s] ; D 3 = 0, 1 1 [s] ; D = 0, 2 1 [s] ; n 1 = 0, 4; n 2 = 0, 2; n 3 = 0, 4; n = 1. Thrd teraton (j = 3 = K: R 1 = 2, 8[s]; R 2 = 3[s]; R 3 = 5, 6[s]; R = 6, 8[s]; D 1 = 0, 29 1 [s] ; D 2 = 0, 12 1 [s] ; D 3 = 0, ; D = 0, 29 [s] [s] ; n 1 = 0, 82; n 2 = 0, 35; n 3 = 0, 82; n = 2. R 1 = 3, 65[s]; R 2 = 3, 39[s]; R 3 = 7, 29[s]; R = 8, 65[s]; D 1 = 0, 35 1 [s] ; D 2 = 0, 14 1 [s] ; D 3 = 0, ; D = 0, 35 [s] [s] ; n 1 = 1, 27; n 2 = 0, 47; n 3 = 1, 27; n = 3. 25

48 CHAPTER 2. STATE OF THE ART 2.2 Tme Augmented Petr Nets The dmenson tme s not consdered n the orgnal Petr Nets by Petr [113] and Holt and Commoner [75]. In ths networks the causal relatonshps between the assocated events or actons are to be nvestgated, as for example n the modelng of (busness processes or the development of asynchronous crcuts. The quanttatve analyss or steady state behavor of servers were not n the focus. Later on the Petr Nets were extended by the aspect tme (as Tme Augmented Petr Nets n order to extend them for the quanttatve analyss as an addtonal am. Compared to Queueng Networks, the analyss of such Tme Augmented Petr Nets s often executed on the level of the reachablty graph, lke llustrated n fgure 2.8. Whle the Queueng Theory often abstracts from ths level of detal through Product Form Solutons and defned equlbrums on the server level, n Tme Augmented Petr Nets the quanttatve behavor of system wth complex (Non Product Form control flows could be predcted wth the trade-off of the possble state-space exploson problem, when consderng the system on the level of contnuous tme Markov chans (CTMCs. A more extensve comparson n provded n secton 4.4. n=λ*r Steady State Relaton for Black Box (Lttle s Law n=(n 1,..,n,..,n N n =n,q +n,s =1,..,N Operatonal State p=(p 0,..,p N Steady State Probablty Vector of M M=(M 0,..,M,.. Reachablty Graph (somorphc to CTMC Fgure 2.8: Range of Steady State Varables [140] In ths secton an overvew of some classfcatons and extensons of Tme Augmented Petr Nets s gven on the bass of [14] Classfcaton The Tme Augmented Petr Nets could be manly dvded nto two classes, dependent whch object type had a specfed sojourn tme [14]: Tmed Places Petr Nets (TPPNs Tmed Transtons Petr Nets (TTPNs In Tmed Places Petr Nets the sojourn tmes are connected to the places of the Petr Net. If a token enters a place, the correspondng transtons, for whch ths place s an nput place, can only fre after a certan tme nterval. On the other hand, n Tmed Transtons Petr Nets the sojourn tmes are connected to the transtons.the transtons become enabled, but fre only after a certan tme nterval has elapsed. Ths group of Tme Augmented Petr Nets could furthermore dvded nto two subclasses [14]: 26

49 2.2. TIME AUGMENTED PETRI NETS TTPNs wth reservaton (preselecton models TTPNs wthout reservaton (race models In preselecton models the transton selects all nput tokens t needs to fre, when the transton s enabled, so that the tokens are unavalable for other possble concurrent transtons. When the correspondng tme nterval of the transton has elapsed, the nput tokens are destroyed and the output tokens are created by the frng of the transton. In race models the tokens are not reserved. In these models concurrent transtons get enabled parallel and the transton, whch fres frst, dsables the other possble concurrent transtons. Another dfferentator s, f the mentoned tmes are defned determnstc or wth a stochastc dstrbuton. In some cases also dfferent frng polces of transtons are dstngushed. Whle the consdered Petr Nets have exponental dstrbuted frng delays, ths dfferentaton s rrelevant [14]. In the followng some mportant Tme Augmented Petr Net types are lsted: In contnuous tme stochastc Petr Nets (SPN [104, 106] (accordng to [14] all transtons are augmented wth a stochastc frng tme nterval. Generalzed Stochastc Petr Nets (GSPN [101, 102] ntroduce mmedate transtons. In Determnstc Stochastc Petr Nets (DSPN [34] the transtons are assocated to determnstc as well as stochastc frng tmes. Queueng Petr Nets [9, 10, 14] combne Queueng Networks and Petr Nets wthn one net and n a mxed TPPN/TTPN Contnuous Tme Stochastc Petr Nets (SPN Contnuous Tme Stochastc Petr Nets (SPN [104, 106] (accordng to [14] extend a Petr Net by connectng a transton rate ω to every transton t. Whle the frng delays n SPNs are exponentally dstrbuted and therefore have a memoryless property, an SPN s somorphc to a contnuous tme Markov chan (CTMC [102]. Furthermore, a k-bounded SPN 3 system s somorphc to a fnte CTMC [102]. Thus the correspondng CTMC of an SPN could be obtaned by applyng the followng rules [102]: The CTMC state space S = s corresponds to the reachablty set RS(M 0 of the Petr Net assocated wth the SPN (M s. The transton rate from state s (correspondng to markng M to state s j (M j s obtaned as the sum of the frng rates of the transtons that are enabled n M and whose frngs generate markng M j. 3 k-bounded [21]: SPN s k-bounded, f the number of tokens n each place s less or equal to k for all markngs n RS(M. 27

50 CHAPTER 2. STATE OF THE ART As a smplfcaton, n ths secton, lke n [102], the transtons are assocated wth sngle server semantcs and a markng ndependent speed, but ths has not to be the case n every system. Based on ths rules, the nfntesmal generator (state transton rate matrx Q of the CTMC could be constructed from the SPN [102]. Every cell n ths matrx s then defned as [102]: ω k = j, q j = T k E j (M q = j, (2.42 where: q = ω k, (2.43 T k E(M wth ω k as frng rate of transton T k and E j (M as the set of transtons whose frngs brng the net from M to M j. If the SPN s ergodc, the steady state probablty dstrbuton vector π could then be found by solvng the lnear system [102]: πq = 0. (2.44 The probablty, that a transton T k E(M fres (frst n markng M has the expresson [102]: The average sojourn tme n markng M s [102]: p(t k M = ω q. (2.45 SJ = 1 q. (2.46 Through the steady state probablty dstrbuton vector π, the probablty of beng n a subset B of markngs s defned as the sum of the steady state probabltes of the dfferent markngs [14]: p(b = π. (2.47 M B The throughput of a tmed transton T k s further defned by the mean number of frngs n steady state as the frng rate ω k multpled by the sum of the steady state probabltes n whch T k s enabled [14]: D k = ω k π. (2.48 T k E(M In order to calculate the mean number of tokens enablng a transton T k, also nterpreted as n k, the steady state probabltes π of all markngs M enablng T k are added: n k = π. (2.49 T k E(M 28

51 2.2. TIME AUGMENTED PETRI NETS The contnuous tme Stochastc Petr Net n fgure 2.9 exemplfes an SPN and wll be analyzed n ths secton. The example orgnates from [102]. pact1 pact2 T req1 λ 1 λ 2 T req2 pdle preq1 preq2 T str1 α 1 α 2 T str2 pacc1 pacc2 T end1 1 2 T end2 Fgure 2.9: SPN Example [102] The performance parameters of ths example are defned n table 2.1. Transton Rate Semantcs T req1 λ 1 = 1 sngle server T req2 λ 2 = 2 sngle server T str1 α 1 = 100 sngle server T str2 α 1 = 100 sngle server T end1 µ 1 = 10 sngle server T end2 µ 2 = 5 sngle server Table 2.1: SPN Example - Parameters [102] 29

52 CHAPTER 2. STATE OF THE ART In the frst step of the calculaton of ths model, the correspondng reachablty graph s set up, as llustrated n fgure Fgure 2.10: SPN Example - Reachablty Graph [102] After that the reachablty set s derved as [102]: M 0 = p act1 + p dle + p act2 M 1 = p req1 + p dle + p act2 M 2 = p acc1 + p act2 M 3 = p acc1 + p req2 M 4 = p req1 + p dle + p req2 M 5 = p act1 + p dle + p req2 M 6 = p act1 + p acc2 M 7 = p req1 + p acc2 Ths leads to the state transton rate dagram of the assocated Markov chan, depcted n fgure M 0 λ2 M 1 M 5 α1λ1 α1 α2 λ2 λ1 M 2 M 4 M λ2 α2λ1 M 3 M Fgure 2.11: State Transton Rate Dagram of the SPN Example [102] 30

53 2.2. TIME AUGMENTED PETRI NETS Knowng ths state transton rate dagram, the state transton rate matrx Q s set up as [102]: Q = λ 1 λ 2 λ λ α 1 λ 2 α 1 0 λ µ 1 0 λ 2 µ 1 λ µ 1 0 µ α 1 α 1 α α λ 1 α 2 λ 1 α 2 0 µ λ 1 µ 2 λ 1 0 µ µ 2 If then πq = 0 and 7 =0 π = 1 and ths lnear system s solved, the steady state probabltes are [102]: π 0 = 0, 61471, π 1 = 0, 00842, π 2 = 0, 07014, π 3 = 0, 01556; π 4 = 0, 00015, π 5 = 0, 01371, π 6 = 0, 22854, π 7 = 0, An exemplary performance value could be the utlzaton of the shared memory. Therefore the expectaton value of the markngs, n whch the place p dle s occuped, s calculated as [102]: E(M(p dle = π 0 + π 1 + π 4 + π 5 = 0, The utlzaton of the shared memory s then defned as [102, 128]: ρ shared memory = 1 E(M(p dle = 0, Another exemplary performance value s the throughput of the transton T req1, whch could be nterpreted as the throughput of the left subnet, calculated as D k = ω k M A k π k, n whch A k s the subset of the result set n whch T j (n ths case T req1 s enabled [102]: D req1 = λ 1 (π 0 + π 5 + π 6 = 0, Compared to a soluton of a closed queueng network calculated wth the help of the Gordon- Newell Theorem, lke shown n secton 2.1.5, n ths calculated example the nterdependences of the dfferent transtons through the semaphore prevented the usage of a Product Form Soluton. Therefore, the network had to be calculated on the level of reachablty graphs. On the other hand, networks whch are accessble to Product Form Solutons could also be calculated on the level of CTMCs, but ths addtonal effort would not result n more precse calculatons on the level of steady state probabltes, as for Product Form Networks, Product Form Solutons are correct. 31

54 CHAPTER 2. STATE OF THE ART Generalzed Stochastc Petr Nets (GSPN Generalzed Stochastc Petr Nets (GSPN [101, 102] extend the SPN by ntroducng mmedate transtons alongsde the tmed transtons. These mmedate transtons fre n zero tme, whle the tmed transtons fre after a random, exponentally dstrbuted enablng tme, lke n the SPN [14]. If transtons n a GSPN are enabled concurrently, whch transton fres s decded as [14]: If the concurrent transtons are tmed transtons, the transton, whch fres frst, dsables the other transtons (lke n an SPN. If both tmed and mmedate transtons are enabled, only the mmedate transton wll be enabled (mmedate transtons have prorty. If only an mmedate transton s enabled, t fres wth probablty 1. If mmedate transtons are enabled n parallel, the random swtch or swtchng dstrbuton s often defned by frng weghts ω. If then, for example, two transtons T 1 and T 2 are enabled n some markng M, the probablty of frng T 1 s ω 1 ω 1 +ω 2. The dfferent states n a GSPN are then parttoned nto tangble states ˆT (all transtons enabled n ths state are tmed and vanshng states ˆV (at least one enabled transton n ths state s mmedate, where ˆT ˆV =. In order to analyze the performance behavor of a GSPN, an embedded Markov chan could be consdered. The correspondng transton state probablty matrx s defned as an augmented matrx wth respect to the sets of vanshng and tangble states [14]: P = [ C E D F ] (2.50 Where n C are the transton probabltes from markng or state M to M j, where M and M j are vanshng states [14]: C = (c j, c j = p(m M j ; M ˆV, M j ˆV. (2.51 D are the transton probabltes from vanshng states to transton states [14]: and E and F accordngly [14]: D = (d j, d j = p(m M j ; M ˆV, M j ˆT (2.52 E = (e j, e j = p(m M j ; M ˆT, M j ˆV (2.53 F = ( f j, f j = p(m M j ; M ˆT, M j ˆT (2.54 Every transton probablty from markng of state M to M j s then defned as the sum of all frng rates resp. frng weghts ω k, whose transtons T k are enabled n markng M and whch 32

55 2.2. TIME AUGMENTED PETRI NETS frng produce M j 4 dvded by the sum of of all frng rates resp. frng weghts ω r of the transtons T r, enabled n state M 5 [14, 102]: p(m M j = ω k T k E j (M (2.55 ω r T r E(M The steady state dstrbuton π of the embedded Markov chan s gven by [14]: and [14]: πp = π (2.56 M ˆT ˆV π = 1. (2.57 The steady state dstrbuton π of the orgnal stochastc process could be derved from the steady state dstrbuton π of the embedded Markov chan by weghtng the probablty π j wth the porton of tme the process spends n markng M j [14]: If M j s a vanshng markng, π j s 0. If M j s a tangble markng, the steady state probablty π j s the fracton of tme the stochastc process spends n markng M j. Ths s characterzed by the mean tme the stochastc process spends n ths state [14]: dvded by the mean cycle tme [14] for ths markng: 1 π j M s ˆT 1 (2.58 ω k T k E(M j Ths leads to the steady state dstrbuton [14]: π s T k E(M s ω k (2.59 π j = 1 π j ω k T k E(M j Ms ˆT πs T k E(Ms ω k M j ˆT, 0 M j ˆV. (2.60 Knowng the steady state dstrbuton π, all other performance values could be derved lke for an SPN. 4 E j (M denotes the set of transtons enabled n M whose frngs produces M j [102] 5 E(M denotes the set of transtons enabled n markng M [102] 33

56 CHAPTER 2. STATE OF THE ART As an exemplary GSPN, the SPN example of subsecton has been slghtly modfed, that t str1 and t str2 are mmedate transtons and therefore the token on place p dle could be consdered as a real semaphore. Ths s llustrated n fgure pact1 pact2 T req1 λ 1 λ 2 T req2 pdle preq1 preq2 t str1 α 1 α 2 t str2 pacc1 pacc2 T end1 1 2 T end2 Fgure 2.12: GSPN Example [102] The performance parameters of ths example are defned n table 2.2. Transton Rate/Weght Semantcs T req1 λ 1 = 1 sngle server T req2 λ 2 = 2 sngle server T str1 α 1 = 1 mmedate T str2 α 1 = 1 mmedate T end1 µ 1 = 10 sngle server T end2 µ 2 = 5 sngle server Table 2.2: GSPN Example - Parameters The new reachablty graph, llustrated n fgure 2.13, has slghtly changed comparng to fgure 2.10, because the old state M 4 s mpossble. Ths state was deleted, because t was reached through the frng of transton T req2 resp. T req1, whle each of the transtons was concurrently enabled wth transton T str1 resp. T str2. The transtons T str1 and T str2 are mmedate transtons n ths model, so the tmed transtons T req2 and T req1 wll be dsabled n ths concurrent enablng. Furthermore, the state numberngs have been changed n order to be sorted n the sets of vanshng ˆV = {M 0, M 1 } and tangble states ˆV = {M 2,.., M 6 }. 34

57 2.2. TIME AUGMENTED PETRI NETS Fgure 2.13: GSPN Example - Reachablty Graph The matrx P s then calculated as: P = [ C E D F ] = λ 1 λ 1 +λ 2 λ 2 λ 1 +λ µ λ λ 2 +µ λ 2 +µ µ 2 λ 1 λ 1 +µ λ 1 +µ If the global balance equatons πp = π and M ˆT ˆV π = 1 are then solved, π s defned as: π 0 = 8 63 ; π 1 = ; π 2 = 5 18 ; π 3 = 8 63 ; π 4 = ; π 5 = ; π 6 = Ths leads to the steady state dstrbuton π (computed through formula 2.60: π 0 = 0; π 1 = 0; π 2 = ; π 3 = ; π 4 = ; π 5 = ; π 6 = Whle ths example s also used as a comparatve example n secton 4.3, further performance values are calculated there Product Form Petr Nets There are several proposals for product form solutons for Tme Augmented Petr Nets n order to address the state-space exploson problem. The approach of Lazar and Robertazz [94] s based on nvestgatons of the reachablty graph of the SPN. Henderson et al. [19, 20, 35, 46, 70, 72, 73] perform a structural analyss on the net level and Florn and Natkn [52] developed a product form soluton for closed synchronzed queueng networks [46, 52]. Furthermore, 35

58 CHAPTER 2. STATE OF THE ART Balbo, Bruell and Sereno [5, 6] proposed product form solutons for GSPNs. In ths subsecton the approaches are brefly explaned. The product form solutons of Queueng Petr Nets [11, 12] are furthermore summarzed n secton Lazar and Robertazz [94] characterze a product form Petr Net through the nvestgaton of the correspondng reachablty graph. They defne an solated crculaton as [46, 94]: Isolated Crculaton An solated crculaton conssts of the probablty flow along a subset of edges of the state transton dagram for whch there s a conservaton of ths flow at each adjacent state. Through the dentfcaton of solated crculatons they defne a product form crtera called Dualty Prncple [46, 94]: Dualty Prncple There s a dualty between the exstence of local balance and the exstence of solated crculatons. That s to say, the exstence of local balance leads to solated crculatons and vce versa. Whle the exploraton of the state-space s a weakness of the approach [46], Lazar and Robertazz also defned a product form crtera on structural level for safe nets 6 [46, 94]: A Safe SPN, consstng of a number of task sequences that are comprsed of a seres of sequental sub-tasks, has product form soluton for the equlbrum state probabltes f the state transton dagram can be naturally assocated wth a Cartesan coordnate systems, and f the transton dagram s comprsed of ntegral buldng blocks (solated crculatons and correspondng consstent set of local balance equatons. Ths crtera could also be formulated n terms of blockng at the state space level [46, 94]: Consder a safe SPN consstng of a number of task sequences that are comprsed of a seres of sequental sub-tasks. The product form soluton for the probabltes at equlbrum exsts f and only f a task sequence s only allowed to proceed f there s a non-zero probablty that t can return to ts current state wthout the need for a state change n other task sequences. Henderson, Lucc and Taylor [73] proposed a structural approach for the dentfcaton of product form Petr Nets. They consder the transtons of the stochastc Petr Net to be themselves states n a Markov chan, called Routng process. Each transton t has a unque nput bag (I(t and the probablstc routng allows for a number of dfferent output bags (O(t. Furthermore they defne the set R(T [46, 73]: R(T = O(t I(t (2.61 t T as the fnte set of all vectors whch are ether output or nput bags for the SPN. Then they defne one step probabltes on the set R(T as [46, 73]: t T p(r, r = p(i(t, t, O j (t (2.62 whenever there exsts a transton t and a j such that r = I(t and r = O j (t (where no vector can be the nput set of two or more transtons. Then they defne the set F as [73]: 6 Safe Net [46]: In a safe net, all places are 1-bounded. 36

59 2.2. TIME AUGMENTED PETRI NETS F = { f ( : f (r > 0, χ(r f (r = χ(r f (r p(r, r, r R} (2.63 r where χ(r = ω(r f r = I(t and χ(r = 1 else [46]. Then, F needs to be nonempty, where [73]: For F to be nonempty, all vectors I(t, t T must be n postve recurrent communcaton classes of the routng process. For F to be nonempty, all r R must be the nput bag for some transton t and all r R must be the output bag for some transton t. If F s nonempty, there exsts a one to one correspondence between dstnct elements of R and elements of F. Through the one to one correspondence of R and T the set F could be defned as [46, 73]: F = { f ( : f (t > 0, ω(t f (t = ω(s f (sp(s, t, t T} (2.64 s where p(s, t = p(i(t, I(s wth s, t T. Ths leads to the product form theorem [46, 73]: Assume, there exsts a functon f ( F and a functon {g( : Z P property, that for all t T and m + I(t n the reachablty graph g(m + I(t g(m + I(s = f (t f (s R} whch have the (2.65 whenever p(t, s > 0. Then the equlbrum dstrbuton of the SPN s gven by: where G s a normalzaton constant. p(m = G g(m (2.66 In order to check f a SPN has a product form soluton t s necessary [46]: to check the nput output crtera and to fnd f g( exsts (analytcally or algorthmcally (reachablty graph. In [72] Henderson and Taylor further extended ther results by analyzng a general dscrete tme model, fndng jont equlbrum dstrbutons at adjacent tme ponts, allowng for arbtrary dstrbutons of frng and enablng tme perods and usng the dscrete tme model as a bass for an embedded SPN analyss. Bouchere and Sereno [19, 20], Donatell and Sereno [46] and Coleman, Henderson and Taylor [35] further extended and generalzed these results. 37

60 CHAPTER 2. STATE OF THE ART Through ths research, a crteron based on mnmal closed support T-nvarants was establshed. Let x 1,.., x h denote the mnmal support T-nvarants 7, then [6, 20]: Closed Set For T T defne R(T, the set of nput and output bags for the transtons n T, as R(T = t T {(t O(t} s a closed set f for any g R(T there exst t, t T such g = I(t, as well as g = O(t, that s f each output bag s also an nput bag for a transton n T. Then the followng structural constrant s the key to the dentfcaton of a product form SPN [6, 20, 70]: Structural Constrant An SPN s sad to be closed ff t T are covered by mnmal closed support T-nvarants,.e. assume, that t T there exsts an {1,.., h} such that t x and x s a closed set. If ths structural constrant s true, there exsts a postve soluton of the traffc equatons [20]. These traffc equatons of the routng process [73] are the global balance equatons of the correspondng Markov chan [70] and calculate the vst ratos v(i(t j as follows [70]: v(i(t j = v(i(t h p(i(t h, I(t j, T j T. (2.67 t h T Followng Haddad et al. [70], a postve soluton for the traffc equatons s not a suffcent condton to assert a product form soluton for an SPN, but a frst step n the fndng of a product form soluton [6]. In order to state, that an SPN has a product form soluton, one other condton [35, 73] has to be fulflled. Product Form for Equlbrum Dstrbuton of SPN [35, 70, 73] Let f = v/µ wth v a soluton for the traffc equatons. The equlbrum dstrbuton for the SPN has the form: π(m = 1 G P =1 y m m RS(M 0 (2.68 f and only f Rank(C = Rank( [ C w f ] where [ C w f ] s the ncdence matrx C augmented wth the row w f and G as a normalzaton constant. In ths case the P component vector r = [ log(y 1,.., log(y P ] satsfes the matrx equaton r.c = w f. wth [70]: w f = [ log ( ( ] f (I(t1 f (I(t T,.., log f (O(t 1 f (O(t T (2.69 So t must be noted [70], that the condton Rank(C = Rank( [ C w f ] depends on the rates and not only on the structure of the net. 7 T-nvarant [20]: A vector x N0 M s a T-nvarant f x = 0 and Ax = 0. The support of a T-nvarant x s the set of transtons correspondng to non-zero entres of x and s denoted by x,.e. x = t T x t > 0. A T-nvarant x s mnmal, f there s no other T-nvarant x such that x m x m m. A support s mnmal f no proper nonempty subset of the support s also a support of a T-nvarant. 38

61 2.2. TIME AUGMENTED PETRI NETS Based on ths results, Haddad, Moreaux, Sereno and Slva [70] establshed a polynomal tme algorthm (O( T 2 to decde whether a SPN has a product form soluton. Furthermore they developed a rate ndependent structural characterzaton of product form SPN. Also, they nvestgated some untmed propertes and some reachablty propertes of product form SPNs. Florn and Natkn [52] developed a product form soluton for a specal class of SPN called closed synchronzed queueng networks. Closed synchronzed queueng networks have the followng propertes [52]: The underlyng Petr Net s a monovaluated place-transton net 8. The underlyng Petr Net s bounded for ts ntal markng 9. The reachablty graph of the underlyng Petr Net s strongly connected (for ergodc propertes. The stochastc Petr Net s a Markov Stochastc Petr Net. The frng rates of the transtons are markng ndependent. The product form soluton for closed synchronzed Queueng Networks s based on results of Gordon-Newell [64] and matrx product form solutons. Balbo, Bruell and Sereno [5, 6] extend the nvestgatons on product form solutons for generalzed Stochastc Petr Nets. They state, that every GSPN that satsfes the known structural constrant: Structural Constrant [6] A GSPN s sad to be closed ff t T there exsts a mnmal T- semflow, such that t x and x s a closed set. and an addtonal crtera: Free-Kllng-Conflct [6] A GSPN s sad to be a free-kllng-conflct f any extended conflct set ECS(t 10, that nvolves mmedate transtons havng the same prorty level has the followng property: t a t b = 0 or t a t b = 0 or t a t b = 0 t a, t b ECS(t (2.70 Whch means, that the frng of an enabled mmedate transton dsables all the other (enabled transtons of ts same ECS(t where t = { p j I(t, p j > 0 } s the preset of the transton t and t = { p j H(t, p j > 0 } s the nhbton set of the transton t admt a soluton for ts traffc equatons and are therefore structurally sutable for a product form soluton. To determne weather a GSPN admt product form soluton, Balbo, Bruell and Sereno [5, 6] apply the results of [35, 70]. 8 Monovaluated Place-Transton Net [52]: In a monovaluated place-transton net, tokens can be added or removed from a place only one by one - non grouped arrval or departure from queue. 9 Bounded [105]: A Petr Net s sad to be bounded, f the number of tokens n each place could not exceed a fnte number k for any markng reachable from the ntal markng. 10 Extended Conflct Set [6]: The extended conflct set ECS(t of a transton t ncludes all the transtons that are n structural conflct wth t as well as those frngs may dsable t (ndrect dsable. 39

62 CHAPTER 2. STATE OF THE ART Queueng Petr Nets (QPN Queueng Petr Nets (QPN [9, 10, 13, 14] combne Tme Augmented Petr Nets and the Queueng Theory from the perspectve of Tme Augmented Petr Nets. In Queueng Petr Nets queues are ntegrated nto colored GSPNs (CGSPN. In ths knd of Petr Nets the type of queueng places s defned n addton to the ordnary places, n order be able to compactly defne dfferent schedulng and queueng strateges. The queueng places consst of a queue and a depostory for tokens, as llustrated n fgure Fgure 2.14: Queueng place and Queueng Place Shorthand Notaton [14] If a token s fred onto ths place, the token s unavalable to all connected transtons untl the servce n the queue was not fnshed. When the servce, accordng to the Kendall Notaton of the queue (lke descrbed n secton 2.1.3, s fnshed, the token s transmtted to the depostory and s then avalable for all connected transtons. Through ths extenson of the colored GSPN, schedulng strateges could be ntegrated nto the defnton of the net wthout the need of a complex GSPN notaton of the defned strategy [14]. An exemplary QPN s llustrated n fgure Fgure 2.15: QPN Example [14] For the analyss of quanttatve system propertes, QPNs must often be analyzed by nspectng the reachablty set of the correspondng colored GSPN where the queue s modeled wth CGSPN elements [14]. Though ths lmtaton, they stll suffer from the state space exploson problem. 40

63 2.2. TIME AUGMENTED PETRI NETS Product Form Queueng Petr Nets One approach to cope the state space exploson problem n QPNs, s to ntroduce product form solutons for Queueng Petr Nets [11, 12]. A QPN s defned by a trple (CGSPN, P 1, P 2 where CGSPN s a colored GSPN, P 1 s the set of queueng places and P 2 s a set of ordnary places (P = P 1 P 2 s the set of places of the CGSPN. Furthermore, the CGSPN s a 4-tuple (CPN, T 1, T 2, W, where CPN s the underlyng Coloured Petr Net, T 1 T s the set of tmed transtons, T 2 T s the set of mmedate transtons (T = T 1 T 2 s the set of transtons of the CPN and W = (ω 1,.., ω T s an array whose entres are possbly markng dependent frng rates of tmed transtons or frng weghts of mmedate transtons [11]. The QPNs, for whch a product form soluton was establshed, s then defned as follows [11]: Product Form QPN [11] A product form QPN s a QPN wth P 1 T 1 = 0 satsfyng the followng restrctons: 1. All transtons are uncolored. 2. The nput bags of mmedate transtons are dsjont or equal and do not ntersect wth the nput bags of tmed transtons. 3. The relatve frng frequences of mmedate transtons do not depend on the markng of the QPN. 4. Output places of mmedate transtons are not nput places of mmedate transtons. 5. No two tmed transtons have the same nput bag. 6. The nput bag of each transtons s the output bag of some other transton and vce versa. 7. The markng dependent frng rate of each tmed transton t T 1 can be expressed as r(m, t = ϕ(m I(tχ(t, where ϕ, χ and φ are arbtrary non-negatve functons. φ(m 8. Input transtons of queueng places have no further output places, output transtons of queueng places have no further nput places and only sngle tokens arrve at and leave from a queue. 9. All queues n queueng places p P 1 are of product form type. The soluton of such product form QPNs follows deas of Henderson et al. [35, 72, 73] and BCMP Networks [8]. Through restrctons 2-4 all mmedate transtons could be elmnated. Restrctons 5-7 are the restrctons for product from SPNs. Through restrcton 8 each queueng place could be replaced by ts smplfed nterpretaton, and through restrcton 9 the global balance equatons reduce to the defnng equatons for functon f [11]: F = { } f : T (0, : χ(t f (t = χ(s f (sp(s, t, t T s T (2.71 So the product form QPN could be transformed nto a product form CGSPN and furthermore an product form SPN (SPN-skeleton, and therefore product form Queueng Networks and product form SPNs are specal cases of product form QPNs [11]. The calculaton of the performance values could then be further optmzed by aggregaton and dsaggregaton methods lke descrbed n [11] and [12]. 41

64 CHAPTER 2. STATE OF THE ART 2.3 Quanttatve Herarchcal Modelng Herarchcal modelng s the key to cope wth complexty n the modelng of quanttatve systems. Therefore ths secton gves an overvew of related work n the area of quanttatve herarchcal modelng. In order to classfy the dfferent approaches and to clarfy the understandng of herarches, some defntons n are gven n advance: There are dfferent knds of herarches: organzatonal herarches wth super-/subordnate relatons and abstracton herarches wth compose/decompose relatons [143]. In organzatonal herarches one nstance s the superordnate to a herarchcally lower nstance. As shown n fgure 2.16(a, n ths knd of herarches the elements are of the same type - both the boss and the employee are persons and they do not compose or decompose to each other. In contrast to ths, n the abstracton herarchy, some parts are composed to a whole. The elements are of dfferent types, as llustrated n fgure 2.16(b. A car s composed by t s parts, so the parts are dfferent from the car and the car s dfferent from the sngle parts. Also n the abstracton herarchy the upper herarchcal nstances not necessarly have to exst (as they could be abstract. (a Elements of same Type Organzatonal Herarchy (b Elements of dfferent Type Abstracton Herarchy Fgure 2.16: Herarches Furthermore, these two herarches could also exst n parallel. As an example, a servce request, handled by a boss could be further decomposed to sub-requests, handled by the employee. In ths case there s a relaton between the dfferent vews, as the doman representaton of the boss as the handler of the request s decomposed nto the sub-requests handlers, represented by the employees. In ths abstracton herarchy the dfferent elements are not the same, as the the boss s nterpreted as the handler and the employees are nterpreted as subhandlers. But nevertheless, the organzatonal herarchy boss-employees stll exsts n parallel, as the dfferent nstances are not only abstract enttes. The decomposablty n subsecton s an abstracton herarchy, as parts of the systems are decomposed nto sub-parts. Norton s Theorem, descrbed n subsecton 2.3.2, could also be used n the development of abstracton herarches (or model abstractons. The (sub-parts or components of the system have to be carefully chosen n order to defne correct herarches and not only a flat refnement of the system or a system wth wrong herarchcal borders or herarchy volatons. The models and technques shown n subsecton and also try to decompose the models represented as Queueng Networks or Petr Nets nto smaller subsystems n order to reduce the complexty of the model analyss. These herarches are model abstracton herarches. The Forced Traffc Flow Law, summarzed n subsecton 2.3.5, as well as the Layered Queueng Networks (LQN, summarzed n subsecton 2.3.6, wth a decomposton of a system nto components or nested requests s also amed to be an abstracton herarchy. But, as the rght herarches are crucal for a correct system understandng, one has to be careful to model the rght herarches and herarchy transformatons. 42

65 2.3. QUANTITATIVE HIERARCHICAL MODELING Decomposablty Smon and Ando [122] ntroduced decomposablty and nearly-decomposable systems n order to support aggregaton of varables n the analyss of large and complex systems. Through the aggregaton of varables on dfferent levels, the overall state-space of the model and therefore the complexty of the evaluaton of such system could be reduced. Completely decomposable systems [38] are systems n whch state varables could be classfed n groups where: nteractons wthn groups can be studed as f nteractons among groups do not exst and nteractons among groups can be analyzed wthout referrng to the nteractons wthn groups. Ths hypothess s correct [38], but often to rgorous. As mentoned, Smon and Ando [122] ntroduced nearly completely decomposable systems. In the analyss of such systems, there are good approxmatons [38] when nteractons among groups are weak compared to the nteractons wthn groups. Smon and Ando [122] showed that n such systems short-run dynamcs and long-run dynamcs can be dstngushed: short-run dynamcs represent the nteractons of the varables wthn each subsystem and long-run dynamcs represent the nteractons among the subsystems. Therefore, the subsystems could be analyzed by local equlbrums and the whole system could be analyzed by aggregate varables and a global equlbrum. Courtos [38, 40] transferred the results of Smon and Ando to Queueng Networks and computer performance analyss. He also ntroduced a herarchy of aggregate varables n order to cope wth the complexty of large systems. Through the dssecton of systems nto subsystems wth dfferent models, the dfferent models could be evaluated separately and through the representaton of the subsystems n aggregate varables, these systems could be analyzed at dfferent levels. Courtos also mentoned that, through ths technque and the related statespace parttonng, t s possble to analyze the dfferent subsystems wth dfferent methods lke Queueng Theory, smulaton and determnstc models. In the herarchcal aggregaton technque of Courtos there remans a known and lmted approxmaton error [39, 126]. Ths error s n the same order of magntude as the maxmum degree of couplng between the dfferent aggregaton varables. As a result of ths the aggregaton varables or systems should be well condtoned and ndecomposable [39]. Courtos also ntroduced multlevel aggregaton [39]. A specal case n the aggregaton of matrces are nearly-completely decomposable matrces whch are also block stochastc [39]. Here the egenvalues of the aggregates matrces are summatons of the egenvalues of the orgnal matrces, whch yelds to exact values for the steady state probabltes. Then the remanng error results only from the aggregaton of stochastc matrces to subsystems. Based on ths results, Vantlborgh [126] ntroduced condtons under whch the aggregaton yelds to exact results. He states and proves that the Perron-Frobenus egenvector 1112 of an 11 Egenvector [22]: For the square matrx A of type (n, n the value λ s the egenvalue and the vector x ( x = 0 s the egenvector, calculated as: A x = λ x. 12 Perron-Frobenus Theorem [111, 123]: The Perron-Frobenus Theorem deals wth postve matrces (A > 0 and ther egenvalues. 43

66 CHAPTER 2. STATE OF THE ART aggregated matrx s correct f, and only f the Perron-Frobenus egenvectors of the non aggregated base matrces are subparallel 13 to the Perron-Frobenus egenvector of the aggregated matrx. The outcome of ths s that the aggregatve analyss of a system through the successve analyss of the subsystems yelds exact results for all steady-state dstrbutons, f and only f the Perron-Frobenus egenvectors of the customer behavor matrces of the subsystems are subparallel to the Perron-Frobenus egenvector of the customer behavor matrx of the aggregated system. Vantlborgh also proves that the equlbrum dstrbuton of the aggregated system s equal to the condtonal dstrbutons on the servers of the subsystems condtoned on the customers served of queued at the servers, f and only f the Perron-Frobenus egenvectors of the subsystems are subparallel to the Perron-Frobenus egenvector of the aggregated system Norton s Theorem Norton s Theorem of the electrcal crcut theory states that n an electrcal crcut of voltage and current sources and resstors (passve components t s possble to replace a subsystem by a sngle current source and a parallel nternal resstance wth the same equvalent behavor, as shown n fgure In ths example fgure 2.17(a shows the orgnal crcut, fgure 2.17(b shows the current source equvalent or Norton Equvalent Crcut and, addtonally n fgure 2.17(c the voltage source equvalent or Thévenn Equvalent Crcut s shown. (a Orgnal Crcut (b Current Source Equvalent (Norton Equvalent Crcut (c Voltage Source Equvalent (Thévenn Equvalent Crcut Fgure 2.17: Norton s Theorem [41] Referrng to [81] ths current source equvalent was publshed n parallel n 1926 by Edward Lawry Norton [107] and Hans Ferdnand Mayer [103], who called ths Ersatzschema or Ersatzschaltung (equvalent crcuts. Both results are based on the studes on Hermann von Helmholz and Léon Charles Thévenn, who also were apparently unaware of the work of each other. Therefore the Norton Theorem sometmes s also called Helmholtz-Thévenn, Helmholz- Norton or Mayer-Norton Theorem [81]. Chandy, Herzog and Woo [30] adapted ths results for the use n Queueng Networks n order to replace a subsystem by a sngle composte queue, as shown n fgure Subparallel [126]: An m-element vector v s subparallel to a vector u = [u 1 u 2..u n ], n > m f there exsts a scalar k such that v = k[u 1 u 2..u n ] 44

67 2.3. QUANTITATIVE HIERARCHICAL MODELING p 1,2=0,5 2 1 I/O 2=0,5 1 4 CPU 1 =1 p 1,3 =0,5 I/O 3 =0,5 (a Orgnal Network 3 CPU 1 =1 Composte I/O (b Transformed Network Fgure 2.18: Norton s Theorem for Queueng Networks - Example [30] They proved Norton s Theorem for closed Gordon-Newell Networks and showed that t s also applcable to open networks and networks whch satsfy local balance. Wth ths results t s possble to replace a subsystem by a composte queue n order to smplfy the analyss of the rest of the system. They proved that for closed networks the queue length dstrbuton for a queue n an equvalent network s the same as n the gven network, and the queue length dstrbuton at arrval for a queue n the equvalent network s also the same as n the gven network. For open networks wth exponental servce tmes and Posson arrvals they state that the queue length dstrbutons for all queues n a subsystem n the equvalent network are the same as n the gven network. They also show that Norton s Theorem holds for the class of networks whch satsfy local balance. Here n a closed network the servce rates at the composte queue s set equal to the throughput to the short wth the same number of servce requests n the short. For open networks all unnterestng queues n the subsystem are replaced by a sngle composte Posson source whch generates servce requests of all classes, where each class s generated ndependently. The generaton rates are set equal to the throughput through the short of the subsystem under consderaton. Balsamo and Iazeolla [7] extended Norton s theorem for the use of any number of queues and arbtrary nterface between the subsystem and the rest of the system. They proposed a soluton for BCMP-Networks [8] wth one class of customers (extenson to several classes s mmedate [7]. They frst constructed a closed network, called the reduced network, where the servers of the subnetwork are shorted (the servce tmes are set to zero and the servce rate of the composte queue s set to the throughput of the orgnal network wth the same number of servce requests n the network. They proved that the margnal probabltes for the queues and the queue length dstrbutons at arrval on the queues are the same for the orgnal and the equvalent network. They also descrbed an algorthm for the soluton of the reduced network and the equvalent network. Walrand [127] proved that Norton s Theorem s also true for multclass quas reversble networks. The results of the transmsson of Norton s Theorem to Queueng Theory are nterestng n the development of aggregaton methods n the herarchcal modelng and analyss, as well as the development of model transformatons. However, n the related work dscussed n ths secton Norton s Theorem s used for the study of a subsystem wthout solvng for the entre network by means of a model transformaton. The dea s to replace the unnterestng parts of the entre model by the equvalent model and to study the behavor of the rest of the model then - herarches are not mentoned there. When consderng herarches, some questons arse. Takng a deeper look n the model understandng, Norton s Theorem s clear to understand for electrcal crcuts, because electrcty knows nothng about servce requests and servce request herarches - t s just a physcal flow balance. But, when consderng Queueng Networks wth 45

68 CHAPTER 2. STATE OF THE ART servce requests and servce request herarches, these herarches and herarchcal transformatons have to be consdered. Servce requests have to be served by specalzed servers and n order to defne sub-requests, servce requests have to be transformed. Also at branches electrcty s drven by resstances of the dfferent paths, whereas the path of the servce requests are defned by control flow decsons (or smplfed by probabltes. But anyhow, the usage of Norton s Theorem n the Queueng Theory could lead to new deas and formulas n the development of transformatons, but, correct model and servce request transformatons always have to be consdered n order to reach to meanngful results Formal Herarches and Combnaton of Models Malhotra and Trved [100] propose a formal expresson of abstracton herarches n model specfcaton and soluton. In ths methodology the models are a composton of dfferent herarchcal models. Every model s treated as a black box, and n the herarchcal overall model the nput and output varables are combned wth the help of an nterconnecton matrx. Though the treatment as a black box and the output and nput connecton many dfferent types of sub models are possble. If parameters are passed from model M to model M j, an order s defned. The relaton further defnes the transtve closure between the models. If the overall model s acyclc (If M M j then not M j M, the soluton of the overall model s acheved through the solvng of the model from the most-nner sub model to the overall model. If the overall model s non-acyclc, an teratve soluton s possble for the soluton of the overall model. Bause and Buchholz [12] use aggregaton and dsaggregaton n ther Product Form Queueng Petr Nets [11], descrbed n secton Ther results for the exact aggregaton are based on Norton s Theorem. Balbo, Bruell and Ghanta [4] propose a technque n whch Queueng Network models and Generalzed Stochastc Petr Nets are combned n a herarchcal modelng approach. In ther approach the herarchcal subsystems, whch do not volate the BCMP-Theorem [8], are modeled as Product Form Queueng Networks (PFQN. Subsystems, whch volate the BCMP assumptons, lke systems wth synchronzaton of processes or smultaneous possesson of resources by a process, are modeled as Generalzed Stochastc Petr Nets (GSPN [102]. Later the dfferent results are combned to an overall soluton. Through ths combnaton t s avoded that the whole system s modeled as a GSPN model whch results n a state space reducton. In ths approach every subsystem s evaluated n solaton and the results are then aggregated. They refer to the decomposablty approach of Courtos [38, 40] and the approxmaton error of ths approach. They [4] state that, through the solvng of the sngle sub models n solaton wth the approprate exact model, only the aggregaton would lead to an approxmatve error Herarches n Tme Augmented Petr Nets Bucc and Vcaro [23] propose Communcatng Tme Petr Nets (CmTPN. CmTPNs augment the basc model of Petr Nets wth nhbtor arcs, the tmng constrants of Tme Augmented Petr Nets, and a module construct whch permts the decomposton of a complex model nto smaller sub models [23]. Ths module construct conssts of wrtng and readng ports connected through communcaton lnks to transtons and places that are referred as wrtng transtons and readng places. An example n gven n fgure 2.19, showng two modules N 1 and N 2, where each module has a readng port left and a wrtng port rght, respectvely up and down. In the 46

69 2.3. QUANTITATIVE HIERARCHICAL MODELING nternal vew t can be seen that through the readng lnks left? and the wrtng lnks rght?, up? and down? these ports are connected to the readng places p 0, p 1 and p 3 as well as the wrtng transtons t 2, t 4 and t 5.! "# "# (a Module N 1 (External Vew (b Module N 1 (Internal Vew (c Module N 2 (External Vew (d Module N 2 (Internal Vew Fgure 2.19: Communcatng Tme Petr Nets Example - Modules [23] Wrtng and readng ports of the dfferent sub models can then be connected through channels to support the communcaton and nteracton between the dfferent sub models, as shown n fgure 2.20, where the two modules of fgure 2.19 are connected and composed n order to ntegrate them n a larger scenaro. (a Connecton Dagram (b Composed Module Fgure 2.20: Communcatng Tme Petr Nets Example - Composton [23] The analyss of such models conssts of two steps: unt analyss and ntegraton analyss. In the unt analyss the dfferent subsystems are analyzed. The dfferent sub models are separated from each other and communcate only through nterfaces. Through that nterfaces the envronment of each sub model s defned. Then every sub model s analyzed smlar to a Tme Augmented Petr Net. In the ntegraton analyss the whole model s computed by aggregatng the results of the dfferent subsystems. Haddad and Moreaux [69] combne aggregaton and decomposton n order to evaluate Petr Nets. In ther understandng aggregaton reduces the state space by groupng states and solvng the correspondng Markov chan on the set of state classes. In ther decomposton method, they follow the decomposton of Plateau and Fourneau [114] (referrng to [69], n whch the state space s a cartesan product of smaller state spaces. In ther artcle they also propose the concept of nternal and external synchronzaton, where nternal synchronzaton means synchronzaton wthn one class and external synchronzaton means synchronzaton between several classes. Buchholz [24] proposes the herarchcal structurng of Superposed Generalzed Stochastc Petr Nets (SGSPN. In hs work he presents a technque n whch a herarchcal structure s generated n a preprocessng step n order to compute a compact generator matrx. Through ths 47

70 CHAPTER 2. STATE OF THE ART approach the problem of unreachable states s avoded. In the approach macro states and the generaton of a macro transton system are proposed. Macro states combne states of a component state space, and a set of macro states defnes a partton of the component state space. Each macro state represents a set of detaled states. In the macro transton system the state transtons of one macro state to another (not locally are defned. Through ths macro transton system a global reachablty analyss s performed. The unreachabllty of a global macro state mples the unreachablty of a detaled state, whch allows the excluson of unreachable states. Frehet and Zmmermann [60] propose a methodology for the automatc decomposton of models. Ther results are based on the decomposton of Stochastc Petr Nets but could be extended to other modelng technques. Ther dvde and conquer approach s dvded nto three steps: In the frst step the system s decomposed nto smaller subsystems. In the second step the dependences between the dfferent subsystems are derved and extracted n low-level subsystems and an aggregatve basc skeleton. In the thrd step the performance values of the system are computed by an teratve approxmaton method. Another approach to reduce the state-space exploson problem n Queueng Petr Nets (QPNs, besde the Product Form Queueng Petr Nets descrbed n secton 2.2.5, are herarchcally combned Queueng Petr Nets (HQPN [15]. In HQPNs a tmed place can contan a whole subnet. Ths subnet has a dedcated nput and output place, where the tokens fred onto places n the subnet are delvered onto the nput place and the output place s smlar to the depostory of a tmed (queueng place. An example s gven n fgure In fgure 2.21(a an solated HQPN subnet s specfed. Ths subnet s then ntegrated nto a larger subnet n fgure 2.21(b. Fnally, ths larger subnet s then represented by a shorthand notaton n fgure 2.21(c. (a Subnet (b Integrated Subnet (c Shorthand Notaton Fgure 2.21: Herarchcally combned Queueng Petr Nets (HQPN - Example [15] For the calculaton of the performance values of the whole net the state spaces and transton matrces for each subnet are generated n solaton [15], and then the overall state space could be determned through a combnaton of the sub states of the subnets. In [15] the authors state that through the herarchcal decomposton the calculaton of the performance values could be reduced by one order of magntude. But, referrng to memory constrants n the calculaton [15], an ntroducton of several herarchcal levels would not extend the sze of the 48

71 2.3. QUANTITATIVE HIERARCHICAL MODELING solvable problems, and so the reducton of the computaton complexty s, as stated, one order of magntude Forced Traffc Flow Law The Forced Traffc Flow Law (also called Forced Flow Law [43, 68, 79, 95] s one of the man laws for herarchcal decomposton. Through the Forced Traffc Flow Law the herarchcal servce request transformaton of the server requests nto subrequests s possble, whch enables the herarchcal decomposton. In the Forced Traffc Flow Law an arrval rate λ wll be transformed nto an arrval rate λ through the use of a traffc flow coeffcent v : λ = v λ (2.72 In Dennng and Buzen [43] the Forced Traffc Flow Law s used for the decomposton of the overall jobs nto requests. They defne unts for the servce request n the defnton of rates lke servce rates. In ther two level models the overall servce request s called job and the sub requests are called request. In ther defnton of the operatonal analyss the traffc flow coeffcent s called vst rato and expresses the mean number of requests per job for a devce. Lazowska et al. [95] extend ths and descrbe the specalzaton of the term request on dfferent levels of detals. For example, a request at a dsk s called dsc access, and a request at the level of the entre system could be defned as a user-level nteracton. They furthermore make use of the Forced Traffc Flow Law n combnaton wth Lttle s Law for the calculaton of performance values on dfferent levels. Whle the man dea of usng the Forced Traffc Flow Law for herarchcal decomposton s very constructve, the man flaw n the approach of Dennng and Buzen [43], as well as the approach of Lazowska et al. [95], s n the modelng only from the vew of the statc server structures. In these networks of servers and queues, as shown n the example n fgure 2.22, the herarchcal control flow and herarchcal servce request transformatons are not modeled, lke they could be modeled usng (Tme Augmented Petr Nets. Through ths nadequate representaton the servce requests have only a dfferent namng at dfferent servers, but there s no real transformaton shown. Fgure 2.22: Memory Constraned System [95] 49

72 CHAPTER 2. STATE OF THE ART Although there are dfferent levels (1-4 n the model n fgure 2.22, the servce request transformatons are not vsualzed. Also the dfferent vst ratos are not represented n ths model. In ths vew the routng and admsson of the overall request to the dfferent servers s modeled. But, ths vew does not necessarly map to the real servce request structures. Furthermore, as the servce request transformatons are not modeled, t s not clearly defned whch knd of (sub-request flows are n whch part of the model. One could argue that nsde every dashed box the correspondng servce request type s exstent, but then, n the example, the dsk request s also n the queue of the CPU (whch could be corrected by extendng the level 1 box around the queue, but furthermore, the dsk request s transfered back to the termnal. Where s the dfference between the requests gong out from the CPU and the requests to the Dsks? In the mathematcal model the flows could be correct, but vewng and questonng the model from the modelng vew rases a lot of queston lke these. Also n Haas and Zorn [68] the Forced Traffc Flow Law (there: Verkehrsflussgesetz s used for the decomposton of a system nto subcomponents. Through the decomposton the overall request s departed nto subrequests at every component through the vst rato V and the throughput D at every component could then be expressed as D = V D. Furthermore, they defne a dstncton of the dfferent throughputs for every servce request class. But as n the former publcatons referenced n ths subsecton, the statc server vew s modeled as Queueng Networks of queues and servers wthout the modelng of the control flows, whch rase the same questons Layered Queueng Networks (LQN Layered queues are an extended queueng network to handle nested multple resource possesson, where the nestng s defned by layers [59]. Referrng to [59], ther central dea s to model networks, where servces may have nested servces executed by another server, wth nestng to any depth. Whle Layered Queueng Networks (LQN [56, 59] subsume a lot of research n ths area [59], s very nterestng to nvestgate. The Layered Queueng Networks (LQN were ntroduced n 1996 by Franks et al. [56]. In [56] the basc LQN model and the LQN prncples were dscussed. Parallelsm effects were added to the approach n [53]. Furthermore, the method of complementary delays [71] was ntegrated and extended to LQNs n [53]. In [54] two-phase servers are ntegrated nto the LQN Mean Values Analyss calculaton approach. Multservers wth several classes n LQNs were ntroduced n [55]. In [108] and [110] the replcaton of structures n LQNs s ntroduced. [109] concentrates on Dependablty and the performance modelng of a quorum pattern. [59] gves an overvew of the whole approach as an entry pont for further nvestgatons. In LQN the layered servce request structure s modeled wth a specfcaton of vsts, customer classes wth arrval rates or populatons and devces wth servce dscplnes. In LQN dagrams, as shown n fgure 2.23, there s a dstncton between logcal servers and multplexers, where the logcal servers are named as software servers or tasks (parallelograms and the multplexers are named as hardware servers and are represented as crcles. Tasks can make requests to any other entty. 50

73 2.3. QUANTITATIVE HIERARCHICAL MODELING Fgure 2.23: LQN Example - Fle Server Applcaton - LQN [56] The LQNs, the actvtes could be modeled n executons graphs whch provde parallelsm, sequence, branchng and loops. An exemplary actvty graph s shown n fgure 2.24.!" # $ % Fgure 2.24: LQN Actvty Graph Example - Quorum Consensus [109] The dynamc behavor could also be modeled n sequence dagrams, as shown n fgure The dfferent nested servce calls to the dfferent servers could also be nterpreted as an organzatonal herarchy, whereas here the servce decomposton s manly done from the vew of the server structures and clent-server / master-slave relatons, whch could be contradctory regardng to the real servce request structures. 51

74 CHAPTER 2. STATE OF THE ART Fgure 2.25: LQN Example - Fle Server Applcaton - Sequence Dagram [59] For the calculatons of the performance values n LQN an LQN Solver (LQNS [57] has been developed [59]. Ths solver derves the performance values by solvng a set of related submodels, where each of the submodels s solved usng a Lnearzer algorthm of the Mean Values Analyss [59]. The algorthm of the LQNS s sketched as follows [59]: LoadModel ExtendModel Topologcal Sort Layerze (create and ntalze layer submodels repeat Solve the layer submodels usng Lnearzer MVA untl converge or teraton lmt Save results Where the solvng of the submodels s sketched as [59]: f or all Clents do Calculate mported servce and thnk tmes. end f or f or all Servers do Calculate mported mean and varance o f servce tmes. end f or solve submodel usng mxed model MVA 52

75 2.4. SUMMARY 2.4 Summary The Queueng Theory, dscussed n secton 2.1, has strengths n the predcton of performance values, especally of Product Form Queung Networks, through a sophstcated and mature mathematcal background wth a well defned set of formulas and fast algorthms. But, as the Queueng Theory focuses on the modelng of server structures and the request flow through ths servers, the control flow s neglected n classcal queueng networks. Ths leads to a lack of expressveness n modelng power f there are a lot of control flows n the modeled system. Tme Augmented Petr Nets, descrbed n secton 2.2, model the systems from the perspectve of the control flow and especally through Colored Generalzed Stochastc Petr Nets or Queung Petr Nets complex systems could be modeled. However, ths often results n a state-spaceexploson problem n the calculaton of the performance values, and even n Product Form Petr Nets complexty s stll a problem. Also through neglectng the server structures and only modelng the dynamc behavor and the control flow, the modelng of shared resources and specfc schedulng strateges s problematc. Herarches are the key to cope wth complexty n the modelng of large systems. As descrbed n secton 2.3, there are some approaches to ntegrate herarches n quanttatve modelng and evaluaton. Anyhow, these approaches are rare and the mportant herarchy law, the Forced Traffc Flow Law, s rather neglected. The new methodology FMC-QE, descrbed n the followng chapters, tres to cope these problems n a herarchcal modelng from the perspectve of the servce requests. In secton 4.4 the author wll come back to some of the questons rased here and dscuss them n comparson to FMC-QE. 53

76

77 Chapter 3 FMC-QE Fundamentals Ths chapter descrbes the fundamentals of FMC-QE, the Fundamental Modelng Concepts for Quanttatve Evaluaton. It starts wth a summary of the foundatons of FMC-QE, FMC (the Fundamental Modelng Concepts and FMC-eCS (the Fundamental Modelng Concepts extended for Communcaton Systems. After that some basc defntons are gven n secton 3.2. Ths ncludes the descrpton of the term servce request whch s of man concern n FMC-QE. Man prncples of herarchcal modelng n FMC-QE are also descrbed besde the ntroducton of the man quanttatve measures, defned n the scope of FMC-QE. In secton 3.3 the graphcal notatons are ntroduced. Ths ncludes the servce request structures, defned n Entty Relatonshp Dagrams, the statc (server structures, defned n Block Dagrams and the dynamc behavor ncludng the control flows, defned n Petr Nets. After that, n secton 3.4, the rules and formulas of the performance measures n the calculus and the derved Tableau are descrbed. Ths ncludes fundamental laws, expermental parameters, the descrpton of the Tableau ncludng a short ntroducton nto model transformatons and a complexty analyss. In secton 3.5 an Open Queueng Network s modeled wth focus on the descrpton of the transformatons and the precson of the predctons for ths class of problems. Model transformatons n FMC-QE are only roughly dscussed, whle ths topc s more focused by the work of Tomasz Porzucek [115, 116], another PhD-Student n the Research Group. Hs work s n the scope of the development of an FMC-QE Tool and n secton 3.6 ths s referenced. 55

78 CHAPTER 3. FMC-QE FUNDAMENTALS 3.1 Foundatons In ths secton the foundatons of FMC-QE, the Fundamental Modelng Concepts (FMC and the Fundamental Modelng Concepts extended for Communcaton Systems (FMC-eCS are summarzed Fundamental Modelng Concepts (FMC The Fundamental Modelng Concepts (FMC are a modelng technque, developed to support the communcaton about nformaton processng systems [92]. The begnnngs of FMC are dated back to a workshop ntated by Segfred Wendt n 1974 and were constantly evolved [92, 124, 130] and used [66, 82, 91]. The author of ths thess has receved hs FMC background through lectures wthn the software systems engneerng studes and hs master thess, analyzng and modelng an Enterprse Resource Plannng (ERP System wth FMC [90]. The human to human communcaton and the support n the understandng of nformaton processng systems are n the focus of FMC and therefore the man concerns for ths modelng technque are [92]: Abstracton, Smplcty, Unversalty, Separaton of concerns and Aesthetcs and secondary notaton. In ths context abstracton means the ablty to descrbe systems on dfferent levels of abstracton n order to reduce the complexty of the models. Smplcty s acheved through restrctng the modelng technque to a few fundamental concepts and notaton elements n order to be able to create models ad-hoc for example n meetngs. Wth FMC t s possble to model a broad range of systems wthout beng bound to a specfc paradgm. Separaton of concerns for reducng the complexty and the ablty of descrbng dfferent aspects of a system are acheved through the three-dmensonal modelng space of compostonal structures, behavor and data / value structures. In FMC aesthetcs and secondary notaton are supported though modelng and vsualzaton gudelnes and easy formaton of the graphcal patterns [2, 92]. In FMC the systems are modeled n three dfferent dmensons [92]: Compostonal Structures, Behavor and Data / Value Structures. The compostonal structures are modeled n Block Dagrams, referrng to [92], based on the German ndustral standard DIN6620 [44]. FMC uses Petr Nets [75, 113] n order to descrbe the behavor of systems. Data and value structures are descrbed n Entty Relatonshp Dagrams, orgnally defned by Chen [31]. In the followng the dfferent dagram types are shortly descrbed: 56

79 3.1. FOUNDATIONS Compostonal Structures - Block Dagram The compostonal (statc structures are represented n Block Dagrams [92]. FMC Block Dagrams are bpartte graphs wth the dstncton of actve and passve system components. Actve components (Agents are represented n rectangles, passve components (channels and storage are represented as round elements. The actve components process nformaton, whle on passve components, nformaton s stored or observed. The passve components are further dstngushed n non-volatle storages and volatle channels. The components are lnked through drected and undrected edges, whch represent wrte, read and read/wrte accesses. An example of an FMC Block Dagram s gven n Fgure 3.1. Fgure 3.1: FMC Block Dagram - Example In the example there are two actve components Agent A and Agent B connected through a channel X. Addtonally, Agent A has a modfyng (read/wrte access to storage Y. Control Structures - Petr Net The control structures of the modeled system are descrbed n Petr Nets [92]. These bpartte Event Condton [75] nets are used to descrbe the sequence of actons and events [92] observed n the system. These actons happen n a certan temporal order (partal order because of concurrency dependent on the state of the system and system specfc rules [92]. Through Petr Nets these rules and the causal order of the events or actons could be descrbed. In FMC actons are represented by a transton of a Petr Net, graphcally represented by a rectangle. The crcles n a Petr Net are the places. Places could be marked (wth a dot nsde or reman empty. Places and transtons are lnked through drected arcs. The correspondng acton of the transton s performed f the transton fres. Ths s possble f [92]: all ts nput places (connected to transton - arc ends at transton are marked and all ts output places (connected to transton - arc starts at transton are unmarked (empty (strct transton rule [105]. After the frng [92]: the correspondng acton has been performed, all nput places are unmarked and all output places are marked. 57

80 CHAPTER 3. FMC-QE FUNDAMENTALS These rules could also be extended to places wth a capacty greater than one and arc weghts greater than one. Therefore see [92, 105]. If, n a case of a conflct (two transtons are ready to fre (enabled and share on nput place, the alternatves depend on a specfc condton (predcate, ths condton s wrtten next to the nput arc and the frng wll depend on ths predcate [92]. In the dstncton of operatonal and control states (see the Petr Net models the control flow and a certan markng represents the control state of the system. Operatonal states of the system (observable on channels or storages are modeled though condtons or descrbed n the actons of the transtons. Fgure 3.2: FMC Petr Net - Example In the example FMC Petr Net, shown n Fgure 3.2, Acton A wll be performed frst. Then, dependng f Condton X s true or not, ether Acton B or Acton C wll be performed. Value Structures - Entty Relatonshp Dagram After descrbng the compostonal structures n Block Dagrams and the behavor and control flow n Petr Nets, the value structures n the storages and channels (operatonal state are defned n FMC n Entty Relatonshp Dagrams [92], adapted from Chen [31]. In ths also bpartte graph enttes are represented as round nodes (wth labels and attrbutes nsde possble and the relatons as rectangles. At the lnks between the relaton and the entty a number (cardnalty defnes how many tmes an nstance of an entty takes part n the relaton. Ths s further supported by arrows n the relatonshp-rectangle (1 to 1: ; 1 to n: ; n to m no arrow [1]. The roles of an entty n a relaton could also be noted at the lnks to the relaton. Entty sets could be parttoned through embeddng the parttonng enttes n the parttoned entty or through a trangular shape connected to the dfferent enttes. In the example n Fgure 3.3 there are the enttes Person, Locaton and Country. A person has the attrbutes Name and Gender, a locaton Name and Area and a country Name. The set of persons s parttoned nto Male and Female and the set of locatons s parttoned nto Ctes, Towns and Vllages. The person lves n up to n dfferent locatons as a Habtant and a locaton has up to m habtants callng ths locaton ther Place of Resdence. Furthermore, the locaton s Part of 1 country whch conssts of n locatons. A locaton could be (0,1 a Captal of a country and every country has 1 Captal. 58

81 3.1. FOUNDATIONS "# $ %! Fgure 3.3: FMC Entty Relatonshp Dagram - Example Operatonal State vs. Control State In FMC [92, 124, 131] the dstncton between operatonal and control states [63, 129] s mportant. The crteron for ths dstncton s based on semantcs (the purpose of the varables [92]: Dstncton Crtera - Operatonal and Control States [92, 131]: Whether a state varable has to be classfed as an operatonal varable or a control varable, depends on how the doman of that varable s defned. Ths s then further specfed by: Crtera - Operatonal State [92, 131]: The doman of an operatonal varable can be specfed wthout temzng each transton between the possble values, explctly. and: Crtera - Control State [92, 131]: The doman of a control varable can only be specfed by explctly temzng each value and each permssble transton between those values. The approprate means of representaton s a graph. [92] and [124] furthermore compare operatonal and control varables as shown n table

82 CHAPTER 3. FMC-QE FUNDAMENTALS Crtera of dstncton Convenent Representaton Power of the doman Extensblty of the doman Namng the varable s doman Operatonal Varables Explct temzaton of each value and each transton between possble values unnecessary Algebrac Notaton, E/R-Dagram Vrtually unlmted Easy: for nstance, extendng the doman of some counter varable Meanngful labels always possble Control Varables Every value and every transton needs to be temzed explctly Graphs lke Petr Nets Lmted by the human need to understand the control flow Dffcult: extensons are modfcatons of the control flow Labelng has no meanng: only sngle states have dedcated meanng Table 3.1: Operatonal vs. Control Varables [92, 124] Fundamental Modelng Concepts extended for Communcaton Systems (FMC-eCS In FMC-eCS, the Fundamental Modelng Concepts extended for Communcaton Systems [135, 145], the herarchcal modelng wth a specal regard on ntegrty, consstency and crtcal actons s n the man concern. It s based on FMC and can be seen as a prelmnary step to FMC-QE, whle the herarchcal modelng and the treatment of servce requests s already n the vewpont. In FMC-eCS there s a great mportance attached to terms and defntons and therefore the man defntons are gven n ths subsecton. As n FMC (descrbed n secton 3.1.1, and later n FMC-QE, the dstncton of operatonal and control states s also regarded n FMC-eCS. Therefore, the extended defntons on operatonal and control states n FMC-eCS are also provded. As the modelng of servce request handlng and herarchcal modelng n FMC-eCS s the man brdge to FMC-QE, ths wll also be provded n ths secton. Besde the references from Werner Zorn [135, 136, 145], ths subsecton s also based on a draft verson of the PhD-thess of Renhard Höllerer [74]. Terms and Defntons Followng the fundamental deas of mutual excluson, loosely connected processes and semaphores of Djkstra [45], n FMC-eCS the ntegrty of content and the handlng and modelng of crtcal actons and crtcal locatons are mportant, therefore these terms are defned as followed [136, 145]: Crtcal Acton / Crtcal Secton Sequence of actons on crtcal contents on a crtcal actonfeld where at least one nconsstent system state could be reached. 60

83 3.1. FOUNDATIONS A crtcal secton s llustrated n fgure 3.4. In an mplementaton, where the crtcal secton s guarded by a semaphore, the transton at the begnnng of the crtcal secton could be assocated to the P-Operaton [45] and the transton at the end could be assocated to the V-Operaton [45]. Another example for a crtcal acton could be a transacton n a database system whch has to be executed completely (or not at all, otherwse the data n the system could be nconsstent. In ths case the database s consstent before and after the transacton and n between t could be n-consstent, whch s guarded by the transacton operatons. Fgure 3.4: Crtcal Secton [145] Crtcal Locaton / Crtcal Actonfeld Locaton (channel or storage, whose content s attackable by a thrd party (agent wth no agreement concernng any nteracton Fgure 3.5 llustrates a crtcal actonfeld. An example for such a crtcal locaton or actonfeld could be a data transmsson channel, where the data s transferred from Agent A to Agent B. In between the agents, the content (data could get corrupted by nose on the channel, whch s the Thrd Party n ths case. Fgure 3.5: Crtcal Actonfeld [145] Crtcal Content / Crtcal Values Content, whose ntegrty s mandatory. Integrty Absence of damages to crucal features of contents ncludng nformaton about such features. In ths noton there could be [136]: 1. Crtcal values on crtcal actonfelds, 2. Crtcal values on un-crtcal actonfelds, 3. Un-crtcal values on crtcal actonfelds, 4. Un-crtcal values on Un-crtcal actonfelds, 61

84 CHAPTER 3. FMC-QE FUNDAMENTALS whereas only n the frst case (crtcal values on crtcal actonfelds crtcal actons are necessary. FMC-eCS furthermore dstngushes between consstent and n-consstent system states as followed [136]: Consstent System State A system state s consstent f nsde the modeled system a specfc subject wll only allow assertons whch do not lead to any objectons. In an FMC-eCS Petr Net, as shown n fgure 3.4, ths state s llustrated by a token on a green place. In-Consstent System State A system state s n-consstent f nsde the modeled system the same subject wll allow assertons whch, when vewed from dfferent perspectves, agent may object to. The objectons may be lodged aganst exstng as well as future system states f ther occurrence s dependent on the nteracton of external agents. In FMC-eCS ths s llustrated by a token on a red place, lke shown n fgure 3.4. Types of Crtcal Actons In FMC-eCS there s a dstncton between unsecured, partally secured and secured crtcal actons as [136]: Unsecured Crtcal Acton Crtcal actons whose all or nothng executon s not guaranteed. Partally Secured Crtcal Acton Crtcal acton whose future system state s known as correct or ncorrect (wthout any correcton Secured Crtcal Acton Crtcal acton whch shall execute completely or not at all. In secured crtcal actons all thrd party actons have to be guarded aganst. (a Unsecured (b Partally Secured Fgure 3.6: Types of Crtcal Sectons [145] (c Secured An unsecured crtcal acton s llustrated n fgure 3.6(a. An example could be a recever of data, who just receves the data, but does no know (undefned f the data s corrupted or not. In a partally secured crtcal acton, lke shown n fgure 3.6(b, the recever would then know (for example through a party bt f the data s correct or not (error. In a secured crtcal acton, lke llustrated n fgure 3.6(c, n the example, the data would be receved correctly (all or the data would be corrupted (known through the party bt and then deleted (nothng or re-sent by the sender. 62

85 3.1. FOUNDATIONS Jont Actons Crtcal actons are often defned among cooperatng agents, where the consderaton of both agents s mportant to know the overall states of the system. One example s the modelng of a clent server scenaro, where one crtcal acton s dependent from another crtcal acton, lke shown n fgure 3.7(a, wth a correspondng abstract short notaton n fgure 3.7(b. (a Clent/Server (b Short Notaton Fgure 3.7: Jont Acton - Clent/Server [135] Another example of jont actons s a producer/consumer, llustrated n fgure 3.8, where agan fgure 3.8(a shows the Petr Net (smplfed and fgure 3.8(b shows the correspondng short notaton. (a Producer/Consumer (b Short Notaton Fgure 3.8: Jont Acton - Producer/Consumer [135] In addton to the clent/server and producer/consumer example, several other scenaros are possble. A summary of the descrbed examples and further scenaros are llustrated n fgure

86 CHAPTER 3. FMC-QE FUNDAMENTALS (a Sngle User (b Independent User (c Clent/Server (d Peer to Peer (e Three Ter Server (f Producer/Consumer (g Ppelne Fgure 3.9: Jont Acton - Short Notatons [135] Operatonal and Control States n FMC-eCS Lke n FMC, as descrbed n secton n FMC-eCS, there s a dstncton between operatonal and control states. In combnaton wth the dstncton nto consstent and nconsstent states ths leads to the followng defntons [74]: Consstent Control State A consstent system state s a state at the begnnng or the end of a crtcal secton, where the correspondng atomc operaton has not started yet or s already fnshed. Inconsstent Control State An nconsstent control state s a state nsde the crtcal secton n whch t s unclear f the crtcal secton could be successfully passed or the system has to be reseted to the consstent start state. The actual behavor could then be dependent from agents whch are not n a protocol-relaton to the actual processng agents (thrd partes. Consstent Operatonal State A system s n a consstent operatonal state f there are no conflctng statements on the system state possble. Inconsstent Operatonal State A system s n an nconsstent operatonal state f conflctng statements on the same system state at the same tme are possble. Furthermore, the resettng of the control automaton of the system mples the resettng of the operatonal automaton of the system. 64

87 3.1. FOUNDATIONS Herarchcal Servce Handlng of Crtcal Actons In FMC-eCS the herarchcal modelng s of man concern as n FMC-QE. A servce request on one herarchcal level could create another servce request on another herarchcal layer, whereas the servce request on one layer s then transformed nto servce requests n the other herarchcal layer. If the servce request at the lower layer s fulflled, the servce response s submtted back to the hgher layer where ths response s then handled. Through ths vew, also the error handlng could be modeled on every fne graned herarchcal level to vsualze t n the model. If ths vew on the system s combned wth the dfferent types of crtcal actons and the dstncton n operatonal and control states, the followng three servce handlng classes are defned [74, 146]: Unrelable Servce In the unrelable servce, as n fgure 3.10 t s unclear, f the result of the crtcal acton on layer n + 1 s correct, therefore the result has to be checked agan at layer n n order to prove f there were no errors and to reset n case of errors. Fgure 3.10: Unrelable Servce [146] Checked Servce In a checked servce, as n fgure 3.11, the lower layer n + 1 can notfy the layer n f the crtcal secton was passed successfully and the result s correct or f there was a problem concernng the servce executon and the result s corrupted. Here the layer n + 1 has no error handlng, but through the error detecton at ths layer there s no need for an error detecton at layer n, only an error handlng s needed. 65

88 CHAPTER 3. FMC-QE FUNDAMENTALS Fgure 3.11: Checked Servce [146] Transactonal Servce In the transactonal servce n fgure 3.12 the crtcal secton n layer n + 1 s passed all or nothng. On layer n the system state s consstent. Only n the case of nothng the state s stll consstent, but the crtcal secton possbly has to be passed agan. Fgure 3.12: Transactonal Servce [146] 66

89 3.1. FOUNDATIONS At the tme the FMC-eCS models n fgures have been developed the development of FMC-QE was already n progress. Therefore, these two developments nfluenced each other and there are a lot of smlartes from these models to the ones later explaned n secton 3.3.3, lke the dstncton of operatonal (gray and control tokens (red - nconsstent state, green - consstent state and whte - control tokens lke dle or busy as well as the ntermedate transtons at the borders of the transtons, here amongst others nterpreted as the semaphore operatons p(s and v(s, n FMC-QE nterpreted as admsson control and departure control. 67

90 CHAPTER 3. FMC-QE FUNDAMENTALS 3.2 Basc Defntons Ths secton provdes basc defntons n FMC-QE. Ths ncludes the descrpton of servce requests, the herarchcal modelng and man quanttatve measures Servce Request The modelng of servce requests s a central concern n FMC-QE. Whle n Queueng Theory the central nterest s the statc structure of the server system and the dynamc behavor s mplct n the statc structure and n Tmed Petr Nets the nterest les n the dynamc behavor and the statc structure beneath s mplctly modeled, n FMC-QE the system s modeled from the three dmensonal vew of the servce request. Comng from the defnton of the servce request structures, both the statc and the dynamc behavor are shown. Because of the specal nterest n the servce requests further defntons are gven n ths secton. Some of the followng defntons do not have to be modeled for every system, but once all of the FMC-QE performance analysts have to thnk about ths topc n order to model ther systems n a rght way. In physcs and engneerng scences nearly all measurements or varables are gven as a tuple of value (here notated n {} and unts ([]. In the Queueng Theory these unts for servce requests are almost gnored. So for a servce rate the unt s one per second, but then t s hard to dstngush between dfferent servce request types and t s not possble to defne herarchcal servce requests. In FMC-QE servce request unts are defned. A servce request SRq s the Servce Request of type. To start wth a smple example, the request SRq could be to fll a car s tank. Then the unt of ths request s the fllng of the tank of car XY. Because of the dfferent szes of the tanks of the dfferent cars, there has to be a unfed servce request N e wth value 1 per defnton. In ths example t would be flled one lter gasolne n a tank of a car (n ths case there s no dstncton between dfferent fuel-types. So wth a capacty of 50 lters the request would be defned as: 1[SRq ] = 50[N e]. So the normalzed servce request N s a multple of the unfed servce request N e. By that defntons t s possble to map dfferent servce requests (cars wth dfferent tank szes to one servce staton. In the modelng and evaluaton of systems the frst transformaton from servce requests to normalzed servce requests s often done ntutvely and s therefore abstracted. When speakng about the evaluaton of systems, the normalzed servce request N s often ntutvely taken and when the dstncton between SRq and N s not of specal nterest, the servce request s taken as synonym to the normalzed servce request. In the evaluaton of FMC-QE models t s mportant to dstngush between servce requests or jobs n the server whch are n servce (N,s and the ones n the queue (N,q. So the servce requests n the servce staton (queue + server are: N = N,q + N,s. (3.1 In the evaluaton t s also useful to defne n = {N } as the number of servce requests n a staton and analogous n = n,q + n,s for the number of requests n the queue and n servce. 68

91 3.2. BASIC DEFINITIONS If a servce request SRq s fulflled, the correspondng servce response s denoted as SRs. An unnormalzed servce response s further defned as: SRs = {SRs } [SRs ]. (3.2 In order to defne a mappng between dfferent servce responses and servers, the normalzed servce request s denoted as: wth the unfed servce response N e,r. N r = {N r } [N r ] ( Herarchcal Servce Requests The herarches of the servce request structures are n the man focus n FMC-QE, because the servce request n the orgn of every servce provsonng process [143] and the herarchcal decomposton s the key to cope wth complex systems, whereas herarches are defned as: Herarchy [143]: A herarchy mples a servce request decomposton on one layer nto dfferent servce requests on a lower layer. The systems are modeled from three dfferent herarchcal vews: the servce request structures, the (logcal server structures and the dynamc control flow behavor. Furthermore, the servce requests are strctly modeled as a tuple of {Value} and [Unt]: SRq = {SRq } [SRq ] (3.4 n order to make the herarchcal decomposton possble through the servce request transformaton on the herarchcal borders of the model. On the bass of canoncal conventons SRq s transformed to N and therefore: N = {N } [N ] SRq = {N e } [Ne ]. (3.5 The fundamental laws of the quanttatve evaluaton used n FMC-QE are Lttle s Law [98, 143]: N = λ and the Forced Traffc Flow Law [43, 68, 79, 95, 143]: R (3.6 λ = v λ [bb 1] parent( (3.7 In the scope of herarches [143] Lttle s Law defnes relatons nsde an herarchcal level 1 (horzontal. Accordng to Lttle s law, the number of servce requests on the herarchcal level 1 The notaton of the herarchcal level n FMC-QE follows the notaton of the lexcographcal level (LL n [25, 26] 69

92 CHAPTER 3. FMC-QE FUNDAMENTALS s a product of the arrval rate λ tme R. of servce requests N per tme unt and the response The Forced Traffc Flow Law defnes nter-herarchcal relatons and defnes therefore vertcal relatons between the herarchcal levels. It s the key to the herarchcal modelng n FMC- QE and was therefore extended [143] wth herarchcal levels (defnton n [143]: λ = v λ [bb 1] parent(, classcal defnton [43, 68, 79, 95]: λ = v λ. It defnes that an arrval rate λ [bb 1] parent( of servce requests N parent( per tme unt on herarchcal level [bb 1] wll be transformed nto an arrval rate λ of servce requests N on the herarchcal level. Ths transformaton s done wth the help of the traffc flow coeffcent v. As already mentoned, n FMC-QE the servce requests are modeled as a tuple of {Value} und [Unt]. Therefore, the Traffc Flow Coeffcent s not only a scalar of {Value} but also the bass for the servce request transformaton from the unt servce request N e[bb 1] to a number of unt servce requests Ne parent( : { v = v } [ v ] { = v } Ne N e[bb 1] parent( { = v } [ ] N e [ ] N e[bb 1] (3.8 parent( Here the servce requests N e[bb 1] on the herarchcal level [bb 1] are the herarchcal parents parent( (parent( of the servce request N e on the herarchcal level. A smple example for a herarchcal servce request could be a grocery lst wth the tems one lter mlk, ten eggs and one loaf of bread. The herarchcal parent s then the servce request Buy everythng from the grocery lst. Ths s then decomposed nto one request Buy one lter mlk, ten requests Buy one egg and one request Buy one loaf of bread wth the correspondng transformatons from the overall request to the sub-requests. In comparson to [43], the traffc flow coeffcent enables now a real herarchcal decomposton of the servce requests nto sub-requests and s not only a vst rato of the mean number of requests per job for a devce n a somehow black box manner. In [43] t was asked: Jobs generate an average of 5 dsk requests and dsk throughput s measured as 10 requests/second. What s the system throughput? Wth the help of FMC-QE the system can be analyzed on multple herarchcal levels and the queston would not only be answered by the rato, but also by questons lke: And from what sub-request(s were these dsk requests requred? Quanttatve Measures n FMC-QE In ths subsecton the quanttatve measures used n FMC-QE are descrbed. The defntons are near to the defntons n the Queueng Theory, gven n secton 2.1, but are redefned here n order to refne them to the herarches and servce requests n FMC-QE. Arrval Rate λ denotes the mean arrval rate of (normalzed servce requests arrvng at server. It s defned as λ = N / t, where t s the mean nter arrval tme. Furthermore, n FMC-QE the overall top level arrval rate λ [1] s calculated as the bottleneck arrval rate λ [1] bott multpled by a correcton factor f (desred bottleneck utlzaton. λ 70

93 3.2. BASIC DEFINITIONS Traffc Flow Coeffcent v As dscussed n the last subsecton, the Forced Traffc Flow Law s a central law to defne herarches n FMC-QE. Therefore the traffc flow coeffcents are also defned n the model. They are defned as v, v,nt as the absolute traffc flow coeffcent of servce request on herarchy level as the relatve traffc flow coeffcent of servce request on herarchy level to the next herarchcal level [bb + 1] and v [bb 1] as the absolute traffc flow coeffcent on the next parent( herarchcal level ([bb 1] relatve to servce request. Multplcty m Wth the parameter multplcty m the number of parallel servers n a server staton s defned. It s possble that m =. In ths case every servce request has a dedcated server. An example could be the user n a clent server scenaro, where every user-nput request has a dedcated user. There are several multplctes defned n an FMC-QE model. In order to support parallelsm on logcal servce request level, the parameters m, m [bb 1] and m parent(,nt are defned. m s the absolute multplcty of the server of servce request on herarchy level. m [bb 1] s the parent( absolute multplcty of the server on the hgher herarchcal level ([bb 1]. m s the relatve multplcty of the server of servce request on herarchy level to the next herarchcal level [bb 1]. So m = m [bb 1] parent( m,nt. On the multplexer server level the number of parallel servers s defned by m j.,nt Multplex Coeffcent m mpx In a multplex scenaro, where one multplexer server s shared among several logcal servers, the multplex coeffcent m,mpx defnes the fracton the logcal server receves form the multplexer. Servce Tme X The servce tme X s the mean tme needed by the logcal server to process a servce request. Ths servce request s located on herarchcal layer. Servce Duraton Y The servce duraton s the mean elapsed tme for the handlng of a servce request. Whle the servce tme X s an nput parameter of the model, the servce duraton, denoted as Y, s a resultng value. If a basc server staton s not handled n a multplex, the servce duraton equals the servce tme, f the basc server staton s multplexed, the servce duraton s longer than the servce tme, because n ths case the servce tme s the tme the servce request s actually handled and the servce duraton s the overall elapsed tme from the begnnng of the processng tll the end (ncludng the breaks for the other multplexed servers. Furthermore, the servce duraton s also a value for herarchcal server statons, whle the servce tme s only a parameter for the basc server statons. 71

94 CHAPTER 3. FMC-QE FUNDAMENTALS Servce Rate µ The servce rate µ s a measure for the maxmal number of servce requests a server could process n a gven tme perod. It wll be represented by µ = SRq / t. Accordng to the dscusson about servce requests and normalzed servce requests n chapter 3.2.1, the normalzed servce rate s denoted by µ e = N e, /X e,. For the sake of smplcty t wll henceforth be assumed normalzaton as default and the superscrpted e wll be omtted, except when needed to remnd of the unts. Utlzaton ρ The utlzaton of a server s denoted by ρ. The Utlzaton s defned as ρ respectvely. = λ µ m Departure Rate D The departure rate s the mean rate fulflled servce requests (servce responses SRs or N r leave the server. Ths s denoted by D = Nr t for mean departure rate of normalzed servce responses leavng the server wth t as mean nter departure tme. In steady state the departure rate equals the arrval rate wth the note of the transformaton of the servce request to the accordng servce response D = λ [N e,r ] [N e]. Mean Number of Servce Requests n The mean number of servce requests n a servce staton s denoted by n wth n { } = N. It s the sum of the servce requests n servce n,s and the servce requests queued n,q. Watng Tme W The watng tme W denotes the mean tme a servce request s queued n a servce staton. Response Tme R The response tme R s defned as the tme nterval between the arrval and the departure of a specfc servce request at a servce staton. So the response tme s the sum of the watng tme and the servce duraton: R = W + Y. 72

95 3.3. GRAPHICAL REPRESENTATION 3.3 Graphcal Representaton FMC-QE follows the dea of FMC of modelng dfferent aspects of the systems n dfferent dagram types. Therefore, the performance analyst models the quanttatve aspects of the the statc archtecture of the systems n Block Dagrams, the quanttatve dynamc behavor n Petr Nets and the relaton between the two plans and the servce request structure n Entty Relatonshp Dagrams Servce Request Structures As already descrbed n chapter 3.2.1, the servce requests are the key to FMC-QE models. The defnton of a servce request tree n FMC-QE s modeled n Entty Relatonshp Dagrams. Referrng to fgure 3.13, a servce request s an entty wth a name and the attrbutes acton and server. The name descrbes the Servce Request n a semantc manner. Due to the dscusson about servce requests and normalzed servce requests n ths dagram all servce requests are consdered to be normalzed and from now on a servce request s normalzed by defnton. Fgure 3.13: Servce Request Entty Besde the defnton of the servce requests the purpose of the Entty Relatonshp Dagram s the mappng between the later descrbed dynamc structure n the Petr Net and the server structure n the Block Dagram. Because of ths the acton whch lnks to the correspondng transton n the Petr Net and the correspondng logcal server n the Block Dagram s stored. Of course, n the process of modelng the three dagrams are constructed n parallel, and so n a frst teraton of the modelng of the servce request sometmes these references are not gven. The strength of FMC-QE s the herarchcal modelng. A global or hgher level servce request can be decomposed nto other servce requests, down to the basc server requests (operatonal servce request, and so a tree structure can be defned. A control servce request s a herarchcal servce request whch controls the process of the chld servce requests. Ths chld servce request can be decomposed nto other servce requests, or t can be an operatonal servce request, whch can be processed by a server. By defnton, control servce requests are tmeless and only the operatonal servce requests consume servce tme. The meta structure of ths tree s shown n fgure dentfes the herarchcal level n the servce request tree. The root has level [1] and chldren or the subtrees have level [bb + 1] and so on. In another layout the herarchcal level could also be noted at the sde of the dagram wth swm lanes separatng the dfferent layers, as llustrated n fgure Fgure 3.14: Entty Relatonshp Dagram Tree Metastructure 73

96 CHAPTER 3. FMC-QE FUNDAMENTALS The traffc flow coeffcent, defned n secton 2.1.2, s a parameter of the servce request and s represented as the cardnalty or the composton relaton n the Entty Relatonshp Dagram as shown n fgure Fgure 3.15: Traffc Flow Coeffcent v The servce request tree s related to an external source and snk (External World - more dscussons n secton 3.4.2, whch s by defnton lnked to the root of the tree. As llustrated n fgure 3.16, both of the requests are on level [1] and the traffc flow coeffcent v 1,nt = v 1 s 1 per defnton. Fgure 3.16: External Servce Request Generator For a better llustraton of ths dagram type a smple barbershop example s modeled n fgure In ths barbershop the customer s served by washng the har, cuttng the har n one of two harcuts (branch n Petr Net, dyng the har or makng a perm and collectng the money.! " # +, $ +&, % & & &' ( ( ( * +-, Fgure 3.17: Barbershop - Servce Request Structure Statc Structures In FMC, as descrbed n secton 3.1.1, the statc structure dagrams (Block Dagrams represent the compostonal structure of systems. Ths s done by usng elementary components of type agent as an actve component and storage as well as channels as passve components, together wth arcs as connectors. In the quanttatve extensons of FMC-QE ths dagram type s also used and extended. In FMC-QE t s another vew on the system, and especally n the statc structure dagrams t s the archtectural or the specfcaton vew. In ths dagram type the logcal servers, followng the servce request structure and the multplexer servers, followng the statc structure of the real systems, as well as queues and channels are consdered. Ths dagram type s closely related to 74

97 3.3. GRAPHICAL REPRESENTATION the Queueng Theory, but n contrast to the Queueng Networks ths dagram only shows the statc structure. The dynamc behavor s not consdered here, ths s the purpose of the Petr Nets. In the Block Dagrams system specfcatons are gven lke servce tmes of servers or buffer szes of queues, whch are partly ndependent from the actual servce request scenaro. Fgure 3.18: Basc Server Staton The most smple combnaton n a Block Dagram s a server together wth a queue shown n fgure Ths combnaton s called basc server staton (BSSt. A net as a combnaton of dfferent basc server statons would be smlar to a Queueng Theory Net. A parameter of the queue would be the capacty of the queue K. In most cases for the sake to smplcty the queue sze s assumed to be nfnte (. A Server would be parameterzed wth a name, ts multplcty and a servce tme or a servce rate. Often the communcaton n real systems s not mplemented va shared memory or fles (Queues. The communcaton s mplemented va channels. For ths fact the queueng staton, shown n fgure 3.19, s defned. The admsson control puts the servce request nto the nternal queue and after the server served the request, the departure control releases the result. Ths vew could be mapped to a programmed envronment, where a procedure call would be represented by the channel. The admsson control s the handler whch takes the call. The server s the procedure and the return value s handled by the departure control. The departure control s tmeless by defnton and s executed mmedately. Fgure 3.19: Queueng Staton The second task of admsson control and the departure control s the herarchcal transformaton of the servce requests. The admsson control transforms the servce request of level [bb 1] nto the servce request n the herarchcal level of the queueng staton. The departure control s responsble for the re-transformaton. In FMC-QE the dspatcher would be tmeless and the workers would consume the servce tmes. For some cases t s helpful to defne a server wth nfnte parallel servers, as shown n Fgure Ths server has no queue because every ncomng request has a dedcated server. In FMC-QE the external source s defned as an nfnte server. 75

98 CHAPTER 3. FMC-QE FUNDAMENTALS Fgure 3.20: Infnte Server The man strength of FMC-QE s the ablty to defne herarches. Often the system descrpton gets complex because herarches neglected and the system s modeled flat. Of course a dagram lke fgure 3.21 looks very complex at the frst glance, but knowng and modelng the system n a fne graned manner, can help for a better understandng of the system and errors and nconsstences n the modelng are found almost mmedately. "!"!! Fgure 3.21: Herarchcal Server Staton These general consderatons about the nner structure of the servers were n the mnds of the FMC-QE developers when mplementng the methodology. Of course for larger systems ths knd of dagrams, as shown n fgure 3.21, would become to large. Therefore, a short notaton was developed. In ths short notaton every logcal server (Herarchcal Server Staton or Basc Server Staton correspondent for the handlng of a servce request s represented by one rectangle annotated by a name and a multplcty m. Also the herarchcal level of the correspondng servce request s denoted. An exemplary herarchcal server staton wth two basc server statons s llustrated n fgure Fgure 3.22: Herarchcal Server Staton - Short Notaton 76

99 3.3. GRAPHICAL REPRESENTATION Whle the logcal servers follow the structure of the servce request, the multplexer servers are also modeled n the Block Dagram. As llustrated n fgure 3.23, the multplexer servers are also named and have a multplcty attrbute. Fgure 3.23: Multplexer Server The mappng between the logcal and the multplexer servers s modeled through a matrx. At the connecton ponts of the matrx the measured servce tme X s notated. Ths s the tme the multplexer server would need to handle a servce request of the logcal basc server when no other servce requests s handled by the multplexer server. An exemplary mappng s llustrated n fgure It s also possble that a multplexer server handles more than one logcal server (multplex.!! # $!!"!!" Fgure 3.24: Mappng between Logcal and Multplexer Servers Fgure 3.25 llustrates the statc structure of the barbershop example. The logcal servers on top are handled by three multplexer servers, the Barber Boss, the Barber and the Apprentce, whereas the boss cuts the harcut 1 and makes the perm, the barber cuts the harcut 2 and dyes the har and the apprentce washes the har and collects the money. 77

100 CHAPTER 3. FMC-QE FUNDAMENTALS * * *,- + *. +!" # $%! # $%! # $%!& # $%!&'( # $% Fgure 3.25: Barbershop - Statc Structures!*' # $% Dynamc Structures The dynamc behavor n FMC-QE s represented n Petr Nets. Wthn a FMC-QE Petr Net the nner most structure s called controlled operatonal transton. Ths transton conssts of an operatonal transton (execute operaton, an nput and an output place for the operatonal transton and a control loop, shown n fgure 3.26 Fgure 3.26: Controlled Operatonal Transton The operatonal transton represents the processng of a basc servce request n a server. The tokens n the nput-place of ths transton represent the servce requests n process and the tokens n the output place represent the servce responses. The places are presented n grey because the tokens are consdered to be colored tokens lke n Colored Petr Nets [80]. The loop at the bottom of the controlled operatonal transton represents the control flow of the server. If the control token (ready s n front of the transton, the server s ready and actve to process a request, and f the token s on the place nsde the transton, the server s busy and not able to process another servce request. The controlled operatonal transton could be ntegrated nto a representaton of the actvtes a queueng staton (admsson control, queue servce requests, process servce requests and the 78

101 3.3. GRAPHICAL REPRESENTATION departure control shown n fgure In ths fgure the controlled operatonal transton s abstracted to the transton called actvty. %!"#$ Fgure 3.27: Dynamc Behavor of a Queueng Staton The admsson control s represented n the black tmeless transton at the left. At ths tme the admsson control dscards no servce requests. In herarchcal systems the admsson control and the departure control perform the servce transformaton. Because of the frng rules of a Tme Augmented Petr Net, there s a dstncton between the servce requests n servce and the ones queued. The token stays before transton n the servce tme, and at the end the transton fres tmelessly and the token jumps from the nput place to the output place. The servce requests n servce are represented n the tokens n the nput place of the operatonal transton. Due to the multplcty m defned n the Block Dagram of the correspondng server, the place would have the capacty m. The queued servce requests are represented n the place top-left. Ths place has the capacty K m also mported from the Block Dagram. The transton between the queue place and the n servce place s a tmeless transton whch represents the start of the servce processng. If the correspondng server has a multplcty m of more than 1, the actvty or the controlled operatonal transton could be actvated more than once. In ths case the nner structure of the actvty could be modeled as shown n fgure A tmeless dspatcher sends the request to the correspondng controlled operatonal transton. Ths fgure s only shown to llustrate the FMC-QE vew on the multplcty and has not to be modeled for every scenaro. Fgure 3.28: Parallel Server - Actvty Refned Comng back to fgure 3.27, t s not necessary to defne the capactes and queue szes here agan, frst the dagrams could be nconsstent, and second the Petr Net gets more complex. 79

102 CHAPTER 3. FMC-QE FUNDAMENTALS But anyway, the performance analyst s free to mport ths parameters here n order to have the nformaton together n one dagram. But as mentoned, the multplcty and the servce tmes are defned n the statc plans and are only mported and not defned here. The black rectangle at the rght n fgure 3.27 represents the departure control. It fres per defnton tmelessly and mmedately, so n the steady state there are zero tokens n the response place. Whle tokens n the fve gray places n fgure 3.27 on the top represent the operatonal states (servce requests and responses, the tokens n the two place at the bottom represent the control states to the system [124, 131, 137]. The tokens n the place n front of the start transton (bottom left represent the capacty of the server. A token n ths place represents that the server s actve to get a request. Ths parameter s also not defned n ths dagram, t s mported from the Block Dagram. The tokens n the place at the bottom mddle represent the number of servce requests n the server. The performance analyst could use Colored Petr Nets n order to defne the operatonal states of the system. Fgure 3.29: Infnte Queues If the queue has an nfnte sze, the staton could enqueue every servce request arrvng at the staton, and therefore, n ths case the actvaton place has an nfnte sze wth an nfnte amount of tokens n t, as depcted n fgure One could argue, that ths nfnte place wth nfnte tokens s rrelevant for the processng of the Petr Net, but nevertheless the place s drawn n order to llustrate the actvaton of the staton. For some cases, lke modelng the external world (servce request generator, there s the need of an nfnte server and an actvty, whch could be actvated nfnte tmes. In ths cases the structure defned n fgure 3.30 could be used. The place n front of the actvty has an nfnte capacty and so the staton s actvated nfnte tmes agan. Fgure 3.30: Infnte Server A herarchcal transton s shown n fgure At the black border transtons at the left and the rght a servce transformaton s processed. At the left border the servce request N s transformed nto v dfferent sub requests N [bb 1]. At the rght border transton the sub servce responses are reassembled. 80

103 3.3. GRAPHICAL REPRESENTATION Fgure 3.31: Herarchcal Transton Agan, the tokens n nfnte capacty places at the bottom represent the number of servce requests n the sub system. Only n exceptonal cases, when a lmted number of servce requests are allowed to be n the sub system, ths place would have a fnte capacty. In the branch, llustated n fgure 3.32, every job n level [bb 1] has to be delvered to one of the actvtes A or B. Insde the level [bb 1] there s no job transformaton. The transformaton s at the ncomng transtons of actvty A and B. Fgure 3.32: Branch An mportant remark to the parallel servce transton n fgure 3.33 s that the traffc on all three outgong arcs of the servce request transformaton transtons at the left are the same. The servce request s delvered to both of the actvtes A and B. The transformaton nto the servce requests of the actvtes A and B (N [bb 1] to N s not done at ths tme, ths s done n the next step, the ncomng transton of the actvtes (admsson control. 81

104 CHAPTER 3. FMC-QE FUNDAMENTALS Fgure 3.33: Parallel Actvtes Besde the parallel actvtes n fgure 3.33, a seral processng of actvtes, shown n fgure 3.34, s also possble. Fgure 3.34: Seral Actvtes One could argue that n a seral processng of servce requests n a herarchcal manner, the servce responses should be redrected to the herarchcal parent after each step and then the herarchcal parent delvers the next sub-request to the next seral handler. For the sake of smplcty ths request-response handlng s smplfed n fgure 3.34 to a sequence n whch every seral sub-request handler delvers the servce response to the next sub-handler. In ths scenaro the servce requests from the herarchcal parent to the herarchcal sub-handlers could also be envsaged as a lst of servce requests n whch every sub-handler marks ts correspondng ser- 82

105 3.3. GRAPHICAL REPRESENTATION vce request as done and hands the lst of servce requests wth ths requests marked as done (servce response to the next sub-handler. In FMC-QE there s a dfferent dstncton between open and closed nets. By defnton the transton on level [1] s lnked to the external load generator, nterpreted as a servce request source and a servce response snk, whch could be mapped to an open net, together wth a tme nterval, whch defnes the tme between a response and a request of a sngle customer. If ths tme nterval would be zero, the soluton could be compared to a closed net. There are some dfferences n the calculaton of closed and open nets and ths hybrd approach, dscussed n secton and 4.1. In fgure 3.35 the most smple FMC-QE Petr Net s shown. A servce request generator and one actvty are connected. In order to get to more complex solutons, the Controlled Operatonal Transton could be replaced by more complex nets. Fgure 3.35: Most Smple FMC-QE Petr Net If parallelsm on logcal server level should be explctly modeled, lke the modelng of threads, a sub-net could be replcated and the multplcty could be llustrated by dots as shown n fgure An example of ths s used n the case study n secton 5.3. Fgure 3.36: Parallelsm on Logcal Server Level - Threads 83

106 CHAPTER 3. FMC-QE FUNDAMENTALS In fgure 3.37 the barbershop s used agan to llustrate an example of the dagram type. In ths model frst the har s washed, then cut n harcut 1 or 2, afterwards n a supplemental step ether dyed, permed or left as t s (NOP and fnally, the money s collected from the customer. ' (! # $!"%&!"!" Fgure 3.37: Barbershop - Dynamc Behavor 84

107 3.4. CALCULUS 3.4 Calculus After modelng the system n the three dmensons of FMC-QE, the servce request structures, the server structures and the dynamc behavor, the predctons of the performance values are calculated wth the help of the FMC-QE Calculus. In ths secton the parameters, values and formulas of the calculus are explaned. Alongsde ths the FMC-QE Tableau, a representaton of the Calculus, s shown for every secton. Ths part s structured as follows: In the frst subsecton the man laws of the calculus, Lttle s Law [98] and the Forced Traffc Flow Law [79], are recaptulated. In the expermental parameters secton of the Calculus the expermental parameters are defned, as the name mples. These are the overall number of servce requests n [1] ges, the derved possble bottleneck throughput λ [1] bott, the desred bottleneck utlzaton f and the overall arrval rate λ [1]. Subsecton descrbes ths Expermental Parameters secton and the external load generaton (load model n FMC-QE. In subsecton the Servce Request Secton s descrbed. In ths secton the performance parameters of the servce request, lke herarchy levels, probabltes, traffc flow coeffcents and arrval rates are defned. In the Server Secton, descrbed n subsecton 3.4.4, the performance parameters of the logcal servers ncludng ther mappngs to the multplexer servers are stored. The man calculatons of the performance values are processed n the Dynamc Evaluaton Secton. Ths secton, dvded nto the calculatons for operatonal (parallel and nfnte and control (herarchcal, seral, parallel, branch and loop servce requests, s explaned n subsecton The performance parameters of the multplexer servers are defned n a separate secton, ths Multplexer Secton n explaned n Subsecton dscusses the complexty of the approach. n ges [1] λ bott [1] f λ [1] Expermental Parameters 30 3,750 0,600 2,250 Table 3.2: Tableau Example (see Appendx - Table B.1 Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Cash 1,00 1,00 1,00 1,00 2,250 Casher ,056 1,000 18,000 0,125 0,018 0,008 0,125 0,056 0,143 0, Blow-Dry 0,50 1,00 1,00 0,50 1,125 Blow-Dryer 1 4 0,350 1,000 2,857 0,000 0,000 0,394 0,350 0,394 0, Dry 1,00 1,00 1,00 1,00 2,250 Dryer ,000 0,000 0,394 0,175 0,394 0, Perm 0,30 1,00 1,00 0,30 0,675 Permer ,350 0,467 1,333 0,506 0,519 0,769 0,506 0,750 1,025 1, Dye 0,20 1,00 1,00 0,20 0,450 Dyer ,333 0,125 0,375 0,600 0,675 1,500 1,200 2,667 1,875 4, Supp. Work 1,00 1,00 1,00 1,00 2,250 Supp ,750 1,194 0,531 1,706 0,758 2,900 1, Cut2 0,60 1,00 1,00 0,60 1,350 Cut2-Cutter ,333 0,750 2,250 0,600 0,900 0,667 0,600 0,444 1,500 1, Cut1 0,40 1,00 1,00 0,40 0,900 Cut1-Cutter ,300 0,533 1,778 0,506 0,519 0,577 0,506 0,563 1,025 1, Cut 1,00 1,00 1,00 1,00 2,250 Cutter ,750 1,419 0,631 1,106 0,492 2,525 1, Wash 1,00 1,00 1,00 1,00 2,250 Washer ,111 1,000 9,000 0,083 0,000 0,000 0,250 0,111 0,250 0, Barbershop Serv. 1,00 1,00 1,00 1,00 2,250 Server ,750 2,631 1,169 3,581 1,592 6,212 2, Job Generaton 1,00 1,00 1,00 1,00 2,250 Generator ,572 0,095 0,000 0,000 23,788 10,572 23,788 10,572 Multplexer Secton j Name j m j [1] X j 1 Appentce 5 0,167 2 Barber 1 0,267 3 Barber Boss 1 0,225 4 Customer Server Secton Dynamc Evaluaton Secton The FMC-QE Tableau s a herarchcal balance sheet used for the performance evaluaton of FMC-QE models. The Tableau conssts of three lnked tables. The mathematcal formulas of the Calculus are used to provde support for the herarchcal modelng. The exemplary Tableau, 85

108 CHAPTER 3. FMC-QE FUNDAMENTALS shown n table 3.2, s the correspondng Tableau of the barbershop example, modeled n secton 3.3. In order to ntegrate the handlng of nfnte servers, ths example s extended by the blowdryng of the har by the customer Fundamental Laws In ths subsecton the two fundamental laws used n FMC-QE, Lttle s Law and the Forced Traffc Flow Law, are descrbed on a hgh level as a recap of the more detaled dscusson on herarches and the fundamental laws n secton as a prelmnary for the FMC-QE Tableau. Also the herarchcal levels ( and ndces ( are ntegrated n the notaton of the laws. Lttle s Law [98] s one of the fundamental laws n FMC-QE. It s the bass of the performance evaluaton wthn a herarchcal level (horzontal. The law defnes the dependence between the mean number of jobs n a system n, the mean arrval rate λ and the mean response tme R [98, 125]: n = λ R (3.9 The Forced Traffc Flow Law (FTFL [43, 68, 79] s the second fundamental law n FMC-QE. It s the key to the herarchcal modelng (vertcal. Through ths law and the use of a traffc flow coeffcent v, the FMC-QE performance analyst s able to transform an arrval rate at a herarchcal border. The nternal arrval rate s computed as a product of the external arrval rate and the traffc flow coeffcent [43, 68, 79]: λ,nt = v λ [bb 1],ext (3.10 Besde the transformaton of the value the servce request tself s transformed (unt-transformaton. The external arrval rate λ [bb 1],ext has the unt [external servce requests per tme unt] and the nternal servce request has the unt [nternal servce requests per tme unt]. A detaled descrpton of ths servce request transformaton could be found n secton Expermental Parameters In ths secton the Expermental Parameters of the Calculus are descrbed. Therefore, frst the parameters and the relatons between the parameters are descrbed, after that the load model n FMC-QE s descrbed. Fnally an exemplary cutout of a tableau s shown. The parameters of ths part are: n [1] ges Overall number of top level ([1] servce requests n the whole model (n the system and external; λ [1] bott Maxmum bottleneck arrval rate; f Desred bottleneck utlzaton; λ [1] Arrval rate of top level ([1] servce requests; 86

109 3.4. CALCULUS n [1] ges defnes the overall number of top level servce requests (SRq [1] n the model. Ths ncludes the servce requests n the system (n [1] sys and the servce requests n the external world (clents preparng the servce request or followng other actvtes (n [1] ext. The parameter λ [1] bott defnes the arrval rate, where the bottleneck of the system would have a utlzaton of 1. Ths parameter s derved as the mnmum servce rate of every logcal server (µ multpled by the number of correspondng parallel logcal servers (m normalzed to top level servce requests ( 1 : v ( λ [1] µ bott = mn m v (3.11 In order to avod arrval rates greater than the maxmum bottleneck arrval rate (whch would be a conflct to the stablty crtera - ρ < 1, the parameter f s establshed. Ths parameter defnes the desred bottleneck utlzaton and allows a smple arrval rate adjustment. The overall arrval rate of top level ([1] servce requests generated by the load generator (Outsde World, Clent, Customer s defned as λ [1]. λ [1] s derved as the product of f and λ [1] bott : λ [1] = f λ [1] bott (3.12 In some cases, due to the modeled systems, λ [1] s drectly defned. Then f and λ [1] used. bott are not External Load Generaton / Load Model In FMC-QE there s no dfferentaton between closed and open models lke n tradtonal Queueng Theory or Tme Augmented Petr Nets. Therefore, the most basc FMC-QE model s represented n fgure Fgure 3.38: Basc FMC-QE Model 87

110 CHAPTER 3. FMC-QE FUNDAMENTALS The external load generaton s always represented by a number of clents or load generators n the Block Dagram whch generate the level [1] servce requests. In the Petr Net the behavor of the load generaton clents are descrbed as nfnte servers because by defnton every servce request has a dedcated clent, so the clent would behave lke an nfnte server. In the Entty Relatonshp Dagram every top level servce request has a 1to1 relaton to a servce request generaton. In comparson to closed systems, the external tme s not defned as zero, because when modelng systems, t nearly always makes no sense to enqueue a servce response mmedately as a new servce request. The model here s dfferent. Frst of all, there s a dfferentaton between the servce request whch enters the system and the servce response. The clent does not only handle the request generaton but also the response. The second dfference s that n FMC-QE models, both, the arrval rate (λ [1] and the mean overall populaton (n [1] ges, could be defned. The relaton between these parameters s defned through the General Response Tme Law [27, 43]: R [1] sys = N =1 and the Interactve Response Tme Formula [43]: R v (3.13 modfed to: R [1] sys = n[1] ges λ [1] X [1] ext (3.14 ext = ( n[1] ges R[1] λ [1] sys (3.15 X [1] Steady State Systems - Notaton In order to explan the dfferent load scenaros, fgure 3.39 shows the notaton of a steady state system llustraton (statonary process. The cyclng servce requests or servce responses are represented as one movng dot on a crcle. The tme the dot s n the system (the servce request s served - R sys, t s movng n the n the lower system part. The tme the servce request s n the external clent or the external clent, t moves n the upper external part (X ext. The dstance between the dots can be nterpreted as the nverse of the overall arrval rate 1/λ [1] (nter arrval tme. X ext 1/λ Servce Request / Servce Response External System R sys Fgure 3.39: Steady State Systems - Notaton 88

111 3.4. CALCULUS Steady State Systems - Scenaros Dependng on what parameters are fxed and whch ones are varable, dfferent scenaros could be defned. Here the dfferent, smple but representatve scenaros are explaned through examples, namely Paternoster, Soup Ktchen and Rollercoaster. In the Paternoster example the arrval rate λ [1] and the response tme R [1] are fxed and the total number of cyclng servce request n ges s varable. When addng a new servce request to the model, the paternoster ellpse (external tme X ext wll be bgger or smaller, as llustrated n fgure 3.40, because the arrval rate and also the nterarrval tme 1/λ [1] s fxed. Fgure 3.40: Steady State Systems - Paternoster An example of ths type could be the stuaton n an agency where you have to take a number n order to be served. The agency would know the response tme R sys and the number of customers n servce n sys, so they would provde the new customers wth the watng tme, for that the new customers could leave the agency and could come back after that tme. In that case the watng tme would be mapped to the external tme X ext. The Soup Ktchen model, shown n fgure 3.41, s dfferent. Here the external tme s always the same (people are recevng lunch every day. In ths model the arrval rate s not a parameter, but a value, when addng new servce requests to the model the arrval rate wll grow. Fgure 3.41: Steady State Systems - Soup Ktchen 89

112 CHAPTER 3. FMC-QE FUNDAMENTALS In the thrd example Roller Coaster the external servce tme X ext s zero. In the example the kds usng the roller coasters mmedately enqueue agan after they took the rde, as llustrated n fgure The model can be compared to closed models. In comparson to the paternoster, n both the Soup Ktchen and the roller coaster model more customers generate a hgher arrval rate and as a result of ths longer queues and longer response tmes. But, there s a sgnfcant dfference between the soup ktchen model and the roller coaster. In the soup ktchen there s stll an external source and the overall number of cyclng servce requests has a mean value (populaton but s not fxed. In the closed roller coaster there s no external source. The populaton s fxed and there s an nternal synchronzaton n the model. An teratve algorthm for the calculaton of closed models has also been developed and s descrbed n secton 4.1. Fgure 3.42: Steady State Systems - Roller coaster For the paternoster and soup ktchen model fgure 3.43 shows an example of the broad range of possble confguratons of the mean populaton n ges and the arrval rate λ. In the chart the twelve graphs are the correspondng curves for 1 n ges 12, where the most left curve s the curve for n ges = 1 and the most rght curve s the curve for n ges = X ext [Tme Unts] n ges =1 n ges =12 Paternoster λ = 0,3 [SRq / TU] Soup Ktchen X ext = 20 [TU] 10 0 Soup Ktchen X ext = 0 [TU] 0 0,2 0,4 0,6 0,8 1 1,2 λ [SRq / Tme Unt (TU] Fgure 3.43: Chart External Servce Tme For an exemplary paternoster (n a paternoster the arrval rate s the free parameter and the external servce tme s a resultng value wth λ = 0, 3 [SRq] [TU], the mnmal populaton s n ges = 3 (then X ext = 0[TU] and for n ges = 12, X ext = 30[TU]. For soup ktchen scenaros the chart s read n the dfferent drecton (X ext s the free parameter and λ s the resultng value. In the example a soup ktchen wth X ext = 20[TU], the resultng arrval rates are n the range from λ = 0, 033 [SRq] for n [TU] ges = 1 to λ = 0, 4 [SRq] for n [TU] ges = 90

113 3.4. CALCULUS 12. Accordngly, for X ext = 0[TU] the correspondng arrval rates are at the correspondng ntersecton ponts wth the λ-axs of the curves 1 to 12. More on the Roller Coaster scenaro and the correspondng dscusson of Closed Queueng Networks could be found later n secton 4.1. An exemplary Expermental Parameters secton of the Tableau s shown n fgure 3.3. Table 3.3: Tableau Example - Expermental Parameters n ges [1] λ bott [1] f λ [1] Expermental Parameters 30 3,750 0,600 2, Servce Request Secton The Servce Request Secton descrbes the servce request herarches wth levels, ndces, names, probabltes traffc flow coeffcents and the arrval rates at the dfferent herarchcal levels. The parameters n ths part are: Herarchcal level Index SRq Servce request labelng p parent(, Routng probablty from servce request parent to to servce request v [bb 1] Absolute traffc flow coeffcent on the hgher herarchcal level (parent parent( v,nt Relatve traffc flow coeffcent to the hgher herarchcal level v λ Absolute traffc flow coeffcent Arrval rate of servce requests The herarchy level s derved from the servce request structure defnton n the Entty Relatonshp Dagram. The top-most servce request and the servce request generator have herarchy level [1]. The herarchy level s then ncremented by 1 for every refnement level. The ndex s a unque ndex for every logcal servce request type. The numberng s establshed through traversng the servce request tree n an adapted postorder traversal (rght, left, root. Through ths traversal method the order of the servce request table s also set up. The job generaton, whch s connected to the root of the servce request tree, s the one wth the hghest ndex ( JobGenerator = Root + 1 n ths notaton. Every servce request s named n the column SRq. Ths name s mported from the correspondng entty n the Entty Relatonshp Dagram. The routng probablty from the herarchcally hgher request (parent( to ths servce request (p parent(, s derved from the Petr Net. It s possble that the sum of the probabltes n one herarchcal part s less than 1, because the NOP transtons n a branch are not shown n the 91

114 CHAPTER 3. FMC-QE FUNDAMENTALS correspondng Tableau. As an example: n the barbershop example, as shown n fgure 3.4, the supplemental work s separated n Dye (p Dye = 0, 3, Perm (p Perm = 0, 2 and do nothng - NOP p NOP = 0, 5, so n ths example, n the Tableau, the probabltes of the chldren of the Supplemental Work sum up to 0, 5. The parameters v [bb 1] parent(, v,nt and v specfy the traffc flow coeffcents of the servce request. These parameters are defned n the correspondng Entty Relatonshp Dagram of the model and then mported n the Tableau. In detal, the parameter v [bb 1] mports the absolute value parent( of the traffc flow coeffcent on the next herarchcal level, the parameter v,nt defnes the value of the traffc flow coeffcent relatve to the next herarchcal level, and the parameter v defnes the absolute value of the traffc flow coeffcent of ths servce request multpled by the probablty of the servce request: v = v [bb 1] parent( v,nt p parent(, (3.16 An example: n a car servce the Change Wheels servce request s decomposed nto four servce requests Change Wheel. If the absolute value of the traffc flow coeffcent of the servce request Change Wheels would be 1 (v ChangeWheels = 1 then the absolute traffc flow coeffcent of the Change Wheel servce request would be 4 (4(v = 1(v [bb 1] parent( 4(v,nt 1(p parent(, (under the assumpton, that p ChangeWheels = 1. The last parameter n ths part s the ndvdual arrval rate of the servce request. The arrval rate λ s computed as the overall arrval rate λ multpled by the traffc flow coeffcent of the servce request: λ = λ v (3.17 The unt of the arrval rate s [Servce Requests /Tme Unt]. The traffc flow coeffcent does not only descrbe the value transformaton, t also descrbes the transformaton of the top level servce request (SRq [ 1] to the servce request on ths herarchcal level (SRq. An exemplary Servce Request Secton of the correspondng Tableau s shown n table 3.4. Table 3.4: Tableau Example - Servce Request Secton Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ 2 1 Cash 1,00 1,00 1,00 1,00 2, Blow-Dry 0,50 1,00 1,00 0,50 1, Dry 1,00 1,00 1,00 1,00 2, Perm 0,30 1,00 1,00 0,30 0, Dye 0,20 1,00 1,00 0,20 0, Supp. Work 1,00 1,00 1,00 1,00 2, Cut2 0,60 1,00 1,00 0,60 1, Cut1 0,40 1,00 1,00 0,40 0, Cut 1,00 1,00 1,00 1,00 2, Wash 1,00 1,00 1,00 1,00 2, Barbershop Serv. 1,00 1,00 1,00 1,00 2, Job Generaton 1,00 1,00 1,00 1,00 2,250 92

115 3.4. CALCULUS Server Secton The Server Secton of the FMC-QE Calculus defnes the performance parameters of the dfferent logcal servers. In ths part logcal multplctes (lke processes and threads and the mappng to the multplexer servers are defned. The parameters n ths part are: Server Labelng of the logcal server m [bb 1] Absolute value of the logcal multplcty on the next herarchcal level (parent parent( m,nt Value of the logcal multplcty relatve to the next herarchcal level m Absolute value of the logcal multplcty Mpx Correspondng multplexer server X Measured servce tme for every request at the correspondng multplexer Server labels the correspondng logcal server of each servce request. Ths parameter s orgnally defned n the Entty Relatonshp Dagram and the Block Dagram of the model. The multplcty of the logcal servce requests (logcal server multplcty s herarchcally defned through the parameters m [bb 1] parent(, m,nt and m. Ths logcal multplcty could be compared to the multplcty of threads and processors. It defnes how many servce requests could be logcally processed n parallel. It addton to ths, the multplcty of the multplexer servers (n the example: processors would be defned n the Multplexer Secton. Ths logcal multplcty of the servce request server structure s orgnally defned n the correspondng Block Dagram of the model. The defnton s very smlar to the defnton of the traffc flow coeffcents. m [bb 1] denotes the multplcty of the herarchcal parent of the logcal server, m parent(,nt specfes the value of the multplcty relatve to the next herarchcal level and m defnes the absolute value of the multplcty: m = m [bb 1] parent( m,nt (3.18 The mappng of a logcal basc server (server of a operatonal servce request - the one whch actually does the work to a multplexer server s defned through the parameter Mpx. Ths parameter s orgnally defned n the Block Dagram through the mappng matrx. Ths parameter s empty for non leaf nodes (Control Servce Requests of the servce request tree. The parameter X defnes the measured servce tme for each servce request processed at the correspondng multplexer server. Ths value s taken under the assumpton that the multplexer server does not process other logcal servce requests at that moment (no multplex. Of course, FMC-QE wth the Tableau s sutable to handle multplex scenaros, but for the sake of normalzaton ths assumpton s made. In multplex scenaros the measurements have to be measured on dedcated (non multplexer servers or have to be normalzed. In the model ths value s also defned n the matrx of the Block Dagram. 93

116 CHAPTER 3. FMC-QE FUNDAMENTALS An exemplary correspondng Server Secton of the Tableau s shown n table 3.5. Table 3.5: Tableau Example - Server Secton Server Casher Blow-Dryer Dryer Permer Dyer Supp. Cut2-Cutter Cut1-Cutter Cutter Washer Server Generator Server Secton [bb-1] m p( m,nt m Mpx X , , , , , , , , Dynamc Evaluaton Secton The performance values of the servce request structures are computed n the Dynamc Evaluaton Secton. The values n ths part are: m,mpx Multplex coeffcent µ ρ n,q Servce rate Utlzaton Mean number of queued servce requests W n,s Mean watng tme Mean number of servce requests n servce Y n R Mean servce duraton Mean number of servce requests n the staton Mean response tme In the next paragraphs the dfferent formulas for the calculaton of the performance values of an operatonal servce request (leaf node - Basc Server Staton BSSt and a control servce request (non leaf node - Herarchcal Server Staton HSSt are descrbed. 94

117 3.4. CALCULUS Multplexer - Operatonal Servce Request In FMC-QE t s possble that a multplexer server s n a role of more than one logcal server (real multplex. The multplexer server then s parttoned nto several fractons, correspondng for every logcal server. Ths fracton s called multplex coeffcent (m,mpx. A multplexer n sgnal transmsson s shown n fgure (a User Vew (b System Vew Fgure 3.44: Multplexer/Demultplexer [130] The correspondng actvtes of each of the multplexed logcal servers are handled then n parallel whch results n a longer server duraton Y (as the servce tme X stays constant. An example: as llustrated n fgure 3.45, the servce duratons Y and Y j for the handlng of the a servce request and servce request j handled by one multplexer Mpx are twce as long as the correspondng servce tmes X and X j as the multplexer handles both actvtes n parallel and each logcal server receves half of the multplexer (m,mpx = m j,mpx = 0, 5. Fgure 3.45: Multplex Example [146] In such a multplexer scenaro, as llustrated n fgure 3.46, only the servce requests n servce are transferred to the multplexers, the dfferent queues for every type of servce request are stll at the dfferent logcal servers. 95

118 CHAPTER 3. FMC-QE FUNDAMENTALS Fgure 3.46: Multplex - Servce Requestor s Vew [146] If the sum of all logcal servers handled by a multplexer server s less or equal than the sum of all multplexer servers ( SRq o f Mpx j m m j, every logcal servce request s handled by a dedcated multplexer server - multplex coeffcent s one (m,mpx = 1. In ths case redundant multplexer servers (m j,dle = m j SRq o f Mpx j m would dle. The scenaro of nfnte servers (both, the logcal and the correspondng multplexer servers s a specal case of equal number of logcal and multplexer servers and the multplex coeffcent s also one for ths scenaro (m,mpx = 1. If the sum of logcal servers handled by a multplexer server s greater than the sum of all multplexer servers ( SRq o f Server j m > m j, the multplexer servers have to work n multplex mode. The servce rates n whch a logcal servce request operates s smaller because the mul- 96

119 3.4. CALCULUS tplexer servers are handlng other servce request types n parallel. The logcal servers receve only a fracton of the multplexer server whch s defned as the multplex coeffcent m,mpx. The multplex coeffcent s calculated as a fracton of ths logcal servce request tme X and the overall amount of servce tme of a multplexer server (multplexer needs to handle all of the correspondng servce request of a top level servce request X j multpled by the fracton of the multplcty of multplexer servers (m j and the number of logcal servers. To summarze the calculaton of the multplex coeffcent: m,mpx = 1 f SRq o f Mpx j m m j X v X j m j m f SRq o f Mpx j m > m j (3.19 SRq o f Mpx j m In ths way of calculatng the multplex coeffcent and the servce rate the utlzaton ρ s equal for every logcal server handled by the same multplexer. There are other possble ways to calculate the multplexer coeffcent f SRq o f Mpx j m > m j, lke dependng only on the number of logcal servers and not on the servce tme: m,mpx = m j. The used calculaton was chosen because t was consdered as far, but for certan scenaros other multplexer coeffcents could be more sutable. Comng back to the Calculus, the servce rate µ then s the recprocal of the servce tme, multpled by the multplex coeffcent: µ = m,mpx X (3.20 For the calculaton of the utlzaton (ρ, the mean number of queued requests (n,q and the mean number of requests n servce (n,s there s a dstncton between parallel logcal servers and nfnte logcal servers. Fgure 3.47: Parallel Server For parallel logcal servers, as shown n fgure 3.47, the calculaton of the values s defned as [67] (M/M/m: ρ = λ m µ (

120 CHAPTER 3. FMC-QE FUNDAMENTALS n,q = ( λ µ m! m ( ρ 1 ρ 2 1 m ( 1 m ρ k ( k=0 k! + m ρ m! m ( 1 1 ρ (3.22 n,s = λ µ (3.23 Fgure 3.48: Infnte Server For nfnte logcal servers, as shown n fgure 3.48, the calculaton of the values s defned as [88] (M/M/: ρ = empty (3.24 n,q = 0 (3.25 n,s = λ µ (3.26 The mean number of servce requests n the staton (n and the tmes: watng tme W, and the response tme R are calculated the same way for every server servce duraton Y type. The mean number of servce requests n the staton (n servce requests (n,q and servce requests n servce(n,s : n = n,q s defned as the sum of queued + n,s (3.27 and the watng tme (W as well as the servce duraton Y and the response tme R are computed through Lttle s Law [98] as mean number of servce requests n the queue (n,q, n servce (n,s or n the staton (n dvded by the arrval rate at the logcal server (λ as: W = n,q λ, Y = n,s λ, R = n,q λ. (

121 3.4. CALCULUS In contrast to the servce tme X, the servce duraton Y s not a parameter, but a resultng value. If the correspondng basc server staton s not multplexed, the servce duraton equals the servce tme, but f the basc server staton s multplexed, the servce duraton s the sum of the servce tme and the tme the multplexer handles the other multplexed basc server statons n parallel (elapsed tme. Control Servce Request For Control Servce Requests (non leaf nodes the FMC-QE Calculus provdes the handlng of: Herarchcal Actvtes, Seral Actvtes, Parallel Actvtes, Branches and Loops. The servce rate of the composed servce request (µ s the mnmum of all servce request rates of the seral chldren dvded by the nternal traffc flow coeffcents and the probabltes: µ = mn The utlzaton ( ρ s empty for all Control Servce Requests: µ[bb+1] chld( m[bb+1] chld(,nt v [bb+1] chld(,nt p chldren o f (3.29,chld( ρ = empty (3.30 The calculaton of mean number of servce requests n the staton (n and the response tme (R s the same for all Control Servce Requests: The mean number of servce requests n the staton (n s defned as the sum of the queued servce requests (n,q and the servce requests n servce (n,s : n = n,q + n,s (3.31 As n the basc server staton the watng tme (W, the servce duraton Y and the response tme R, accordng to Lttle s Law [98], are calculated as the mean number of servce requests n the queue (n,q logcal server (λ as:, n servce (n,s or n the staton (n dvded by the arrval rate at the W = n,q λ, Y = n,s λ, R = n,q λ. (3.32 The other dfferent formulas for the calculaton of the mean number of queued servce requests (n,q and the mean number of servce requests n servce(n,s are provded n the next paragraphs. 99

122 CHAPTER 3. FMC-QE FUNDAMENTALS Herarchcal Actvty A herarchcal actvty wth one actvty nsde, as shown n fgure 3.49, s just a specal case of the other herarchcal decompostons lke seral actvtes or parallel actvtes, but nevertheless the formulas are shown here. Fgure 3.49: Herarchcal Actvtes n,q = n [bb+1] chld(,q (3.33 n,s = n [bb+1] chld(,s (3.34 Seral Actvty In a seral actvty servce request, as shown n fgure 3.50, the servce request s herarchcally decomposed nto dfferent seral actvtes. Fgure 3.50: Seral Actvtes The number of queued servce requests (n,q and the number of servce requests n servce n a seral actvty servce request s the sum of the correspondng values of the sngle seral actvtes: (n,s n,q = n [bb+1] (3.35 chld(,q chldren o f 100

123 3.4. CALCULUS n,s = n [bb+1] (3.36 chld(,s chldren o f Parallel Actvtes In a parallel actvty servce request, as shown n fgure 3.51, the servce request s herarchcally decomposed nto dfferent actvtes whch are processed n parallel. Fgure 3.51: Parallel Actvtes The number of queued servce requests (n,q and the number of servce requests n servce (n,s n a parallel actvty servce request are: n,q = max(n [bb+1] chldren o f (3.37 chld(,q n,s = max(n [bb+1] chldren o f (3.38 chld(,s 101

124 CHAPTER 3. FMC-QE FUNDAMENTALS Branch In a branch actvty servce request, as shown n fgure 3.51, the servce request s herarchcally decomposed nto dfferent actvtes, where each of t s processed wth the probablty p. Fgure 3.52: Branch The number of queued servce requests (n,q and the number of servce requests n servce n a branch actvty servce request s the sum of the correspondng values of the sngle seral actvtes: (n,s n,q = n [bb+1] (3.39 chld(,q chldren o f n,s = n [bb+1] (3.40 chld(,s chldren o f Loops In FMC-QE the system s modeled from the perspectve of the herarchcal servce request structures processed by the system. In order to evaluate the model through the FMC-QE Calculus, the model must follow a tree shaped structure. If the modeled system does not follow ths assumpton, due to modelng wth a dfferent modelng technque or specal system behavor, the model s transformed nto ths tree shape structure. Examples for ths behavor are loops. Loops are transformed, and after the transformaton the behavor of the transformed model s equvalent to the behavor of the orgnal loop but analyzable wth the FMC-QE Calculus. 102

125 3.4. CALCULUS Whle Loop: In FMC-QE a whle loop s transformed n order to compute the performance values of the model. Fgure 3.53 shows an FMC-QE model of a whle loop. The loop body s represented by the executon of the servce request SRq. Fgure 3.53: Orgnal Whle Loop In the servce request structures the loop s represented by a herarchy of servce requests. The servce request SRq sup( represents the whole executon of all teratons, whle the servce request SRq represents one sngle loop body executon. The traffc flow coeffcent v,nt represents the mean number of loop teratons. In the Petr Net the nner servce request loop represents the whle loop and the correspondng multplexer server has the multplcty m and the Servce tme X. In the frst step of the transformaton the loop s transformed to a sequence of executons of the loop body (SRq, as shown n fgure Fgure 3.54: Whle Loop Transformaton (Seralzaton 103

126 CHAPTER 3. FMC-QE FUNDAMENTALS Then the sngle executons of the loop body are combned to one executon, n whch the servce tme s the sum of the sngle servce tmes. Ths step s shown n fgure 3.55.! Fgure 3.55: Whle Loop Transformaton (Combnaton In ths transformaton the arrval rate of the servce requests SRq s not changed: λ = λ, (3.41 but the servce tme s the sum of the sngle executons of the loop bodes, respectvely the product of the mean loopcount v,nt and the orgnal servce tme X : X = v,nt X (3.42 Feed Backward Loop: In FMC-QE feed backward loops, as shown n fgure 3.56, are transformed nto a feed forward executon, shown n fgure Fgure 3.56: Feed Backward Loop In the feed backward executon there s a probablty p ret that the executon servce request SRq has to be repeated (e.g. falure. 104

127 3.4. CALCULUS Fgure 3.57: Feed Forward In the feed backward - feed forward transformaton not the servce tme X but the arrval rate λ s transformed. The ncreased arrval rate better corresponds to the real system than the extenson of the servce rate. The new arrval rate λ s the product of the traffc flow coeffcent v,nt and the old arrval rate λ : λ = v,nt λ (3.43 The traffc flow coeffcent s calculated due to the formula of the geometrc sum [16, 65]: as: s = a 1 q n 1 = a 1 1 q n=1 (3.44 v,nt = v,nt 1 p return (3.45 In comparson to the transformaton of the whle loop the servce tme X s not transformed: X = X (3.46 After the transformaton the performance values are calculated lke a normal herarchcal actvty. An exemplary Dynamc Evaluaton Secton of the Tableau s shown n table 3.6. Table 3.6: Tableau Example - Dynamc Evaluaton Secton m,mpx μ Dynamc Evaluaton Secton ρ n,q W n,s Y n R 1,000 18,000 0,125 0,018 0,008 0,125 0,056 0,143 0,063 1,000 2,857 0,000 0,000 0,394 0,350 0,394 0,350 0,000 0,000 0,394 0,175 0,394 0,175 0,467 1,333 0,506 0,519 0,769 0,506 0,750 1,025 1,519 0,125 0,375 0,600 0,675 1,500 1,200 2,667 1,875 4,167 3,750 1,194 0,531 1,706 0,758 2,900 1,289 0,750 2,250 0,600 0,900 0,667 0,600 0,444 1,500 1,111 0,533 1,778 0,506 0,519 0,577 0,506 0,563 1,025 1,139 3,750 1,419 0,631 1,106 0,492 2,525 1,122 1,000 9,000 0,083 0,000 0,000 0,250 0,111 0,250 0,111 3,750 2,631 1,169 3,581 1,592 6,212 2,761 0,095 0,000 0,000 23,788 10,572 23,788 10,

128 CHAPTER 3. FMC-QE FUNDAMENTALS Multplexer Secton In the Multplexer Secton the overall parameters of the multplexer servers are defned to support the calculaton of the ndvdual performance values of the multplexed logcal servers, defned n the operatonal servce request part of the dynamc evaluaton secton, explaned n secton The parameters n ths part are: j Index Name j Multplexer name m j Multplcty X [1] j Aggregated normalzed servce tme In Server j a server name s defned. Ths parameter s mported from the correspondng Block Dagram of the model. The parameter m j s also mported from the Block Dagram and defnes the multplcty of every multplexer server. s the servce tme a server needs to serve ts parts of the top level servce request. It s defned as the normalzed sum of all measured correspondng servce tmes. X [1] j X [1] j = X v (3.47 mpxed SRq For nfnte multplexer servers (m j = the aggregated normalzed servce tme X [1] j s empty and the multplexer coeffcent m,mpx wll be 1 for each logcal server connected to the nfnte server j. In fgure 3.7 an exemplary Multplexer Secton of a Tableau s shown. Table 3.7: Tableau Example - Multplexer Secton Multplexer Secton j Name j m j [1] X j 1 Appentce 5 0,167 2 Barber 1 0,267 3 Barber Boss 1 0,225 4 Customer 106

129 3.4. CALCULUS Computaton Algorthm / Complexty Analyss In order to analyze the complexty of the FMC-QE performance predctons and the FMC-QE Calculus wth the correspondng Tableau, the algorthm of the dervaton of the values wll be sketched. In the frst step the dfferent parameters of the servce request structure tree and the correspondng server structures and dynamc behavor model are mported. Therefore, the tree(s - multple trees n multclass scenaros - are traversed n an adapted postorder traversal (rght left root. These parameters are: n [1] ges and f as expermental parameters,, SRq and v,nt from the Entty Relatonshp Dagram, Server, m,nt, Mpx and X from the Block Dagram and p parent(, and the executon order from the Petr Net. The parameters v parent(, v, m and m parent( have only top-down nterconnectons and could therefore computed as a by-product of the tree traversal. Also the ndex s defned n the traversal algorthm (order of traversal. Also as a part of the mport procedure, the multplexer parameters Name j and m j are extracted from the Block Dagram and the ndex j s defned. Therefore, the lst of multplexers n the Block Dagram has to be traversed, whch s by defnton shorter than the number of nodes n the servce request tree (there can not be more multplexers than logcal servers. Whle n the frst step the model tree s created and traversed, the complexty of ths step s O(n where n s the number of servce request nodes. In the second step the servce request table s traversed n order to aggregate the servce tmes for every multplexer to compute X j. Therefore, every row n the servce request table has to be traversed (O(n - n=number of servce request nodes and the multplexer secton has also to be traversed once (O(n. X In the thrd step the multplex coeffcent m,mpx and the servce rates µ are computed. Therefore, the servce request table s traversed from = 1 to = n. In order to compute the multplex coeffcent, the multplex secton has also be traversed for every basc logcal server. In ths servce request table traversal the mnmum of all normalzed logcal servce rates (see 3.11 s saved n order to compute λ [1] bott. Whle n ths step the servce request table has to be traversed twce and the multplexer table has to be traversed once, the complexty s agan O(n. In the forth and last step the servce request table s traversed one tme from = 1 to = n n order to compute λ, ρ, n,q, n,s, n and R. The values of the logcal basc servers (leaf nodes n servce request tree can be computed drectly and the logcal control servers have to aggregate the values of the sub servce requests. In ths step the servce request table has also to be traversed once and for every node some calculatons (only parameters of the node have to be performed. Therefore, the complexty s agan O(n. Whle all steps n the calculaton of the Tableau are performed wth a complexty of O(n, where n s the number of servce request nodes, the calculaton of the whole Tableau s also performed wth a lnear complexty of O(n. 107

130 CHAPTER 3. FMC-QE FUNDAMENTALS 3.5 FMC-QE Example - Open Queueng Network In the followng example an open Queueng Network (taken from [18] - Example 7.4, page 287f s modeled and analyzed. Ths example has been chosen n order to provde a full example for FMC-QE and to compare the performance predctons of an exemplary open Queueng Network, computed through Queueng Theory methods wth the performance predctons of FMC-QE. Another goal of ths example s an exemplary descrpton of the transformaton of a Queueng Network nto an FMC-QE model Orgnal Model and Calculaton The orgnal model s shown n fgure p 1,2 = 0,5 Prnter 2 p 2,1=1 Source 4 1 I/ODevce CPU p 1,3 = 0,5 Dsk 3 p 3,5=0,4 Snk p 3,1 =0,6 Fgure 3.58: Open Queueng Example - Orgnal Model [18] In ths example four sngle server FCFS nodes wth exponentally dstrbuted servce tmes are connected. The mean servce rates are [18]: µ 1 = 25 [Jobs] ; µ 2 = 33 1 [s] 3 [Jobs] ; µ 3 = 16 2 [s] 3 [Jobs] ; µ [s] 4 = 20 [Jobs] [s] The example s an open Queueng Network wth an exponentally dstrbuted arrval rate of [18]: λ = λ 0,4 = 4 [Jobs]. [s] In [18] the performance values are calculated usng Jacksons Theorem (see In the frst step the arrval rates are calculated [18]: λ 1 = λ 2 p 2,1 + λ 3 p 3,1 + λ 4 p 4,1 = 20 [Jobs] ; λ 2 = λ [s] 1 p 1,2 = 10 [Jobs] [s] λ 3 = λ 1 p 1,3 = 10 [Jobs] [s], λ 4 = λ 0,4 = 4 [Jobs]. [s] 108

131 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK In the second step the dfferent performance values are derved. The utlzatons λ are calculated as ρ = λ µ [18]: ρ 1 = λ 1 µ 1 = 0, 8; ρ 2 = λ 2 µ 2 = 0, 3; ρ 3 = λ 3 µ 3 = 0, 6; ρ 4 = λ 4 µ 4 = 0, 2. The mean number of jobs at a node for an M/M/1 node are calculated as n = A.3 and therefore [18]: ρ 1 ρ (see table n 1 = 4 [Jobs]; n 2 = 0, 429 [Jobs]; n 3 = 1, 5 [Jobs]; n 4 = 0, 25 [Jobs]. The mean response tmes of the servers are computed wth the help of Lttle s Law [98] as R = n λ [18]: R 1 = 0, 2 [s]; R 2 = 0, 043 [s]; R 3 = 0, 15 [s]; R 4 = 0, 0625 [s]. The mean overall response tme s calculated by aggregatng the mean number of jobs at every node and Lttle s Law [18]: R = n λ = 1 λ The mean queue lengths are calculated as n,q = 4 n = 1, 545 [s] =1 ρ2 1 ρ (see table A.3 [18]: n 1,q = 3, 2 [Jobs]; n 2,q = 0, 129 [Jobs]; n 3,q = 0, 9 [Jobs]; n 4,q = 0, 05 [Jobs]. Therefore, the mean watng tmes (calculated as W = ρ µ λ - (see table A.3 are [18]: W 1 = 0, 16 [s]; W 2 = 0, 013 [s] W 3 = 0, 09 [s]; W 4 = 0, 0125 [s]. In the orgnal example n [18] the margnal probabltes and the state probabltes are also computed, but n FMC-QE these values are not consdered and so ths s omtted here. 109

132 CHAPTER 3. FMC-QE FUNDAMENTALS Transformaton In order to compare these results wth the correspondng FMC-QE model, the Queueng Theory model wll be transformed nto the correspondng FMC-QE model. In the frst step an ntal Petr Net, shown n fgure 3.59, was set up. In ths frst Petr Net the behavor of the dfferent servers s represented. The transtons are now labeled wth the actons (Retreve Input, Compute Results, Prnt, Save to Dsk of the dfferent servers and not wth the names (I/O-Devce, CPU, Prnter, Dsk as n the Queueng Theory server structures, because the Petr Net represents the behavoral vew of the model, whch s only mplctly modeled n the Queueng Theory. Ths frst Petr Net has no herarches (flat and s stll an open network.! Fgure 3.59: Open Queueng Example - Intal Petr Net 110

133 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK Due to the fact that n FMC-QE models there s no dstncton between open and closed queueng networks and the outsde world s always modeled (see 3.4.2, n the second transformaton step the load generaton, represented by the acton Generate Request, s added as shown n fgure $! "# Fgure 3.60: Open Queueng Example - Load Generaton An open network wth a fxed arrval rate s then represented by a modfed paternoster model (see n whch the external tme s calculated as a result from the parameters mean overall number of servce requests n [1] ges, arrval rate λ [1] and overall mean response tme R [1] sys: ext = ( n[1] ges R[1] λ [1] sys. (3.48 X [1] The mean overall number of servce requests n [1] ges therefore have to be greater than one and thus have to be greater than the mean number of servce requests n the system n [1] sys: n [1] ges n [1] sys = R [1] sysλ [1]. (

134 CHAPTER 3. FMC-QE FUNDAMENTALS In the thrd step a frst herarchy, the Persst Data Branch, s ntroduced. Ths herarchy abstracts from the decson f the results are Prnted or Saved to the Dsk. Though ths abstracton the return probablty p ret and the fnsh probablty p f nsh = 1 p ret where adjusted n order to add the 50% of the prnted and then returned servce requests: p ret = p 1,2 p 2,1 + p 1,3 p ret = p 1,2 p 2,1 + p 1,3 p 3,1 = 0, 8. The new Petr Net s shown n fgure 3.61.!" # Fgure 3.61: Open Queueng Example - Persst Data Branch 112

135 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK In the next step, shown n fgure 3.62, the seral executon of the Result Computaton and the Persst Data Request are abstracted to the Unrelable Executon. It s an unrelable executon because a return loop envelopes ths executon. $ %& %" ' (# %!"# Fgure 3.62: Open Queueng Example - Unrelable Executon 113

136 CHAPTER 3. FMC-QE FUNDAMENTALS The return feed backward loop around the Unrelable Executon s resolved through transformng the feed backward loop nto an feed forward executon, as shown n fgure The new Relable Executon has an nternal traffc flow coeffcent v,nt calculated as the geometrc sum of the returns: v,nt = v,nt 1 p return. (3.50!" #! &%' # $% " Fgure 3.63: Open Queueng Example - Feed Forward - Feed Backward 114

137 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK Fnally the seral executon of Input and Relable Executon s abstracted to Request, whch s the top-level ([1] servce request generated by the clents. The fnal transformed Petr Net s shown n fgure "#!! " &%'!! $% # Fgure 3.64: Open Queueng Example - Transformed Petr net 115

138 CHAPTER 3. FMC-QE FUNDAMENTALS Servce Request Structure and Statc Structure The correspondng servce request structures of the example are shown n fgure The servce request s parttoned nto 5 herarchcal levels. In ths dagram the ncreased traffc flow between the Relable Executon and the Unrelable Executon (v nt = 5 [UnrelableExecutonRequests] s [RelableExecutonRequests] vsualzed.!!! &!!! ' " # " # #" ( " $ " " $ # % Fgure 3.65: Open Queueng Example - Servce Request Structures 116

139 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK The thrd dagram of the model, the server structures are represented n the Block Dagram, as shown n fgure In ths model the fve herarches of the logcal server structures and the mappngs to the four multplexer servers (CPU, Prnter, Dsk and I/O-Devce are defned. The dfferent servce tmes for the basc servers are also defned n ths dagram. /0 % +, *!% *!" # $% # #, &!%' ((* '' & ((+. & # ((- # & % (( %' Fgure 3.66: Open Queueng Example - Server Structures 117

140 CHAPTER 3. FMC-QE FUNDAMENTALS Summary FMC-QE delvers exact solutons for ths open Product From Queueng Networks. Through transformatons the flat example could be transformed to a herarchcal model. After the transformaton a broad range of performance values could be calculated n the FMC-QE Tableau, as shown n table 3.8. [1] n ges [1] λ bott f λ [1] Expermental Parameters 80 5,0000 0,9000 4,5000 Table 3.8: Open Queueng Example - Tableau (see Appendx - Table B.2 Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 5 1 Savng Request 0,50 5,00 1,00 2,50 11,250 Request Saver ,060 1,000 16,667 0,675 1,402 0,125 0,675 0,060 2,077 0, Prntng Request 0,50 5,00 1,00 2,50 11,250 Request Prnter ,030 1,000 33,333 0,338 0,172 0,015 0,338 0,030 0,509 0, Persst Data Request 1,00 5,00 1,00 5,00 22,500 Data Persster ,333 1,574 0,070 1,013 0,045 2,586 0, Computaton Request 1,00 5,00 1,00 5,00 22,500 Request Computer ,040 1,000 25,000 0,900 8,100 0,360 0,900 0,040 9,000 0, Unrelable Executon 1,00 1,00 5,00 5,00 22,500 Unrelable Executer ,000 9,674 0,430 1,913 0,085 11,586 0, Relable Executon 1,00 1,00 1,00 1,00 4,500 Relable Executer ,000 9,674 2,150 1,913 0,425 11,586 2, Input 1,00 1,00 1,00 1,00 4,500 Input Retrever ,050 1,000 20,000 0,225 0,065 0,015 0,225 0,050 0,290 0, Request 1,00 1,00 1,00 1,00 4,500 Request Handler ,000 9,739 2,164 2,138 0,475 11,877 2, Request Generaton 1,00 1,00 1,00 1,00 4,500 Clent ,139 0,066 0,000 0,000 68,123 15,139 68,123 15,139 Multplexer Secton j Name j m j [1] X j 1 CPU 1 0,200 2 Prnter 1 0,075 3 Dsk 1 0,150 4 I/O-Devce 1 0,050 Server Secton Dynamc Evaluaton Secton One example could be the dependency of overall response tme R ges from the overall arrval rate λ, as shown n fgure In ths chart t can be seen that for an arrval rate from approxmately λ > 4, 3 [SRq] [s] for λ = λ bott = 5 [SRq] [s]. the overall response tme R ges begns to grow rapdly and goes to nfnty 8 λ bott = R ges [s] λ [SRq]/[s] Fgure 3.67: Open Queueng Example - Chart: Response Tme - Arrval Rate Another example s the dependency of the external servce tme X ext on the mean overall number of servce requests n the model (mean populaton n ges, as depcted n fgure Ths type of chart could help n drawng conclusons to the behavor of the clents. In an example of 118

141 3.5. FMC-QE EXAMPLE - OPEN QUEUEING NETWORK a mean populaton of n ges = 80[SRq] every sngle clent could request the system every 15, 1s n a confguraton of a desred mean overall arrval rate of λ = 4, 5 [SRq] [s] (resp. f = 0, X ext [s] n ges [SRq] (f=0,9; λ=4,5 [SRq]/[s] Fgure 3.68: Open Queueng Example - Chart: External Servce Tme - Populaton Fgure 3.69 shows the changes of the utlzaton of the dfferent servers ρ j dependng on the arrval rate λ. Whle the CPU s the bottleneck, the utlzaton of the CPU grows untl ρ CPU = 1 for λ = λ bott = 5 [SRq]. The Dsk, the Prnter and the I/O Devce are the respectvely lower utlzed [s] servers wth a utlzaton of ρ Dsk = 0, 8, ρ Prnter = 0, 375 and ρ I/O Devce = 0, 25 for λ = λ bott. In addton to the utlzaton of the dfferent servers the correspondng overall response tme R ges s also plotted. ρ j 1 ρ CPU 25 0,9 R ges 0,8 20 ρ Dsk 0,7 0,6 15 0,5 0,4 ρ Prnter 10 0,3 ρ I/O-Devce 0,2 5 0, λ [SRq]/[s] Fgure 3.69: Open Queueng Example - Chart: Utlzaton, Response Tme - Arrval Rate R ges [s] 119

142 CHAPTER 3. FMC-QE FUNDAMENTALS 3.6 FMC-QE Tool The development of an FMC-QE Tool s n the man focus of Tomasz Porzucek, who s also a member of the Research Group of Prof. Dr.-Ing. Werner Zorn. Whle ths thess s focused on the theoretcal background and the further development of FMC-QE, hs thess wll be focused on the development of the FMC-QE framework and research questons n the development of the Tool, lke model transformatons. Through ths topc there are close cooperatons between the author and Mr. Porzucek, as n [115]. Fgure 3.70 shows a screenshot of a prototype of the tool. Fgure 3.70: FMC-QE Tool - Screenshot [115] Wth the help of the Tool the performance analyst s able to model the quanttatve behavor of the analyzed system. From ths model a Tableau s generated, whch could be evaluated wth a broad range of dfferent system and load parameters n order to predct many dfferent system scenaros. Addtonally, exstng models, whch are not modeled usng FMC-QE and are possbly not herarchcal, could also be mported n ths tool, as descrbed n [116]. 120

143 Chapter 4 FMC-QE Extensons Ths chapter provdes extensons of FMC-QE, explaned wth the help of representatve basc examples as well as comparsons of FMC-QE to related work. In secton 4.1 FMC-QE s extended to handle Closed Queueng Networks. In ths secton, the ntegraton of the summaton method [17] nto FMC-QE s explaned n order to support the approxmatve evaluaton for the class of Closed Queueng Systems. Secton 4.2 llustrates the handlng of multclass scenaros n FMC-QE. In secton 4.3 a classcal Tme Augmented Petr Net problem, the semaphore synchronzaton, s modeled and the methodology s extended to handle such scenaros. In secton 4.4 related work s compared to FMC-QE. Ths ncludes the Queueng Theory, Tme Augmented Petr Nets, Layered Queueng Networks (LQN and performance smulatons. 121

144 CHAPTER 4. FMC-QE EXTENSIONS 4.1 Closed Queueng Networks In the followng secton the extenson of FMC-QE for the handlng of closed queueng networks s descrbed. Therefore, frst the general model of closed queueng networks s dscussed. Then the performance predctons for exemplary closed queueng networks, modeled and computed through Queueng Theory approaches, are compared to the predctons of the correspondng FMC-QE model and Tableau and the extensons of FMC-QE are explaned wth the help of these models General Dscusson In closed queueng networks there s a short crcut from the system output to the system nput. In the moment the servce request s fulflled and the servce response s delvered from the system, ths response s mmedately nserted n the system, agan. In secton ths model was referenced as the Roller Coaster (the chldren nstantly queue up agan. From the vewpont of FMC-QE the common model of closed queung networks rases some questons. The frst queston comes from the dstncton of servce requests and servce responses. In the steady state of closed queueng networks the numercal value of the throughput (servce response rate equals the arrval rate (servce request rate, but the servce request s not the same as the servce response. The handlng of the servce response was done for some reason and therefore t makes no sense to mmedately queue up the response agan. Even n the roller coaster model the response (chld dd the rde and s happy has to be changed nto the servce request (chld wants to rde agan. Therefore, the consderaton of the external (outsde world s mportant, because even f the external servce tme would be zero, the external world s to be consdered for the generaton of the servce request and the recevng of the servce response. Another queston concernng the classcal closed model also results from the neglecton of the outsde world and the servce request generaton: Who nserted the servce requests n the system? or Who ntalzed the system?. In the standard model there s no servce request generaton and therefore, the sngle servers are not of type M/M/m (or smlarly but -/M/m. The output (throughput of the system s transformed (or not - as dscussed nto the arrval rate. Therefore, there s no servce request generaton M/M/m. The arrvals are a result of the network throughput -/M/m. As the arrval rate λ s not a free parameter n ths model, also the basc laws, lke Lttle s Law [98] (n = λ R, have to be nterpreted dfferently. Whle n standard FMC-QE models the number of servce requests n the system (n sys s a result, here n sys s the parameter and the arrval rate (or throughput s the result. From the vewpont of FMC-QE closed queueng networks and the not modelng of the request generaton and outsde world s questonable, but as closed queueng networks are used as a standard model n a broad range of lterature, ths class of models wll also be handled n FMC- QE. The extensons of FMC-QE for closed networks are descrbed n ths secton. The extensons wll be dscussed on examples n order to exemplary compare the dfferent approaches. In ths examples frst the networks are modeled and analyzed wth Queueng Theory technques, and later on an FMC-QE model and Tableau are set up. It wll be seen that the standard FMC- QE performance evaluaton technques are not a good approxmaton for ths class of models. Therefore, the methodology wll be extended n order to solve ths class of problems. 122

145 4.1. CLOSED QUEUEING NETWORKS Closed Tandem Network As a smple but sgnfcant example a tandem network consstng of two servers connected to each other n a closed network wll be examned. Frst the example wll be calculated usng standard Queueng Theory approaches, calculatng the global balance equatons and solvng a lnear equaton system. Then the example s calculated usng the standard FMC-QE load model and settng the external tme to zero. It wll be seen that there are large approxmaton errors n ths model. In a second Tableau the performance values are calculated usng an M/M/1/K model. Ths model leads to correct solutons for ths specal case of a closed network, but n order to compute the performance values, some results of the calculatons had to be known n advance. In the thrd model the summaton method [17] s adapted to FMC-QE. The adapton wll be explaned and t wll be seen that ths s a good approxmaton method. Orgnal Example -/M/1 The orgnal model, shown n fgure 4.1, s defned n [18]. 1 2 Fgure 4.1: Closed Tandem Network - Orgnal Model [18] In ths example [ the two ] servers have exponentally [ ] dstrbuted servce tmes wth mean values of 5s (µ 1 = 1 SRq1 5 s and 2,5s (µ 1 = 1 SRq1 2,5 s and a FCFS servce dscplne. There are 3 servce requests n the network (n ges = 3, whch lead to the state transton dagram shown n fgure 4.2 [18]: ,0 2,1 1,2 0, Fgure 4.2: Closed Tandem Network - State Transton Dagram [18] In [18] the global balance equatons are set up as: p(3, 0µ 1 = p(2, 1µ 2 ; p(2, 1(µ 1 + µ 2 = p(3, 0µ 1 + p(1, 2µ 2 ; p(1, 2(µ 1 + µ 2 = p(2, 1µ 1 + p(0, 3µ 2 ; p(0, 3µ 2 = p(1, 2µ 1. Ths leads to the steady state probabltes [18]: p(3, 0 = 0, 5333; p(2, 1 = 0, 2667; p(1, 2 = 0, 1333; p(0, 3 = 0,

146 CHAPTER 4. FMC-QE EXTENSIONS Usng ths steady state probabltes, the margnal probabltes are computed as [18]: p 1 (0 = p 2 (3 = p(0, 3 = 0, 0667; p 1 (1 = p 2 (2 = p(1, 2 = 0, 1333; p 1 (2 = p 2 (1 = p(2, 1 = 0, 2667; p 1 (3 = p 2 (0 = p(3, 0 = 0, After computng these probabltes, the performance values, startng wth the utlzaton ρ, could be derved as [18]: ρ 1 = 1 p 1 (0 = 0, 9333; ρ 2 = 1 p 2 (0 = 0, The arrval rates (throughput n the closed network are then derved as [18]: λ = λ 1 = λ 2 = rho 1 µ 1 = rho 2 µ 2 = 0, 1867 The mean number of servce requests n are [18]: [ SRq s ]. n 1 = 3 3 k p 1 (k = 2, 2667 [SRq 1 ] ; n 2 = k p 2 (k = 0, 7333 [SRq 2 ]. k=1 k=1 The mean response tmes R are [18]: R 1 = n 1 λ 1 = 12, 1429[s]; R 2 = n 2 λ 2 = 3, 9286[s]. FMC-QE Model The correspondng FMC-QE model s shown n fgure The servce request structure s defned n fgure 4.3. The actons of the two connected servers are now defned as Sub-Request 1 and Sub-Request 2, whch are parts of an overall Request. Furthermore, n ths model there s also a request generaton assocated to the overall request. Fgure 4.3: Closed Tandem Network - Servce Request Structure The server structure of ths model s shown n fgure 4.4. The two servers Server 1 and Server 2 are connected to the two actons, executed by the logcal servers Sub-Request 1 Executer and Sub-Request 2 Executer. 124

147 4.1. CLOSED QUEUEING NETWORKS " #!" # Fgure 4.4: Closed Tandem Network - Server Structure Fnally, the dynamc behavor and the control flow are descrbed n fgure 4.5. Besde the defnton of two herarchcal layers (one for Execute request and one for the two Sub-Requests the man dfference between ths model and the orgnal model n fgure 4.1 s the ntroducton of a request generaton, as usual n FMC-QE. In order to extend FMC-QE to closed networks, the thnk tme of ths external server (Clent/Request Generator has to be reduced to zero ("shortcrcut" and furthermore, the overall number of servce requests n the system has to be adjusted to a constant natural number. Fgure 4.5: Closed Tandem Network - Dynamc Behavor In the followng some approaches for the extenson of FMC-QE to closed networks are descrbed wth the help of ths example. 125

148 CHAPTER 4. FMC-QE EXTENSIONS M/M/1 Approxmaton Method In a frst approxmaton approach for a closed network the performance values are derved n a standard FMC-QE Tableau wth M/M/1 servers (formulas are also n table A.3: n, q = ρ2 1 ρ n, s = ρ n = ρ 1 ρ (4.1 and an external tme X ext = 0. In ths model the network s handled lke an open network, ncludng an external servce request generaton (M/M/1 servers, whereas the servce request generaton and the handlng of the servce response (outsde world s handled wth a servce tme of zero (X ext = 0. In order to acheve an external servce tme of zero, the arrval rate λ s adjusted n an teratve approxmaton approach untl X ext = 0 for n ges = 3. In fgure 4.6 the functonal dependency of the external servce tme X ext from the desred bottleneck utlzaton f, where λ = f λ bott and 0 < f < 1, s shown, whch s then later adjusted though the teratve approach (values of frst teratve steps are plotted. For a small f the external servce tme s large, because the arrval rate s small and therefore only a few servce requests are n the system and the external servce tme X ext for the rest of the servce requests (n ext = n ges n sys has to be large. If the desred bottleneck utlzaton f s too large and therefore the arrval rate λ s also too large, there are too many servce requests n the system (n sys > n ges and therefore the external servce tme X ext converges aganst (the external world s a "tme machne". 400,0 300,0 200,0 X ext [s] 100,0 0, ,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0, ,0-200,0-300,0-400,0 f (n ges = 3[SRq] Fgure 4.6: Closed Tandem Network - M/M/1 - Chart: Adjustment External Servce Tme - f The correspondng FMC-QE Tableau s shown n table 4.1. The desred bottleneck utlzaton f wth the value 0, was calculated through an teratve approxmaton approach n whch ths value was adjusted untl n [1] 2 = 3 (overall number of requests n the system as predefned n the parameter n ges of the orgnal model. 126

149 4.1. CLOSED QUEUEING NETWORKS f λ [1] Table 4.1: Closed Tandem Network - M/M/1 Tableau (see Appendx - Table B.3 Expermental Parameters 3 0,2000 0, ,1420 n ges [1] λ bott [1] [bb-1] p p(, v p( v [bb-1] SRq,nt v λ Server m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,142 Executer ,500 1,000 0,400 0,355 0,195 1,376 0,355 2,500 0,551 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,142 Executer ,000 1,000 0,200 0,710 1,739 12,247 0,710 5,000 2,449 17, Request 1,00 1,00 1,00 1,00 0,142 Executer ,200 1,935 13,624 1,065 7,500 3,000 21, Req. Generaton 1,00 1,00 1,00 1,00 0,142 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton j Name j m j [1] X j 1 Server 1 1 2,500 2 Server 2 1 5,000 Servce Request Secton Server Secton Dynamc Evaluaton Secton The approxmaton errors (relatve error δx = x x arrval rate: 100 [22] n the predcton of the overall λ GlobalBalanceCalculaton = 0, 1867; λ M/M/1 Approx. = 0, 1420 δ λ = λ GlobalBalanceCalculaton λ M/M/1 Approx. λ GlobalBalanceCalculaton = 23, 9% and the response tmes of the servers (especally R 1 : R 1,GlobalBalanceCalculaton = 12, 1429; R 1,M/M/1 Approx. = 17, 2474; δ R1 = R 1,GlobalBalanceCalculaton R 1,M/M/1 Approx. R 1,GlobalBalanceCalculaton = 29, 6%; R 2,GlobalBalanceCalculaton = 3, 9286; R 2,M/M/1 Approx. = 3, 8763; δ R2 = R 2,GlobalBalanceCalculaton R 2,M/M/1 Approx. R 2,GlobalBalanceCalculaton = 1, 3% are very hgh, because n ths soluton the number of servce requests n the system s only a mean number derved by calculatons for open networks and not a constant natural number as usually for closed networks. M/M/1/K Method In a second calculaton approach the two servers are represented by M/M/1/K servers wth a capacty of K = 3 and so the server formulas are (formulas are also n table A.6: ρ = λ µ ρ 1 ρ n,q = ρ (Kρ K+1 ρ 1 ρ K+1 = 1 K(K 1 ρ 2(K+1 = 1 { 1 1 ρ ρ n,s = 1 ρ K+1 = K+1 ρ = 1 { ρ 1 ρ K+1 ρ K+1 ρ = 1 n = 1 ρ K+1 K 2 ρ = 1 (

150 CHAPTER 4. FMC-QE EXTENSIONS The problem s that n ths method the arrval rates are an nput parameter of the model and not a result, as usual for closed models. Furthermore, n M/M/1/K models there s a dstncton between the arrval rates λ and the effectve arrval rates λ,e f f whch s dependent on the utlzaton of the dfferent servers. So n order to adjust the rght effectve arrval rates, not only the overall arrval rate but also the dfferent traffc flow coeffcents had to be adjusted. For ths proof-of-concept the known effectve arrval rates from the orgnal example, calculated n the begnnng of ths secton, are used to solve ths problem. So n order to receve the correct results, n the Tableau shown n table 4.2, the dfferent effectve arrval rates λ,e f f are adjusted by the overall arrval rate λ [1] and the traffc flow coeffcents v n order to ft wth the value 0, 1867 of the orgnal example [18], calculated va the global balance equaton. Table 4.2: Closed Tandem Network - M/M/1/K Tableau (see Appendx - Table B.4 Expermental Parameters [1] n ges 3 λ [1] 0,4000 [bb-1] p p(, v p( v [bb-1] SRq,nt v λ λ,eff Server m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 0,50 0,5 0,200 0,187 Executer ,500 1,000 0,400 0,500 0,267 1,429 0,467 2,500 0,733 3, Sub-Request 1 1,00 1,00 1,00 1 0,400 0,187 Executer ,000 1,000 0,200 2,000 1,333 7,143 0,933 5,000 2,267 12, Request 1,00 1,00 1,00 1 0,400 0,187 Executer ,200 1,600 8,571 1,400 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1 0,400 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton j Name j m j [1] X j 1 Server 1 1 5,000 2 Server 2 1 1,250 Servce Request Secton Server Secton Dynamc Evaluaton Secton In the specal case of the tandem network the performance values are exactly the same for the M/M/1/K model and the calculaton va the global balance equatons, but ths n not true for every closed network (n the second example of ths secton the values are not the same. Furthermore, for the calculaton of ths model the effectve arrval rates had to be known n advance n order to adjust the traffc flow coeffcents for ths model or a more complex equaton system or mult dmensonal teraton approach had to be solved n order to retreve the results. Because n ths closed tandem model the effectve arrval rates: ( λ 1 1 ρ 1 λ,e f f = ρ K ρ λ ( 1 1 K+1 ρ = 1 ρ = 1 had to be the same for every server n the tandem network, so the arrval rates λ had to be adjusted through the traffc flow coeffcent λ = v λ and also the overall number of servce requests n the system n ges, wth: (4.3 and n = { ρ 1 ρ K+1 ρ K+1 1 ρ K+1 ρ = 1 K 2 ρ = 1 (4.4 n ges = 2 n (4.5 =1 had to be 3[SRq] (n ges = 3[SRq]. For ths proof-of-concept tandem network the M/M/1/K model was calculable, but for larger networks ths model and calculaton s not feasble. 128

151 4.1. CLOSED QUEUEING NETWORKS Summaton Method Whle n FMC-QE the arrval rate s an nput parameter and the number of servce requests n the system s a result, n closed systems ths normally s the opposte, as the number of servce requests n the system s an nput parameter and the arrval rate or throughput s a result. The summaton method [17] s an excepton and fts to ths model of the arrval rate as nput and the number of servce requests n the system as a result and so t solves problems of the M/M/1/K model. In the summaton method the mean number of servce requests n each node s a functon of the throughput of the node [18]: [17] propose the followng formulas for f (λ [18]: n = f (λ. (4.6 ρ, Type 1, 2, 4 (m 1 K 1 K ρ = 1, ρ f (λ = m ρ + p 1 K m 1 (m, Type 1 (m > 1, K m ρ λ µ, Type 3. (4.7 wth the utlzaton [18]: ρ = and the watng probabltes (for Type-3 Server, m > 1 [18]: λ m µ (4.8 m! (1 ρ m 1 k=0 p (m = (m ρ k k! (m ρ m (4.9 The functon f (λ s correct for Type-3 servers (nfnte servers and an approxmaton for Type-1,2 and 4 [18]. If f s gven for every basc server staton (number of basc server statons = I n the network, the overall number of servce requests n the system s gven by [18]: I I n = f (λ = K (4.10 =1 =1 and ncludng the traffc flow coeffcents v, the overall number of servce requests n the system s a functon of the arrval rate [18]: I f (v λ = g(λ = K. (4.11 =1 129

152 CHAPTER 4. FMC-QE EXTENSIONS For the usage of the summaton method n the FMC-QE calculatons the basc server formulas are substtuted by the summaton formulas (4.7, and then the soluton s derved n an teratve calculaton, modfed from [18]: Whle the desred bottleneck utlzaton f (λ = f λ bott s 0 for the lower bound of the arrval rate and 1 for the upper bound (arrval rate λ = bottleneck throughput λ bott, the bounds are f l = 0 and f u = 1 n the frst step (desred bottleneck utlzaton f = f. Then n the second step f = f l+ f u 2 and the Tableau s solved. If the overall number of servce requests n the system s K ± ɛ, then the soluton s found, else the bounds are set to f u = f l+ f u 2 f the overall number of servce requests n the system > K and f l = f l+ f u 2 f the overall number of servce requests n the system < K, and then the next teraton s started wth the second step. The correspondng chart, whch shows the dependency of the desred bottleneck utlzaton ( f and the number of servce requests n the system (n system for the example, s depcted n fgure 4.7 (ncludng the frst teratve steps. In ths confguraton the desred bottleneck utlzaton s adjusted to f = 0, 9144 to acheve an overall number of servce requests n the system of 3 servce requests (n system = 3[SRq]. 4,0 3,5 0,9144 3,0 3,0000 n system (n 3 [SRq] 2,5 2,0 1,5 1,0 0,5 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 f (λ bott = 0,2[SRq]/[s] Fgure 4.7: Closed Tandem Network - Summaton Method Chart: n System - f 130

153 4.1. CLOSED QUEUEING NETWORKS Table 4.3 shows the correspondng Tableau of the closed tandem network example. Table 4.3: Closed Tandem Network - Summaton Method Tableau (see Appendx - Table B.5 Expermental Parameters 3 0,2000 0,9144 0,1829 n ges [1] λ bott [1] f λ [1] [bb-1] p p(, v p( v [bb-1] SRq,nt v λ Server m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,183 Executer ,500 1,000 0,400 0,457 0,200 1,096 0,457 2,500 0,658 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,183 Executer ,000 1,000 0,200 0,914 1,428 7,808 0,914 5,000 2,342 12, Request 1,00 1,00 1,00 1,00 0,183 Executer ,200 1,628 8,904 1,372 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1,00 0,183 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton j Name j m j [1] X j 1 Server 1 1 2,500 2 Server 2 1 5,000 Servce Request Secton Server Secton Dynamc Evaluaton Secton The approxmaton error (relatve error δx = x x 100 [22] n the predcton of the overall arrval rate and the response tmes of the servers are: λ GlobalBalanceCalculaton = 0, 1867; λ SUM Approx. = 0, 1829 δ λ = λ GlobalBalanceCalculaton λ SUM Approx. λ GlobalBalanceCalculaton = 2, 04% R 1,GlobalBalanceCalculaton = 12, 1429; R 1,SUM Approx. = 12, 8078; δ R1 = R 1,GlobalBalanceCalculaton R 1,SUM Approx. R 1,GlobalBalanceCalculaton = 5, 48%; R 2,GlobalBalanceCalculaton = 3, 9286; R 2,SUM Approx. = 3, 5961; δ R2 = R 2,GlobalBalanceCalculaton R 2,SUM Approx. R 2,GlobalBalanceCalculaton = 8, 46% Although there are two approxmatons n the new FMC-QE algorthm extenson - one through the teraton and the second through the summaton method tself [18], the approxmaton errors for ths example are qute small. Furthermore, as dscussed later n the summary of ths secton, the computatonal complexty of the summaton method n FMC-QE s small for a large overall number of servce requests (populaton n comparson to the Mean Value Analyss Central Server Network The second closed network example s a classcal CPU - Dsk(s closed model (Central Server. In ths part the soluton followng the Theorem of Gordon and Newell [64], already dscussed n secton 2.1.5, s compared to the FMC-QE summaton method. Orgnal Model The orgnal example s the example n [68]. It was descrbed usng a graph, shown n fgure 4.8, and a table, shown n table

154 CHAPTER 4. FMC-QE EXTENSIONS Fgure 4.8: Central Server Model - Orgnal Model [68] Table 4.4: Central Server Model - Orgnal Parameters [68] Name Index B p,1 p,2 p,3 N = 3 CPU 1 0,50 0,10 0,40 0,50 Platte1 2 0,40 1,00 0,00 0,00 Platte2 3 0,25 1,00 0,00 0,00 In order to have consstent Queueng Network models n ths thess, the model was reengneered to a more convenent model, shown n fgure 4.9. CPU 1 p 1,1 =0,1 2 3 Dsk1 Dsk2 p 1,2 =0,4 p 1,3=0,5 Fgure 4.9: Central Server - Orgnal Model Reengneered Ths model has the vst ratos: e 1 = 1, 000; e 2 = 0, 400; e 3 = 0, 500. In [68] the model was calculated usng the Theorem of Gordon and Newell [64], further explaned n secton For a system wth three Servce Requests nsde (n ges = 3[SRq] the calculatons lead to the followng results [68] (recalculated manually and usng WnPEPSY 1 [86]: ρ 1 = 0, 6939; ρ 2 = 0, 3469; ρ 3 = 0, 6939; D 1 = 0, 3469; D 2 = 0, 1388; D 3 = 0, 1735; n 1,q = 0, 5714; n 2,q = 0, 1224; n 3,q = 0, 5714; n 1,s = 0, 6939; n 2,s = 0, 3469; n 3,s = 0, 6939; n 1 = 1, 2653; n 2 = 0, 4693; n 3 = 1, 2653; R 1 = 3, 6471; R 2 = 3, 3824; R 3 = 7, 2941; R = 8, 6471 D = 0, 3469 ( WnPESPY, Webste: January

155 4.1. CLOSED QUEUEING NETWORKS FMC-QE Model In the correspondng FMC-QE model, shown n fgure 4.10, fgure 4.11 and fgure 4.12, there have also been done some transformatons, as n the example n secton 3.5. The dfferent steps wll not be shown here n detal, but these were a transformaton to a Petr Net, the ntegraton of the external load generaton, a feed-forward - feed-backward transformaton for the loop around the CPU, a herarchy for the dsk branch and another herarchy for the whole servce request. For the sake of smplcty and a smaller model the branch of f the data has to be wrtten (p wrte = 0, 9 or not (p out = 0, 1 - request fnshed has no addtonal herarchy. Ths probablty s ntegrated nto the model and correspondng tableau wthout an addtonal herarchcal level. &'! &#' $ % $ $ &(' " " # "# "# &' Fgure 4.10: Central Server - FMC-QE Model (Servce Request Structure $ *+ $! $ $ " $ ' " # $%& ' " #$ ( ' $ Fgure 4.11: Central Server - FMC-QE Model (Server Structure 133

156 CHAPTER 4. FMC-QE EXTENSIONS ' $#%!" # $#% % % &# % Fgure 4.12: Central Server - FMC-QE Model (Dynamc Behavor The correspondng Tableau n table 4.5 shows the performance predctons of the FMC-QE model. [1] n ges [1] λ bott f λ [1] Expermental Parameters 3 0,0500 0,6895 0,0345 Table 4.5: Central Server - Tableau (see Appendx - Table B.7 Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 4 1 Wrte Request 2 0,56 9,00 1,00 5,00 0,172 Dsk2 Wrter ,000 1,000 0,250 0,690 0,587 3,403 0,690 4,000 1,276 7, Wrte Request 1 0,44 9,00 1,00 4,00 0,138 Dsk1 Wrter ,500 1,000 0,400 0,345 0,103 0,746 0,345 2,500 0,448 3, Data Wrtng Request 0,90 10,00 1,00 9,00 0,310 Data Wrter ,450 0,690 2,222 1,034 3,333 1,724 5, Calculaton Request 1,00 10,00 1,00 10,00 0,345 Calculator ,000 1,000 0,500 0,690 0,587 1,702 0,690 2,000 1,276 3, Internal Request 1,00 1,00 10,00 10,00 0,345 Internal Exec ,500 1,276 3,702 1,724 5,000 3,000 8, Transacton 1,00 1,00 1,00 1,00 0,034 Trans. Exec ,050 1,966 57,016 1,724 50,000 3,000 87, Transacton Generaton 1,00 1,00 1,00 1,00 0,034 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 0,000 Multplexer Secton j Name j m j [1] X j 1 CPU 1 20,000 2 Dsk1 1 20,000 3 Dsk2 1 10,000 Server Secton Dynamc Evaluaton Secton 134

157 4.1. CLOSED QUEUEING NETWORKS When comparng the results of FMC-QE n table 4.5 to the results of the Queueng Theory method n equaton 4.12, the approxmaton errors (relatve error δx = x x 100 [22] at the server level are very small (except the outler ("Ausreßer" n 2,q for whch the error s stll small: δ ρ1 = 0, 63%; δ ρ2 = 0, 61%; δ ρ3 = 0, 63%; δ D1 = 0, 61%; δ D2 = 0, 65%; δ D3 = 0, 63%; δ n1,q = 2, 66%; δ n2,q = 15, 93%; δ n3,q = 2, 66%; δ n1,s = 0, 63%; δ n2,s = 0, 61%; δ n3,s = 0, 63%; δ n1 = 0, 86%; δ n2 = 4, 60%; δ n3 = 0, 86%; δ R1 = 1, 49%; δ R2 = 4, 03%; δ R3 = 1, 49%. On the net level (D and R the approxmaton error seems to be very hgh at the frst moment (D Queueng Theory = 0, 3469 SRq s vs. D FMC QE = 0, 0345 SRq s and R Queueng Theory = 8, 6471s vs. R FMC QE = 87, 016s, but n the FMC-QE model the repetton of the nternal requests s transformed nto a hgher traffc flow wthn the Internal Request. The throughput at the hghest level s 10 tmes lower than n the orgnal model, whch s just another nterpretaton of the model and not an error. - In the orgnal model the CPU (ncludng the repettons of the nternal requests s the reference for the network throughput and the the new model the real net level throughput s the reference (loop above CPU. - Analogcal, the overall response tme has to be changed n order to consder these repettons. Wth ths "normalzaton" the approxmaton errors are: δ D = 0, 63%, δ R = 0, 63%. The problems wth the normalzaton and the dfferent understandng reveal another already dscussed problem of the classcal models: no dstncton between the dfferent servce request types. Through the flat orgnal model there s no dstncton between the nternal requests and the transactons (real system requests and therefore these nterpretaton problems arse. In [68] n example they also defne ths dfferent net-throughput, whch then leads to the normalzed values of the orgnal calculaton of D Queueng Theory = 86, 47s (resp. 85, 71s - rounded values n the calculaton of [68]. R Queueng Theory = 0, 0347 SRq s After settng up the model and the correspondng Tableau, system and performance parameters could be changed n order to predct a broad range of scenaros. An exemplary parameter change s depcted n fgure 4.13, where the populaton (n ges s changed and the derved throughput of the system (λ [1] s plotted. For comparatve values the network was also modeled and evaluated usng WnPEPSY 2 [86]. In the comparson of the predcted throughput of FMC-QE and the MVA [118] calculatons of WnPEPSY the predcton s very precse, wth a maxmal error of δ = 0, 75% for n ges = and 2 WnPESPY, Webste: January

158 CHAPTER 4. FMC-QE EXTENSIONS 0,06 0,05 λ bott =0,05 λ [SRq]/[s] 0,04 0,03 0,02 FMC-QE WnPEPSY MVA λbott 0,01 0, n ges [SRq] Fgure 4.13: Central Server - Chart: Throughput - Populaton Summary Three methods have been presented: The frst two approaches of the ntegraton of closed Queueng Networks nto FMC-QE, more precsely the approxmaton usng M/M/m servers and an external servce tme of zero tme unts as well as the second model of M/M/m/K servers, were ether to mprecse or napplcable. But the thrd approach, the ntegraton of the summaton method [17, 18] nto FMC-QE, explaned and exemplfed n ths secton, extends FMC-QE to the class of closed Queueng Networks, provdng an teratve algorthm and good approxmaton for closed systems 3. The approxmatve errors n the examples, evaluated usng the summaton method, are, except for the outler n 2,q n the second example, wthn the maxmal error of 15% descbed n [17] and often even better than the average error of 5% for the number of servce requests and the response tme, as also descrbed n [17]. The values for the throughput also delver precse values, as descbed n [17]. Referrng to [118], the operatons count of the fast, smple and wdely dstrbuted closed Queueng Network analyss algorthm, the algorthm of the Mean Value Analyss (MVA s 5MR per servce center and recursve step, where M s the number of parallel servers at the servce center and R s the number of chans whch vst the servce center (number of classes. Therefore, the overall complexty s O(n m, where n s the number of servce centers and m s the overall number of servce requests (n ges. Comparng to ths, the complexty of the FMC-QE summaton method s ndependent from the overall number of servce requests, whle every value s calculated ndependently n contrast to the teraton over the number of servce requests n the MVA. In the summaton method n FMC-QE there s an teraton n order to adjust the desred bottleneck utlzaton respectvely the throughput, here the number of teratons could be seen as a constant 4 and therefore the complexty s stll O(n, where n s the number of servce centers (statons. Therefore, for a small overall number of servce requests n ges MVA s faster than FMC-QE, but n ths regon both algorthms are very effcent. For a large number of servce requests FMC-QE outperforms MVA, whle t s ndependent from the overall number of servce requests n the system. 3 The teratve algorthm converges to the overall number of servce requests n the system, but there s stll an approxmatve error for servers of Type-1,2 and 4 n the system [18]. 4 In the approxmaton of f, where (0 f < 1, the search space s bsected n every teraton, so f has a accuracy of ɛ 10 4,5 after 15 steps. 136

159 4.2. HANDLING OF MULTICLASS SCENARIOS 4.2 Handlng of Multclass Scenaros If there are dfferent knds of servce requests and t s not reasonable to combne the dfferent arrvals to a sngle class of servce requests, multclass models are the way of modelng such systems. In FMC-QE the multclass scenaro s an extenson of the multplex case. In multclass models the logcal structures dffer from each other, so there s a dfferent model for every class. The dfferent logcal structures meet at the server level, were the the dfferent logcal servers of the dfferent classes are handled by overlappng multplexers. The model s then parttoned by the multplex coeffcents and calculated separately. The handlng of multclass scenaros s a pre-step for the handlng of semaphores, explaned n the next secton, where there are more nter relatons between the dfferent sub-nets. The handlng of multclass scenaros s explaned on an example of two classes A and B as follows: In fgure 4.14 the servce request structures and n fgure 4.15 dynamc behavor of servce request class A are shown. In class A a servce request A s decomposed nto a seral executon for two servce requests A.1 and A.2. Fgure 4.14: Multclass Example - Class A - Servce Request Structures Fgure 4.15: Multclass Example - Class A - Dynamc Behavor 137

160 CHAPTER 4. FMC-QE EXTENSIONS In the other class B (descrbed n fgure 4.16 and 4.17, there s also a decomposton nto two servce requests B.1 and B.2. The servce request B.2 s furthermore decomposed nto two parallel executed servce requests B.2.1 and B.2.2. Fgure 4.16: Multclass Example - Class B - Servce Request Structures Fgure 4.17: Multclass Example - Class B - Dynamc Behavor 138

161 4.2. HANDLING OF MULTICLASS SCENARIOS In addton to the servce request structures and the dynamc behavor of the two classes A and B, the statc structures are defned n fgure There s a multplex of server X for the logcal servers Request A.1 Server and Request B.1 Server and a multplex of server Y for the logcal servers Request A.1 Server and Request B.2.1 Server. The server Z s a dedcated server for the servce request B.2.2 n class B.!"# $% $% $% Fgure 4.18: Multclass Example - Statc Structures $% $% The correspondng Tableau of of the multclass example s shown n table 4.6. In comparson to a non multclass model ths table conssts of more than one expermental parameter and servce request sectons. Expermental Parameters n ges 30 λ bott 0,9434 f 0,8000 λ A 0,7547 Table 4.6: Multclass Example - Tableau (see Appendx - Table B.9 SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Servce Request A.2 1,00 1,00 2,00 2,00 1,509 Req. A.2 Srv ,330 0,623 1,887 0,800 3,200 2,120 0,800 0,530 4,000 2, Servce Request A.1 1,00 1,00 1,00 1,00 0,755 Req. A.1 Srv ,200 0,500 2,500 0,302 0,131 0,173 0,302 0,400 0,432 0, Servce Request A 1,00 1,00 1,00 1,00 0,755 Req. A.2.2 Srv ,887 3,331 4,413 1,102 1,460 4,432 5, Generate Request A 1,00 1,00 1,00 1,00 0,755 Clent A ,877 0,030 0,000 25,568 33,877 25,568 33,877 Expermental Parameters n ges 30 λ bott 0,9434 f 0,5000 λ B 0,4717 Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 5 Servce Request B.2.2 1,00 2,00 2,00 4,00 1,887 Req. B.2.2 Srv ,100 1,000 10,000 0,189 0,044 0,023 0,189 0,100 0,233 0, Servce Request B.2.1 1,00 2,00 1,00 2,00 0,943 Req. B.2.1 Srv ,200 0,377 1,887 0,500 0,500 0,530 0,500 0,530 1,000 1, Servce Request B.2 1,00 1,00 2,00 2,00 0,943 Req. B.2 Srv ,887 0,544 0,577 0,689 0,730 1,233 1, Servce Request B.1 1,00 1,00 1,00 1,00 0,472 Req. B.1 Srv ,200 0,500 2,500 0,189 0,044 0,093 0,189 0,400 0,233 0, Servce Request B 1,00 1,00 1,00 1,00 0,472 Req. B Srv ,887 0,588 1,246 0,877 1,860 1,465 3, Generate Request B 1,00 1,00 1,00 1,00 0,472 Clent B ,494 0,017 0,000 28,535 60,494 28,535 60,494 Multplexer Secton Servce Request Secton j Name j m j [1] X j 1 Server X 1 0,400 2 Server Y 1 1,060 3 Server Z 1 0,400 Server Secton Server Secton Dynamc Evaluaton Secton Dynamc Evaluaton Secton 139

162 CHAPTER 4. FMC-QE EXTENSIONS Through the change of parameters n the Tableau and the correspondng plottng of the results nterestng observatons could be llustrated. Fgure 4.19 shows the dependency of the response tmes R A and R B on the arrval rate of one class (λ A R A [s] R ges [s] R B [s] 0 0 0,05 0,1 0,15 0,2 0,25 λ A [SRq]/[s] (λ B =0,4717[SRq]/[s] Fgure 4.19: Multclass Example - Chart: Response Tmes A, B - Arrval Rate A In ths fgure t can be seen that the change of the arrval rate n one class has no nfluence on the response tme of the other class. Ths s the case because the multplex coeffcent s precomputed ndependent from the actual arrval rate, as descrbed n secton 3.4.5: m,mpx = 1 f SRq o f Mpx j m m j X v X j m j m f SRq o f Mpx j m > m j (4.13 The dependency of the dfferent classes s defned through ths multplex coeffcent, and therefore, a change of the servce tme of one logcal server n one class changes the response tmes of both classes, as shown n fgure 4.20, where the servce tme of server Request A.2 Server (X A.2 s varated and the correspondng overall response tmes of both classes (R A and R B are recorded. If the servce tme X A.2 grows, the overall response tmes of the correspondng class A also grows as well as the response tme of the other classes, n whch a logcal server s handled by the same multplexer as Request A.2 Server - here Server Y also multplexes the Request B.2.1 Server of class B, and therefore, the response tme of class B also grows for a larger servce tme X A.2 for Request A.2 Server wth constant arrval rates of λ A λ B = 0, 4717 [SRq] [s]. = 0, 7547 [SRq] [s] and 140

163 4.2. HANDLING OF MULTICLASS SCENARIOS 25 R A [s] 20 R ges [s] R B [s] ,5 1 1,5 2 2,5 X A.2 [s] (λ A =0,7547[SRq]/[s]; λ B =0,4717[SRq]/[s] Fgure 4.20: Multclass Example - Chart: Response Tmes A, B - Servce Tme Req. A.2 Srv. As a further extenson of the methodology the dfferent arrval rates of the classes could nfluence the multplex coeffcent and therefore the dvson of the subnets. The next secton on the ntegraton of semaphore synchronzaton scenaros nto FMC-QE addresses ths ssue. 141

164 CHAPTER 4. FMC-QE EXTENSIONS 4.3 Semaphore Synchronzaton The performance modelng of processes, whch have to synchronze, or the modelng of semaphores s a classcal Tme Augmented Petr Net task. In ths secton an exemplary semaphore synchronzaton scenaro s modeled and evaluated usng FMC-QE. Whle ths problem s a Non Product Form problem, an approxmaton for the calculaton has been developed and presented n ths secton. For comparatve reasons ths system s also modeled usng Generalzed Stochastc Petr Nets (GSPN [102]. In Tme Augmented Petr Nets the mutual exclusve usage of resources could be synchronzed va semaphores and drlls down to the modelng and smulaton or calculaton of the dscrete steady state dstrbuton ncludng the state space exploson problem of ths knd of calculatons. In FMC-QE the mutual exclusve usage of the resources and therefore the nter-server control flows are approxmated by shared resources and a stochastc model wthout the synchronzaton and therefore wthout nter-server control flows. The example wll show that although the real synchronzaton through the semaphore s neglected, ths smplfcaton leads to qute accurate results wthout the state space exploson problem. Whle early deas of modelng semaphore synchronzaton scenaros through shared resources and multplexers were already publshed by Zorn n 2007 [143], the method n secton gves a clearer defnton of the nterdependences of the dfferent sub-nets through a pre-calculaton step extendng the deas of the last secton about multclass scenaros. Furthermore, the usage of the summaton method, dscussed n secton 4.1, ncludng the teratve calculaton of closed networks, brngs the ntegraton of semaphore synchronzaton scenaros nto FMC-QE to a round fgure GSPN Model In fgure 4.21 a semaphore synchronzaton scenaro s modeled. Ths model was already descrbed n secton where ths network was used as an exemplary Generalzed Stochastc Petr Net. p act1 p act2 T req1 λ 1 λ 2 T req2 p dle p req1 p req2 t str1 α 1 α 2 t str2 p acc1 p acc2 T end1 1 2 T end2 Fgure 4.21: Semaphore Synchronzaton - GSPN Model [102] The performance parameters of ths semaphore synchronzaton model are defned n table

165 4.3. SEMAPHORE SYNCHRONIZATION Transton Rate/Weght Semantcs T req1 λ 1 = 1 sngle server T req2 λ 2 = 2 sngle server T str1 α 1 = 1 mmedate T str2 α 1 = 1 mmedate T end1 µ 1 = 10 sngle server T end2 µ 2 = 5 sngle server Table 4.7: Semaphore Synchronzaton - Parameters The reachablty graph of ths Petr Net s llustrated n fgure Fgure 4.22: Semaphore Synchronzaton - Reachablty Graph The transton state probablty matrx P s defned as: P = λ 1 λ 1 +λ 2 λ 2 λ 1 +λ µ λ λ 2 +µ λ 2 +µ µ 2 λ 1 λ 1 +µ λ 1 +µ Ths results n the steady state dstrbuton π of: 143

166 CHAPTER 4. FMC-QE EXTENSIONS π 0 = 8 63 ; π 1 = ; π 2 = 5 18 ; π 3 = 8 63 ; π 4 = ; π 5 = ; π 6 = Fnally, the steady state dstrbuton π s calculated as: π 0 = 0; π 1 = 0; π 2 = ; π 3 = ; π 4 = ; π 5 = ; π 6 = The throughput of the dfferent tmed transton T k s calculated as D k = ω k π and T k E(M therefore: D req1 = λ 1 (π 2 + π 5 = ; D req2 = λ 2 (π 2 + π 3 = ; D end1 = µ 1 (π 3 + π 4 = ; D end2 = µ 2 (π 5 + π 6 = The throughput of the left subnet equals: and the throughput rght subnet s: D req1 = D end1 = 240 0, D req2 = D end2 = 390 1, FMC-QE Model In the correspondng FMC-QE model, llustrated n fgure 4.23 and 4.24, the transtons T req1 and T req2 were nterpreted as the request generaton denoted as Generate Request of Type 1 and Generate Request of Type 2. The transtons T end1 and T end2, whch are guarded by the semaphore, are denoted as Execute Crtcal Acton 1 and Execute Crtcal Acton 2. Ths model ncludes nterserver control flows n order to mplement the semaphore synchronzaton.!" Fgure 4.23: Semaphore Sync. wth Inter-Server Control Flows (Statc Structures [147] 144

167 4.3. SEMAPHORE SYNCHRONIZATION # $!" Fgure 4.24: Semaphore Sync. wth Inter-Server Control Flows (Dynamc Structures [147] In order to elmnate the nter-server control flows, n fgure 4.25 and 4.26, the model has been transformed. Instead of synchronzng two ndependent crtcal resources, n ths model the crtcal actons are handled by one crtcal resource modeled as a multplexer. Ths model s an approxmatve model because the dfferent servce requests are now handled n parallel by the 145

168 CHAPTER 4. FMC-QE EXTENSIONS multplexer and are not really synchronzed, lke n the semaphore synchronzaton model n fgure Fgure 4.25: Semaphore Sync. wthout Inter-Server Control Flows (Statc Structures Fgure 4.26: Semaphore Sync. wthout Inter-Server Control Flows (Dynamc Structures 146

169 4.3. SEMAPHORE SYNCHRONIZATION n ges [1] λ bott [1] f λ [1] Expermental Parameters 1 1,0000 0,8800 0,8800 Table 4.8: Semaphore Synchronzaton - Tableau (see Appendx - Table B.8 Servce Request Secton Server Secton Dynamc Evaluaton Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Crtcal Acton 1 1,00 1,00 1,00 1,00 0,880 CA 1 Executer ,100 0,733 7,333 0,120 0,000 0,000 0,120 0,136 0,120 0, Request 1 1,00 1,00 1,00 1,00 0,880 Executer ,333 0,000 0,000 0,120 0,136 0,120 0, Req. Generaton 1 1,00 1,00 1,00 1,00 0,880 Clent ,000 1,000 1,000 0,880 0,000 0,000 0,880 1,000 0,880 1,000 ##### Expermental Parameters [1] n ges 1 [1] λ bott 2,0000 f 0,7073 λ [1] 1,4146 Servce Request Secton Server Secton Dynamc Evaluaton Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Crtcal Acton 2 1,00 1,00 1,00 1,00 1,415 CA 2 Executer ,200 0,967 4,833 0,293 0,000 0,000 0,293 0,207 0,293 0, Request 2 1,00 1,00 1,00 1,00 1,415 Executer ,833 0,000 0,000 0,293 0,207 0,293 0, Req. Generaton 2 1,00 1,00 1,00 1,00 1,415 Clent ,500 1,000 2,000 0,707 0,000 0,000 0,707 0,500 0,707 0,500 ##### Multplexer Secton j Name j m j [1] X j 1 Crtcal Resource 1 0,300 In the Tableau n table 4.8 the performance values of the multplexer semaphore synchronzaton model are calculated. Whle the orgnal model s an closed Petr Net, n ths Tableau the summaton method formulas, explaned n secton 4.1, have been used. As depcted n the model, the crtcal actons Crtcal Acton 1 and Crtcal Acton 2 are executed by one server Crtcal Resource as a multplexer, lke proposed n [147]. Here the multplexer coeffcent formula: wth 1 f SRq o f Mpx j m m j m,mpx = X v m (4.14 j m f SRq o f Mpx j m > m j X j = X j X v, (4.15 SRq o f Crtcal Resource j from secton 3.4 has been adapted for the semaphore case. When the two crtcal actons are handled by one crtcal resource, the crtcal resource has to be multplexed f the respectvely opposte crtcal acton k s executed. The respectvely opposte crtcal acton k s executed wth the probablty ρ k and not executed wth the probablty (1 ρ k. So wth a probablty of ρ k the multplex coeffcent of formula 4.14 s used and wth a probablty of (1 ρ k the semaphore multplex coeffcent s 1 (the crtcal acton s the only acton on the crtcal resource. So the semaphore multplex coeffcent m,sem mpx s defned as: 1 f m m j SRq o f Crtcal Resource j m,sem mpx = ( X (1 ρ k + ρ v k X j m j m f m > m j, SRq o f Crtcal Resource j (4.16 where ρ k s the probablty that the respectvely opposte crtcal acton k s actve (utlzaton. For more than two crtcal actons competng for one server ths probablty could be adapted. 147

170 CHAPTER 4. FMC-QE EXTENSIONS These values could then be calculated teratvely. In the Tableau n table 4.8 ths teratve calculaton has been omtted because ths does not results n a sgnfcant mprovement of the approxmaton value whle resultng n a more complex calculaton. Here the utlzatons ρ = λ µ are used, where λ 1 = 1, λ 2 = 2, µ 1 = 10 and µ 2 = 5, as defned n the model. The approxmaton errors (relatve error δx = x x 100 [22] for the throughputs D are: δd req1 = δd end1 = δd req2 = δd end2 = 0, , , , , , 4079 = 1, 55%; = 0, 48%. For the comparson of the predcton of the mean number of servce requests n, the mean number of tokens n the places p act1, p act2 n the GSPN model s nterpreted as n Clent 1 resp. n Clent 2 and calculated as: p act1 = π 2 + π 5 = 240 0, 8664; 277 p act2 = π 2 + π 3 = 195 0, The mean numbers of servce requests executed n the crtcal actons n Crtcal Acton 1 resp. n Crtcal Acton 2 are the sums of the mean number of tokens n the places p req1 (watng and p acc1 (executed as well as p req2 (watng and p acc2 (executed because n the multplexer model the requests are handled n parallel: p req1 + p acc1 = π 3 + π 4 + π 6 = 37 0, 1336; 277 p req2 + p acc2 = π 5 + π 6 + π 4 = 82 0, Ths results n the followng approxmaton errors: 0, , 8800 δ nclent 1 = = 1, 55%; 0, , , 7073 δ nclent 2 = = 0, 47%; 0, , , 1200 δ ncrtcal Acton 1 = = 10, 18%; 0, , , 2927 δ ncrtcal Acton 2 = = 1, 11%. 0, 2960 After settng up the FMC-QE model and the correspondng Tableau, several dfferent system and load scenaros could be predcted, as shown n fgure In ths chart the servce tme of the second crtcal acton X CA2 s vared from 0, 001s to 0, 8s, wth a constant servce tme of 0, 1s for the frst crtcal acton (X CA1 = 0, 1. Then the correspondng throughputs of subnet 1 (λ 1 and subnet 2 (λ 2 are observed. For comparatve reasons the network was also modeled and evaluated usng the Tme Augmented Petr Net Tool TmeNET 5 [133, 134]. In the chart t can be seen that for the upper throughput, the throughput λ 2 of the network wth the crtcal acton 5 TmeNET, Webste: January

171 4.3. SEMAPHORE SYNCHRONIZATION 2, the performance predctons of FMC-QE and the Tme Augmented Petr Net predctons of TmeNET are nearly dentcal. For the lower curve λ 1, the predctons are fne for X CA2 < 0, 5. For X CA2 0, 5 the precomputed utlzaton ρ CA2, needed for the calculaton of m sem mpxca1, whch s defned as ρ CA2 = λ 2 µ 2, would be 1 (λ 2 = 0, 5, µ 2 = 1 X CA2 0, 5 and s therefore set to 1. Ths s the case f the crtcal acton becomes the bottleneck of the subnet because ths pre-defned utlzaton calculates the utlzatons of the servers f the bottleneck server s fully utlzed (ρ bott = 1. However, ntalzng ρ CA2 0, 9, as also shown n fgure 4.27, leads to more precse results. In future work the pre-calculaton of ths value could be further optmzed. 2,5 2,0 λ2 FMC-QE λ2 TmeNET λ1 FMC-QE λ1 TmeNET λ1 FMC-QE ρca2<=0.9 λ [SRq]/[s] 1,5 1,0 λ 2 0,5 λ 1 0,0 0 0,2 0,4 0,6 0,8 1 X CA2 [s] (X CA1 = 0,1[s] Fgure 4.27: Semaphore Synchronzaton - Chart: Throughput - Crtcal Acton 2 Servce Tme Summary For ths exemplary semaphore synchronzaton problem the ntegraton of the handlng of the semaphore by a multplexed resource and therefore elmnaton of the control flow [147], the summaton method [17, 18] and deas from the method of complementary delays [71] result n a very precse approxmaton of the performance values, as seen n fgure 4.27, wthout the need of the calculaton of the steady state dstrbuton. Therefore, the complexty reduces from the exponental calculaton of the ( n k states, where n = number of Petr Net - places ( number of queues number of servce statons and k = number of tokens (servce requests to the lnear complexty calculaton (O (n of FMC-QE, where n s the number of basc and herarchcal servce statons. 149

172 CHAPTER 4. FMC-QE EXTENSIONS 4.4 Comparsons In ths secton FMC-QE s compared to the foundatons descrbed n chapter 2. As an ntroducton to ths secton fgure 4.28 relates the compared methods to the range of the consdered steady state varables. Black Box n=λ*r Steady State Relaton for Black Box Lttle s Law Queueng Networks Tme,Augmented Petr Nets LQN FMC,QE Local Balance n=(n 1,..,n,..,n N n =n,q +n,s =1,..,N p=(p 0,..,p N Operatonal State Steady State Probablty Vector of M Product Form Network Open Closed exact exact Global Balance M=(M 0,..,M,.. Reachablty Graph (somorphc to CTMC Non Product Form approx.* *exact f solved va CTMCs exact approx. + Fgure 4.28: Range of Steady State Varables [140] and Methods approx. + Algorthm based on Lnearzer Algorthm The range of steady state varables n fgure 4.28 can be manly dvded nto three sectons: black box, local balance and global balance. On the black box level the steady state relaton of the black box s computed on the bass of Lttle s Law [98]. If the consdered system s of type product form, where the overall state of the system could be computed as a product of the sngle states of the dfferent statons and the states of the dfferent systems are ndependent from each other, the dfferent statons are n local balance. Ths leads to an effcent computaton of the steady state probablty vector and the correspondng operatonal states through product form solutons. The operatonal state and the steady state probabltes are n a close relaton as for example the number of servce requests n a staton maps onto the steady state probablty of that staton. If the steady state probabltes of the dfferent statons are not ndependent from each other, the underlyng problem s of type non product form. Then the performance values of the system have to be calculated on the bass of global balance through reachablty graphs or contnuous tme Markov chans (CTMCs. Tme Augmented Petr Nets, further compared n subsecton 4.4.2, analyze the systems on the bass of reachablty graphs respectvely the correspondng contnuous tme Markov chan (CTMC. Therefore, Non-Product Form problems can be addressed wth the drawback of hgh computatonal complexty due to state-space exploson. There are also product form solutons defned for Tme Augmented Petr Nets whch are further dscussed n secton and n subsecton Methods of the Queueng Theory, further compared n subsecton 4.4.1, and especally methods for Product Form Queueng Networks could abstract from ths level and solve the networks on the bass of ndependent steady state probabltes of the dfferent servers. Product form solutons of the Queueng Theory are therefore powerful n terms of computatonal complexty n solvng ths knd of questons. If the underlyng problem s of type Non-Product Form, 150

173 4.4. COMPARISONS they also analyze the network on the level of CTMCs (as Tme Augmented Petr Nets or use approxmatons [18]. Layered Queueng Networks (LQN, further compared n subsecton 4.4.3, evaluate the models on the bass of an approxmatve Lnearzer algorthm of the Mean Value Analyss [29, 59]. They consder open as well as closed workloads where an open workload can also generate a closed sub-behavor [58]. In FMC-QE the performance values of every basc server s computed and then the performance values of the network are computed n a bottom up approach. There are exact solutons for open product form networks as well as approxmatve solutons for closed networks (see secton 4.1 and selected non product form problems as the semaphore synchronzaton (see secton 4.3. Besde the calculaton of the performance values there are also many dfferences n the modelng of the dfferent methods where fundamental dfferences are summarzed n table 4.9. Queueng Networks Tme Augmented Layered Queueng FMC-QE Petr Nets Networks System Math. System Math. System Math. System Math. Modeled Type of Structures Modelng Model Modelng Model Modelng Model Modelng Model Server Structures (statc flat Yes Yes No Yes Yes Yes Yes Yes herarchcal No 1 Yes 2 No Yes Layered 3 Yes Yes 4 Yes Control Structures (dynamc flat No 5 Yes Yes Yes Yes Yes Yes Yes herarchcal No 1,5 Yes 2 Yes 6 Yes Layered 3 Yes Yes 4 Yes Servce Req. Structure (content flat Yes Yes Yes Yes Yes Yes Yes Yes herarchcal No 1 Yes 2 Yes 6 Yes Layered 3 Yes Yes 4 Yes 1: Only early deas lke n Dennng, Buzen [43] (secton and : Herarches are dstngushed through names of statons = ndces 3: Layered flat Network - no mandatory usage of the Forced Traffc Flow Law and unts of servce requests, no herarchcal transformaton (secton : Herarchcal transformatons through unts of servce requests and the Forced Traffc Flow Law (secton : Data Flow - only exceptons lke Fork/Jon or Synchronzaton Nodes (secton : Through Colored Petr Nets [80] Table 4.9: Comparson of Modelng Aspects Table 4.9 dfferentates the compared methodologes by ther type of modeled structures (server, control and servce request and f the structures are defned flat or herarchcal n the system modelng and the underlyng mathematcal analytcal model. Amongst others, ths s further descrbed n subsecton to where FMC-QE s further compared to the other methodologes. An mportant defnton n ths dstncton s: Herarchy [143]: A herarchy mples a servce request decomposton on one layer nto dfferent servce requests on a lower layer. In FMC-QE the decomposton s done by control servce requests, the fnal servce by operatonal servce requests. 151

174 CHAPTER 4. FMC-QE EXTENSIONS Besde mathematcal analytcal approaches, performance smulaton s another method to predct the quanttatve behavor of systems. In subsecton FMC-QE and smulatve approaches are shortly compared to each other Queueng Theory The Queueng Theory, descrbed n secton 2.1, delvers the man mathematcal bass for the FMC-QE Calculus. The herarchcal calculatons n the Calculus are based on the Forced Traffc Flow Law [68, 79, 95] for relatons between herarchcal layers (vertcal and Lttle s Law [98] for relaton wthn a herarchcal layer (horzontal. For the calculatons of the performance values of the logcal and multplexer servers FMC-QE also uses the mathematcal background of the Queueng Theory [67, 79, 83, 88, 89, 125]. Despte the Forced Traffc Flow Law and Lttle s Law, n the calculatons of whole nets the nvestgatons of Jackson [77, 78] and Baskett, Chandy, Muntz and Palacos [8] on Product Form Queueng Networks and the ndependence of the dfferent servers were used. In the calculatons of closed networks the summaton method [17] was ntegrated nto the calculus and compared to results examples calculated through the results of Gordon and Newell [64] and the Mean Value Analyss (MVA [118]. The summaton method n combnaton wth the dea of modelng the synchronzaton through a multplexer [147] and complementary delays [71] was also used for the performance estmaton of semaphore synchronzaton problems. Whle the Queueng Theory has strengths n the calculatons of performance values, especally of Product Form Queueng Networks, the weaknesses are n the lack of expressveness modelng power. Whle Queueng Networks focus on the modelng of server structures and servce request flows, the modelng of control flows s not possble. There are extensons, dscussed later, n whch Fork-Jons or Synchronzatons are modeled but n the classcal case control flows could not be modeled. Furthermore, n ths extensons the control flow s ntegrated nto the statc structures. Whle n FMC-QE the modelng of server structures, whch s related to Queueng Theory, s only one vew n the three-dmensonal modelng space, n Queueng Theory ths leads to ths control flow modelng problem and therefore less expressveness n the modelng of complex systems. Furthermore, n FMC-QE the modelng of servce requests as a tuple of value and unt, whch enables the transformaton of servce requests at herarchcal borders and therefore the herarchcal modelng, has roots n the Queueng Theory - especally through the nvestgatons of Dennng and Buzen [43]. But there ths approach was not systematcally used n order to defne herarches of servce requests on arbtrary herarchcal levels, n [43] there were only two levels of job and requests. As already descrbed n secton 2.3.5, there was no defnton of servce request transformatons n [43] and [95] and therefore real herarches could not be defned. Queueng Networks wth Fork-Jon and Synchronzaton Nodes There are extensons of classcal Queueng Networks to nclude Fork-Jon Queues [3, 84, 96] or Synchronzaton nodes [117]. Fork-Jon queues [3] are queueng systems wth parallel servers and a synchronzed arrval and departure stream. Servce requests could arrve n a batch and are then mmedately dspatched to one server each (fork. The dfferent servce requests are then queued nto nfnte buffers and ndvdually handled by the correspondng server. If all servce requests of the batch are fnshed, the batch s recomposed and departed (jon, as shown n fgure

175 4.4. COMPARISONS 1 λ Fork Pont 2 N Jon Pont Fgure 4.29: Fork-Jon Queueng Model [96] Synchronzaton nodes consst of M nfnte capacty buffers where servce requests arrvng on M dstnct random nput flows are stored (one buffer per flow, as depcted n fgure λ 1 λ M Fgure 4.30: SM/M/1 Queue [117] The servce requests are stored untl each buffer contans at least one servce request. Then the servce requests are grouped to one servce request and nstantaneously released n a synchronzaton departure [117]. In [117] t s shown that the synchronzaton of ndependent Posson flows at a synchronzaton node converge to a Posson process wth a rate whch s equal to the mnmum of all nput flows. They also ntegrate the synchronzaton nodes nto Jackson networks [77, 78] and state that [117]: If n a generalzed Jackson network wth an acyclc synchronzaton process (every synchronzaton node s dsconnected from tself - routng probablty p = 0 and fnte synchronzaton buffers, the jont dstrbuton of the equlbrum queue length process at all exponental server nodes s asymptotcally product form and the equlbrum departure process converges weakly to ndependent Posson processes, as the sze of the synchronzaton buffers goes to nfnty. The results of the nvestgatons of the Fork-Jon Queues and especally the Synchronzaton nodes support the performance predctons of the herarchcal (parallel actvtes of FMC-QE, descrbed n secton 3.4.5, whle n these predctons the overall servce rate s also equal to the mnmum of all rates Tme Augmented Petr Nets For the modelng of the dynamc behavor and the control flow the tme augmented adaptatons of the FMC Petr Nets n FMC-QE also consst of tmed and mmedate transtons lke Generalzed Stochastc Petr Nets (GSPN [101, 102], descrbed n secton The dstncton of operatonal and control flow n FMC-QE Petr Nets has smlartes to Colored Petr Nets [80] where attrbutes are assocated to tokens lke n Colored GSPNs [33] (referrng to [12]. Followng the classfcaton of secton 2.2.1, FMC-QE Petr Nets are n the class of Tmed Transtons Petr Nets (TTPN wthout reservaton whle the reservaton s often rrelevant because conflcts are mostly modeled usng mmedate transtons. In networks wth several nter-server dependences, the performance estmatons of Tme Augmented Petr Nets are more precse than the FMC-QE approxmatons because n Tme Augmented Petr Nets the global balance state of a Non-Product Form Network s computed, whle 153

176 CHAPTER 4. FMC-QE EXTENSIONS n FMC-QE these Non-Product Form problems are approxmated though transformatons nto product form solutons. But, n contrast to FMC-QE wth t s lnear complexty n the calculaton of the performance values, the algorthms for the calculaton of the performance values of Tme Augmented Petr Nets suffer from the state space exploson problem through the calculatons of the global balance equatons and the assocated lnear equaton systems. In FMC-QE the steady state behavor of the systems s of nterest. In contrast to Tme Augmented Petr Nets and Petr Nets n general, the transent behavor as well as an analyss for deadlocks s not n the focus n FMC-QE. In the followng some extensons of the Tme Augmented Petr Nets are brefly compared to FMC-QE. Product Form Petr Nets In comparson to Non Product Form SPNs, stochastc Petr Nets wth product form solutons, descrbed n secton 2.2.4, can calculate larger Petr Nets by avodng the state-space exploson problem through product form solutons. But stll, n comparson to the lnear complexty n FMC-QE, the dfferent approaches need complex calculatons. In the approach of Lazar and Robertazz [94] there s a need for the nvestgaton of the reachablty state whch s the man drawback n ths approach. In the approach of Henderson et al., even n the hghly optmzed verson of Haddad et al. [70], only the recognton f an SPN could have a product form soluton has a complexty of O( T 2 steps [70] where T s the set of transtons. The approach of Florn and Natkn [52] s lmted to closed synchronzed queueng networks and there are effcency problems n the solvng of the lnear system for the determnaton of the constant vector of ther matrx product form approach [52]. Furthermore, Robertazz [119] states that the general lmtatons of product form solutons for SPNs n concurrent resource sharng and synchronzaton lmt the modelng space of SPNs. For resource sharng models Robertazz [119] states that blockng excluded an SPN from product form solutons and only f resources are mmedately avalable or requests for resources whch are not mmedately avalable are mmedately cleared from the system, for whch he states that ths knd of resource sharng s not so wdely dstrbuted n real systems. From the vewpont of solated crculatons [94] Robertazz [119] also conjectures that synchronzaton could prevent hgher dmensonal models of havng a product form soluton. Furthermore, the product form SPNs stll only model the behavor of the systems and not the server structures, lke FMC-QE, and therefore ths clear dstncton for a complexty reducton and larger modelng capablty s not ncluded. Queueng Petr Nets (QPN Whle the foundatons of FMC-QE are manly FMC and the Queueng Theory and mathematcal foundaton reles on the Queueng Theory, the Queueng Petr Nets [9, 10, 13, 14], further descrbed n secton 2.2.5, are an extenson of a colored GSPN, through the ntegraton of schedulng strateges nto the places. Ths s the bass for the man dfferences between QPNs and FMC-QE. Whle n FMC-QE the quanttatve system propertes are modeled n three vews, n a QPN both quanttatve and qualtatve (logcal propertes are mxed n one network. The analyss of qualtatve propertes, lke tmeless traps or boundness, s not n the scope of FMC-QE. For the analyss of quanttatve system propertes QPNs must often be analyzed by nspectng the reachablty set of the correspondng colored GSPN where the queue s modeled wth CGSPN elements [14]. Though ths lmtaton QPNs could descrbe queues n a 154

177 4.4. COMPARISONS more convenent way wthout havng to specfy queues by pure Petr Net elements [14] but n the calculatons they stll suffer from the state space exploson problem of the Tme Augmented Petr Nets. Product form QPNs [11, 12] could address some problems of the state-space exploson problem n QPNs but snce the networks are transformed to an equvalent SPN, the problems of product form SPNs are stll the same and the complexty n the calculaton of the performance values s stll hgh whle the modelng power of the orgnal QPNs s reduced. The HQPNs [15] delver exact solutons for large networks and reduce the state pace exploson problem by one order of magntude but stll n contrast to FMC-QE wth t s lnear complexty n the calculaton of the performance values, the HQPN suffers form the state space exploson problem. Also n HQPNs the dfferent subnets have to be solated through the nput and output places whle n the dfferent vews n FMC-QE and a clearer dstncton of logcal and multplexer servers a logcal server could be handled by an arbtrary multplexer. But of course, f there are a lot of control flows and synchronzaton ssues n the modeled system, an HQPN could be more attractve than an FMC-QE model f the HQPN s stll computable whle FMC- QE uses approxmatons for synchronzatons Layered Queueng Networks (LQN In [59] the LQN approach, further descrbed n secton 2.3.6, especally the LQN Solver (LQNS, was compared to other layered queueng system approaches where as a result LQN covers more system features than any other attempt. In table 4.10 ths comparson s mported and extended by a comparson to FMC-QE. Feature LQNS FMC-QE MOL SRVN TDA Ramesh MOD Fontenot WSQN-HQN Kurasug APERA FULL-ACCESS yes yes no yes yes no no no no no yes Devce Schedulng FHPSh Mpx FPHS FPH F FP FP F FP FP FP Task Schedulng FH FPSh F F F F F F F F F Open arrvals (OPEN yes yes no yes no?? yes yes yes? Infnte-servers yes yes yes yes yes yes yes yes yes yes yes SERV-PATTERN SD SD S SD S SD D S SD SD SD VAR yes yes yes yes no yes no? yes no no PAR yes yes no no no no no no no no no REPL yes yes yes no no no no no no no? ASYNC yes yes no yes no yes? no no yes? Forwardng yes yes no no no no no no no no no FAST, INTERLOCK yes - no yes no no no no no no no Where F: FIFO, P: Preemptve Prorty, H: Head-of-Lne Prorty, R: Random, Sh: Processor Sharng, Mpx: Multplex S: Stochastc Phase, D: Determnstc Phase,?: Unclear from the reference Table 4.10: Comparson of LQNS to FMC-QE and other Layered Queueng Systems [59] FULL-ACCESS means that a server can ssue a request to any server at a lower layer, rather than just to the layer below t [59]. In FMC-QE, through the dstncton of logcal and multplexer servers, ths s also possble but herarchcally embedded. Whle the logcal servers are defned through the servce request structures, the relatons through logcal server layers are by defnton only to the lower layer but ths s no restrcton whle the servers are the logcal layers followng the servce request structure. From the logcal to multplexer servers a connecton from every layer to every layer s possble. 155

178 CHAPTER 4. FMC-QE EXTENSIONS For the modelng of devce schedulng n LQN the strateges FIFO, Preemptve Prorty, Headof-Lne Prorty and Processor Sharng and for task schedulng FIFO and Head-of-Lne Prorty are realzed. In FMC-QE the devce (multplexer server s modeled as a multplexer and therefore the schedulng strateges are molded at Task (logcal server level. There, n FMC-QE, all possble schedulng strateges of the Kendall Notaton, for whch steady-state formulas have been set up, could be used. By now, FIFO, Preemptve Prorty and Processor Sharng as well as determnstc server and nfnte server have been realzed but the calculus could be easly extended to other strateges by mportng the formulas at the basc server staton level. The LQN Solver supports open Posson arrval processes as well as closed models [59]. The standard hybrd load model n FMC-QE, where the arrval rate as well as the overall number of servce requests could be defned, as descrbed n secton 3.4.2, follows the calculatons of open queueng networks where the ntegraton of the summaton method wth an teratve soluton, descrbed n secton 4.1, enhances the modelng power to closed models. So n FMC-QE open and closed models are also handled. The handlng of nfnte-servers for the modelng of for example clent behavor s ntegrated nto the LQN Solver as well as n FMC-QE. In the LQN Solver both stochastc and determnstc patterns of request for lower-layer servce (SERV-PATTERN [59] are ntegrated. Whle n the FMC-QE Tableau the formulas for the dfferent basc servers could be exchanged n order to ft to the real behavor of the system, also both stochastc and determnstc patterns are ntegrated. An arbtrary varance of CPU demands (VAR, possble n LQN, s also possble to model and evaluate n FMC-QE. Parallelsm (PAR and replcaton (REPL used n LQN to model parallel scenaros as well as ncreasng the scalablty of solutons by replcatng servers or structures [59, 108] s also supported n FMC-QE. In FMC-QE parallelsm n the handlng of servce requests, lke the handlng of sub-request 1 by server X and handlng of sub-request 2 by server Y, s modeled n the Petr Net and then the dfferent sub-requests are represented by one lne each n the Tableau. Replcaton s supported by the ablty of defnng multplcty coeffcents for both logcal and multplexer servers on all herarchcal levels. Asynchronous messages (ASYNC and the forwardng of servce requests (Forwardng have not been explctly modeled n ths thess but could also be modeled and evaluated n FMC-QE. The solver features FAST (a fast-couplng correcton for multclass FIFO servers wth dfferent servce tmes [59] and INTERLOCK (a correcton for correlated requests due to shared resources n generatng arrvals [59] are ntegrated nto the LQN Solver for the mprovement of the approxmatons. Whle the calculaton of the performance values n FMC-QE s dfferent to LQNS, these mprovements have not been consdered. Whereas LQN and FMC-QE address smlar problems, as descrbed n ths secton, the calculaton of the performance values dffer fundamentally. The FMC-QE Tableau s manly based on Lttle s Law and the Forced Traffc Flow Law. Although the vst ratos are mentoned, the Forced Traffc Flow Law s not a central concern. The LQN Solver s manly based on Mean Value Analyss [59]. Furthermore, when herarches are defned through servce request transformatons and servce request unt transformatons [143], the Layered Queueng Networks are consdered as flat networks wth layers and no herarches because the Forced Traffc Flow Law s used n the detecton of bottlenecks n LQNs [58] but t s not mandatory n the system modelng and the transformaton of servce requests and unts of servce requests lke n FMC-QE. Also n FMC-QE there are many other fundamental dfferences to LQN lke that n FMC-QE 156

179 4.4. COMPARISONS s a strcter modelng from the vew of the servce request structures. But overall, the whole approach s nterestng and n later collaboratons to the LQN Research Group, where a frst contact has already been ntated, further dfferences and smlartes could be found Performance Smulatons The results of the cooperaton wth the Enterprse Platform and Integraton Concepts Group of Prof. Dr. h.c. Hasso Plattner n the comparson of the Perfact Smulaton envronment [42] to FMC-QE and a cooperaton [116] wth Mathas Frtzsche, SAP Research CEC Belfast, for a couplng of FMC-QE and the Model-Drven Performance Engneerng (MDPE [61, 62] show that performance smulatons and analytcal methods lke FMC-QE are more supplementatons to each other than rvals. If n a system there are a lot of control flows or sde effects, an analytcal model, lke n FMC- QE s sometmes hard to derve. Therefore, a smulaton and especally a smulaton whch s embedded n the real envronment, lke Perfact, s easer to establsh and sometmes more precse. So analytcal models and smulatons could complete each other through comparsons for a hgher precson and also through the ntegraton of the respectvely other methodology. The comparson of Perfact and FMC-QE through a case study accelerated the development of both approaches. In addton, the herarchcal modelng and calculatons n FMC-QE would allow an ntegraton of smulated performance values nto the Tableau and thereby extendng the modelng and evaluaton power of FMC-QE. Through the ntegraton of FMC-QE nto smulaton envronments, lke n the cooperaton wth SAP Research CEC Belfast, FMC-QE could provde other performance values lke the smulaton envronment. Addtonally, through the lnear complexty of the performance value calculaton a broad range of possble parameters could be covered n order to support senstvty analyses whch are very tme consumng f every smulaton takes some tme. If nterestng parameters are found, the smulaton of the system wth ths parameters could support and refne the computed performance values. In the next chapter some case studes wth FMC-QE are summarzed. The results of the comparson of the Perfact Smulaton envronment and FMC-QE are a part of ths and are summarzed n secton

180

181 Chapter 5 FMC-QE Case Studes Ths chapter provdes case studes modeled and evaluated wth FMC-QE. Each case study focuses on dfferent FMC-QE related questons. In the frst case study n secton 5.1 a servce-based search portal was modeled and evaluated usng FMC-QE. The focus s on modelng a larger system wth, n ths case, 62 logcal servers - 44 basc server statons and 18 herarchcal server statons on 7 herarchcal levels wth a man nterest n the computatonal complexty of the correspondng Tableau. The second example n secton 5.2, the Emergency Rsk Management Framework (ERMF, focuses on the comparson of the performance predctons of an FMC-QE performance model, a smulaton and the correspondng measured values n the real system. The Emergency Rsk Management Framework (ERMF s a system n the SAP NetWeaver and SAP WebAS Envronment 1, orgnally developed by SAP Research France. Ths case study was also a case study n the development of the performance smulaton envronment Perfact [42]. For the valdaton of Perfact and the FMC-QE predctons the performance values of the real system were compared to the performance predctons of the Perfact smulaton and the mathematcal-analytcal predctons of FMC-QE. Therefore, n ths case study, also publshed n [120], the focus s on the dfferences respectvely complances between the appled methods. In the thrd example n secton 5.3 the Axs2 web servce framework 2 s nvestgated. Ths case study focuses on the modelng of an herarchcal protocol stack and the performance predcton of multplexers and synchronzatons n the context of threads and processes. In [37] ths case study was also publshed. 1 SAP AG, SAP NetWeaver, Webste: August Apache Software Foundaton, Apache Axs2 Archtecture Gude, Webste: Axs2ArchtectureGude.html, August

182 CHAPTER 5. FMC-QE CASE STUDIES 5.1 HPI Search Portal - a Servce based Case Study HPI Search Portal was a jont project of the "Servce-Orented Systems Engneerng" Research School at the Hasso-Plattner-Insttute 3. It has been ntated to serve as a common scenaro for research actvtes n the doman of Servce-based systems. One of the major goals of the HPI Search Portal project was to provde a common playground for need fndng purposes, problem dentfcatons, technology testng and proof of concepts. A second mportant goal of the HPI Search Portal project s to show and present the benefts of ths SOA-based approach and the dstrbuted development and to ntegrate many dfferent technques and programmng languages and usng only nterface descrptons for the ntegraton. Whle n ths jont project every partcpant benefted from a dfferent perspectve, the gan of the author was n the performance - as well as archtecture modelng of the system. From the vewpont of FMC-QE ths case study was a proof-of-concept for a larger system wth many herarchcal layers as well as a performance test for a larger Tableau. The author wants the thank the other Research School members contrbuted to ths project, especally Flavus Copacu, Benjamn Hagedorn, Frank Kaufer, Harald Meyer, Hagen Overdck, Mchael Schöbel and Matthas Uflacker Archtecture The HPI Search Portal s a Servce-based applcaton whch provdes a dstrbuted search engne nsde a research nsttute n order to fnd nformatons about lectures, lecturers, locatons and mportant events. Ths composte applcaton aggregates nformaton, gathered and combned from dfferent servces, n order to offer a unque search nterface combnng several other (external search servces. An overvew on the archtecture of ths system s provded n fgure 5.1. ( 8 *1/9 <8( ( (,*,..( 8 9;&&& 8 9; 19 : 9 8 ( 8 ( (-./ 9;* & &$ & *1!" #$% ($ * 21 *3 *+,*6, &&7 *5 *+ *5 *+, * 21 *4 *00,1! $ 21 * & 6 &$ "2( & "2(. &'(( &$'(( &'(( (-./ Fgure 5.1: HPI Search Portal - Archtecture (see Appendx - Fgure C.1 3 HPI Research School on Servce-Orented Systems Engneerng, Webste: February

183 5.1. HPI SEARCH PORTAL - A SERVICE BASED CASE STUDY In the HPI Search Portal Applcaton the search for nformaton s dvded nto 5 sub searches, the Super Search, the Fle Search, the People Search, the Room Search and the Events of the Day. The Super Search ntegrates dfferent publc avalable search engnes by usng dfferent technques. The frst search engne s ntegrated by connectng to the SOAP 4 nterface of the engne, the second s connected va REST 5 Web Servces and the thrd s ntegrated by a HTML Webste wrapper. The dfferent search responses then are aggregated and ranked and further enrched by prevews of the dfferent results. The Fle Search performs a search for fles at the local fle servers (Index Search and also enrches the results wth prevews. Names, mal addresses and other contact nformaton of the lecturers and the other nsttute staff-members could be searched va the People Search. The contact nformaton s further enrched by a photo of the person and a 3D path to the person s offce. The Room Search could also be drectly accessed n order to fnd a path to a semnar room or an offce of a known person. The search s further complemented by newsworthy event schedules, provded by the Events of the Day servce. From a hgh level the overall servce request s composed as follows: Portal Search Request Super Search Request - Search Request - Prevew Request Fle Search Request - Index Search - Prevew Request People Search Request - Lst Search Request - People Detal Search Request Room Search Request Event Search Request In the next subsecton on the FMC-QE model ths servce request structure s further refned and embedded nto the techncal envronment FMC-QE Model The fgures 5.2 and 5.3 show an excerpt (servce request structures and behavor of the FMC- QE Model of the HPI Search Portal scenaro. Due to the sze of the dagram the author wll not gve a closer descrpton of the model. Ths fgures shall only llustrate that larger FMC-QE models (62 logcal servers (44 basc, 18 herarchcal on 7 herarchcal levels are also possble and the models and especally the Tableau (table scale. 4 W3C, SOAP Verson 1.2, W3C Recommendaton 27 Aprl 2007, February Representatonal State Transfer, REST, In: Roy Thomas Feldng, Archtectural Styles and the Desgn of Network-based Software Archtectures, Dssertaton, Unversty of Calforna, Irvne, 2000, Webste: cs.uc.edu/~feldng/pubs/dssertaton/top.htm, February The performance parameters n ths Tableau are test values to show the capabltes of the Tableau and do not reflect the performance behavor of the real system. 161

184 CHAPTER 5. FMC-QE CASE STUDIES *$ *$ +!"!#$" #$"! #"""$ """$ # #"""$ #%&'("""$ ""% #%&'( """$ 1234'$" '$1234 # 1234'$ '( 1$!'( '(*$ *$'( '(1$! '(*$ 1234*$ *$ *$ """$&*$ ""% %/*%'( """$&*$.'7($##" 7($#.'# $-"1$!" 1$!$-" $-"1$!.'7($## $-"$- $-" $-" #$- #$-.'*$ *$.'# #$-.'*$ 27($##" 7($# 2 $- $- $- $-"1$!" 1$!$-" $-"1$! 27($## $-"$- $-" $-" 2*$ *$ 2 2*$ $-"1$!" 1$!$-" $-"1$! 1234'$" '$1234 # 1234'$ $-"$- $-" $-" 1234*$ *$ *$ %&.'*$ *$ %$!.' %&.'*$ """$ ""% #%&'( """$ 1234'$" '$1234 # 1234'$ '( 1$!'( '(1$! '(*$ *$'( '(*$ $- $- $- 1234*$ *$ *$ '$ '$ '$ '$$- %'$$-% '$$- """$&*$ ""% %/*%'( """$&*$ 0!$- 0!$- 0!$- $- %*% *%$- % *$ #!"! # #! #1$!" 1$! # #1$! '$- $-' '$- '"$- $-'" '" '#5$$- %'#5$$- '#5$$-1$! *$ *$ *$ #1$!" 1$! # #1$! 6$- $- 6 5'$-*$ *$5'$- '#$- $-'# 6$- 5'$-*$ '#$- $$!*$ % 605&*% % $$!*$ """$&*$ ""% %/*%%, % """$&*$ #'$" '$ # #'$ 45' 1$!45' % 45'1$! '#$- #$- '#$- 45' *$ *$45' 45' %*$ '#$- %*% *%'#$- % *$ #1$!" 1$! # #1$! 6$- $- 6 5'$-*$ *$5'$- 6$- 5'$-*$ $$!*$ % 605&*% % $$!*$ #1$!" 1$! # #1$! $- $- $- +$!$$- $-+$!$ +$!$, *$ *$,, *$ *$ *$ *$ #$" #$" #$" Fgure 5.2: HPI Search Portal - Servce Request Structure (see Appendx - Fgure C.2 162

185 5.1. HPI SEARCH PORTAL - A SERVICE BASED CASE STUDY 2..- /2. /!" $ "!" # $ # ( ( #%& #%& " " ' #%& #%& ' " * + ( ( #%& #%& ( " " " / +, / &, ( &,( (- (.-( 0 + 0*, ( ', /0 +0*, ( ', / 3 ( " 2. $ $ ' ' ' $ Fgure 5.3: HPI Search Portal - Behavor (see Appendx - Fgure C.3 163

186 CHAPTER 5. FMC-QE CASE STUDIES n ges [1] λ bott [1] f λ [1] Expermental Parameters ,9313 0, ,6382 Table 5.1: HPI Search Portal - Tableau (see Appendx - Table B.12 SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Renderng Request 1,00 1,00 1,00 1,00 11,638 Webpage Renderer ,0005 0,007 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Response Aggregaton 1,00 1,00 1,00 1,00 11,638 Response Agg ,0002 0,002 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Resp. + Prevews Agg. 1,00 1,00 1,00 1,00 11,638 Aggregator ,0002 0,003 12,931 0,000 0,000 0,900 0,077 0,900 0, HTML Parsng 1,00 1,00 1,00 1,00 11,638 HTML Parser ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew Generaton 1,00 1,00 20,00 20,00 232,764 Prevew Generator ,0001 1, ,769 0,004 0,000 0,000 0,012 0,000 0,012 0, HTML Generaton 1,00 1,00 1,00 1,00 11,638 HTML Generator ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew 1,00 1,00 1,00 1,00 11,638 Prevew Handler ,931 16,200 1,392 1,812 0,156 18,012 1, Aggregaton + Generaton 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, SOAP Unwrappng 1,00 5,00 1,00 5,00 58,191 SOAP Unwrapper ,0001 0, ,152 0,013 0,000 0,000 0,038 0,001 0,038 0, Search Engne 1 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0087 1,000 64,657 0,000 0,000 0,938 0,016 0,938 0, SOAP Generaton 1,00 5,00 1,00 5,00 58,191 SOAP Generator ,0001 0, ,152 0,013 0,000 0,000 0,038 0,001 0,038 0, Search Engne 1 Handlng 1,00 1,00 5,00 5,00 58,191 SE 1 Handler ,152 0,000 0,000 1,015 0,017 1,015 0, REST Unwrappng 1,00 5,00 1,00 5,00 58,191 REST Unwrapper ,0008 0,053 65,789 0,221 0,000 0,000 0,885 0,015 0,885 0, Search Engne 2 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0090 1, ,111 0,000 0,000 0,885 0,015 0,885 0, REST Generaton 1,00 5,00 1,00 5,00 58,191 REST Generator ,0030 0,197 65,789 0,221 0,004 0,000 0,885 0,015 0,888 0, Search Engne 2 Handlng 1,00 1,00 5,00 5,00 58,191 SE 2 Handler ,789 0,004 0,000 2,654 0,046 2,657 0, HTML Parsng 1,00 5,00 1,00 5,00 58,191 HTML Parser ,0024 0,102 42,553 0,273 0,008 0,000 1,367 0,024 1,375 0, Search Engne 3 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0008 1, ,000 0,000 0,000 0,047 0,001 0,047 0, HTML Generaton 1,00 5,00 1,00 5,00 58,191 HTML Generator ,0023 0,098 42,553 0,273 0,005 0,000 1,367 0,024 1,373 0, Search Engne 3 Handlng 1,00 1,00 5,00 5,00 58,191 SE 3 Handler ,553 0,013 0,000 2,782 0,048 2,794 0, Search Request 1,00 1,00 1,00 1,00 11,638 Searcher ,553 0,013 0,001 2,782 0,239 2,794 0, Req. + SOAP Generaton 1,00 1,00 1,00 1,00 11,638 Req. + SOAP Gen ,0008 0,010 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Super Search Request 1,00 1,00 1,00 1,00 11,638 Super Search Srv ,931 32,413 2,785 7,294 0,627 39,707 3, Aggregaton 1,00 1,00 1,00 1,00 11,638 Aggregator ,0006 0,008 12,931 0,900 32,413 2,785 0,900 0,077 33,313 2, HTML Parsng 1,00 1,00 1,00 1,00 11,638 HTML Parser ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew Generaton 1,00 1,00 20,00 20,00 232,764 Prevew Generator ,0050 1, ,000 0,388 0,083 0,000 1,164 0,005 1,247 0, HTML Generaton 1,00 1,00 1,00 1,00 11,638 HTML Generator , ,931 0,900 8,183 0,703 0,900 0,077 9,083 0, Prevew 1,00 1,00 1,00 1,00 11,638 Prevew Handler ,931 16,366 1,406 2,964 0,255 19,330 1, Aggregaton + Gen. 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Index Search 1,00 1,00 1,00 1,00 11,638 Index Searcher ,0050 1, ,000 0,058 0,004 0,000 0,058 0,005 0,062 0, Fle Search Req. Gen. 1,00 1,00 1,00 1,00 11,638 Generator ,0008 0,010 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Fle Search 1,00 1,00 1,00 1,00 11,638 Fle Searcher ,931 64,983 5,584 5,722 0,492 70,705 6, Response Sendng 1,00 1,00 1,00 1,00 11,638 Response Sender ,0054 0,070 12,931 0,900 0,000 0,000 0,900 0,077 0,900 0, Response Handlng 1,00 10,00 1,00 10,00 116,382 Response Handler ,0015 0, ,313 0,900 0,000 0,000 0,900 0,008 0,900 0, Pcture Engne Search 1,00 10,00 1,00 10,00 116,382 Pcture Engne ,0040 1, ,000 0,466 0,405 0,003 0,466 0,004 0,871 0, Request Generaton 1,00 10,00 1,00 10,00 116,382 Request Generator ,0057 0, ,313 0,900 8,100 0,070 0,900 0,008 9,000 0, Pcture Search 0,40 5,00 5,00 10,00 116,382 Pcture Searcher ,313 8,505 0,073 2,266 0,019 10,771 0, Response Handlng 1,00 5,00 1,00 5,00 58,191 Response Handler ,0025 0,162 64,657 0,900 8,100 0,139 0,900 0,015 9,000 0, D Path Generaton 1,00 5,00 1,00 5,00 58,191 3D Path Generator ,0025 1, ,000 0,073 0,001 0,000 0,145 0,003 0,146 0, Extracton and Generaton 1,00 5,00 1,00 5,00 58,191 Extractor and Gen ,0021 0,136 64,657 0,900 8,101 0,139 0,900 0,015 9,001 0, Room Search 1,00 5,00 1,00 5,00 58,191 Room Searcher ,657 16,202 0,278 1,945 0,033 18,147 0, People Detal Search 1,00 1,00 5,00 5,00 58,191 People Search H ,657 24,707 0,425 4,211 0,072 28,918 0, Aggregaton + Generaton 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0098 0,127 12,931 0,900 0,000 0,000 0,900 0,077 0,900 0, Response Parsng 1,00 1,00 1,00 1,00 11,638 Response Parser ,0090 1, ,111 0,105 0,000 0,000 0,105 0,009 0,105 0, People Search 1,00 1,00 1,00 1,00 11,638 People Searcher ,0024 1, ,667 0,028 0,001 0,000 0,028 0,002 0,029 0, LDAP Request Generaton 1,00 1,00 1,00 1,00 11,638 LDAP Req. Gen ,0025 1, ,000 0,029 0,001 0,000 0,029 0,003 0,030 0, LDAP Servce Request 1,00 1,00 1,00 1,00 11,638 LDAP Handler ,000 0,002 0,000 0,162 0,014 0,163 0, People Search Req. Gen. 1,00 1,00 1,00 1,00 11,638 Request Generator ,0080 0,103 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, People Search 1,00 1,00 1,00 1,00 11,638 People Searcher ,931 32,809 2,819 7,073 0,608 39,881 3, Response Handlng 1,00 3,00 1,00 3,00 34,915 Response Handler ,0012 0,047 38,794 0,900 40,909 1,172 0,900 0,026 41,809 1, Calendar Search Req. 1,00 3,00 1,00 3,00 34,915 Calendar ,0090 1, ,111 0,314 0,144 0,004 0,314 0,009 0,458 0, Request Generaton 1,00 3,00 1,00 3,00 34,915 Request Generator ,0090 0,349 38,794 0,900 8,100 0,232 0,900 0,026 9,000 0, Event Search Request 1,00 1,00 3,00 3,00 34,915 Event Searcher ,794 49,153 1,408 2,114 0,061 51,267 1, Response Handlng 1,00 3,00 1,00 3,00 34,915 Response Handler ,0001 0,005 38,794 0,900 49,153 1,408 0,900 0,026 50,053 1, D Path Generaton 1,00 3,00 1,00 3,00 34,915 3D Path Generator ,0025 1, ,000 0,044 0,000 0,000 0,087 0,003 0,087 0, Extracton and Generaton 1,00 3,00 1,00 3,00 34,915 Extractor and Gen ,0002 0,006 38,794 0,900 8,100 0,232 0,900 0,026 9,000 0, Room Search 1,00 1,00 3,00 3,00 34,915 Room Searcher ,794 57,253 1,640 1,887 0,054 59,140 1, Portal Search Serv. Req. 1,00 1,00 1,00 1,00 11,638 Portal Searcher ,931 64,983 5,584 7,294 0,627 72,277 6, Sub-Request Generaton 1,00 1,00 1,00 1,00 11,638 Sub-Request Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Portal Request 1,00 1,00 1,00 1,00 11,638 Portal Server ,931 81,183 6,976 9,094 0,781 90,277 7, Webpage Request 1,00 1,00 1,00 1,00 11,638 Webpage Req ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Webrequest 1,00 1,00 1,00 1,00 11,638 Webrequest Exec ,931 97,383 8,368 10,894 0, ,277 9, Job Generaton 1,00 1,00 1,00 1,00 11,638 Generator Clent334,3927 0,003 0,000 0, , , , ,393 Multplexer Secton Servce Request Secton j Name j m j [1] X j 1 HPI B Man 2 0,155 2 HPI C Prevew Srv. 8 0,101 3 HPI V SE 1 Serv.Srv. 1 0,001 4 Search Engne 1 Server 5 HPI C1.1 - SE 2 Serv. Srv. 1 0,019 6 Search Engne 2 Server 7 HPI C SE 3 Serv. Srv. 1 0,024 8 Search Engne 3 Server 9 HPI C Index Serv.Srv. 4 0, HPI C Pc. Serv. Srv. 2 0, HPI A DBIMServce 4 0, HPI C1.3 - LDAP Serv. Srv. 2 0, HPI LDAP Server 4 0, HPI C Event Srv. 4 0,027 Server Secton Dynamc Evaluaton Secton Wth the help of the Tableau some performance predctons, examples n fgure fgure 5.6, could be performed. The chart n fgure 5.4 shows the overall response tme R ges dependency of the overall arrval rate λ. Whle n the ntal system confguraton n table 5.1 the bottleneck throughput s λ bott 12, 93 [SRq], the overall response tme grows sgnfcant when λ > 10 [SRq]. The Tableau shows [s] [s] that the Man Server (HPI B Man s the bottleneck because all assocated logcal servers have the hghest utlzaton of ρ = 0, 9, and therefore, an mprovement of the Man Server could 164

187 5.1. HPI SEARCH PORTAL - A SERVICE BASED CASE STUDY speed up the whole system. Three confguratons, one wth one more CPU (m 1 = 3, one wth two more CPUs (m 1 = 4 and one wth three more CPUs (m 1 = 5, are also shown n fgure 5.4. It can be seen that the upgrade of the Man Server results n a better overall performance of the system for m 1 = 3 and m 1 = 4. A further upgrade of the Man Server to 5 CPUs does not result n such a huge performance gan as then the Pcture Engne s the new bottleneck λ bott, m=2 = 12,93 λ bott, m=3 = 19,40 λ bott, m=4=5 = R ges [s] m 1 =2 m 1 =3 m 1 =4 m 1 = λ [SRq]/[s] Fgure 5.4: HPI Search Portal - Chart: Overall Response Tme - Arrval Rate Fgure 5.5 shows the relaton between the overall arrval rate λ and the external servce tme X ext for the ntal system confguraton of table 5.1 wth a mean overall populaton of n ges = 4.000[SRq]. In ths confguraton a desred overall arrval rate λ n a range from λ = 1 [SRq] [s] to λ = 10 [SRq] results n an external servce tme from approxmately 7 to approxmately 66 [s] mnutes (more precsely 396s to 3.965s. If then, n ths confguraton, the mean external servce tme would be longer that 7 mnutes, referrng to fgure 5.4, the overall response tme R ges s stll small (λ 10 [SRq] for the ntal confguraton of a Man Server wth 2 CPUs. In the example [s] everyone of the students could request the portal (n ges = 4.000[SRq] every 7 mnutes X ext [mn] λ [SRq]/[s] (n ges =4000 [SRq] Fgure 5.5: HPI Search Portal - Chart: External Servce Tme - Arrval Rate 165

188 CHAPTER 5. FMC-QE CASE STUDIES Fgure 5.6 shows the relaton of the overall response tme R ges from dfferent confguratons of the number of processors n the man server (HPI B-1.3 (Man, whch s referred as multplexer 1 (m j=1. For two dfferent overall arrval rates (λ = 8 [SRq] (+ and λ = 12 [SRq] (x [s] [s] the correspondng overall response tmes are plotted. Ths chart then shows that for the confguraton wth an arrval rate of λ = 8 [SRq], two or three processors (m [s] j=1 = 2 would be the best choce because the change from three processors to more processors does not sgnfcantly gan n more performance n ths specfc scenaro. For the confguraton wth an arrval rate of λ = 12 [SRq] t s hghly apprecated to nstall at least three processors because there s a huge [s] performance ncrease from two to three processors λ=12 [SRq/s] 10 R ges [s] λ=8 [SRq/s] m j=1 (HPI B-1.3 (Man Fgure 5.6: HPI Search Portal - Chart: Response Tme - Number of Parallel Man Processors Summary In ths secton s was shown that FMC-QE s also sutable for larger models (62 logcal servers (44 basc, 18 herarchcal on 7 herarchcal levels. Wth the help of the Tableau and the exemplary charts n fgures 5.4 to 5.6 dfferent system and load confguratons could be evaluated n order to have ndcatons for future system dmensonng of the modeled system. Furthermore, ths secton gave further nsghts nto FMC-QE, as fgure 5.2 shows a slghtly modfed, more compact layout for larger servce request structures. Also ths case study shows that the calculatons n the Tableau are very fast. For a performance evaluaton study parameterschanges (recalculaton of the whole Tableau + value loggng n the Tableau n table 5.1 were performed n 3,0 seconds 7, whch means that a performance predcton for one system confguraton of the performance parameters n the Tableau s calculated n approxmately 3 mllseconds. In another case study [116], whch s not a part of ths thess, another scenaro wth 46 actons n two classes was evaluated wth a prototype verson of the FMC-QE Tool by Tomasz Porzucek n approxmately 1 mllsecond 8. 7 Tableau developed n Mcrosoft Offce Excel 2003 ( SP3 n Mcrosoft Wndows XP Professonal (Verson Servce Pack 3 Buld 2600 on Dell OptPlex GX620, Intel Pentum 4 HT 3192 Mhz (Famly 15 Model 4 Steppng 3 wth 2 GB DDR2 RAM 8 FMC-QE Tool developed under Java 1.6 and deployed on Mcrosoft Wndows XP Professonal (Servce Pack 3 on Apple MacBook Pro, Intel Core2Duo 2,16 GHz wth 2 GB DDR2 RAM 166

189 5.2. MODELING OF A SERVICE BASED SYSTEM: ERMF 5.2 Modelng of a Servce based System: ERMF In a jont Bachelor s Project of the Research Groups of Prof. Dr. Hasso Plattner / Dr. Alexander Zeer and Prof. Dr.-Ing. Werner Zorn at the Hasso-Plattner-Insttute n 2006/2007, called Perfact,too, the performance smulaton framework Perfact, developed n prelmnary Bachelor s Project, was further developed and extended. Wth the help of Perfact the performance of a servce-orented system could be smulated through mplementng dummy-components n the real system envronment n order to detect bottlenecks and performance problems n the early desgn phase of the development. As a proof-of-concept for the performance predctons of the Perfact framework and the FMC-QE predctons, a benchmark of an exstng system was compared to the correspondng Perfact smulaton and the FMC-QE model. For ths case study the Emergency Rsk Management Framework (ERMF was used, a system n the SAP NetWeaver and SAP WebAS Envronment 9, orgnally developed by SAP Research France. Besdes the modelng part the man nterest for the FMC-QE development was the comparson and valdaton of the performance values of the three approaches and the analyss of a Servce based system. As a result of ths work the paper "Comparson of performance modelng and smulaton - a case study" was wrtten and presented at the 15 th IEEE Internatonal Conference on Engneerng of Computer-Based Systems (ECBS 2008 n Belfast, UK [120]. Ths secton s based on ths paper, especally on the FMC-QE part. Ths secton s structured as follows: After a short ntroducton n the servce request structures and the behavor of the system s modeled n The measurements taken are explaned n In the correspondng FMC-QE Tableau s shown. In secton the Perfact Smulaton s explaned. Fnally, some results are shown n secton The author would lke to thank the co-authors of the paper "Comparson of performance modelng and smulaton - a case study" and the other project members of the "Perfact, too" Bachelor s Project. In addton, the author wants to thank SAP Research France, especally Cedrc Ulmer, Volker Gersabeck and Martn Grund, provdng the case study and supportng the project Introducton The system modeled and analyzed n the case study s a Servce-based applcaton for the automatc generaton of alerts based on pre-defned rules, web servce calls and an Event- Condton-Acton (ECA rule engne. The sketch of the flow n the system s as follows: Due to dynamcally defned rules some web servces are called from the system. Ths web servces delver envronment data wth whch alerts could be generated SAP AG, SAP NetWeaver, Webste: August 167

190 CHAPTER 5. FMC-QE CASE STUDIES Fgure 5.7 shows the statc structures of the system and ts envronment. The Tmer Applcaton generates the servce requests for the system. The dfferent servce requests are receved then by the Control Unt and dspatched for further processng. The man part of the system s the ECA Engne (Event-Condton-Acton Engne. Ths component s responsble for the applyng of the rules n order to generate the rsk estmaton. Ths s done by to choosng the most relevant web servces and evaluatng the web servce responses together wth the rules. In ths case study publc avalable weather and traffc servces were used. The responses of the ECA Engne are passed to the Acton Performer through the Control Unt. Fnally the alert s generated n the Acton Performer.!"" #$ % & (! (( ' Fgure 5.7: ERMF - Statc Structure The test platform was provded by SAP Research France and conssts of the SAP NetWeaver Developer Studo 2004s as a development envronment and the SAP Web Applcaton Server 6.40 as the applcaton server Servce Request Structure and Dynamc Behavor Fgures 5.8 and 5.9 llustrate the servce request structure and the dynamc behavor of the system. These two dagrams have the same herarches and complement each other. Whle the whole structure wth traffc flow coeffcents and the correspondng servers are shown n the Entty Relatonshp Dagram, the dynamc flow wth parallelsm and the rght order of executon s defned n the Petr Net. As shown n fgure 5.8 the Make Forecast request s the topmost request whch s then herarchcally decomposed nto Intalze Request, Evaluate Rules and Call Web Servces and Generate Alert requests. Accordngly the Make Forecast transton n the dynamc structure (fgure 5.9 s de- 168

191 5.2. MODELING OF A SERVICE BASED SYSTEM: ERMF composed nto a sequence of three correspondng transtons. Furthermore, the Evaluate Rules and Call Web Servces request was decomposed, as shown n fgure 5.8 and fgure 5.9../!! "# &"'( "" $%.0/ * " '( "'( "'($+ ".1/ %% %% %%,& '- '- '-,&.2/ Fgure 5.8: ERMF - Servce Request Structure! " #$ "! % & ' (( Fgure 5.9: ERMF - Dynamc Behavor 169

192 CHAPTER 5. FMC-QE CASE STUDIES Measurements The orgnal ERMF source code was modfed by ncludng a measurement mechansm n order to provde measurements as bass for the smulaton and analyss as well as for the comparson of the obtaned results. The events handled by the ERMF framework are artfcally generated by the Tmer Applcaton. These events are traced throughout the system on a lmted number of measurement ponts where the trace data s asynchronous wrtten to a database for later evaluaton. For measurements on the real system and durng the smulaton the same measurement ponts where used. Thereby, any nfluence on the systems performance through the tracng s equal on both results. The charts n fgures 5.10 and 5.11 show the response tmes of the traffc and weather servces for 720 respectvely 56 measured values. The measured response tmes have an average of ± 1.09ms and ± 8.21ms wth the correspondng confdence ntervals of (95%. The dscrete values are a result of measurement resoluton. R Traffc Servce [s] 0,35 0,34 0,33 0,32 0,31 0,3 0,29 0,28 0,27 0,26 0, Experment Number Fgure 5.10: ERMF - Chart: Traffc Servce - Response Tme R Weather Servce [s] 0,6 0,58 0,56 0,54 0,52 0,5 0,48 0,46 0,44 0,42 0, Experment Number Fgure 5.11: ERMF - Chart: Weather Servce - Response Tme 170

193 5.2. MODELING OF A SERVICE BASED SYSTEM: ERMF Analyss The mean values obtaned from the expermental data were used as nput parameters for the model evaluaton. In ths case study the statc structure s smplfed to the ERMF Server wth two processors and the web servers wth nfnte capacty. The correspondng Tableau s shown n table 5.2. In ths Tableau the multplexer s also exemplfed: the queues are at the multplexers and the logcal servers see a fracton of ther servce request n the multplexer queue as defned by the multplexer coeffcent: m mpx, = v X X j m j m. Expermental Parameters [1] n ges 3 λ [1] 1,0000 Table 5.2: ERMF - Tableau (see Appendx - Table B.10 Servce Request Secton Server Secton Dynamc Evaluaton Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Alert Request 1,00 1,00 1,00 1,00 1,000 Acton Performer ,010 0,014 1,408 0,355 0,001 0,001 0,010 0,010 0,011 0, Ruleset Eva. Req. 1,00 1,00 1,00 1,00 1,000 ECA-E. Rule Engne ,150 0,211 1,408 0,355 0,022 0,022 0,150 0,150 0,172 0, Weather Servce 1,00 1,00 1,00 1,00 1,000 Weather Serv. Hdl ,470 1,000 2,128 0,000 0,000 0,470 0,470 0,470 0, Traffc Servce 1,00 1,00 1,00 1,00 1,000 Traffc Servce Hdl ,300 1,000 3,333 0,000 0,000 0,300 0,300 0,300 0, Web Servces Req. 1,00 1,00 1,00 1,00 1,000 ECA-E. WS Proxy ,000 0,000 0,770 0,770 0,770 0, Mappng Request 1,00 1,00 1,00 1,00 1,000 ECA-E. Rule Engne ,200 0,282 1,408 0,355 0,029 0,029 0,200 0,200 0,229 0, Evaluaton Request 1,00 1,00 1,00 1,00 1,000 ECA-Engne Cont ,817 0,050 0,050 1,120 1,120 1,170 1, Intalzaton Request 1,00 1,00 1,00 1,00 1,000 Control Unt ,350 0,493 1,408 0,355 0,050 0,050 0,350 0,350 0,400 0, Forecast Request 1,00 1,00 1,00 1,00 1,000 ERMF-System ,817 0,102 0,102 1,480 1,480 1,582 1, Request Generaton 1,00 1,00 1,00 1,00 1,000 Tmer Applcaton ,418 0,705 0,000 0,000 1,418 1,418 1,418 1,418 Multplexer Secton M/M/m rsp. M/M/ j Name j m j [1] X j λ j μ j ρ n,q n,s n 1 ERMF Server 2 0,710 1,000 1,408 0,355 0,102 0,710 0,812 2 Weather Server 0,470 1,000 2,128 0,000 0,470 0,470 3 Traffc Server 0,300 1,000 3,333 0,000 0,300 0, Smulaton For the smulaton the Perfact framework s used [42]. In order to run the smulaton the followng steps are taken teratvely. Frst buldng blocks and ther connectons are defned. Ths ncludes already exstng components as well as ther nteractons. For ths case study the external web servces exst, whereas the whole ERMF system s smulated. Communcaton between both sdes s takng place on the actual applcaton server. The smulated components also communcate wth each other (see fgure 5.7. Next, the system s enrched wth setup nformaton needed for the smulaton. Connecton parameters for the external web servces are defned, and the behavor of the generc smulaton components (GSC s customzed. In order to run the smulaton the dynamc archtecture s defned by use cases. The model s based on the UML 2.0 Sequence Charts 10. All components defned n the frst stage can be used n the scenaros. For each actvty the performance relevant behavor can be defned by choosng a scenaro from a predefned set. Work on storages, algorthmc computaton, and man memory consumng processes as the core parts of performance relevant behavor has been dentfed. 10 Object Management Group, Unfed Modelng Language Specfcaton v. 2.0, OMG UML 2.0 Superstructure Specfcaton, formal/ , July

194 CHAPTER 5. FMC-QE CASE STUDIES Durng the smulaton phase generc smulaton components are deployed and confgured accordng to ths setup. After executng the smulaton the system components are trggered accordng to the workload ntensty and performance measurements are traced. The trace data s collected and aggregated then nto smulaton results where they can be used for comparson and further evaluaton of the system Summary Fgure 5.12 shows a comparson between smulated results, calculated results and measurements of the real system. The values are computed by changng expermental data n the FMC-QE Tableau and runnng experments n the real system and the Perfact smulaton. The calculated results are comparable as long as the system s operatng under normal condtons (ρ < 1. Runnng at ts lmts (ρ 1, the calculaton predcts nfnte response tmes due to the mathematcal formulas of Queueng Theory, whereas the real system crashes because of buffer overflows. Smlarly the smulaton predcts nvald values. In ths case the calculated results of FMC-QE helped n understandng the system crashes due to overload and n fndng and correctng an error n the mplementaton of Perfact. +,-. ' ( *(!"#$ %& (-/0. Fgure 5.12: ERMF - Chart: Result Comparson 172

195 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 5.3 Modelng of nteractng herarchcal Protocol Stacks - Axs2 As a result of a collaboraton nsde the Research Group, the case study "Analyss and Modelng of the Axs2 Web Servce Framework wth FMC-QE" was prepared. In ths case study the Axs2 Web Servce framework was modeled usng FMC-QE. Ths case study was a Proof-of-Concept for the herarchcal modelng and the performance predcton of models wth synchronzaton and multplexers. Another specalty was the modelng of worker threads and ppelnes of sequental tasks. On the bass of ths work the paper "Herarchcal Modelng of the Axs2 Web Servces Framework wth FMC-QE" was wrtten and presented at the 3 rd Internatonal Conference on COMmuncaton Systems software and mddleware (COMSWARE 2008 n Bangalore, Inda n 2008 [37]. Ths secton s based on ths paper and structured as follows: secton provdes an overvew on the man components of Axs2, the Apache framework for Web Servces and some detals of the SOAP processng. In secton the Axs2 FMC-QE model ncludng the Input- and Output-Flows s descrbed. In order to ntalze the FMC-QE Tableau n secton 5.3.4, performance measurements of an Axs2 nstallaton are conducted n a test envronment. Ths testbed s descrbed n secton Fnally, some results are summarzed n secton The author wants to thank the co-authors of the paper "Herarchcal Modelng of the Axs2 Web Servces Framework wth FMC-QE", especally Flavus Copacu, for the cooperaton Axs2 Web Servces Framework Axs2 s a hghly modular web servce framework developed under the gudance of the Apache Software Foundaton 11. Orgnally the frst mplementaton was developed by IBM under the name soap4j 12, later donated to the Apache Software Foundaton and offered to the publc as Apache SOAP 13. Ths mplementaton was a proof of concept wth the goal to promote web servces and to offer to the users a frst contact wth ths, at that tme new, technology. The second teraton, called Apache Axs 14, was developed wth the goal of mprovng the support for the web servces specfcatons (collectvely known as WS- that appeared n the communty. Axs2 15, the thrd teraton, s a redesgn of the orgnal Axs code wth the purpose of makng t easer to provde support for the WS- standards as well as mprovng the performance ssues that were notced wth the prevous mplementatons. The Axs2 archtecture can be splt nto two man parts: the core components and the non-core components. Most of the non-core components are avalable as plug-n modules wth multple mplementatons avalable to choose from. In a system buld usng Axs2 the framework can be deployed on the servce consumer sde, the servce provder sde or both sdes of the system. One of such system bult wth Axs2 s presented n fgure The clent applcaton encapsulates the clent logc and makes use of the Axs2 lbrares to nvoke servces offered by the servce provder. On the server sde, Axs2 s deployed as a web applcaton nsde a Tomcat 16 applcaton contaner. Axs2 does not 11 Apache Software Foundaton, Webste: February IBM, SOAP for Java, Webste: August Apache Software Foundaton, Apache SOAP, Webste: August Apache Software Foundaton, Web Servces - Axs, Webste: August Apache Software Foundaton, Apache Axs2 Archtecture Gude, Webste: Axs2ArchtectureGude.html, August Apache Software Foundaton, Apache Tomcat Servlet Contaner, Webste: August

196 CHAPTER 5. FMC-QE CASE STUDIES depend on Tomcat, t can be deployed as an web applcaton on any J2EE complant servlet contaner. The servces themselves, contanng the busness logc, are deployed nsde Axs2. For an extensve dscusson regardng fgure 5.13 as well as the other dagrams that form the model please refer to secton '( *!'(!! $%! &!! "! " '( *!'( #! Fgure 5.13: Axs2 Based System - Block Dagram More then a SOAP 17 processng stack Axs2 provdes the functonalty requred to address many of aspects that appear when buldng a servce-orented envronment [112]: 1. Framework to develop, deploy, nvoke and manage Web Servces. 2. Extensble SOAP processng model. 3. Framework for supportng dfferent Message Exchange Patterns (MEPs, ncludng synchronous as well as asynchronous servce nvocaton. 4. Modular transports and data bndngs. 17 W3C, SOAP Verson 1.2, W3C Recommendaton 27 Aprl 2007, February

197 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 5. WS- support va pluggable modules, e.g. WS-Addressng, WS-RelableMessagng, WS- Coordnaton or WS-Securty [50]. 6. Support for Message Transmsson and Optmzaton Mechansm (MTOM Representatonal State Transfer (REST 19 support. SOAP Processng n Axs2 The Axs2 SOAP processng engne and the dfferent processng stages are n the focus of ths case study and wll be presented n detal n the followng sectons. In Axs2 the SOAP processng s mplemented as a three layer archtecture consstng of handlers, phases and flows [32]. The processng handlers are stuated at the lowest abstracton level n ths three layer archtecture. These handlers are the smallest components n the SOAP processng flow, encapsulatng well defned functonalty. All handlers mplement a well defned nterface. The processng phases are located on the second abstracton level. Each phase manages a specfc, logcally ndependent task. The phases mplement the same nterface as the handlers, and ths makes t possble to mplement phases through a sngle handler. In order to realze complex functonalty, multple handlers are chaned together mplementng a phase. The poston of a sngle handler nsde a phase can be specfed va a set of rules n one of the confguraton fles of Axs2. The SOAP processng engne n Axs2 s based on a set of predefned phases and can be extended by user defned phases. The most mportant phases n Axs2 are 20 : Transport - responsble for processng transport related nformaton, e.g. transport headers, and addng ths data to the Axs2 message context; n the output flow ths phase s responsble for nvokng the assocated transport handler. Pre-Dspatch - n ths phase data used for dspatchng the message s gathered, e.g. from the HTTP or SOAP headers. Dspatch - responsble for matchng the ncomng message wth one of the servces deployed n the Axs2 nstance. User Defned Phases - used for enhancng the capabltes of the framework. The processng flows belong to the hghest herarchcal level n the processng model. These flows are bult by channg multple processng phases together. Each ncomng or outgong servce request s processed by one of the ncomng or outgong flows (InFlow, OutFlow. The SOAP processng model of Axs2 can be extended n several ways: Addng user defned phases: New phases, mplemented va sngle or multple handlers, can be nserted n the ncomng and outgong flows. Ths way new functonalty s added to the system. 18 W3C, SOAP Message Transmsson Optmzaton Mechansm, W3C Recommendaton 25 January 2005, http: // February Representatonal State Transfer, REST, In: Roy Thomas Feldng, Archtectural Styles and the Desgn of Network-based Software Archtectures, Dssertaton, Unversty of Calforna, Irvne, 2000, Webste: cs.uc.edu/~feldng/pubs/dssertaton/top.htm, February Apache Software Foundaton, Web Servces - Axs, Webste: August

198 CHAPTER 5. FMC-QE CASE STUDIES Addng new handlers: Ths s done to change or enhance the functonalty of one of the exstng phases, and these changes can not be done n a user specfed phase. Addng a new module: Complex Axs2 extensons can also be mplemented as modules. These usually are employed when mplementng WS- extensons (e.g. WS-Addressng and are a set of new modules and/or user phases logcally grouped together. The modules are realzed as jar fles contanng a set of handlers and an XML descrptor that specfes the placement of each handler n the processng flows. The modules have to be deployed frst and then enabled n the Axs2 confguraton fles Axs2 Model Axs2 can be used n two dfferent ways, a standalone mode or deployed nsde an applcaton server. Due to the lmtatons of the standalone mplementaton t s usually deployed on applcaton servers as a web applcaton. Ths second case s already shown n fgure Ths fgure 5.13 descrbes the mechansm employed when new servce requests are receved. After the requests pass the Admsson Control, they are queued n the Request Queue. Then the Dspatcher s responsble for allocatng avalable threads from the Thread Pool for processng the servce requests. The allocated thread performs all the processng requred by the servce request and s returned to the pool as soon as the processng has ended. The sze of the thread pool s not statc, t grows durng the start phase, then t stablzes between a mnmum and a maxmum thread count. New threads can be spawned to deal wth usage peeks, and dle threads are removed from the pool after a certan perod of tme. As soon as the processng of a servce request has been completed, the servce response s sent. The Departure Control shows the processng performed upon outgong servce responses. #!! " $ %& ' Fgure 5.14: Axs2 Dynamc Behavor - Petr Net 176

199 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 The dynamc behavor of the system s presented n fgure The external world representng the clents generatng servce requests s modeled va the Generate Requests transton. Ths transton s at herarchcal level [1] just as the Supervse and Execute Servce Request transton used to model the Axs2 nstance. The processng threads are represented usng the herarchcal level [2] transton Execute Servce, whle the processng flows and the busness logc assocated wth the servce are represented va the correspondng level [3] transtons. In secton the two processng flows are presented n detal. The servce requests queue s represented by the nfnte place between level [1] and level [2] transtons. By default, Axs2 uses an nfnte queue, as shown n the pcture. If a specfc queue sze s set, the model has to be adapted by replacng the nfnte place by a fnte, mult-token place.! "#$ % "#$ "#$%! & ' " & ' " & ' "!!, -', -', -'! ( ' * ( ( " "(+ * " "(+ "*"(+!! ".(+ * ".(+ "*.(+!!.#$ %.#$.#$%! + + +!! *. *.. + %//(-* * *,*%//(-* %//(-*+ "(&',* * *,*"(&', "(&',+!!! 0."** * *,*0."* 0."*+ * * *,* +! ( + ( + ( +!! / / /!! "(* * *,*"( "(+ 0.* * *,* * Fgure 5.15: Axs2 Herarchcal Servce Request Structure - Entty Relatonshp Dagram 177

200 CHAPTER 5. FMC-QE CASE STUDIES The servce request structure s presented n fgure Ths fgure descrbes the herarchcal structure of the modeled Axs2 Servce Request. Ths dagram descrbes among others the mappng between Petr Net transtons and agents n the Block Dagram, for example the Execute Servce s handled by the Worker Thread. Besdes ths, the herarchcal levels and traffc flow coeffcents are presented. Axs2 Flows In the Axs2 framework, mportant parts are the four SOAP processng flows that are part of the framework core: InFlow, OutFlow, InFaultFlow and OutFaultFlow. The frst two are responsble for nput and respectvely output processng, whle the last two are responsble for nput and output processng n the case that errors have appeared. The error processng flows are qute smlar to the regular ones and no longer consdered. The Petr Net detalng the nput flow s presented n fgure Ths flow s the most complex one n ths case study, spawnng over three herarchy levels, from level [3] to level [5]. &'! $!!! "#$! % Fgure 5.16: Axs2 Input Flow - Petr Net The output flow, presented n fgure 5.17, s smpler than the nput flow and has only two herarchcal levels. Ths smplfcaton s explaned by the fact that the output flow has to take nto account parameters that have been set durng the nput flow and by ths the range of possble changes that can appear has been reduced. For example, f durng the nput flow the transport protocol has been decded as HTTP, the output flow wll use ths nformaton and wll not have to do any extra processng n order to determne a transport stack. 178

201 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 Fgure 5.17: Axs2 Output Flow - Petr Net When comparng the graphcal representaton from fgure 5.14 and 5.16 wth the ones found n secton 3.3.3, t can be notced that the transtons have been smplfed. Ths has been done by removng the place representng the queued servce requests watng to be served. Ths smplfcaton s possble because there are no queues n the processng flows. All the transtons belongng to the same thread wll start executng as soon as the operaton correspondng to the prevous transton has fnshed. Insde Axs2 the only place where servce requests are beng queued s the queue n front of the Thread Pool. Extendng the Axs2 Model One of the strengths of Axs s ts flexblty and the possbltes to customze ts behavor. The core of the system, presented n secton 5.3.2, can be extended n order to ncorporate other WS- extenson, deployed as modules, that can be enabled or dsabled ndvdually for each servce or group of servces. When a new module s added to the handler chan and all servces make use of that module, the model could be extended. Such changes could be reflected n the dynamc structure of the model by addng new handlers or phases n the exstng handler chans, accordng to the specfcaton of the newly actvated module. It s also possble to have specfc modules actvated and used only by some of the deployed web servces, as mentoned before. In order to cover such a case a decson-makng part has to be ntegrated n the model, as llustrated n fgure 5.18, where the Encrypt handler s not mandatory but optonal. If encrypton s actvated, the servce responses wll be processes va the Encrypt handler. If encrypton s not used, no processng wll be performed, as ndcated by the no-operaton (NOP transton. For ths stuaton t s necessary to determne the usage probablty of the optonal handler and to extend the model to ncorporate ths, as n fgure Ths soluton s approprate f most of the servces follow one branch and only a small percent follows the other. The drawback of ths soluton s the fact that when there are many optonal handlers, treatng them through averages leads to a loss of representatvty of the fnal results. Ths can be consdered acceptable when such cases account only for a small amount of the total servce requests. 179

202 CHAPTER 5. FMC-QE CASE STUDIES!"# $!"# $ Fgure 5.18: Axs2 - Optonal Handlers The second soluton to ths problem mples transformng the scenaro nto a multclass one. Servces that use the same path or closely related paths through the handler chan can be grouped together n servce classes and the modelng can be done for these classes. Both of these approaches can be modeled and evaluated wth FMC-QE Testbed Descrpton In order to perform the performance evaluaton and predcton usng FMC-QE a benchmark of a server runnng Axs2 has been done. The benchmarkng has been done by usng the Java method call System.getNano(. Accordng to JAVA API documentaton 21 the method returns the current value of the most precse avalable system tmer n nanoseconds and provdes nanosecond precson, but not necessarly nanosecond accuracy. On Apple Mac OS X ths method delvers results wth mcro second precson. The Axs2 code has been extended wth measurng ponts connected to each handler. The server nstance runs on an Apple MacBook Pro machne wth an Intel Core 2 Duo CPU at 2.16 GHz and 2 GB RAM. The machne runs Axs2 verson 1.2 nsde a Tomcat applcaton server on a Mac OS X wth Java 1.5.0_07. The experments were repeated tmes, and the mean value correspondng to each handler has been calculated. For the experments the servce Verson avalable by default wth each Axs2 dstrbuton has been nvoked FMC-QE Tableau Wth the the help of the Tableau, shown n table 5.3, performance predctons could be derved. In ths Tableau, the specal behavor of worker threads and a worker pool, as descrbed n ths case study, s modeled. The worker threads and the correspondng logcal servers (level [2] and 21 Oracle Corporaton, Java 2 Platform Standard Edton 5.0 API Specfcaton, docs/ap/, February Apple Inc., Mac OS X, newer Verson 10.6 Snow Leopard, Webste: February

203 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 below are modeled as nfnte servers wth a multplex coeffcent of m,mpx = #Workers #CPUs. Whle the queued servce requests are queued at the worker pool, the system s modeled as an M/M/n server wth the correspondng queue and the aggregated servce rates of the worker threads. n ges [1] λ bott [1] f λ [1] Expermental Parameters ,526 0,800 8,421 Table 5.3: Axs2 - Tableau (see Appendx - Table B.11 Servce Request Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 4 1 Encrypton Req. 1,00 1,00 1,00 1,00 8,42 Encrypter ,0006 0, ,66 0,002 0,000 0,000 0,012 0,001 0,012 0, MessageOut Req. 1,00 1,00 1,00 1,00 8,42 MessageOut Srv ,0045 0,400 89,89 0,019 0,000 0,000 0,094 0,011 0,094 0, Polcy Det. Req. 1,00 1,00 1,00 1,00 8,42 Polcy Det ,0004 0, ,00 0,002 0,000 0,000 0,008 0,001 0,008 0, OpOut Phase Req. 1,00 1,00 1,00 1,00 8,42 OpOut Phase Srv ,0004 0, ,23 0,002 0,000 0,000 0,009 0,001 0,009 0, OutFlow Request 1,00 1,00 1,00 1,00 8,42 OutFlow Handler ,65 0,000 0,000 0,123 0,015 0,123 0, Busness Logc R. 1,00 1,00 1,00 1,00 8,42 Busness Logc ,0325 0,400 12,33 0,137 0,000 0,000 0,683 0,081 0,683 0, OpIn Phase Req. 1,00 1,00 1,00 1,00 8,42 OpIn Phase Srv ,0006 0, ,16 0,003 0,000 0,000 0,013 0,002 0,013 0, Dsp. Instance Req. 1,00 1,00 1,00 1,00 8,42 Instance Dsp ,0329 0,400 12,15 0,139 0,000 0,000 0,693 0,082 0,693 0, HTTP Loc. Dsp. R. 1,00 1,00 1,00 1,00 8,42 HTTP Loc. Dsp ,0003 0, ,48 0,001 0,000 0,000 0,006 0,001 0,006 0, SOAP MB. Dsp. R. 1,00 1,00 1,00 1,00 8,42 SOAP MB Dsp ,0001 0, ,00 0,000 0,000 0,000 0,002 0,000 0,002 0, URI Op. Dsp. Req. 1,00 1,00 1,00 1,00 8,42 URI Op. Dsp ,0001 0, ,00 0,000 0,000 0,000 0,002 0,000 0,002 0, Address Dsp. Req. 1,00 1,00 1,00 1,00 8,42 Address Dsp ,0287 0,400 13,93 0,121 0,000 0,000 0,604 0,072 0,604 0, Dspatch Request 1,00 1,00 1,00 1,00 8,42 Dspatcher ,10 0,000 0,000 1,308 0,155 1,308 0, PreDspatch Req. 1,00 1,00 1,00 1,00 8,42 PreDspatcher ,0050 0,400 80,00 0,021 0,000 0,000 0,105 0,013 0,105 0, Decrypt Req. 1,00 1,00 1,00 1,00 8,42 Decrypter ,0005 0, ,33 0,002 0,000 0,000 0,010 0,001 0,010 0, SOAP Act. Dsp. R. 1,00 1,00 1,00 1,00 8,42 SOAP Act. Dsp ,0397 0,400 10,07 0,167 0,000 0,000 0,836 0,099 0,836 0, Req.t URI Dsp. R. 1,00 1,00 1,00 1,00 8,42 Req. URI Dsp ,0437 0,400 9,14 0,184 0,000 0,000 0,921 0,109 0,921 0, Transport Request 1,00 1,00 1,00 1,00 8,42 Transporter ,98 0,000 0,000 1,757 0,209 1,757 0, InFlow Request 1,00 1,00 1,00 1,00 8,42 InFlow Handler ,59 0,000 0,000 3,193 0,379 3,193 0, Executon Request 1,00 1,00 1,00 1,00 8,42 Servce Exec ,26 0,000 0,000 4,000 0,475 4,000 0, Sup. and Ex. Req. 1,00 1,00 1,00 1,00 8,42 System ,400 2,11 0,800 2,216 0,263 4,000 0,475 6,216 0, Generaton Request 1,00 1,00 1,00 1,00 8,42 External Source ,0118 8,42 993, , , ,012 Multplexer Secton j Name j m j [1] X j 1 CPU 2 0,190 Server Secton Dynamc Evaluaton Secton Fgure 5.19 shows the predcton of the response tme of the whole system R ges n relaton to the overall arrval rate λ for a confguraton wth 5 worker threads and 2 CPUs. Whle n ths confguraton the bottleneck throughput s λ bott 10, 5 [SRq], the response tmes grow rapdly [s] for λ > 9, 5 [SRq]. In addton to ths, three other system confguratons wth more processors [s] are also shown n ths fgure as an example for the support n future system confguraton questons wth dfferent ranges of possble arrval rates λ bott, 2 CPUs = 10,526 λ bott, 3 CPUs = 15,789 λ bott, 4 CPUs = 21,053 λ bott, 5 CPUs = 26,316 8 R ges [s] CPUs 3 CPUs 4 CPUs 5 CPUs λ [SRq]/[s] (5 Worker Fgure 5.19: Axs2 - Chart: Response Tme - Arrval Rate 181

204 CHAPTER 5. FMC-QE CASE STUDIES Fgure 5.20 shows the dependence on the total number of servce requests (n ges and the external servce tme (X ext as a result of the Response Tme Law X ext = (n ges /A R sys. For the ntal confguraton of 5 workers and 2 CPUs and an desred bottleneck utlzaton f = 0, 8 (λ = 8, 421 [SRq], clents could request the server every 2 mnutes (118s each. Dagrams [s] lke ths could lead to a better understandng of the clents behavor and the relaton between the number of clents and ther external servce tme (Thnk Tme [43] X ext [s] n ges [SRq] (5 Worker; 2 CPU; f=0,8; λ=8,421 Fgure 5.20: Axs2 - Chart: External Servce Tme - Populaton In fgure 5.21 the number of avalable CPUs s ncreased for the confguraton of 5 workers and an overall arrval rate of λ = 8, 421 [SRq] as an example for decson support of a system confguraton queston under a specfc load scenaro. It can be seen that the ncrease to 3 CPUs results [s] n a strong mprovement of the system performance, whle n ths specal confguraton more CPUs do not result n a sgnfcant performance mprovement. A confguraton of 1 CPU s not shown here because an arrval rate of λ = 8, 421 [SRq] [s] throughput for ths confguraton (λ bott, 1CPU = 5, 262 [SRq] [s]. s larger than the maxmum bottleneck 0,8 0,7 0,6 0,5 R ges [s] 0,4 0,3 0,2 0, Number of CPUs (5 Worker; λ=8,421 Fgure 5.21: Axs2 - Chart: Response Tme - Number of CPUs 182

205 5.3. MODELING OF INTERACTING HIERARCHICAL PROTOCOL STACKS - AXIS2 Durng the evaluaton dfferences have been notced between the herarchcally estmated servce tmes and some measurements done at the phase level. The hypothess s that the dfferences are due to Java object nstantaton tme, partally not taken nto account n the benchmark Summary In ths secton an analyss on Axs2 has been done and based on ths, an FMC-QE model of the framework has been developed. A complex system has been modeled usng FMC-QE and the methodology has shown to be sutable for modelng such systems and depctng ther herarchcal structure. A performance model of Axs2 has been done usng measurements from a test system. The results have confrmed the methodology and the usablty of FMC-QE for performance estmatons. The experence and knowledge gathered through ths process have provded a better understandng of Axs2 and systems based on Axs2, and s to be of further use n the mplementaton an adaptve system for servce-based Systems by Flavus Copacu [36]. After settng up the FMC-QE model and Tableau of Axs2, performance predctons could be performed n a fast and smple way. A set of parameters, e.g probabltes, servce tme or number of CPUs, can be changed n the Tableau and allow to nvestgate the behavor of the system under a broad range of possble confguratons, as depcted n fgures 5.19 to

206 CHAPTER 5. FMC-QE CASE STUDIES & &,, ( /.+ ( -,, ( / #-' - (.+#/, &!"+$"% ( ( ( / #-' *, & '( ( /!' 0 & # + ', ( + & # # ' & ( '"!"# $"% & & * # Fgure 5.22: Axs2 - Dynamc - All (see Appendx - Fgure C.4 184

207 Chapter 6 Conclusons Ths chapter summarzes the man contrbutons and gves an outlook on future work. Contrbutons of FMC-QE The new methodology FMC-QE contrbutes through: The extenson of the modelng technque FMC for quanttatve modelng and performance predctons as a powerful approach to predct the quanttatve behavor of systems adaptng FMCs 3-dmensonal modelng space wth servce request structures, server structures and dynamc control flow. A consstent modelng from the perspectve of herarchcal servce requests wth a herarchcal mult-level modelng through the usage of the Forced Traffc Flow Law and the modelng of servce requests as a tuple of value and [unt] lke physcal unts. Through ths herarchcal modelng the model transformatons and the complexty reducton through dstncton of operatonal and control states allow the modelng and analyss of complex herarchcal systems n steady state. A calculus based on herarchcal equlbrums n steady state wth Lttle s Law for relatons wthn a herarchcal layer (horzontal and the Forced Traffc Flow Law for relatons among herarchcal layers (vertcal, deducng a system of equatons used n development of nterpretable Tableau wth easy to change system- and load-parameters whch scales for complex systems, wthout the state space exploson problem through achevng an algorthmcal complexty of the Tableau of approxmately O(n > (n = number of servce statons. Wth the dfferent levels of parallelsm n both logcal and multplexer server structures, FMC-QE gves the ablty to compute a very broad range of possble system confguratons nstantly. Ths number of possble system confguratons s extended by the ablty to change probabltes and expermental parameters, lke arrval rate and number of crculatng servce requests through a strctly modelng of the outsde world as servers wth a handlng of open and closed system as specal cases. The ablty to model and compute multplexer servers that serve many logcal servers (multplex whch enables FMC-QE to model complex scenaros and the handlng of multclass scenaros ntegrated through multplexers and parttonng. 185

208 CHAPTER 6. CONCLUSIONS Contrbutons of the Author Whle early deas of FMC-QE were developed by Prof. Dr.-Ing. Werner Zorn, the author contrbuted through completng and valdatng the methodology and makng t accessble to a broader communty through ths thess ncludng foundatons, a descrpton of the whole methodology and the extenson and usage of ths methodology n examples and case studes. These contrbutons are n partcular: In the chapters 2 and 3 the author contrbuted through valdatng, completng and updatng the results of Prof. Dr.-Ing. Werner Zorn through: The further development of the FMC-QE graphcal notatons. The development of a more mplementaton orented Tableau n cooperaton wth Tomasz Porzucek and Flavus Copacu wth a specfcaton for parallel and branch request handlng and a generalzaton and standard ntegraton of the multplexer n the Tableau as well as the further development and ntegraton of parallelsm n both logcal and multplexer servers. The chapter 4 contans the exclusve contrbuton of the author. These contrbutons are: The handlng of closed networks through the ntegraton of the summaton method n an teratve computaton whch complexty s ndependent from the overall number of servce requests n the system. The handlng of semaphore synchronzaton scenaros through extendng the dea of modelng the crtcal resource through a multplexer and combne ths approach wth the summaton method and deas of the method of complementary delays wth the goal of precson and the reducton of computatonal complexty. The comparson of FMC-QE to other performance modelng and evaluaton approaches. The ntegraton of multclass problems through a modfed multplexer. In chapter 5, n cooperatons wth others, the author appled FMC-QE through case studes as a proof of concept, mostly n the context of servce based systems. The contrbutons of the author were: The development of a larger FMC-QE model n order to show the scalablty of the approach. The support n the quanttatve modelng of a servce based system as a case study to compare the performance values of a real system, a smulaton and the FMC-QE predctons. The FMC-QE part n the modelng of the Axs2 Web Servce Framework n order to mprove the herarchcal modelng dagrams and to focus on performance predcton of multplexers and synchronzatons. 186

209 Future Work The most mportant future work s actually the mplementaton of an FMC-QE Tool n order to open the methodology to a broader usage. Ths ncludes the development of a FMC-QE tableaux nterpreter as well as the development of transformatons of other, possbly non herarchcal, models to FMC-QE n order to further broaden the addressed systems, avalable for modelng wth FMC-QE. Furthermore, ths ncludes the extenson of statstcal evaluatons, e.g. quantles and approxmaton of mathematcal dstrbutons through preprocessng, lke Webull or log-normal for the servce tmes and arrval rates. Suggested papers from the area of Hgh Performance Computng are: [76, 85] (Bags-of-Tasks, dstrbuted systems, power awareness, [97] (cluster job start tme predctons through traces and smulaton and [47] (workload model for parallel computers. A further nvestgaton of performance bounds, lke n [48, 49], s also nterestng. Another feld of nterest s a systematc comparson of the algorthmc complexty of FMC-QE wth dfferent other methodologes. Also the nvestgaton of further usage domans and case studes s of nterest. In the area of dependable systems [99] the ntegraton of performance modelng and performance predcton nto the research of dependable systems, called performablty, s one possble topc. The predcton of the nfluence n the performance for a gan n avalablty s an mportant queston and addressable. Related work n ths area s also done n the research of Layered Queueng Networks [59] (Dependable-LQN where especally the performance modelng of a quorum pattern n [109] s nterestng. Another possble area of nterest are mathematcal/analytcal models for producton plannng n comparson to exstng smulaton methods. An nterestng doman s also the usage of the performance predctons of FMC-QE n the negotaton and guaranteeng of servce level agreements (SLAs. 187

210

211 Publcatons Journal Artcles Stephan Kluth, Tomasz Porzucek, Flavus Copacu, Werner Zorn: Quanttatve Modellerung und Analyse mt FMC-QE, PIK Specal Issue on Servce-orented Computng, PIK - Praxs der Informatonsverarbetung und Kommunkaton, 2008/4, pages , Specal Issue Edtors: Ncolas Repp, Sebastan Hudert, Steffen Bleul, Edtor: Hans Meuer, K. G. Saur Verlag, December 2008, ISSN Prnt , ISSN Onlne Conference Contrbutons Tomasz Porzucek, Mathas Frtzsche, Stephan Kluth and Davd Redlch: Combnaton of a Dscrete Event Smulaton and an Analytcal Performance Analyss through Model-Transformatons, In Roy Sterrt, Brandon Eames, Jonathan Sprnkle (Ed.: Proceedngs of the 17th IEEE Internatonal Conference and Workshop on the Engneerng of Computer Based Systems (ECBS Oxford, , March 2010, IEEE Computer Socety, Los Alamtos, CA, USA, ISBN: Tomasz Porzucek, Stephan Kluth, Flavus Copacu and Werner Zorn: Modelng and Evaluaton Framework for FMC-QE, In: Ted Bapty, Brandon Eames (Ed.: Proceedngs of the 16th IEEE Internatonal Conference and Workshop on the Engneerng of Computer Based Systems (ECBS San Francsco, , IEEE Computer Socety, Los Alamtos, CA, USA, Aprl, 2009, ISBN: Marcel Seelg, Stephan Kluth, Tomasz Porzucek, Flavus Copacu, Nco Naumann, Steffen Kühn: Comparson of Smulaton and Performance Modelng - A Case Study, n: Davd W. Bustard and Roy Sterrtt (Ed.: Proceedngs of the 15th IEEE Internatonal Conference and Workshop on the Engneerng of Computer Based Systems (ECBS Belfast, 49-56, IEEE Computer Socety, Los Alamtos, CA, USA, March 2008, ISBN: Flavus Copacu, Stephan Kluth, Tomasz Porzucek, Werner Zorn: Herarchcal Modelng of the Axs2 Web Servces Framework wth FMC-QE, In: 3rd Internatonal Conference on COMmuncaton Systems software and mddleware (COMSWARE Bangalore, 74-81, IEEE Computer Socety Press, Los Alamtos, CA, USA, January, 2008, ISBN: Doctoral Symposa Contrbutons Stephan Kluth: Quanttatve Modellerung des Lestungsverhaltens SOA-baserter Systeme mt FMC- QE, In: Thomas Kühne, Wolfgang Resg and Fredrch Stemann: Modellerung 2008 (Proceedngs of the Modellerung 2008 Doctoral Symposum, , In Seres: Lecture Notes n Informatcs (LNI, Volume P-127, Gesellschaft für Informatk e.v. (GI, Bonn, Germany, March 2008, ISBN:

212 PUBLICATIONS Reports Stephan Kluth: Sprng 2010 Actvty Report, Presented at the Sprng 2010 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso-Plattner-Insttute for Software Systems Engneerng, Potsdam, Germany, Aprl 2010 Stephan Kluth: Handlng of Closed Networks n FMC-QE, Presented at the Fall 2009 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso-Plattner-Insttute for Software Systems Engneerng, Döllnsee, Germany, October In: Chrstoph Menel, Hasso Plattner, Jürgen Döllner, Mathas Weske, Andreas Polze, Robert Hrschfeld, Felx Naumann and Holger Gese (edt.: "Proceedngs of the 4th Ph.D. Retreat of the HPI Research School on Servce-orented Systems Engneerng", Technsche Berchte des Hasso-Plattner-Insttuts für Softwaresystemtechnk an der Unverstät Potsdam, Vol. 31, Unverstätsverlag Potsdam, Potsdam, 2010, ISBN: Stephan Kluth: Sprng 2009 Actvty Report, Presented at the Sprng 2009 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso-Plattner-Insttute for Software Systems Engneerng, Potsdam, Germany, Aprl 2009 Stephan Kluth: FMC-QE - Herarches, Transformatons and Rules, Presented at the Fall 2008 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso Plattner Insttute for Software Systems Engneerng, Potsdam, Germany, October In: Chrstoph Menel, Hasso Plattner, Jürgen Döllner, Mathas Weske, Andreas Polze, Robert Hrschfeld, Felx Naumann and Holger Gese (edt.: "Proceedngs of the 3rd Ph.D. Retreat of the HPI Research School on Servce-orented Systems Engneerng", Technsche Berchte des Hasso-Plattner-Insttuts für Softwaresystemtechnk an der Unverstät Potsdam, Vol. 27, Unverstätsverlag Potsdam, Potsdam, 2009, ISBN: Stephan Kluth: FMC-QE - Calculus, Presented at the Sprng 2008 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso-Plattner-Insttute for Software Systems Engneerng, Potsdam, Germany, Aprl 2008 Stephan Kluth: FMC-QE - Case Studes, Presented at the Fall 2007 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso Plattner Insttute for Software Systems Engneerng, Potsdam, Germany, October In: Profs. Dres. Chrstoph Menel, Andreas Polze, Mathas Weske, Jürgen Döllner, Robert Hrschfeld, Felx Naumann, Holger Gese and Hasso Plattner (edt.: "Proceedngs of the 2. Ph.D. retreat of the HPI Research School on Servce-orented Systems Engneerng", Technsche Berchte des Hasso-Plattner-Insttuts für Softwaresystemtechnk an der Unverstät Potsdam, Vol. 23, Unverstätsverlag Potsdam, Potsdam, 2008, ISBN: Stephan Kluth: FMC-QE - Postonng, Basc Defntons and Graphcal Representaton, Presented at the Sprng 2007 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, Hasso-Plattner-Insttute for Software Systems Engneerng, Potsdam, Germany, Aprl

213 PUBLICATIONS Other Publcatons Stephan Kluth: Quanttatve Modelng and Analyss wth FMC-QE, In: Ka Bollue, Domnque Gückel, Ulrch Loup, Jacob Spönemann, Melane Wnkler (Ed.: Dagstuhl 2010: Proceedngs of the Jont Workshop of the German Research Tranng Groups n Computer Scence, DFG Research Tranng Group 1298 AlgoSyn, RWTH Aachen Unversty, Verlagshaus Manz GmbH, Aachen, 2010, p. 198, ISBN: Stephan Kluth: Performance Modelng and Performance Predcton wth FMC-QE, In: Artn Avanes, Drk Fahland, Joanna Gebg, Samak Haschem, Sebastan Heglmeer, Danel A. Sadlek, Falko Thesselmann, Gudo Wachsmuth, Stephan Weßleder (Ed.: Dagstuhl 2009: Proceedngs des gemensamen Workshops der Informatk-Graduertenkollegs und Forschungskollegs, Graduertenkolleg METRIK, Insttut für Informatk, Humboldt Unverstät zu Berln, GITO mbh - Verlag für Industrelle Informatonstechnk und Organsaton, Berln, Germany, June 2009, p , ISBN: Stephan Kluth: Quanttatve Modelng and Analyss of Servce-Based Systems, In: Malte Dehl, Henrk Lpskoch, Roland Meyer, Chrstan Storm (Ed.: Proceedngs des gemensamen Workshops der Graduertenkollegs 2008, In: Trustworthy Software Systems, Graduertenkolleg vertrauenswürdger Software - Systeme (Trustsoft, Unversty of Oldenburg, GITO mbh - Verlag für Industrelle Informatonstechnk und Organsaton, Berln, Germany, May 2008, p. 96, ISBN: Stephan Kluth: Quanttatve Modelng and Analyss of Servce-orented Archtectures, In: Dagstuhl zehn plus ens, Volume 1, Verlagshaus Manz GmbH, Aachen, Germany, June 2007, p. 198, ISBN:

214

215 Bblography [1] Rémy Apfelbacher and Anne Roznat. FMC Notaton Reference Sheets. Onlne, May URL [2] Rémy Apfelbacher, Andreas Knöpfel, Peter Aschenbrenner, and Sebastan Preetz. FMC Vsualzaton Gudelnes. Hasso-Plattner-Insttute, Potsdam, Germany, January URL [3] Francos Baccell, Armand M. Makowsk, and Adam Shwartz. The Fork-Jon Queue and Related Systems wth Synchronzaton Constrants: Stochastc Orderng and Computable Bounds. Advances n Appled Probablty, 21(3: , September ISSN URL [4] Ganfranco Balbo, Steven C. Bruell, and Subbarao Ghanta. Combnng Queueng Networks and Generalzed Stochastc Petr Nets for the Soluton of Complex Models of System Behavor. IEEE Transactons on Computers, 37(10: , October ISSN DOI: [5] Ganfranco Balbo, Matteo Sereno, and Steven C. Bruell. Embedded Processes n Generalzed Stochastc Petr Nets. In Proceedngs of the 9th IEEE Internatonal Workshop on Petr Nets and Performance Models (PNPM 01, Los Alamtos, CA, USA, IEEE Computer Socety. ISBN DOI: [6] Ganfranco Balbo, Steven C. Bruell, and Matteo Sereno. Product Form Soluton for Generalzed Stochastc Petr Nets. IEEE Transactons on Software Engneerng (TSE, 28(10: , October ISSN DOI: [7] Smonetta Balsamo and Guseppe Iazeolla. An Extenson of Norton s Theorem for Queueng Networks. IEEE Transactons on Software Engneerng (TSE, 8(4: , July ISSN DOI: [8] Forest Baskett, K. Man Chandy, Rchard R. Muntz, and Fernando G. Palacos. Open, Closed, and Mxed Networks of Queues wth Dfferent Classes of Customers. Journal of the ACM (JACM, 22(2: , ISSN DOI: / [9] Falko Bause. Queueng Petr Nets: A Formalsm for the Combned Qualtatve and Quanttatve Analyss of Systems. In Proceedngs of the 5th Internatonal Workshop on Petr Nets and Performance Models (PNPM93 - Toulouse, France, pages 14 23, Los Alamtos, CA, USA, October IEEE Computer Socety. ISBN DOI: http: //do.eeecomputersocety.org/ /pnpm

216 BIBLIOGRAPHY [10] Falko Bause and Henz Belner. Ene Modellwelt zur Integraton von Warteschlangenund Petr-Netz-Modellen. In Günther Stege and J. S. Le, edtors, Proceedngs of the 5. GI/ITG-Fachtagung: Messung, Modellerung und Bewertung von Rechensystemen, Braunschweg, September 1989, volume 218 of Informatk-Fachberchte, pages Sprnger, September ISBN [11] Falko Bause and Peter Buchholz. Product Form Queueng Petr Nets: A Combnaton of Product Form Queueng Networks and Product Form Stochastc Petr Nets. Techncal Report 529, Fachberech Informatk Unverstät Dortmund, Dortmund, Germany, [12] Falko Bause and Peter Buchholz. Aggregaton and Dsaggregaton n Product Form Queueng Petr Nets. In Proceedngs of the 7th IEEE Internatonal Workshop on Petr Nets and Performance Models (PNPM 97, pages 16 25, Los Alamtos, CA, USA, IEEE Computer Socety. DOI: [13] Falko Bause and Peter Kemper. Queueng Petr Nets. In E. Schneder, edtor, Proceedngs of the 3. Fachtagung "Entwurf komplexer Automatserungssystem, Braunschweg (Germany, In: Methoden, Anwendungen und Tools auf Bass von Petr-Netzen, pages Verlag Technsche Un Braunschweg Inst. f. Regelungs- und Automatserungstechnk, May [14] Falko Bause and Peter S. Krtznger. Stochastc Petr Nets. Fredr. Veweg & Sohn Verlagsgesellschaft mbh, Braunschweg / Wesbaden, Germany, 2 edton, ISBN [15] Falko Bause, Peter Buchholz, and Peter Kemper. Quanttatve Evaluaton of Computng and Communcaton Systems - 8th Internatonal Conference on Modellng Technques and Tools for Computer Performance Evaluaton Performance Tools 95, volume 977 of Lecture Notes n Computer Scence (LNCS, chapter QPN-Tool for the specfcaton and analyss of herarchcally combned Queueng Petr nets, pages Sprnger, Berln / Hedelberg, Germany, ISBN DOI: [16] Frank-Mchael Becker, Gunter Boortz, Volkmar Detrch, Lutz Engelmann, Chrstne Ernst, Günter Fanghänel, Hen Höhne, Rud Lenertat, Günter Lesenberg, Lothar Meyer, Chrsta Pews-Hocke, Gerd-Detrch Schmdt, Renhard Stamm, and Karlhenz Weber. Formeln und Tabellen für de Sekundarstufen I und II. Peatec, Gesellschaft für Bldung und Technk mbh, Berln, Germany, 6 edton, ISBN [17] Gunter Bolch, Georg Fleschmann, and R. Schreppel. En funktonales Konzept zur Analyse von Warteschlangennetzen und Optmerung von Lestungsgrößen. In Ulrch Herzog and Martn Paterok, edtors, Messung, Modellerung und Bewertung von Rechensystemen, 4. GI/ITG-Fachtagung, Erlangen, 29. September - 1. Oktober 1987, Proceedngs, volume 154 of Informatk-Fachberchte, pages , Berln / Hedelberg, Germany, September - October Sprnger. ISBN (Sprnger Berln, Hedelberg, New York, (Sprnger New York, Berln, Hedelberg. [18] Gunter Bolch, Stefan Grener, Hermann de Meer, and Kshor Shrdharbha Trved. Queueng Networks and Markov Chans : Modelng and Performance Evaluaton Wth Computer Scence Applcatons. John Wley & Sons, Inc., ISBN [19] Rchard J. Bouchere and Matteo Sereno. A structural charactersaton of product form stochastc Petr nets. Techncal Report BS-R9402, Department of Operatons Reasearch, Statstcs, and System Theory, Centrum voor Wskunde en Informatca, Amsterdam, Amsterdam, The Netherlands,

217 BIBLIOGRAPHY [20] Rchard J. Bouchere and Matteo Sereno. On Closed Support T-Invarants and the Traffc Equatons. Journal of Appled Probablty, 35(2: , ISSN URL http: // [21] Nkolaos Bourbaks and Anya Tascllo. An SPN-Neural Plannng Methodology for Coordnaton of two Robotc Hands wth Constraned. Journal of Intellgent and Robotc Systems - Sprnger Netherlands, 19(3: , July DOI: [22] Ilja N. Bronsten, Konstantn A. Semendjajew, Gerhard Musol, and Hener Muehlg. Taschenbuch der Mathematk. Verlag Harr Deutsch, Thun, Frankurt am Man, 2 edton, ISBN [23] Gacomo Bucc and Enrco Vcaro. Compostonal Valdaton of Tme-Crtcal Systems Usng Communcatng Tme Petr Nets. IEEE Transactons on Software Engneerng (TSE, 21(12: , December ISSN DOI: org/ / [24] Peter Buchholz. Herarchcal Structurng of Superposed GSPNs. IEEE Transactons on Software Engneerng (TSE, 25(2: , March/Aprl ISSN DOI: http: //do.eeecomputersocety.org/ / [25] Burroughs B 6700 Handbook - Volume I Hardware (Form No Burroughs Cooperaton, Cty of Industry, CA, USA, January [26] Burroughs B 6700 / B 7700 System Software Handbook (Replaces Volume II of Form No Burroughs Cooperaton, Cty of Industry, CA, USA, July [27] Jeffrey P. Buzen. Fundamental laws of computer system performance. In SIGMETRICS 76: Proceedngs of the 1976 ACM SIGMETRICS conference on Computer performance modelng measurement and evaluaton, pages , New York, NY, USA, ACM. DOI: http: //do.acm.org/ / [28] Chrstos G. Cassandras and Stéphane Lafortune. Introducton to Dscrete Event Systems. Kluver Academc Publshers, ISBN [29] K. Man Chandy and Doug Neuse. Lnearzer: A Heurstc Algorthm for Queueng Network Models of Computng Systems. Communcatons of the ACM, 25(2: , February ISSN DOI: [30] K. Man Chandy, Ulrch Herzog, and Ln S. Woo. Parametrc Analyss of Queung Networks. IBM Journal of Research and Development, 19(1:36 42, January [31] Peter Pn-Shan Chen. The entty-relatonshp model toward a unfed vew of data. ACM Transactons on Database Systems (TODS, 1(1:9 36, March ISSN DOI: [32] Eran Chnthaka. Web Servces and Axs2 Archtecture. IBM developerworks - SOA and Web servces - Techncal Lbrary, November URL developerworks/webservces/lbrary/ws-apacheaxs2/. [33] Govann Chola, G. Bruno, and T. Demara. Introducng a Color Formalsm nto Generalzed Stochastc Petr Nets. In Proceedngs of the 9th European Workshop on Applcaton and Theory of Petr Nets, Veneza, Italy, June

218 BIBLIOGRAPHY [34] Ganfranco Cardo and Chrstoph Lndemann. Analyss of determnstc and stochastc Petr Nets. In 5th Internatonal Workshop on Petr Nets and Performance Models, pages , Toulouse, France, October IEEE Computer Socety. [35] J. L. Coleman, Wllam Henderson, and Peter G. Taylor. Product form equlbrum dstrbutons and a convoluton algorthm for stochastc Petr nets. Performance Evaluaton, 26(3: , ISSN DOI: [36] Flavus Copacu. A Framework for Adaptve Transport n Servce-Orented Systems based on Performance Predcton. In Proceedngs of the Fall 2006 Workshop of the HPI Research School on Servce-Orented Systems Engneerng, volume 18. Hasso Plattner Insttute for Software Systems Engneerng, [37] Flavus Copacu, Stephan Kluth, Tomasz Porzucek, and Werner Zorn. Herarchcal Modelng of the Axs2 Web Servces Framework wth FMC-QE. In 3rd Internatonal Conference on COMmuncaton Systems software and mddleware (COMSWARE 2008, Bangalore, Inda, pages IEEE Computer Socety, January ISBN DOI: [38] Perre-Jacques Courtos. Decomposablty, nstabltes, and saturaton n multprogrammng systems. Communcatons of the ACM, 18(7: , July ISSN DOI: [39] Perre-Jacques Courtos. Error Analyss n Nearly-Completely Decomposable Stochastc Systems. Econometrca, 43(4: , July ISSN URL org/stable/ [40] Perre-Jacques Courtos. Decomposablty, Queueng and Computer System Applcatons. ACM Monograph Serces. Academc Press, Inc., New York, San Francsco, London, ISBN X. [41] Horst Czchos, edtor. Hütte - De Grundlagen der Ingeneurwssenschaften. Akademscher Veren Hütte e.v., Berln, Sprnger Verlag, Berln Hedelberg, 31 edton, ISBN [42] Gero Decker, Volker Gersabeck, Jan Schaffner, and Marcel Seelg. Archtecture-Based Performance Smulaton. In So Iong Ao, Oscar Castllo, Crag Douglas, Davd Dagan Feng, and Jeong-A. Lee, edtors, Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts (IMECS 2007, Lecture Notes n Engneerng and Computer Scence, pages , Hong Kong, Chna, March IMECS, Newswood Lmted. ISBN , [43] Peter J. Dennng and Jeffrey P. Buzen. The Operatonal Analyss of Queueng Network Models. ACM Computng Surveys (CSUR, 10(3: , September ISSN DOI: [44] DIN66200: Betreb von Rechensystemen - Begrffe, Auftragsbezehungen. Deutsches Insttut für Normung e.v., March [45] Edsger Wybe Djkstra. Cooperatng sequental processes. In F. Genuys, edtor, Programmng Languages: NATO Advanced Study Insttute, pages Academc Press,

219 BIBLIOGRAPHY [46] Susanna Donatell and Matteo Sereno. On the Product Form Soluton for Stochastc Petr Nets. In Kurt Jensen, edtor, Proceedngs of the 13th Internatonal Conference on Applcaton and Theory of Petr Nets, number 616 n Lecture Notes n Computer Scence, pages , London, UK, June Sprnger. ISBN DOI: / _9. [47] Allen B. Downey. A parallel workload model and ts mplcatons for processor allocaton. Cluster Computng, 1(1: , May ISSN (Prnt (Onlne. DOI: [48] Derek L. Eager and Kenneth C. Sevck. Performance bound herarches for queueng networks. ACM Transactons on Computer Systems (TOCS, 1(2:99 115, ISSN DOI: [49] Derek L. Eager and Kenneth C. Sevck. Bound herarches for multple-class queung networks. Journal of the ACM (JACM, 33(1: , January ISSN DOI: [50] Jalya Ekanayake and Denns Gannon. Common Archtecture for Functonal Extensons on Top of Apache Axs2. Techncal report, Indana Unversty Bloomngton, [51] Muhammad El-Taha. Lecture Notes - Queueng Networks (ncompete classnotes. Department of Mathematcs and Statstcs Unversty of Southern Mane, August [52] Gerard Florn and Stéphane Natkn. Generalzaton of Queueng Network Product Form Solutons to Stochastc Petr Nets. IEEE Transactons on Software Engneerng (TSE, 17 (2:99 107, ISSN DOI: [53] Greg Franks and C. Murray Woodsde. Performance of Mult-level Clent-Server Systems wth Parallel Servce Operatons. In Proceedngs of the Frst Internatonal Workshop on Software and Performance (WOSP98 - Santa Fe, New Mexco, Unted States, pages , New York, NY, USA, October ACM. ISBN DOI: http: //do.acm.org/ / [54] Greg Franks and C. Murray Woodsde. Effectveness of Early Reples n Clent-Server Systems. Performance Evaluaton, 36-37: , August ISSN DOI: [55] Greg Franks and C. Murray Woodsde. Multclass Multservers wth Deferred Operatons n Layered Queueng Networks, wth Software System Applcatons. In Proceedngs of the 12th IEEE / ACM Int. Symp. on Modelng, Analyss, and Smulaton of Computer and Telecommuncaton Systems (MASCOTS 2004, pages , Los Alamtos, CA, USA, IEEE Computer Socety. DOI: MASCOT [56] Greg Franks, Shkharesh Majumdar, John E. Nelson, Dorna C. Petru, Jerome A. Rola, and C. Murray Woodsde. Performance Analyss of Dstrbuted Server Systems. In Proceedngs of the Sxth Internatonal Conference on Software Qualty, pages 15 26, Ottawa, Canada, October [57] Greg Franks, Peter Maly, Murray Woodsde, Dorna C. Petru, and Alex Hubbard. Layered Queueng Network Solver and Smulator User Manual. Department of Systems and 197

220 BIBLIOGRAPHY Computer Engneerng, Carleton Unversty, Ottawa, Canada, 6840 edton, December [58] Greg Franks, Dorna Petru, Murray Woodsde, Jng Xu, and Peter Tregunno. Layered Bottlenecks and Ther Mtgaton. In Proceedngs of the 3rd Internatonal Conference on Quanttatve Evaluaton of Systems (QEST2006, pages , Los Alamtos, CA, USA, IEEE Computer Socety. ISBN DOI: org/ /qest [59] Greg Franks, Tarq Al-Omar, C. Murray Woodsde, Olva Das, and Salem Dersav. Enhanced Modelng and Soluton of Layered Queueng Networks. IEEE Transactons on Software Engneerng (TSE, 35(2: , ISSN DOI: eeecomputersocety.org/ /tse [60] Jörn Frehet and Armn Zmmermann. A Dvde and Conquer Approach for the Performance Evaluaton of Large Stochastc Petr Nets. In Proceedngs of the 9th IEEE Internatonal Workshop on Petr Nets and Performance Models (PNPM 01, Los Alamtos, CA, USA, IEEE Computer Socety. DOI: [61] Mathas Frtzsche and Jendrk Johannes. Puttng Performance Engneerng nto Model- Drven Engneerng: Model-Drven Performance Engneerng. In Holger Gese, edtor, MoDELS 2007 Workshops n: Models n Software Engneerng, volume 5002/2008 of Lecture Notes n Computer Scence (LNCS, pages , Berln / Hedelberg, Germany, Sprnger. ISBN DOI: [62] Mathas Frtzsche, Mchael Pcht, Wasf Glan, Ivor Spence, John Brown, and Peter Klpatrck. Extendng BPM Envronments of your choce wth Performance related Decson Support. In U. Dayal et al., edtor, Busness Process Management (BPM2009, volume 5701/2009 of Lecture Notes n Computer Scence (LNCS, pages , Berln / Hedelberg, Germany, Sprnger. ISBN DOI: / _8. [63] Vctor M. Glushkov. Automata theory and structural desgn problems of dgtal machnes. Cybernetcs and Systems Analyss, 1(1:3 9, January ISSN (Prnt (Onlne. DOI: [64] Wllam J. Gordon and Gordon F. Newell. Closed queueng systems wth exponental servers. Operatons Research, 15(2: , March - Aprl ISSN X. URL [65] Baerbel Grmm, Wll Woerstenfeld, Peter Pfel, and Karlhenz Martn. Das grosse Tafelwerk. Volk und Wssen Verlag GmbH, Berln, Germany, frst edton, ISBN [66] Bernhard Gröne. Konzeptonelle Patterns und hre Darstellung. PhD thess, Hasso Plattner Insttut für Softwaresystemtechnk an der Unverstät Potsdam, Potsdam, Germany, August [67] Donald Gross and Carl M. Harrs. Fundamentals of Queueng Theory. John Wley & Sons, Inc., New York, NY, USA, 3rd edton, ISBN

221 BIBLIOGRAPHY [68] Martn Haas and Werner Zorn. Methodsche Lestungsanalyse von Rechensystemen. R. Oldenbourg Verlag GmbH, München, Germany / Venna, Austra, ISBN [69] Serge Haddad and Patrce Moreaux. Evaluaton of Hgh Level Petr nets by Means of Aggregaton and Decomposton. In Proceedngs of the 6th IEEE Internatonal Workshop on Petr Nets and Performance Models (PNPM 95, pages 11 20, Los Alamtos, CA, USA, IEEE Computer Socety. DOI: [70] Serge Haddad, Patrce Moreaux, Matteo Sereno, and Manuel Slva. Structural Characterzaton and Qualtatve Propertes of Product Form Stochastc Petr Nets. In J.- M. Colom and M. Koutny, edtors, Applcatons and Theory of Petr Nets 2001 (ICATPN 2001, volume 2075/2001 of Lecture Notes n Computer Scence (LNCS, pages , Berln / Hedelberg, Germany, Sprnger. ISBN DOI: http: //dx.do.org/ / _11. [71] Phlp Hedelberger and Kshor S. Trved. Analytc Queueng Models for Programs wth Internal Concurrency. IEEE Transactons on Computers, 32(1:73 82, January ISSN DOI: [72] Wllam Henderson and Peter G. Taylor. Embedded Processes n Stochastc Petr Nets. IEEE Transactons on Software Engneerng (TSE, 17(2: , ISSN DOI: [73] Wllam Henderson, D. Lucc, and Peter G. Taylor. A Net Level Performance Analyss of Stochastc Petr Nets. J. Australan Math. Soc. Seres B, 31(2: , [74] Renhard Höllerer. Modellerung und Optmerung von Bürgerdensten am Bespel der Stadt Landshut. PhD thess, Hasso-Plattner-Insttute at the Unversty of Potsdam, Potsdam, Germany, June Draft Verson. [75] Anatol Holt and Frederc Commoner. Events and Condtons. In Jack B. Denns, edtor, Record of the Project MAC conference on concurrent systems and parallel computaton, pages 3 52, New York, NY, USA, ACM. [76] Alexandru Iosup, Ozan Sonmez, Shanny Anoep, and Dck Epema. The performance of bags-of-tasks n large-scale dstrbuted systems. In HPDC 08: Proceedngs of the 17th nternatonal symposum on Hgh performance dstrbuted computng, pages , New York, NY, USA, ACM. ISBN DOI: [77] James R. Jackson. Networks of watng lnes. Operatons Reseach, 5(4: , August ISSN X. URL [78] James R. Jackson. Jobshop-lke Queung Systems. Management Scence, 10(1: , October ISSN URL [79] Raj Jan. The Art of Computer Systems Performance Analyss: Technques for Expermental Desgn, Measurement, Smulaton, and Modelng. John Wley & Sons, New York, NY, USA, ISBN [80] Kurt Jensen. Coloured Petr Nets, Volume 1, Basc Concepts, Analyss Methods and Practcal Use. Sprnger, Berln, Germany, 2nd edton, Februar ISBN

222 BIBLIOGRAPHY [81] Don H. Johnson. Orgns of the equvalent crcut concept: the voltage-source equvalent. Proceedngs of the IEEE, 91(4: , Aprl ISSN DOI: eeecomputersocety.org/ /jproc [82] Frank Keller. Über de Rolle von Archtekturbeschrebungen m Software-Entwcklungsprozess. PhD thess, Hasso Plattner Insttut für Softwaresystemtechnk an der Unverstät Potsdam, Potsdam, Germany, August [83] Davd George Kendall. Stochastc Processes Occurrng n the Theory of Queues and ther Analyss by the Method of the Imbedded Markov Chan. The Annals of Mathematcal Statstcs, 24(3: , September ISSN URL stable/ [84] Cheeha Km and Ashok K. Agrawala. Analyss of the Fork-Jon Queue. IEEE Transactons on Computers, 38(2: , February ISSN DOI: eeecomputersocety.org/ / [85] Kyong Hoon Km, Rajkumar Buyya, and Jong Km. Power Aware Schedulng of Bag-of- Tasks Applcatons wth Deadlne Constrants on DVS-enabled Clusters. In Proceedngs of the Seventh IEEE Internatonal Symposum on Cluster Computng and the Grd (CCGRID 07 -Ro De Janero, pages , Washngton, DC, USA, May IEEE Computer Socety. ISBN DOI: CCGRID [86] Matthas Krschnck. The Performance Evaluaton and Predcton SYstem for Queueng NetworkS PEPSY-QNS. Techncal Report TR-I , Computer Scence Department Operatng Systems - IMMD IV, Fredrch-Alexander-Unversty, Erlangen-Nürnberg, Erlangen, Germany, June [87] Leonard Klenrock. Communcaton Nets: Stochastc Message Flow and Delay. Dover Publcatons, New York, NY, USA, ISBN [88] Leonard Klenrock. Queueng Systems Volume I: Theory. John Wley & Sons, New York, NY, USA, ISBN [89] Leonard Klenrock. Queueng Systems Volume II: Computer Applcatons. John Wley & Sons, New York, NY, USA, ISBN X. [90] Stephan Kluth. Analyse und Modellerung des ERP-Systems Sage Offce Lne. Master s thess, Hasso Plattner Insttute for Software Systems Engneerng, Potsdam, Germany, November [91] Andreas Knöpfel. Konzepte der Beschrebung nteraktver Systeme. PhD thess, Hasso Plattner Insttut für Softwaresystemtechnk an der Unverstät Potsdam, Potsdam, Germany, August [92] Andreas Knöpfel, Bernhard Gröne, and Peter Tabelng. Fundamental Modelng Concepts: Effectve Communcaton of IT Systems. John Wley & Sons, März ISBN X. [93] Stephen S. Lavenberg and Martn Reser. Statonary State Probabltes at Arrval Instants for Closed Queueng Networks wth Multple Types of Customers. Journal of Appled Probablty, 17(4: , December ISSN URL org/stable/

223 BIBLIOGRAPHY [94] Aurel A. Lazar and Thomas G. Robertazz. Markovan Petr Net protocols wth product form soluton. Performance Evaluaton, 12(1:67 77, January DOI: org/ / ( v. [95] Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevck. Quanttatve System Performance: Computer System Analyss Usng Queueng Network Models. Prentce-Hall, Inc., Englewood Clffs, NJ, USA, Februar ISBN [96] Abgal S. Lebrecht and Wllam J. Knottenbelt. Response Tme Approxmatons n Fork- Jon Queues. In Proceedngs of the 23rd Annual UK Performance Engneerng Workshop (UKPEW, Edge Hll Unversty, Ormskrk, Lancashre, UK, June [97] Hu L, Davd Groep, Jeff Templon, and Lex Wolters. Predctng job start tmes on clusters. In Proceedngs of the 2004 IEEE Internatonal Symposum on Cluster Computng and the Grd (CCGRID 04 - Chcago, IL, pages , Los Alamtos, CA, USA, IEEE Computer Socety. ISBN X. DOI: CCGrd [98] John D. C. Lttle. A Proof of the Queueng Formula L = λ W. Operatons Research, 9(3: , May - June ISSN X. URL [99] Mroslaw Malek, Bratslav Mlc, and Nkola Mlanovc. Analytcal Avalablty Assessment of IT Servces. In T. Nanya et al., edtor, Servce Avalablty - Proceedngs of the 5th Internatonal Servce Avalablty Symposum (ISAS 2008 Tokyo, Japan, volume 5017/2008 of Lecture Notes n Computer Scence, pages , Berln / Hedelberg, Germany, May Sprnger. ISBN DOI: _16. [100] Mansh Malhotra and Kshor S. Trved. A methodology for formal expresson of herarchy n model soluton. In Proceedngs on the 5th Internatonal Workshop on Petr Nets and Performance Models, pages , Toulouse, France, October IEEE Computer Socety. DOI: [101] Marco Ajmone Marsan, Gann Conte, and Ganfranco Balbo. A class of generalzed stochastc Petr nets for the performance evaluaton of multprocessor systems. ACM Transactons on Computer Systems (TOCS, 2(2:93 122, ISSN DOI: http: //do.acm.org/ / [102] Marco Ajmone Marsan, Ganfranco Balbo, Gann Conte, Susanna Donatell, and Gulana Franceschns. Modellng wth Generalzed Stochastc Petr Nets. Wley Seres n Parallel Computng. John Wley & Sons, Inc., New York, NY, USA, ISBN [103] Hans Ferdnand Mayer. Über das Ersatzschema der Verstärkerröhre [On equvalent crcuts for electronc amplfers]. Telegraphen- und Fernsprech-Technk, 15: , [104] Mchael Karl Molloy. On the ntegraton of delay and throughput measures n dstrbuted processng models. PhD thess, Unversty of Calforna, Los Angeles, [105] Tadao Murata. Petr Nets: Propertes, Analyss and Applcatons. In Proceedngs of the IEEE, volume 77-4, pages IEEE Computer Socety, Aprl DOI: eeecomputersocety.org/ / [106] Stéphane Natkn. Les Reseaux de Petr Stochastques et leur Applcaton a L evaluaton des Systemes Informatques. PhD thess, Le Conservatore natonal des arts et méters (Cnam, Pars, France,

224 BIBLIOGRAPHY [107] Edward Lawry Norton. Desgn of fnte networks for unform frequency characterstc. Techncal Report TM , Bell Laboratores, [108] Tarq Omar, Greg Franks, C. Murray Woodsde, and Amy Pan. Solvng Layered Queueng Networks of Large Clent Server Systems wth Symmetrc Replcaton. In Proceedngs of the 5th Internatonal Workshop on Software and Performance (WOSP 2005, pages , New York, NY, USA, July ACM. ISBN DOI: [109] Tarq Omar, Salem Dersav, Greg Franks, and C. Murray Woodsde. Performance Modelng of a Quorum Pattern n Layered Servce Systems. In Proceedngs of the 4th Internatonal Conference on Quanttatve Evaluaton of SysTems (QEST Ednburgh, pages , Los Alamtos, CA, USA, September IEEE Computer Socety. ISBN X. DOI: [110] Tarq Omar, Greg Franks, C. Murray Woodsde, and Amy Pan. Effcent performance models for layered server systems wth replcated servers and parallel behavor. Journal of Systems and Software, 80(4: , Aprl ISSN DOI: org/ /j.jss [111] Athanasos Papouls and S. Unnkrshna Plla. Probablty, Random Varables and Stochastc Processes. McGraw-Hll, New York, NY, USA, 4 edton, ISBN [112] Srnath Perera, Chathura Herath, Jalya Ekanayake, Eran Chnthaka, Ajth Ranabahu, Deepal Jayasnghe, Sanjva Weerawarana, and Glen Danels. Axs2, Mddleware for Next Generaton Web Servces. In Proceedngs of the IEEE Internatonal Conference on Web Servces (ICWS 06, pages , Washngton, DC, USA, September IEEE Computer Socety. DOI: [113] Carl Adam Petr. Kommunkaton mt Automaten. PhD thess, Insttut für nstrumentelle Mathematk, Bonn, Germany, [114] Brgtte Plateau and Jean-Mchel Fourneau. A methodology for solvng Markov models of parallel systems. Journal of Parallel and Dstrbuted Computng, 12(4: , ISSN DOI: [115] Tomasz Porzucek, Stephan Kluth, Flavus Copacu, and Werner Zorn. Modelng and Evaluaton Framework for FMC-QE. In Proceedngs of the 16th IEEE Internatonal Conference on the Engneerng of Computer-Based Systems (ECBS2008, pages , Los Alamtos, CA, USA, Aprl IEEE Computer Socety. ISBN DOI: [116] Tomasz Porzucek, Mathas Frtzsche, Stephan Kluth, and Davd Redlch. Combnaton of a Dscrete Event Smulaton and an Analytcal Performance Analyss through Model-Transformatons. In Proceedngs of the 17th IEEE Internatonal Conference on the Engneerng of Computer-Based Systems (ECBS2010, pages , Los Alamtos, CA, USA, March IEEE Computer Socety. ISBN DOI: eeecomputersocety.org/ /ecbs [117] Balaj Prabhakar, Ncholas Bambos, and T. S. Mountford. The Synchronzaton of Posson Processes and Queueng Networks wth Servce and Synchronzaton Nodes. Advances n Appled Probablty, 32(3: , September ISSN URL jstor.org/stable/

225 BIBLIOGRAPHY [118] Martn Reser and Stephen S. Lavenberg. Mean-Value Analyss of Closed Multchan Queung Networks. Journal of the ACM (JACM, 27(2: , ISSN DOI: [119] Thomas G. Robertazz. Why most stochastc Petr nets are non-product form networks. Techncal report, Stony Brook, N.Y.: State Unversty of New York at Stony Brook, College of Engneerng, New York, NY, USA, URL [120] Marcel Seelg, Stephan Kluth, Flavus Copacu, Tomasz Porzucek, Nco Naumann, and Steffen Kühn. Comparson of Performance Modelng and Smulaton - a Case Study. In Davd W. Bustard and Roy Sterrtt, edtors, Proceedngs of the 15th IEEE Internatonal Conference on Engneerng of Computer-Based Systems (ECBS Belfast, UK, pages 49 56, Los Alamtos, CA, USA, March IEEE Computer Socety. DOI: http: //do.eeecomputersocety.org/ /ecbs [121] Kenneth C. Sevck and Is Mtran. The Dstrbuton of Queung Network States at Input and Output Instants. Journal of the ACM (JACM, 28(2: , ISSN DOI: [122] Herbert A. Smon and Albert Ando. Aggregaton of Varables n Dynamc Systems. Econometrca, 29(2: , Aprl ISSN URL stable/ [123] Wllam J. Stewart. Introducton to the Numercal Soluton of Markov Chans. Prnceton Unversty Press, Prnceton, NJ, USA, ISBN [124] Peter Tabelng. Softwaresysteme und hre Modellerung. Sprnger, Berln / Hedelberg, Germany, ISBN [125] Phuoc Tran-Ga. Enführung n de Lestungsbewertung und Verkehrstheore. Oldenbourg Wssenschaftsverlag GmbH, München, Germany, 2 edton, Oktober ISBN [126] Hendrk Vantlborgh. Exact Aggregaton n Exponental Queueng Networks. Journal of the ACM (JACM, 25(4: , ISSN DOI: / [127] J. Walrand. A Note on Norton s Theorem for Queung Networks. Journal of Appled Probablty, 20(2: , June ISSN URL [128] Jacum Wang. Tmed Petr Nets - Theory and Applcaton. Kluwer Academc Publshers, Boston / Dordrecht / London, ISBN [129] Segfred Wendt. Ene Methode zum Enfwurf komplexer Schaltwerke unter Verwendung spezller Ablaufdagramme. Elektronsche Rechenanlagen, 12(6: , December [130] Segfred Wendt. Nchtphyskalsche Grundlagen der Informatonstechnk. Interpreterte Formalsmen. Sprnger, Berln, Germany, 2 edton, ISBN [131] Segfred Wendt. Operatonszustand versus Steuerzustand - ene äußerst zweckmäßge Unterschedung. Techncal report, Unversty of Kaserslautern, Kaserslautern, Germany, Februar

226 BIBLIOGRAPHY [132] Andreas Wllg. Lecture Notes - Performance Evaluaton Technques. Hasso-Plattner- Insttute, Unversty of Potsdam, Potsdam, Germany, Aprl [133] Armn Zmmermann, Renhard German, Jörn Frehet, and Günther Hommel. TmeNET 3.0 Tool Descrpton. In Proceedngs of the Internatonal Conference on Petr Nets and Performance Models (PNPM 99, [134] Armn Zmmermann, Jörn Frehet, Renhard German, and Günter Hommel. Petr Net Modellng and Performablty Evaluaton wth TmeNET 3.0. In Proceedngs of the 11th Internatonal Conference on Computer Performance Evaluaton: Modellng Technques and Tools (TOOLS 00, pages , London, UK, Sprnger. ISBN [135] Werner Zorn. Kommunkatonssysteme - durch Abstrakton zum Durchblck, am Bespel enes der mestgenutzen Protokolle. Inaugural Lecture at the Hasso-Plattner- Insttue at the Unversty of Potsdam, October [136] Werner Zorn. A Dfferent Approach to Network Modellng. Presentaton, March [137] Werner Zorn. Dstngushng between Control States and Operatonal States - a well Proven Paradgm to Cope wth the Complexty of Dscrete Dynamc Systems. Hasso- Plattner-Insttute, Unversty of Potsdam, [138] Werner Zorn. Lecture Sldes: Quanttatve Modelng. Hasso-Plattner-Insttute, Unversty of Potsdam, Potsdam, Germany, [139] Werner Zorn. Lecture Sldes: Communcaton Systems I & II. Hasso-Plattner-Insttute, Unversty of Potsdam, Potsdam, Germany, [140] Werner Zorn. Quanttatve Modelng as a Specal Abstracton of Systems Modelng. Submtted to the 3rd Internatonal Conference on the Quanttatve Evaluaton of SysTems (QEST2006, [141] Werner Zorn. Herarchcal Modelng based on Servce Requests. Hasso-Plattner- Insttute, Unversty of Potsdam, [142] Werner Zorn. Calculus Paper. Internal, status , March [143] Werner Zorn. FMC-QE - A New Approach n Quanttatve Modelng. In Hamd R. Arabna, edtor, Proceedngs of the Internatonal Conference on Modelng, Smulaton and Vsualzaton Methods (MSV 2007 wthn WorldComp 07, pages , Las Vegas, NV, USA, June CSREA Press. ISBN [144] Werner Zorn. Herarchsche Modellerung baserend auf Bedenanforderungen. Presented at the 21.DFN- Arbetstagung über Kommunkatonsnetze, Techncal Unversty Kaserslautern, Germany, May [145] Werner Zorn. Lecture Sldes: Communcaton Systems I. Hasso-Plattner-Insttute, Unversty of Potsdam, Potsdam, Germany, [146] Werner Zorn. FMC-QE - Introducton wth Examples. Presentaton, Aprl Internal Presentaton at the Research Group of Prof. Dr.-Ing. Werner Zorn at the Hasso-Plattner- Insttute at the Unversty of Potsdam. [147] Werner Zorn. Herarchsche Modellerung auf Bass von Bedenanforderungen. Presented at the Insttut für Operatons Research, Humboldt-Unverstät zu Berln, Aprl

227 Glossary A B BSSt D f HSSt K K λ λ [1] λ [1] bott λ m M Arrval rate wth determnstc nter-arrval tmes Servce Rate, determnstc Basc server staton Throughput Desred bottleneck utlzaton Herarchcal server staton Queue-capacty (ncludng servce requests n servce Overall number of servce Requests n a Closed System - also n ges Arrval rate wth stochastc nter-arrval tmes Arrval rate of top level ([1] servce requests Saturaton arrval rate of bottleneck (Unt: [Top level Servce requests / Tmeunt] Arrval rate of servce request SRq on herarchy level (Unt: [SRq /Tmeunt] Number of parallel servers (multplcty Populaton µ Servce Rate, stochastc m m,mpx Absolute multplcty of the server of servce request on herarchy level Multplex coeffcent of logcal server m [bb 1] parent( Absolute multplcty of the server of on the next herarchcal level ([bb 1] m,nt m j µ N Relatve multplcty of the server of servce request on herarchy level to the next herarchcal level [bb 1] Number of parallel (Multplex-Servers Servce rate of the server of servce request on herarchy level Normalzed servce request on herarchcal level 205

228 GLOSSARY n N N e n ges n N,q N r n q n,q N e,r N,s n s n,s ρ R ρ R Server Server j SRq SRq SRs U v v ext v nt v v [bb 1] parent( Mean number of servce requests n the staton Overall number of servce statons n the network Normalzed unty servce request on herarchcal level Overall number of crculatng servce requests Mean number of servce request on herarchy level Queued normalzed servce request on herarchcal level Normalzed servce response on herarchcal level Mean number of queued servce requests Mean number of queued servce request on herarchy level Normalzed unty servce response on herarchcal level Normalzed servce request on herarchcal level n servce Mean number of servce requests n servce Mean number of servce request on herarchy level n servce Utlzaton, stochastc Response tme Utlzaton of the server of queued servce request on herarchy level Response tme of the server of the servce request on herarchy level Correspondng (Multplex-Server of servce request Non-ambguous name of a (Multplex-Server Unnormalzed servce request on herarchcal level Non-ambguous name of the servce request on herarchy level Unnormalzed servce response on herarchcal level Utlzaton, determnstc Absolute traffc flow coeffcent External traffc flow coeffcent Internal traffc flow coeffcent Absolute traffc flow coeffcent of servce request on herarchy level Absolute traffc flow coeffcent on the next herarchcal level ([bb 1] relatve to servce request 206

229 GLOSSARY v,nt W W X X Y Y Relatve traffc flow coeffcent of servce request on herarchy level to the next herarchcal level [bb + 1] Queue tme Watng tme for servce request on herarchy level Servce tme Servce tme for servce request on herarchy level Servce duraton Servce duraton for servce request on herarchy level 207

230

231 Index A Abstracton Herarchy Approxmaton Error , 135, 148 Arrval Rate , 70 Arrval Theorem Axs B Basc Defntons Basc Server Staton BCMP Theorem Black Box Block Dagram Bottleneck Bounded Branch , 102 C Calculus Checked Servce Clent/Server Closed Queueng Networks Closed Set Closed Tandem Network Communcatng Tme Petr Nets (CmTPN.47 Comparson Complexty Analyss Consstent Control State Consstent Operatonal State Consstent System State Cont. Tme Stochastc Petr Net (SPN Control Servce Request , 151 Controlled Operatonal Transton Crtcal Acton Crtcal Actonfeld Crtcal Content Crtcal Locaton Crtcal Secton Crtcal Values D D/D/ D/D/m Decomposablty Departure Rate , 72 Dvde and Conquer Approach Dualty Prncple Dynamc Evaluaton Secton Dynamc Structures E Egenvalue Egenvector Entty Relatonshp Dagram , 73 Erlang s loss formula ERMF Expermental Parameters Extended Conflct Set External Load Generaton External Servce Tme F Feed Backward Loop Feed Forward Frng Delay Frng Rate , 41 Frng Tme Frng Weght , 41 Frst Come, Frst Served (FCFS , 17 Frst In, Frst Out (FIFO FMC-eCS FMC-QE Tool Forced Traffc Flow Law , 49, 69, 86 Fork-Jon Queue Formal Herarches Free-Kllng-Conflct Fundamental Laws Forced Traffc Flow Law , 49, 69, 86 Lttle s Law , 69, 86 Fundamental Modelng Concepts

232 INDEX G Behavor Compostonal Structures Man Concepts Operatonal and Control State Value Structures G/G/ G/G/m General System Generalzed Stochastc Petr Nets (GSPN.. 32 Global Balance Gordon-Newell Theorem Graphcal Representaton Dynamc Structures Servce Request Structures Statc Structures H Herarchcal Actvty Herarchcal Modelng , 69 Decomposablty Forced Traffc Flow Law Formal Herarches Norton s theorem Tme Augmented Petr Nets Herarchcal Server Staton Herarchcal Servce Request Herarchcally combned QPN (HQPN Herarchy , 151 HPI Search Portal I In-Consstent System State Inconsstent Control State Inconsstent Operatonal State Independent User Infnte Server Infnte Server (IS Integrty Inter-Server Control Flow Interactve Response Tme Formula Isolated Crculaton J Jackson s Theorem K k-bounded Kendall-Notaton L Last Come, Frst Served (LCFS , 17 Last In, Frst Out (LIFO Layered Queueng Networks (LQN.. 50, 155 Lexcographcal Level Lttle s Law , 69, 86 Local Balance Logcal Server , 97 Loop Loosely Connected Processes LQN Actvty Graph LQN Algorthm LQN Sequence Dagram LQN Solver M M/M/ M/M/ M/M/1/K M/M/m M/M/m/K M/M/m/K/M M/M/m/m Mean Value Analyss (MVA , 52, 135 Model Transformaton Monovaluated Place-Transton Net Multclass Multplex Coeffcent , 97 Multplexer , 97, 106 Multplexer Secton Multplexer Server Multplcty , 71 Mutual Excluson N Non Product Form Norton s theorem O Open Queueng Network Operatonal and Control State Operatonal Servce Request , 151 Organzatonal Herarchy

233 INDEX P Parallel Actvtes , 101 Partally Secured Crtcal Acton Peer to Peer Perron-Frobenus Egenvector Perron-Frobenus Theorem Petr Net Ppelne Preselecton Model Prorty Based Queung (PR Processor Sharng (PS Producer/Consumer Product Form Product Form Petr Nets , 154 Closed Set Dualty Prncple Free-Kllng-Conflct Isolated Crculaton Structural Constrant Product Form QPN Q Queue Sze Queueng Petr Nets (QPN , 154 Herarchcally combned QPN Product Form QPN Queueng Staton Queueng Theory , 152 R Race Model Random Selecton for Servce (RSS Reachablty Graph , 35 Reachablty Set , 30, 154 Reser/Lavenberg-Theorem Relatve Error , 135, 148 Response Tme , 10, 72, 88 Response Tme Law REST , 175 Routng Probablty S Safe Net Secured Crtcal Acton Semaphore Semaphore Multplex Coeffcent Semaphore Synchronzaton Seral Actvtes , 100 Server Secton Servce Duraton Servce Rate , 72 Servce Request Servce Request Secton Servce Request Structures Servce Response , 83 Servce Tme , 71 Smulaton , 167 Sngle User SM/M/ SOAP , 174 Sojourn Tme SPN-skeleton Statc Structures Steady State , 88 Stochastc Petr Nets (SPN Subparallel Summaton Method Superposed GSPN (SGSPN Synchronzaton Node T T-nvarant Tangble State Thnk Tme Three Ter Server Throughput Tme Augmented Petr Nets , 153 GSPN QPN SPN TPPN TTPN Tmed Places Petr Nets (TPPN Tmed Transtons Petr Nets (TTPN Traffc Flow Coeffcent , 74 Transactonal Servce U Unrelable Servce Unsecured Crtcal Acton Utlzaton , 10, 72 V Vanshng State W Watng Tme ,

234 INDEX Whle Loop

235 Appendx A Server Performance Values 213

236 APPENDIX A. SERVER PERFORMANCE VALUES Table A.1: D/D/1 D/D/1 - Sngle Server, Determnstc Servce Tme Model: Arrval Rate [68]: A Servce Tme [68]: X Servce Rate [68]: B = 1 X Utlzaton: U = A B U 1 Exp. Number of Queued SRqs: n q = 0 Exp. Number of SRqs n Servce: Exp. Number of SRqs n the Staton: n s = A B n = n s = A B Exp. Watng Tme: W = 0 Exp. Response Tme: R = 1 B 214

237 SERVER PERFORMANCE VALUES Table A.2: D/D/m D/D/m - Parallel Server, Determnstc Servce Tmes Model: Arrval Rate [68]: A Servce Tme [68]: X Servce Rate: B = 1 mx Utlzaton: U = A B U 1 Exp. Number of Queued SRqs: n q = 0 Exp. Number of SRqs n Servce: Exp. Number of SRqs n the Staton: n s = m A B n = n s = m A B Exp. Watng Tme: W = 0 Exp. Response Tme: R = 1 B 215

238 APPENDIX A. SERVER PERFORMANCE VALUES Table A.3: M/M/1 M/M/1 - Sngle Server, Exponental Dstrbuted Servce Tme Model: Arrval Rate [88]: λ k = λ k = 0, 1, 2,.. λ Servce Rate [88]: µ k = µ k = 0, 1, 2,.. Utlzaton [67]: ρ = λ µ < 1 Markov Chan [88]: λ λ λ λ k1 k k+1 State Probabltes [67, 87 89]: p k = (1 ρ ρ k k = 0, 1, 2,.. Exp. Number of Queued SRqs [67, 79]: Exp. Number of SRqs n Servce: Exp. Number of SRqs n the Staton [79, 87, 88]: n q = ρ2 1 ρ n s = ρ n = ρ 1 ρ Exp. Watng Tme [67, 88]: W = n q λ = ρ2 λ(1 ρ = Exp. Response Tme [67, 79, 88]: R = n λ = ρ λ(1 ρ = 1 µ λ ρ µ λ 216

239 SERVER PERFORMANCE VALUES Table A.4: M/M/m M/M/m - Parallel Server, Exponental Dstrbuted Servce Tmes Model: Arrval Rate [88]: λ 1 2 m λ k = λ k = 0, 1, 2,.. Servce Rate [67]: { kµ 1 k < m µ k = mµ m k Utlzaton [67, 88, 125]: ρ = λ mµ < 1 Markov Chan [88]: λ λ λ λ λ m2 m1 m m+1 2 (m1 m m State Probabltes [88]: p 0 = 1 m 1 k=0 p k = Exp. Number of Queued SRqs [67, 125]: n q = λ µ (mρ k k! + ( (mρ m m! ( 1 1 ρ p 0 (mρ k k! 1 k m ρ p k m m 0 m ρ p m!(1 ρ 2 0 Exp. Number of SRqs n Servce [125]: n s = mρ = λ µ Exp. Number of SRqs n the Staton [67]: n = mρ + λ m µ Exp. Watng Tme [67, 125]: Exp. Response Tme [67]: W = n q m! k m p m!(1 ρ 2 0 λ = λ m µ p m!(mµ(1 ρ 2 0 λ = λ m µ R = n q λ + n s m!(mµ(1 ρ 2 p µ 217

240 APPENDIX A. SERVER PERFORMANCE VALUES Table A.5: M/M/ M/M/ - Infnte Server, Exponental Dstrbuted Servce Tmes Model: Arrval Rate [88]: λ k = λ k = 0, 1, 2,.. 1 Servce Rate[88]: λ 2 3 µ k = kµ k = 1, 2, 3,.. Utlzaton: - Markov Chan [88]: λ λ λ λ k1 k k+1 2 k (k+1 State Probabltes [88]: Exp. Number of Queued SRqs: n q = 0 Exp. Number of SRqs n Servce: n s = λ µ Exp. Number of SRqs n the Staton [88]: n = λ µ Exp. Watng Tme: W = 0 Exp. Response Tme [88]: R = 1 µ p k = (λ/µk k! e λ/µ k = 0, 1, 2,.. 218

241 SERVER PERFORMANCE VALUES Table A.6: M/M/1/K M/M/1/K - Sngle Server, Exponental Dstrbuted Servce Tme, Fnte Storage Model: Arrval Rate [88]: { λ k < K λ k = 0 k K λ K Effectve Arrval Rate [67]: ( λ 1 1 ρ 1 λ e f f = λ (1 p K = ρ K ρ λ ( 1 1 Servce Rate [88]: K+1 ρ = 1 ρ = 1 µ k = µ k = 1, 2,.., K Markov Chan [88]: λ λ Traffc Intensty [67]: ρ = λ µ λ K1 K State Probabltes [18, 67, 79, 88]: { (1 ρρ 1 ρ ρ = 1 k 0 k K; ρ = 1 1 ρ K+1 1 ρ p 0 = K+1 p 1 k = 1 K+1 ρ = 1 K+1 0 k K; ρ = 1 0 else ρ Exp. Number of Queued SRqs [67, 79]: n q = 1 ρ ρ(kρk +1 ρ = 1 1 ρ K+1 K(K 1 ρ = 1 2(K+1 { 1 1 ρ ρ = 1 1 ρ Exp. Number of SRqs n Servce [67]: n s = (1 p 0 = K K+1 ρ = 1 Exp. Number of SRqs n the Staton [18, 67, 79]: Exp. Watng Tme [67]: n = { ρ 1 ρ K+1 K W = n q λ e f f = Exp. Response Tme [67]: R = n λ e f f = n λ(1 p K 1 ρ K+1 ρ K+1 ρ = 1 2 ρ = 1 1 ρ K 2 Kρ+ρ2 (K 1 λ(1 ρ 2 ρ = 1 K 1 2λ ρ = 1 219

242 APPENDIX A. SERVER PERFORMANCE VALUES Table A.7: M/M/m/K M/M/m/K - Parallel Server, Exponental Dstrbuted Servce Tmes, Fnte Storage Model: Arrval Rate [67]: { λ k < K λ k = 0 k K λ K Markov Chan [79]: λ λ 1 2 m λ Effectve Arrval Rate [67]: λ e f f = λ (1 p K Servce Rate [88]: { kµ 0 k m µ k = mµ m k Traffc Intensty [67, 79]: ρ = λ mµ λ λ λ m2 m1 m m+1 K1 K 2 (m1 m m m State Probabltes [67, 79]: ( p 0 = ( m 1 k=0 m 1 k=0 ( λµ k k! + ( λµ k p k = k! + ( λµ m m! ( λµ m 1 ρ K m+1 1 ρ 1 ρ = 1 m! (K n ρ = 1 λ k p k!µ k 0 λ k p m k m m!µ k 0 1 k m c k K Exp. Number of Queued SRqs [51, 67]: ( p λµ mρ 0 ( 1 ρ n q = K m+1 (1 ρ (K m + 1 ρ K m ρ = 1 m!(1 ρ 2 m m 1 (K m(k m+1 2 ρ = 1 Exp. Number of SRqs n Servce [67]: Exp. Number of SRqs n the Staton [67]: Exp. Watng Tme [67, 79]: n s = λ e f f µ = λ(1 p K µ n = n q + λ e f f µ = n q + λ(1 p K µ W = R 1 µ = n q λ e f f Exp. Response Tme [67, 79]: R = n λ e f f = n λ(1 p K 220

243 SERVER PERFORMANCE VALUES Table A.8: M/M/m/K/M M/M/m/K/M - Parallel Server, Exp. Dst. Servce Tmes, Fnte Storage, Fnte Populaton Model: Populaton[67, 88]: K Markov Chan [88]: Mλ (M1λ λ λ λ 1 2 M 1 2 m (Mm+2λ M Arrval Rate [67, 88]: { λ (M k 0 k K 1 λ k = 0 else Effectve Arrval Rate [67]: λ e f f = M 1 (M n λp k = λ (M n k=0 Servce Rate[67, 88]: { kµ 0 k m µ k = mµ k m Traffc Intensty: ρ = λ mµ (Mm+1λ (Mmλ (MK+1λ m2 m1 m m+1 K1 K 2 (m1 m m m State Probabltes [28, 88]: Exp. Number of Queued SRqs [67]: p 0 = 1 Exp. Number of SRqs n Servce [67]: 1+ m 1 ( ( M k λµ k+( ( m 1 M m 1 λµ K k=1 k=m Exp. Number of SRqs n the Staton [67]: ( (M m+1! k m+1 λ (M k! µm ( k p λ 0 µ ( M k p k = ( 0 k m 1 k p λ 0 µ ( M k k! m! mm k m k K Exp. Watng Tme [67]: W = n q = n λ e f f µ = n λ µ (M n n s = λ e f f µ = λ µ (M n n = M kp k k=1 n q λ(m n q Exp. Response Tme [67]: R = n λ(m n 221

244 APPENDIX A. SERVER PERFORMANCE VALUES Table A.9: M/M/m/m M/M/m/m - Parallel Server, Exponental Dstrbuted Servce Tmes, no Storage Model: Arrval Rate [88]: { λ k < m λ k = 1 0 k m λ 2 Effectve Arrval Rate [67]: λ e f f = λ (1 p m m Servce Rate[88]: { kµ k m µ k = 0 k > m Traffc Intensty: ρ = λ mµ Markov Chan [88]: λ λ λ λ m2 m1 m State Probabltes [67, 88]: p 0 = ( m Exp. Number of Queued SRqs: n q = 0 Exp. Number of SRqs n Servce [125]: Exp. Number of SRqs n the Staton [125]: Exp. Watng Tme: W = 0 2 (m1 ( λ µ m k 1 k! 1 k=0 ( k p λ 1 p k = 0 µ k! k m 0 k > m n s = λ e f f µ = λ(1 p m µ n = n s = λ e f f µ = λ(1 p m µ Exp. Response Tme [125]: R = 1 µ Erlang s loss formula [67, 79, 88, 125]: p m = m k=0 ( λ µ m m! ( λ µ k k! 222

245 SERVER PERFORMANCE VALUES Table A.10: Server Performance Values - Overvew D/D/1 D/D/m M/M/1 M/M/m M/M/ M/M/1/K M/M/m/K M/M/m/K/M M/M/m/m 1 Model 1 2 m K 1 2 K m 2 1 M 2 1 m 2 K m Arrval Rate A A λ k = λ k = 0, 1, 2,.. λ = λ k = 0, 1, 2,.. λ k = λ k k = 0, 1, 2,.. λ k = { λ k < K 0 k K λ k = { λ k < K 0 k K λ k = { λ (M k 0 k K 1 0 else λ k = { λ k < m 0 k m Effectve Arrval Rate Servce Rate B = 1 X B = 1 mx µ k = µ k = 0, 1, 2,.. µ k = { kµ 1 k < m mµ m k µ k = kµ k = 1, 2, 3,.. λ e f f = λ (1 pk ( = λ 1 1 ρ 1 ρ K ρ ρ = 1 λ ( 1 1 K+1 ρ = 1 = µ k =.., K M 1 λ e f f = λ (1 pk λe f f = (M n λp k = λ (M n λe f f = λ (1 pm k=0 kµ 0 k m mµ m k µ k = { kµ 0 k m mµ k m µ k = { kµ k m 0 k > m Utlzaton U = A B 1 U = A B 1 ρ = λ µ < 1 ρ = λ mµ < 1 Traffc Intensty State Probabltes Exp. Number of Queued SRqs p k = (1 ρ ρ k k = 0, 1, 2,.. nq = 0 nq = 0 nq = ρ2 1 p0 = m 1 ( (mρ k k! + (mρ m m! k=0 p k = 1 ρ nq = λ µ ( 1 1 ρ p0 (mρk k! 1 k m p0 ρk m m m ρ m! k m p k = (λ/µ k k! e λ/µ k = 0, 1, 2,.. 2 p0 nq = 0 nq = m!(1 ρ ρ = λ µ ρ = λ mµ ρ = λ mµ ρ = λ mµ { 1 ρ 1 ρ K+1 ρ = 1 m 1 p0 = ( λ µ k ( ρ = 1 λ µ m 1 ρ k! + K m+1 1 m! 1 ρ K+1 ρ = 1 p0 = k=0 p0 = 1+ (1 ρρ m 1 ( k ( m 1 K ( ( k 1 ρ K+1 0 k K; ρ = 1 m 1 ( λ µ k ( ρ = 1 M λ µ m k λµ +( M m 1 λµ (M m+1! k m+1 λ (M k! µm k=1 k=m ( k! + m! (K n+1 k λ p k = k=0 K+1 0 k K; ρ = 1 p0 µ ( M k λ k p 0 k m 1 0 else p k = k!µ k p0 1 k m k = ( k p k = λ p0 λ k µ ( M k k! m! mm k m k K m k m m!µ k p0 c k K m p0( λµ ρ ( ρ 1 ρ ρ(kρk +1 1 ρ K+1 ρ = 1 1 ρ K m+1 m!(1 ρ 2 K(K 1 nq = ρ = 1 (1 ρ (K m + 1 ρ K m ρ = 1 nq = n λ e f f µ = n λ µ (M n nq = 0 2(K+1 m m 1 (K m(k m+1 2 ρ = 1 ( m p0 = k=0 ( λ µ k 1 k! 1 p0 ( λ µ k 1 k! k m 0 k > m Exp. Number of SRqs n Servce ns = A B ns = m A B ns = ρ ns = mρ = λ µ ns = λ µ ns = { 1 1 ρ 1 ρ K+1 ρ = K+1 ρ = 1 ns = λ(1 pk µ ns = λ µ λ(1 pm (M n ns = µ Exp. Number of SRqs n Staton Exp. Watng Tme n = A B n = m A B n = ρ n = mρ + λ µ 1 ρ m!(1 ρ 2 p0 n = λ µ W = 0 W = 0 W = ρ W = µ λ λ µ m m n = m!(mµ(1 ρ 2 p0 W = 0 W = { ρ 1 ρ K+1 1 ρ K+1 ρk+1 ρ = 1 K 2 ρ = 1 ρ K 2 Kρ+ρ2 (K 1 ρ = 1 λ(1 ρ 2 K 1 2λ ρ = 1 n = nq + λ(1 pk µ W = nq W = λ e f f n = M kp k n = λ(1 pm µ k=1 nq λ(m nq W = 0 Exp. Response Tme Erlang s Loss Formula R = 1 B R = 1 B R = 1 R = µ λ λ µ m m!(mµ(1 ρ 2 p0 + 1 µ R = 1 µ R = n λ(1 pk R = n R = n R = 1 λ(1 pk λ(m n µ pm = ( λ µ m m! m ( λ µ k k! k=0 λ λ λ λ λ 1 1 λ λ λ λ 223

246

247 Appendx B Tables 225

248 APPENDIX B. TABLES Table B.1: Tableau Example (Table 3.2 Expermental Parameters [1] n ges 30 3,750 f 0,600 λ [1] 2,250 λ bott [1] Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Cash 1,00 1,00 1,00 1,00 2,250 Casher ,056 1,000 18,000 0,125 0,018 0,008 0,125 0,056 0,143 0, Blow-Dry 0,50 1,00 1,00 0,50 1,125 Blow-Dryer 1 4 0,350 1,000 2,857 0,000 0,000 0,394 0,350 0,394 0, Dry 1,00 1,00 1,00 1,00 2,250 Dryer ,000 0,000 0,394 0,175 0,394 0, Perm 0,30 1,00 1,00 0,30 0,675 Permer ,350 0,467 1,333 0,506 0,519 0,769 0,506 0,750 1,025 1, Dye 0,20 1,00 1,00 0,20 0,450 Dyer ,333 0,125 0,375 0,600 0,675 1,500 1,200 2,667 1,875 4, Supp. Work 1,00 1,00 1,00 1,00 2,250 Supp ,750 1,194 0,531 1,706 0,758 2,900 1, Cut2 0,60 1,00 1,00 0,60 1,350 Cut2-Cutter ,333 0,750 2,250 0,600 0,900 0,667 0,600 0,444 1,500 1, Cut1 0,40 1,00 1,00 0,40 0,900 Cut1-Cutter ,300 0,533 1,778 0,506 0,519 0,577 0,506 0,563 1,025 1, Cut 1,00 1,00 1,00 1,00 2,250 Cutter ,750 1,419 0,631 1,106 0,492 2,525 1, Wash 1,00 1,00 1,00 1,00 2,250 Washer ,111 1,000 9,000 0,083 0,000 0,000 0,250 0,111 0,250 0, Barbershop Serv. 1,00 1,00 1,00 1,00 2,250 Server ,750 2,631 1,169 3,581 1,592 6,212 2, Job Generaton 1,00 1,00 1,00 1,00 2,250 Generator ,572 0,095 0,000 0,000 23,788 10,572 23,788 10,572 Multplexer Secton [1] j j X j Name j m 1 Appentce 5 0,167 2 Barber 1 0,267 3 Barber Boss 1 0,225 4 Customer 226

249 TABLES Table B.2: Open Queueng Example - Tableau (Table 3.8 [1] n ges [1] λ bott f λ [1] Expermental Parameters 80 5,0000 0,9000 4,5000 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] SRq p p(, v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 5 1 Savng Request 0,50 5,00 1,00 2,50 11,250 Request Saver ,060 1,000 16,667 0,675 1,402 0,125 0,675 0,060 2,077 0, Prntng Request 0,50 5,00 1,00 2,50 11,250 Request Prnter ,030 1,000 33,333 0,338 0,172 0,015 0,338 0,030 0,509 0, Persst Data Request 1,00 5,00 1,00 5,00 22,500 Data Persster ,333 1,574 0,070 1,013 0,045 2,586 0, Computaton Request 1,00 5,00 1,00 5,00 22,500 Request Computer ,040 1,000 25,000 0,900 8,100 0,360 0,900 0,040 9,000 0, Unrelable Executon 1,00 1,00 5,00 5,00 22,500 Unrelable Executer ,000 9,674 0,430 1,913 0,085 11,586 0, Relable Executon 1,00 1,00 1,00 1,00 4,500 Relable Executer ,000 9,674 2,150 1,913 0,425 11,586 2, Input 1,00 1,00 1,00 1,00 4,500 Input Retrever ,050 1,000 20,000 0,225 0,065 0,015 0,225 0,050 0,290 0, Request 1,00 1,00 1,00 1,00 4,500 Request Handler ,000 9,739 2,164 2,138 0,475 11,877 2, Request Generaton 1,00 1,00 1,00 1,00 4,500 Clent ,139 0,066 0,000 0,000 68,123 15,139 68,123 15,139 Multplexer Secton [1] j Name j m j X j 1 CPU 1 0,200 2 Prnter 1 0,075 3 Dsk 1 0,150 4 I/O-Devce 1 0,

250 APPENDIX B. TABLES Table B.3: Closed Tandem Network - M/M/1 Tableau (Table 4.1 Expermental Parameters [1] n ges [1] λ bott f λ [1] 3 0,2000 0, ,1420 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,142 Executer ,500 1,000 0,400 0,355 0,195 1,376 0,355 2,500 0,551 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,142 Executer ,000 1,000 0,200 0,710 1,739 12,247 0,710 5,000 2,449 17, Request 1,00 1,00 1,00 1,00 0,142 Executer ,200 1,935 13,624 1,065 7,500 3,000 21, Req. Generaton 1,00 1,00 1,00 1,00 0,142 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j X j Name j m j 1 Server 1 1 2,500 2 Server 2 1 5,

251 TABLES Table B.4: Closed Tandem Network - M/M/1/K Tableau (Table 4.2 Expermental Parameters [1] n ges 3 0,4000 λ [1] Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ λ,eff Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 0,50 0,5 0,200 0,187 Executer ,500 1,000 0,400 0,500 0,267 1,429 0,467 2,500 0,733 3, Sub-Request 1 1,00 1,00 1,00 1 0,400 0,187 Executer ,000 1,000 0,200 2,000 1,333 7,143 0,933 5,000 2,267 12, Request 1,00 1,00 1,00 1 0,400 0,187 Executer ,200 1,600 8,571 1,400 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1 0,400 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j Name j m j X j 1 Server 1 1 5,000 2 Server 2 1 1,

252 APPENDIX B. TABLES Table B.5: Closed Tandem Network - Summaton Method Tableau (Table 4.3 Expermental Parameters [1] n ges [1] λ bott f λ [1] 3 0,2000 0,9144 0,1829 Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,183 Executer ,500 1,000 0,400 0,457 0,200 1,096 0,457 2,500 0,658 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,183 Executer ,000 1,000 0,200 0,914 1,428 7,808 0,914 5,000 2,342 12, Request 1,00 1,00 1,00 1,00 0,183 Executer ,200 1,628 8,904 1,372 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1,00 0,183 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j X j Name j m j 1 Server 1 1 2,500 2 Server 2 1 5,

253 TABLES Table B.6: Closed Tandem Network - Tableaux - Comparson (a M/M/1 (b M/M/1/K (c Summaton Method Expermental Parameters [1] n ges [1] λ bott f λ [1] 3 0,2000 0, ,1420 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,142 Executer ,500 1,000 0,400 0,355 0,195 1,376 0,355 2,500 0,551 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,142 Executer ,000 1,000 0,200 0,710 1,739 12,247 0,710 5,000 2,449 17, Request 1,00 1,00 1,00 1,00 0,142 Executer ,200 1,935 13,624 1,065 7,500 3,000 21, Req. Generaton 1,00 1,00 1,00 1,00 0,142 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j X j Name j m j 1 Server 1 1 2,500 2 Server 2 1 5,000 Expermental Parameters [1] n ges 3 λ [1] 0,4000 Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ λ,eff Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 0,50 0,5 0,200 0,187 Executer ,500 1,000 0,400 0,500 0,267 1,429 0,467 2,500 0,733 3, Sub-Request 1 1,00 1,00 1,00 1 0,400 0,187 Executer ,000 1,000 0,200 2,000 1,333 7,143 0,933 5,000 2,267 12, Request 1,00 1,00 1,00 1 0,400 0,187 Executer ,200 1,600 8,571 1,400 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1 0,400 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j Name j m j X j 1 Server 1 1 5,000 2 Server 2 1 1,250 Expermental Parameters [1] n ges [1] λ bott f λ [1] 3 0,2000 0,9144 0,1829 Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 3 1 Sub-Request 2 1,00 1,00 1,00 1,00 0,183 Executer ,500 1,000 0,400 0,457 0,200 1,096 0,457 2,500 0,658 3, Sub-Request 1 1,00 1,00 1,00 1,00 0,183 Executer ,000 1,000 0,200 0,914 1,428 7,808 0,914 5,000 2,342 12, Request 1,00 1,00 1,00 1,00 0,183 Executer ,200 1,628 8,904 1,372 7,500 3,000 16, Req. Generaton 1,00 1,00 1,00 1,00 0,183 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j X j Name j m j 1 Server 1 1 2,500 2 Server 2 1 5,

254 APPENDIX B. TABLES Table B.7: Closed Central Server Example - Tableau (Table 4.5 [1] n ges [1] λ bott f λ [1] Expermental Parameters 3 0,0500 0,6895 0,0345 Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 4 1 Wrte Request 2 0,56 9,00 1,00 5,00 0,172 Dsk2 Wrter ,000 1,000 0,250 0,690 0,587 3,403 0,690 4,000 1,276 7, Wrte Request 1 0,44 9,00 1,00 4,00 0,138 Dsk1 Wrter ,500 1,000 0,400 0,345 0,103 0,746 0,345 2,500 0,448 3, Data Wrtng Request 0,90 10,00 1,00 9,00 0,310 Data Wrter ,450 0,690 2,222 1,034 3,333 1,724 5, Calculaton Request 1,00 10,00 1,00 10,00 0,345 Calculator ,000 1,000 0,500 0,690 0,587 1,702 0,690 2,000 1,276 3, Internal Request 1,00 1,00 10,00 10,00 0,345 Internal Exec ,500 1,276 3,702 1,724 5,000 3,000 8, Transacton 1,00 1,00 1,00 1,00 0,034 Trans. Exec ,050 1,966 57,016 1,724 50,000 3,000 87, Transacton Generaton 1,00 1,00 1,00 1,00 0,034 Clent ,000 #### 0,000 0,000 0,000 0,000 0,000 0,000 Multplexer Secton [1] j Name j m j X j 1 CPU 1 20,000 2 Dsk1 1 20,000 3 Dsk2 1 10,

255 TABLES Table B.8: Semaphore Synchronzaton - Tableau (Table 4.8 [1] n ges [1] λ bott f λ [1] Expermental Parameters 1 1,0000 0,8800 0,8800 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Crtcal Acton 1 1,00 1,00 1,00 1,00 0,880 CA 1 Executer ,100 0,733 7,333 0,120 0,000 0,000 0,120 0,136 0,120 0, Request 1 1,00 1,00 1,00 1,00 0,880 Executer ,333 0,000 0,000 0,120 0,136 0,120 0, Req. Generaton 1 1,00 1,00 1,00 1,00 0,880 Clent ,000 1,000 1,000 0,880 0,000 0,000 0,880 1,000 0,880 1,000 ##### [1] n ges [1] λ bott f λ [1] Expermental Parameters 1 2,0000 0,7073 1,4146 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Crtcal Acton 2 1,00 1,00 1,00 1,00 1,415 CA 2 Executer ,200 0,967 4,833 0,293 0,000 0,000 0,293 0,207 0,293 0, Request 2 1,00 1,00 1,00 1,00 1,415 Executer ,833 0,000 0,000 0,293 0,207 0,293 0, Req. Generaton 2 1,00 1,00 1,00 1,00 1,415 Clent ,500 1,000 2,000 0,707 0,000 0,000 0,707 0,500 0,707 0,500 ##### Multplexer Secton [1] j Name j m j X j 1 Crtcal Resource 1 0,

256 APPENDIX B. TABLES Table B.9: Multclass Example - Tableau (Table 4.6 n ges λ bott f λ A Expermental Parameters 30 0,9434 0,8000 0,7547 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v SRq,nt v Mpx X Server 2 1 Servce Request A.2 1,00 1,00 2,00 2,00 1,509 Req. A.2 Srv ,330 0,623 1,887 0,800 3,200 2,120 0,800 0,530 4,000 2, Servce Request A.1 1,00 1,00 1,00 1,00 0,755 Req. A.1 Srv ,200 0,500 2,500 0,302 0,131 0,173 0,302 0,400 0,432 0, Servce Request A 1,00 1,00 1,00 1,00 0,755 Req. A.2.2 Srv ,887 3,331 4,413 1,102 1,460 4,432 5, Generate Request A 1,00 1,00 1,00 1,00 0,755 Clent A ,877 0,030 0,000 25,568 33,877 25,568 33,877 λ m p( [bb-1] m,nt m m,mpx μ ρ n,q W n,s Y n R n ges λ bott f λ B Expermental Parameters 30 0,9434 0,5000 0,4717 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] p p(, v p( v [bb-1] SRq,nt v λ m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R Server 3 5 Servce Request B.2.2 1,00 2,00 2,00 4,00 1,887 Req. B.2.2 Srv ,100 1,000 10,000 0,189 0,044 0,023 0,189 0,100 0,233 0, Servce Request B.2.1 1,00 2,00 1,00 2,00 0,943 Req. B.2.1 Srv ,200 0,377 1,887 0,500 0,500 0,530 0,500 0,530 1,000 1, Servce Request B.2 1,00 1,00 2,00 2,00 0,943 Req. B.2 Srv ,887 0,544 0,577 0,689 0,730 1,233 1, Servce Request B.1 1,00 1,00 1,00 1,00 0,472 Req. B.1 Srv ,200 0,500 2,500 0,189 0,044 0,093 0,189 0,400 0,233 0, Servce Request B 1,00 1,00 1,00 1,00 0,472 Req. B Srv ,887 0,588 1,246 0,877 1,860 1,465 3, Generate Request B 1,00 1,00 1,00 1,00 0,472 Clent B ,494 0,017 0,000 28,535 60,494 28,535 60,494 Multplexer Secton [1] j Name j m j X j 1 Server X 1 0,400 2 Server Y 1 1,060 3 Server Z 1 0,

257 TABLES Table B.10: ERMF - Tableau (Table 5.2 n ges [1] λ [1] Expermental Parameters 3 1,0000 Servce Request Secton Server Secton Dynamc Evaluaton Secton [bb-1] SRq p p(, v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Alert Request 1,00 1,00 1,00 1,00 1,000 Acton Performer ,010 0,014 1,408 0,355 0,001 0,001 0,010 0,010 0,011 0, ,00 1,00 1,00 1,00 1,000 ECA-E. Rule Engne ,150 0,211 1,408 0,355 0,022 0,022 0,150 0,150 0,172 0, ,00 1,00 1,00 1,00 1, ,470 1,000 2,128 0,000 0,000 0,470 0,470 0,470 0,470 Ruleset Eva. Req. Weather Servce Traffc Servce Weather Serv. Hdl. Traffc Servce Hdl ,00 1,00 1,00 1,00 1, ,300 1,000 3,333 0,000 0,000 0,300 0,300 0,300 0, Web Servces Req. 1,00 1,00 1,00 1,00 1,000 ECA-E. WS Proxy ,000 0,000 0,770 0,770 0,770 0, Mappng Request 1,00 1,00 1,00 1,00 1,000 ECA-E. Rule Engne ,200 0,282 1,408 0,355 0,029 0,029 0,200 0,200 0,229 0, Evaluaton Request 1,00 1,00 1,00 1,00 1,000 ECA-Engne Cont ,817 0,050 0,050 1,120 1,120 1,170 1, Intalzaton Request 1,00 1,00 1,00 1,00 1,000 Control Unt ,350 0,493 1,408 0,355 0,050 0,050 0,350 0,350 0,400 0, Forecast Request 1,00 1,00 1,00 1,00 1,000 ERMF-System ,817 0,102 0,102 1,480 1,480 1,582 1, Request Generaton 1,00 1,00 1,00 1,00 1,000 Tmer Applcaton ,418 0,705 0,000 0,000 1,418 1,418 1,418 1,418 Multplexer Secton M/M/m rsp. M/M/ [1] j Name j m j X j λ j μ j ρ n,q n,s n 1 ERMF Server 2 0,710 1,000 1,408 0,355 0,102 0,710 0,812 2 Weather Server 0,470 1,000 2,128 0,000 0,470 0,470 3 Traffc Server 0,300 1,000 3,333 0,000 0,300 0,

258 APPENDIX B. TABLES Table B.11: Axs2 - Tableau (Table 5.3 Expermental Parameters [1] n ges ,526 f 0,800 λ [1] 8,421 λ bott [1] Servce Request Secton Server Secton Dynamc Evaluaton Secton p p(, v p( [bb-1] SRq v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 4 1 Encrypton Req. 1,00 1,00 1,00 1,00 8,42 Encrypter ,0006 0, ,66 0,002 0,000 0,000 0,012 0,001 0,012 0, MessageOut Req. 1,00 1,00 1,00 1,00 8,42 MessageOut Srv ,0045 0,400 89,89 0,019 0,000 0,000 0,094 0,011 0,094 0, Polcy Det. Req. 1,00 1,00 1,00 1,00 8,42 Polcy Det ,0004 0, ,00 0,002 0,000 0,000 0,008 0,001 0,008 0, OpOut Phase Req. 1,00 1,00 1,00 1,00 8,42 OpOut Phase Srv ,0004 0, ,23 0,002 0,000 0,000 0,009 0,001 0,009 0, OutFlow Request 1,00 1,00 1,00 1,00 8,42 OutFlow Handler ,65 0,000 0,000 0,123 0,015 0,123 0, Busness Logc R. 1,00 1,00 1,00 1,00 8,42 Busness Logc ,0325 0,400 12,33 0,137 0,000 0,000 0,683 0,081 0,683 0, OpIn Phase Req. 1,00 1,00 1,00 1,00 8,42 OpIn Phase Srv ,0006 0, ,16 0,003 0,000 0,000 0,013 0,002 0,013 0, Dsp. Instance Req. 1,00 1,00 1,00 1,00 8,42 Instance Dsp ,0329 0,400 12,15 0,139 0,000 0,000 0,693 0,082 0,693 0, HTTP Loc. Dsp. R. 1,00 1,00 1,00 1,00 8,42 HTTP Loc. Dsp ,0003 0, ,48 0,001 0,000 0,000 0,006 0,001 0,006 0, SOAP MB. Dsp. R. 1,00 1,00 1,00 1,00 8,42 SOAP MB Dsp ,0001 0, ,00 0,000 0,000 0,000 0,002 0,000 0,002 0, URI Op. Dsp. Req. 1,00 1,00 1,00 1,00 8,42 URI Op. Dsp ,0001 0, ,00 0,000 0,000 0,000 0,002 0,000 0,002 0, Address Dsp. Req. 1,00 1,00 1,00 1,00 8,42 Address Dsp ,0287 0,400 13,93 0,121 0,000 0,000 0,604 0,072 0,604 0, Dspatch Request 1,00 1,00 1,00 1,00 8,42 Dspatcher ,10 0,000 0,000 1,308 0,155 1,308 0, PreDspatch Req. 1,00 1,00 1,00 1,00 8,42 PreDspatcher ,0050 0,400 80,00 0,021 0,000 0,000 0,105 0,013 0,105 0, Decrypt Req. 1,00 1,00 1,00 1,00 8,42 Decrypter ,0005 0, ,33 0,002 0,000 0,000 0,010 0,001 0,010 0, SOAP Act. Dsp. R. 1,00 1,00 1,00 1,00 8,42 SOAP Act. Dsp ,0397 0,400 10,07 0,167 0,000 0,000 0,836 0,099 0,836 0, Req.t URI Dsp. R. 1,00 1,00 1,00 1,00 8,42 Req. URI Dsp ,0437 0,400 9,14 0,184 0,000 0,000 0,921 0,109 0,921 0, Transport Request 1,00 1,00 1,00 1,00 8,42 Transporter ,98 0,000 0,000 1,757 0,209 1,757 0, InFlow Request 1,00 1,00 1,00 1,00 8,42 InFlow Handler ,59 0,000 0,000 3,193 0,379 3,193 0, Executon Request 1,00 1,00 1,00 1,00 8,42 Servce Exec ,26 0,000 0,000 4,000 0,475 4,000 0, Sup. and Ex. Req. 1,00 1,00 1,00 1,00 8,42 System ,400 2,11 0,800 2,216 0,263 4,000 0,475 6,216 0, Generaton Request 1,00 1,00 1,00 1,00 8,42 External Source ,0118 8,42 993, , , ,012 Multplexer Secton [1] j Name j m j X j 1 CPU 2 0,

259 TABLES n ges [1] λ bott [1] f λ [1] Expermental Parameters ,9313 0, ,6382 Table B.12: HPI Search Portal - Tableau (Table 5.1 Servce Request Secton Server Secton Dynamc Evaluaton Secton SRq p p(, [bb-1] v p( v,nt v λ Server [bb-1] m p( m,nt m Mpx X m,mpx μ ρ n,q W n,s Y n R 2 1 Renderng Request 1,00 1,00 1,00 1,00 11,638 Webpage Renderer ,0005 0,007 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Response Aggregaton 1,00 1,00 1,00 1,00 11,638 Response Agg ,0002 0,002 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Resp. + Prevews Agg. 1,00 1,00 1,00 1,00 11,638 Aggregator ,0002 0,003 12,931 0,000 0,000 0,900 0,077 0,900 0, HTML Parsng 1,00 1,00 1,00 1,00 11,638 HTML Parser ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew Generaton 1,00 1,00 20,00 20,00 232,764 Prevew Generator ,0001 1, ,769 0,004 0,000 0,000 0,012 0,000 0,012 0, HTML Generaton 1,00 1,00 1,00 1,00 11,638 HTML Generator ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew 1,00 1,00 1,00 1,00 11,638 Prevew Handler ,931 16,200 1,392 1,812 0,156 18,012 1, Aggregaton + Generaton 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, SOAP Unwrappng 1,00 5,00 1,00 5,00 58,191 SOAP Unwrapper ,0001 0, ,152 0,013 0,000 0,000 0,038 0,001 0,038 0, Search Engne 1 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0087 1,000 64,657 0,000 0,000 0,938 0,016 0,938 0, SOAP Generaton 1,00 5,00 1,00 5,00 58,191 SOAP Generator ,0001 0, ,152 0,013 0,000 0,000 0,038 0,001 0,038 0, Search Engne 1 Handlng 1,00 1,00 5,00 5,00 58,191 SE 1 Handler ,152 0,000 0,000 1,015 0,017 1,015 0, REST Unwrappng 1,00 5,00 1,00 5,00 58,191 REST Unwrapper ,0008 0,053 65,789 0,221 0,000 0,000 0,885 0,015 0,885 0, Search Engne 2 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0090 1, ,111 0,000 0,000 0,885 0,015 0,885 0, REST Generaton 1,00 5,00 1,00 5,00 58,191 REST Generator ,0030 0,197 65,789 0,221 0,004 0,000 0,885 0,015 0,888 0, Search Engne 2 Handlng 1,00 1,00 5,00 5,00 58,191 SE 2 Handler ,789 0,004 0,000 2,654 0,046 2,657 0, HTML Parsng 1,00 5,00 1,00 5,00 58,191 HTML Parser ,0024 0,102 42,553 0,273 0,008 0,000 1,367 0,024 1,375 0, Search Engne 3 Search 1,00 5,00 1,00 5,00 58,191 Search Engne ,0008 1, ,000 0,000 0,000 0,047 0,001 0,047 0, HTML Generaton 1,00 5,00 1,00 5,00 58,191 HTML Generator ,0023 0,098 42,553 0,273 0,005 0,000 1,367 0,024 1,373 0, Search Engne 3 Handlng 1,00 1,00 5,00 5,00 58,191 SE 3 Handler ,553 0,013 0,000 2,782 0,048 2,794 0, Search Request 1,00 1,00 1,00 1,00 11,638 Searcher ,553 0,013 0,001 2,782 0,239 2,794 0, Req. + SOAP Generaton 1,00 1,00 1,00 1,00 11,638 Req. + SOAP Gen ,0008 0,010 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Super Search Request 1,00 1,00 1,00 1,00 11,638 Super Search Srv ,931 32,413 2,785 7,294 0,627 39,707 3, Aggregaton 1,00 1,00 1,00 1,00 11,638 Aggregator ,0006 0,008 12,931 0,900 32,413 2,785 0,900 0,077 33,313 2, HTML Parsng 1,00 1,00 1,00 1,00 11,638 HTML Parser ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Prevew Generaton 1,00 1,00 20,00 20,00 232,764 Prevew Generator ,0050 1, ,000 0,388 0,083 0,000 1,164 0,005 1,247 0, HTML Generaton 1,00 1,00 1,00 1,00 11,638 HTML Generator , ,931 0,900 8,183 0,703 0,900 0,077 9,083 0, Prevew 1,00 1,00 1,00 1,00 11,638 Prevew Handler ,931 16,366 1,406 2,964 0,255 19,330 1, Aggregaton + Gen. 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Index Search 1,00 1,00 1,00 1,00 11,638 Index Searcher ,0050 1, ,000 0,058 0,004 0,000 0,058 0,005 0,062 0, Fle Search Req. Gen. 1,00 1,00 1,00 1,00 11,638 Generator ,0008 0,010 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Fle Search 1,00 1,00 1,00 1,00 11,638 Fle Searcher ,931 64,983 5,584 5,722 0,492 70,705 6, Response Sendng 1,00 1,00 1,00 1,00 11,638 Response Sender ,0054 0,070 12,931 0,900 0,000 0,000 0,900 0,077 0,900 0, Response Handlng 1,00 10,00 1,00 10,00 116,382 Response Handler ,0015 0, ,313 0,900 0,000 0,000 0,900 0,008 0,900 0, Pcture Engne Search 1,00 10,00 1,00 10,00 116,382 Pcture Engne ,0040 1, ,000 0,466 0,405 0,003 0,466 0,004 0,871 0, Request Generaton 1,00 10,00 1,00 10,00 116,382 Request Generator ,0057 0, ,313 0,900 8,100 0,070 0,900 0,008 9,000 0, Pcture Search 0,40 5,00 5,00 10,00 116,382 Pcture Searcher ,313 8,505 0,073 2,266 0,019 10,771 0, Response Handlng 1,00 5,00 1,00 5,00 58,191 Response Handler ,0025 0,162 64,657 0,900 8,100 0,139 0,900 0,015 9,000 0, D Path Generaton 1,00 5,00 1,00 5,00 58,191 3D Path Generator ,0025 1, ,000 0,073 0,001 0,000 0,145 0,003 0,146 0, Extracton and Generaton 1,00 5,00 1,00 5,00 58,191 Extractor and Gen ,0021 0,136 64,657 0,900 8,101 0,139 0,900 0,015 9,001 0, Room Search 1,00 5,00 1,00 5,00 58,191 Room Searcher ,657 16,202 0,278 1,945 0,033 18,147 0, People Detal Search 1,00 1,00 5,00 5,00 58,191 People Search H ,657 24,707 0,425 4,211 0,072 28,918 0, Aggregaton + Generaton 1,00 1,00 1,00 1,00 11,638 Aggregator + Gen ,0098 0,127 12,931 0,900 0,000 0,000 0,900 0,077 0,900 0, Response Parsng 1,00 1,00 1,00 1,00 11,638 Response Parser ,0090 1, ,111 0,105 0,000 0,000 0,105 0,009 0,105 0, People Search 1,00 1,00 1,00 1,00 11,638 People Searcher ,0024 1, ,667 0,028 0,001 0,000 0,028 0,002 0,029 0, LDAP Request Generaton 1,00 1,00 1,00 1,00 11,638 LDAP Req. Gen ,0025 1, ,000 0,029 0,001 0,000 0,029 0,003 0,030 0, LDAP Servce Request 1,00 1,00 1,00 1,00 11,638 LDAP Handler ,000 0,002 0,000 0,162 0,014 0,163 0, People Search Req. Gen. 1,00 1,00 1,00 1,00 11,638 Request Generator ,0080 0,103 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, People Search 1,00 1,00 1,00 1,00 11,638 People Searcher ,931 32,809 2,819 7,073 0,608 39,881 3, Response Handlng 1,00 3,00 1,00 3,00 34,915 Response Handler ,0012 0,047 38,794 0,900 40,909 1,172 0,900 0,026 41,809 1, Calendar Search Req. 1,00 3,00 1,00 3,00 34,915 Calendar ,0090 1, ,111 0,314 0,144 0,004 0,314 0,009 0,458 0, Request Generaton 1,00 3,00 1,00 3,00 34,915 Request Generator ,0090 0,349 38,794 0,900 8,100 0,232 0,900 0,026 9,000 0, Event Search Request 1,00 1,00 3,00 3,00 34,915 Event Searcher ,794 49,153 1,408 2,114 0,061 51,267 1, Response Handlng 1,00 3,00 1,00 3,00 34,915 Response Handler ,0001 0,005 38,794 0,900 49,153 1,408 0,900 0,026 50,053 1, D Path Generaton 1,00 3,00 1,00 3,00 34,915 3D Path Generator ,0025 1, ,000 0,044 0,000 0,000 0,087 0,003 0,087 0, Extracton and Generaton 1,00 3,00 1,00 3,00 34,915 Extractor and Gen ,0002 0,006 38,794 0,900 8,100 0,232 0,900 0,026 9,000 0, Room Search 1,00 1,00 3,00 3,00 34,915 Room Searcher ,794 57,253 1,640 1,887 0,054 59,140 1, Portal Search Serv. Req. 1,00 1,00 1,00 1,00 11,638 Portal Searcher ,931 64,983 5,584 7,294 0,627 72,277 6, Sub-Request Generaton 1,00 1,00 1,00 1,00 11,638 Sub-Request Gen ,0003 0,004 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Portal Request 1,00 1,00 1,00 1,00 11,638 Portal Server ,931 81,183 6,976 9,094 0,781 90,277 7, Webpage Request 1,00 1,00 1,00 1,00 11,638 Webpage Req ,0002 0,003 12,931 0,900 8,100 0,696 0,900 0,077 9,000 0, Webrequest 1,00 1,00 1,00 1,00 11,638 Webrequest Exec ,931 97,383 8,368 10,894 0, ,277 9, Job Generaton 1,00 1,00 1,00 1,00 11,638 Generator Clent334,3927 0,003 0,000 0, , , , ,393 Multplexer Secton j Name j m j [1] X j 1 HPI B Man 2 0,155 2 HPI C Prevew Srv. 8 0,101 3 HPI V SE 1 Serv.Srv. 1 0,001 4 Search Engne 1 Server 5 HPI C1.1 - SE 2 Serv. Srv. 1 0,019 6 Search Engne 2 Server 7 HPI C SE 3 Serv. Srv. 1 0,024 8 Search Engne 3 Server 9 HPI C Index Serv.Srv. 4 0, HPI C Pc. Serv. Srv. 2 0, HPI A DBIMServce 4 0, HPI C1.3 - LDAP Serv. Srv. 2 0, HPI LDAP Server 4 0, HPI C Event Srv. 4 0,

260

261 Appendx C Fgures 239

262 8 8 APPENDIX C. FIGURES ( 8 *1/9 <8( ( (,*,..( 9;&&& 8 9; 19 : 9 ( 8 ( ;* & &$ & 1. 9.!" #$% ($ * 21 *3 *+,*6, &&7 *5 *+ *5 *+, * 21 *4 *00,1! $ & 6 &$ "2( & "2(. &'(( &$'(( &'(( (-./ 8 (-./ 21 8 *1 21 *4 8 Fgure C.1: HPI Search Portal - Archtecture (Fgure

263 FIGURES *$ *$ +!"!#$" #$"! #"""$ """$ # #"""$ #%&'("""$ ""% #%&'( """$ 1234'$" '$1234 # 1234'$ '( 1$!'( '(*$ *$'( '(1$! '(*$ 1234*$ *$ *$ """$&*$ ""% %/*%'( """$&*$.'7($##" 7($#.'# $-"1$!" 1$!$-" $-"1$!.'7($## $-"$- $-" $-" #$- #$-.'*$ *$.'# #$-.'*$ 27($##" 7($# 2 $- $- $- $-"1$!" 1$!$-" $-"1$! 27($## $-"$- $-" $-" 2*$ *$ 2 2*$ $-"1$!" 1$!$-" $-"1$! 1234'$" '$1234 # 1234'$ $-"$- $-" $-" 1234*$ *$ *$ %&.'*$ *$ %$!.' %&.'*$ """$ ""% #%&'( """$ 1234'$" '$1234 # 1234'$ '( 1$!'( '(1$! '(*$ *$'( '(*$ $- $- $- 1234*$ *$ *$ '$ '$ '$ '$$- %'$$-% '$$- """$&*$ ""% %/*%'( """$&*$ 0!$- 0!$- 0!$- $- %*% *%$- % *$ #!"! # #! #1$!" 1$! # #1$! '$- $-' '$- '"$- $-'" '" '#5$$- %'#5$$- '#5$$-1$! *$ *$ *$ #1$!" 1$! # #1$! 6$- $- 6 5'$-*$ *$5'$- '#$- $-'# 6$- 5'$-*$ '#$- $$!*$ % 605&*% % $$!*$ """$&*$ ""% %/*%%, % """$&*$ #'$" '$ # #'$ 45' 1$!45' % 45'1$! '#$- #$- '#$- 45' *$ *$45' 45' %*$ '#$- %*% *%'#$- % *$ #1$!" 1$! # #1$! 6$- $- 6 5'$-*$ *$5'$- 6$- 5'$-*$ $$!*$ % 605&*% % $$!*$ #1$!" 1$! # #1$! $- $- $- +$!$$- $-+$!$ +$!$, *$ *$,, *$ *$ *$ *$ #$" #$" #$" Fgure C.2: HPI Search Portal - Servce Request Structure (Fgure

264 APPENDIX C. FIGURES 2..- /2. /!" $ "!" # $ # ( ( #%& #%& " " ' #%& #%& ' " * + ( ( #%& #%& ( " " " / +, / &, &,( (- ( (.-( 0 + 0*, ', ( /0 +0*, ', ( / 3 ( " 2. $ $ ' ' ' $ Fgure C.3: HPI Search Portal - Behavor (Fgure

265 FIGURES & &,, ( /.+ ( -,, ( / #-' - (.+#/, &!"+$"% ( ( ( / #-' *, & '( ( /!' 0 & # + ', ( + & # # ' & ( '"!"# $"% & & * # Fgure C.4: Axs2 - Dynamc - All (Fgure