# Structure Preserving Model Reduction for Logistic Networks

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Structure Preserving Model Reduction for Logistic Networks Fabian Wirth Institute of Mathematics University of Würzburg Workshop on Stochastic Models of Manufacturing Systems Einhoven, June 24 25, joint work with: Michael Schönlein Sergey Dashkovskiy, Michael Kosmykov Thomas Makuschewitz, Bernd Scholz-Reiter

2 Motivation Queueing Systems Ranking in Graphs Heuristics Conclusions

3 Motivation UK SAT (separaters) Brook Hansen (motors) Flender couplings London B NL D Metaproduct (baseplates coupling guards) Siemens (motors) F local supplier (baseplates) Leroy Somer (motors) E local supplier (baseplates) baseplates and coupling guards Metaproduct, local suppliers (F, E) motors Leroy Somer motors Siemens motors Brook Hansen couplings Flender LRVP pumps IND pumps SC pumps

4

5

6 Queueing Systems Queueing systems provide a stochastic framework for the modelling of logistic systems

7 Queueing Systems Consider a set of servers which are able to treat different classes of jobs. Server 1 Server 2 Server 5 Server 3 Server 4

8 Queueing Systems Without loss of generality each class only receives service at one given server. Unserved jobs wait in a queue. Server 1 Server 2 Server 5 Server 3 Server 4

9 Queueing Systems In open systems jobs arrive from the outside according to some stochastic process. Server 1 Server 2 Server 5 Server 3 Server 4

10 Queueing Systems After service jobs leave the network or with a certain probability they go to another station to receive service there. Server 1 Server 2 Server 5 Server 3 Server 4

11 Queueing Systems - The Maths Classes k = 1,..., K. Interarrival times: ξ k (n), n = 0, 1, 2,... i.i.d. E(ξ k (0)) < Service times: η k (n), n = 0, 1, 2,... i.i.d. E(µ k (0)) < routing matrix P = (p ij ), p ij - probability that job of class i becomes a job of class j Assumption: r(p) < 1.

12 Queueing Systems - Balance Equations Classes k = 1,..., K Interarrival times: ξ k (n), n = 0, 1, 2,... i.i.d. E(ξ k (0)) < Service times: η k (n), n = 0, 1, 2,... i.i.d. E(η k (0)) < routing matrix P = (p ij ), p ij - probability that job of class i becomes a job of class j Assumption: r(p) < 1. with Q(t) = Q(0) + A(t) + P T S(t) S(t) A j (t) = max { n } n ξ k (m) t m=0 S(t) is the service process - depends on the service discipline. The state space X is very often countable, but also depends on the service discipline.

13 Why Model Reduction For large scale manufacturing processes a complete description of the process as a queueing system can become too large to be accessible to analysis or reliable simulation. Thus reduced order models are of interest, that capture some of the essential features of the original network. We argue that in logistic networks structure of the network is an essential feature so classical reduction techniques from control theory do not really help.

14 Ranking in Graphs Ranking schemes try to extract information about the importance/relevance of a vertex from graph properties.

15 Ranking in Graphs - Eigenvector Centrality Given a weighted adjecency matrix A corresponding to a directed graph, the following steps are performed Colums are rescaled to have column sum 1 (where possible). A is made colum stochastic by adding artificial entries in zero columns. A is made irreducible e.g. by considering for some α (0, 1) Ã := αa + (1 α)ee T Then Perron-Frobenius theory says that there is an eigenvector r > 0 (unique up to scaling) such that Ar = r The entries of r quantify the importance of the nodes.

16 Ranking in Graphs: Eliminating zero columns

17 Ranking in Graphs: Eliminating zero columns

18 Ranking in Graphs - Ensuring irreducibility Given the weighted adjecency matrix A R n n consider the enlarged matrix [ A ] R (n+m) (n+m)

19 Ranking in Graphs Consider the graph described by A as weakly coupled with a larger network. Coupling described by vectors v and w. [ ] αa + (1 α)vn e T n w n e T m B = (1 α)v m e T n w m e T m Notation: e := [ ] T

20 Ranking in Graphs B = [ αa + (1 α)vn e T n w n e T m (1 α)v m e T n w m e T m Proposition Assume that B is irreducible, then if x = [ xn T xm T ] T is an eigenvector corresponding to the eigenvalue 1 of B then x n is an eigenvector corresponding to the eigenvalue 1 of the matrix ( ) A α (v, w) := αa + (1 α) v n + et mv m 1 e T w n e T n. mw m Furthermore, A α (v, w) is irreducible. Interpretation: The added vector describes a weak coupling of the network under consideration to a larger network, that has been neglected. ]

21 Dynamic Large-Scale A simple example Logistics Networks Important locations of the test case 3rd-tier supplier (Production of A) v 1 (35) v 2 (58) v 3 (47) v 4 (41) v 5 (44) v 6 (25) 2nd-tier supplier (Production of B) v 7 (70) v 8 (70) v 9 (82) v 10 (28) 1st-tier supplier (Production of C) v 11 (93) v 12 (47) v v 13 (73) 14 (37) OEM locations (Production of D) v 15 (140) v 16 (110) Local warehouses (Storage and shipping of D) v 17 v 18 High importance Low importance Retailer (Distribution of D) v 19 v 20 v 21 v 22 Frontiers in network science - advances and applications Volkswagen Symposium: September 2009, Berlin 10/17

22 Ranking in Queueing Networks How can we evaluate if a reduced order model is sufficiently close to the original one? The invariant probability distributions should be close to the original one.

23 Particular case: Jackson Networks In Jackson networks each server serves exactly one class of jobs. The arrival process is Poisson and the service times are exponentially distributed. Vector of external arrivals: α Traffic equation for effective load of a server: λ = P T λ + α. Without the embedding in a larger network λ is the ranking vector. λ determines the stationary probability distribution. So in this case there is no problem.

24 Heuristics A straightforward reduction approach is now: 1. Compute rank vector. 2. Delete unimportant nodes from the network. 3. Use this as the basis for simulation. Error estimates for the load of the reduced order networks can be obtained, but are conservative. A more refined method can be described: Theorem The following procedures for reduction do not change the rank of unaffected nodes. Interpretation: The load of the untouched nodes remains the same, so reduced order model is more useful.

25 Heuristics - Parallel Servers

26 Heuristics - Consecutive Servers

27 Heuristics - Subnetworks

28 Conclusions We have discussed reduction techniques for queueing networks, based on eigenvector centrality, which quantifies load of nodes of the network. There are other graph ranking algorithms. The benefits or disadvantages of these need to be investigated in this context. Approximation properties concerning the dynamics of the reduced order model still needs to be investigated. In particular, stability perserving model reduction is of interest - this is still an open question.

29 Thank you

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Random access protocols for channel access. Markov chains and their stability. Laurent Massoulié.

Random access protocols for channel access Markov chains and their stability laurent.massoulie@inria.fr Aloha: the first random access protocol for channel access [Abramson, Hawaii 70] Goal: allow machines

### UNIT 2 QUEUING THEORY

UNIT 2 QUEUING THEORY LESSON 24 Learning Objective: Apply formulae to find solution that will predict the behaviour of the single server model II. Apply formulae to find solution that will predict the

### M/M/1 and M/M/m Queueing Systems

M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

### Load Balancing and Switch Scheduling

EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Analysis of an Artificial Hormone System (Extended abstract)

c 2013. This is the author s version of the work. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purpose or for creating

### Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

### Power Rankings: Math for March Madness

Power Rankings: Math for March Madness James A. Swenson University of Wisconsin Platteville swensonj@uwplatt.edu March 5, 2011 Math Club Madison Area Technical College James A. Swenson (UWP) Power Rankings:

### Performance Analysis of a Telephone System with both Patient and Impatient Customers

Performance Analysis of a Telephone System with both Patient and Impatient Customers Yiqiang Quennel Zhao Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### INTEGRATED OPTIMIZATION OF SAFETY STOCK

INTEGRATED OPTIMIZATION OF SAFETY STOCK AND TRANSPORTATION CAPACITY Horst Tempelmeier Department of Production Management University of Cologne Albertus-Magnus-Platz D-50932 Koeln, Germany http://www.spw.uni-koeln.de/

### Analysis of a Production/Inventory System with Multiple Retailers

Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

### Conclusions and Suggestions for Future Research

6 Conclusions and Suggestions for Future Research In this thesis dynamic inbound contact centers with heterogeneous agents and retrials of impatient customers were analysed. The term dynamic characterises

### BRAESS-LIKE PARADOXES FOR NON-COOPERATIVE DYNAMIC LOAD BALANCING IN DISTRIBUTED COMPUTER SYSTEMS

GESJ: Computer Science and Telecommunications 21 No.3(26) BRAESS-LIKE PARADOXES FOR NON-COOPERATIVE DYNAMIC LOAD BALANCING IN DISTRIBUTED COMPUTER SYSTEMS Said Fathy El-Zoghdy Department of Computer Science,

### Optimal Dynamic Resource Allocation in Multi-Class Queueing Networks

Imperial College London Department of Computing Optimal Dynamic Resource Allocation in Multi-Class Queueing Networks MEng Individual Project Report Diagoras Nicolaides Supervisor: Dr William Knottenbelt

### A Case Study of Joint Online Truck Scheduling and Inventory Management for Multiple Warehouses

Technische Universität Chemnitz Fakultät für Mathematik D-09107 Chemnitz, Germany A Case Study of Joint Online Truck Scheduling and Inventory Management for Multiple Warehouses C. Helmberg, S. Röhl Preprint

### Modeling and Performance Evaluation of Computer Systems Security Operation 1

Modeling and Performance Evaluation of Computer Systems Security Operation 1 D. Guster 2 St.Cloud State University 3 N.K. Krivulin 4 St.Petersburg State University 5 Abstract A model of computer system

### Social Media Mining. Network Measures

Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### LECTURE 4. Last time: Lecture outline

LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random

### Web Hosting Service Level Agreements

Chapter 5 Web Hosting Service Level Agreements Alan King (Mentor) 1, Mehmet Begen, Monica Cojocaru 3, Ellen Fowler, Yashar Ganjali 4, Judy Lai 5, Taejin Lee 6, Carmeliza Navasca 7, Daniel Ryan Report prepared

### Queueing Networks with Blocking - An Introduction -

Queueing Networks with Blocking - An Introduction - Jonatha ANSELMI anselmi@elet.polimi.it 5 maggio 006 Outline Blocking Blocking Mechanisms (BAS, BBS, RS) Approximate Analysis - MSS Basic Notation We

### Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring

Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring Qiang Zheng, Student Member, IEEE, and Guohong Cao, Fellow, IEEE Department of Computer Science and Engineering

### STABILITY OF LU-KUMAR NETWORKS UNDER LONGEST-QUEUE AND LONGEST-DOMINATING-QUEUE SCHEDULING

Applied Probability Trust (28 December 2012) STABILITY OF LU-KUMAR NETWORKS UNDER LONGEST-QUEUE AND LONGEST-DOMINATING-QUEUE SCHEDULING RAMTIN PEDARSANI and JEAN WALRAND, University of California, Berkeley

### A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIX-EXPONENTIAL DISTRIBUTIONS

Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 1532-6349 print/1532-4214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL

### THE \$25,000,000,000 EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE

THE \$5,,, EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE KURT BRYAN AND TANYA LEISE Abstract. Google s success derives in large part from its PageRank algorithm, which ranks the importance of webpages according

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### A QUEUEING-INVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND. bhaji@usc.edu

A QUEUEING-INVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND Rasoul Hai 1, Babak Hai 1 Industrial Engineering Department, Sharif University of Technology, +98-1-66165708, hai@sharif.edu Industrial

### Search engines: ranking algorithms

Search engines: ranking algorithms Gianna M. Del Corso Dipartimento di Informatica, Università di Pisa, Italy ESP, 25 Marzo 2015 1 Statistics 2 Search Engines Ranking Algorithms HITS Web Analytics Estimated

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### October 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix

Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,

### Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

### Least-Squares Intersection of Lines

Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a

### Conditional Tail Expectations for Multivariate Phase Type Distributions

Conditional Tail Expectations for Multivariate Phase Type Distributions Jun Cai Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L 3G1, Canada jcai@math.uwaterloo.ca

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### UNCOUPLING THE PERRON EIGENVECTOR PROBLEM

UNCOUPLING THE PERRON EIGENVECTOR PROBLEM Carl D Meyer INTRODUCTION Foranonnegative irreducible matrix m m with spectral radius ρ,afundamental problem concerns the determination of the unique normalized

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Manifold Learning Examples PCA, LLE and ISOMAP

Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition

### Lectures on Stochastic Processes. William G. Faris

Lectures on Stochastic Processes William G. Faris November 8, 2001 2 Contents 1 Random walk 7 1.1 Symmetric simple random walk................... 7 1.2 Simple random walk......................... 9 1.3

### Dynamic Load Balancing in Parallel Queueing Systems: Stability and Optimal Control

Dynamic Load Balancing in Parallel Queueing Systems: Stability and Optimal Control Douglas G. Down Department of Computing and Software McMaster University 1280 Main Street West, Hamilton, ON L8S 4L7,

### 5 Scheduling. Operations Planning and Control

5 Scheduling Operations Planning and Control Some Background Machines (resources) are Machines process jobs (molding machine, x ray machine, server in a restaurant, computer ) Machine Environment Single

### Evaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem

International Journal of Applied Science and Technology Vol. 3 No. 8; December 2013 Evaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem D. R. Aremu O. A. Gbadamosi

### Rank one SVD: un algorithm pour la visualisation d une matrice non négative

Rank one SVD: un algorithm pour la visualisation d une matrice non négative L. Labiod and M. Nadif LIPADE - Universite ParisDescartes, France ECAIS 2013 November 7, 2013 Outline Outline 1 Data visualization

Insensitive Load Balancing T Bonald, M Jonckheere and A routière France Telecom R&D 38-4 rue du Général Leclerc 92794 Issy-les-Moulineaux, France {thomasbonald,matthieujonckheere,alexandreproutiere}@francetelecomcom

### THE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:

THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces

### Supplement to Call Centers with Delay Information: Models and Insights

Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290

### Machine Learning and Pattern Recognition Logistic Regression

Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.

Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine Genon-Catalot To cite this version: Fabienne

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### NETZCOPE - a tool to analyze and display complex R&D collaboration networks

The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots NETZCOPE - a tool to analyze and display complex R&D collaboration networks L. Streit & O. Strogan BiBoS, Univ.

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### Forecasting methods applied to engineering management

Forecasting methods applied to engineering management Áron Szász-Gábor Abstract. This paper presents arguments for the usefulness of a simple forecasting application package for sustaining operational

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### On the D-Stability of Linear and Nonlinear Positive Switched Systems

On the D-Stability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on D-stability of positive switched systems. Different

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### CHAPTER 3 CALL CENTER QUEUING MODEL WITH LOGNORMAL SERVICE TIME DISTRIBUTION

31 CHAPTER 3 CALL CENTER QUEUING MODEL WITH LOGNORMAL SERVICE TIME DISTRIBUTION 3.1 INTRODUCTION In this chapter, construction of queuing model with non-exponential service time distribution, performance

### 1: B asic S imu lati on Modeling

Network Simulation Chapter 1: Basic Simulation Modeling Prof. Dr. Jürgen Jasperneite 1 Contents The Nature of Simulation Systems, Models and Simulation Discrete Event Simulation Simulation of a Single-Server

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

### The Exponential Distribution

21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

### Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

### Stochastic Models for Inventory Management at Service Facilities

Stochastic Models for Inventory Management at Service Facilities O. Berman, E. Kim Presented by F. Zoghalchi University of Toronto Rotman School of Management Dec, 2012 Agenda 1 Problem description Deterministic

### Inner products on R n, and more

Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

### NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES

NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES Silvija Vlah Kristina Soric Visnja Vojvodic Rosenzweig Department of Mathematics

### TEST 2 STUDY GUIDE. 1. Consider the data shown below.

2006 by The Arizona Board of Regents for The University of Arizona All rights reserved Business Mathematics I TEST 2 STUDY GUIDE 1 Consider the data shown below (a) Fill in the Frequency and Relative Frequency

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### Bonus-malus systems and Markov chains

Bonus-malus systems and Markov chains Dutch car insurance bonus-malus system class % increase new class after # claims 0 1 2 >3 14 30 14 9 5 1 13 32.5 14 8 4 1 12 35 13 8 4 1 11 37.5 12 7 3 1 10 40 11

### Optimization of Supply Chain Networks

Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal

### Experiments on the local load balancing algorithms; part 1

Experiments on the local load balancing algorithms; part 1 Ştefan Măruşter Institute e-austria Timisoara West University of Timişoara, Romania maruster@info.uvt.ro Abstract. In this paper the influence

### The Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback

The Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback Hamada Alshaer Université Pierre et Marie Curie - Lip 6 7515 Paris, France Hamada.alshaer@lip6.fr

### Introduction to Probability Theory for Graduate Economics. Fall 2008

Introduction to Probability Theory for Graduate Economics Fall 2008 Yiğit Sağlam December 1, 2008 CHAPTER 5 - STOCHASTIC PROCESSES 1 Stochastic Processes A stochastic process, or sometimes a random process,

### Big Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time

Big Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time Edward Bortnikov & Ronny Lempel Yahoo! Labs, Haifa Class Outline Link-based page importance measures Why

### Decision-making with the AHP: Why is the principal eigenvector necessary

European Journal of Operational Research 145 (2003) 85 91 Decision Aiding Decision-making with the AHP: Why is the principal eigenvector necessary Thomas L. Saaty * University of Pittsburgh, Pittsburgh,

### Scheduling Algorithm for Delivery and Collection System

Scheduling Algorithm for Delivery and Collection System Kanwal Prakash Singh Data Scientist, DSL, Housing.com Abstract Extreme teams, large-scale agents teams operating in dynamic environments are quite

### Extended warranty pricing considering the time of money. Hui-Ming Teng

Extended warranty pricing considering the time of money Hui-Ming Teng Department of Business Administration Chihlee Institute of Technology Panchiao 22005 Taiwan R.O.C. Abstract Warranty usually plays

### 2014-2015 The Master s Degree with Thesis Course Descriptions in Industrial Engineering

2014-2015 The Master s Degree with Thesis Course Descriptions in Industrial Engineering Compulsory Courses IENG540 Optimization Models and Algorithms In the course important deterministic optimization

### Optimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani. EPEW 2014, Florence

Optimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani EPEW 2014, Florence Scenario How many cloud instances should be hired? Requests Host hiring servers The number of active servers is controlled

### Fluid Approximation of a Priority Call Center With Time-Varying Arrivals

Fluid Approximation of a Priority Call Center With Time-Varying Arrivals Ahmad D. Ridley, Ph.D. William Massey, Ph.D. Michael Fu, Ph.D. In this paper, we model a call center as a preemptive-resume priority

### Using Ant Colony Optimization for Infrastructure Maintenance Scheduling

Using Ant Colony Optimization for Infrastructure Maintenance Scheduling K. Lukas, A. Borrmann & E. Rank Chair for Computation in Engineering, Technische Universität München ABSTRACT: For the optimal planning

### Spatial Statistics Chapter 3 Basics of areal data and areal data modeling

Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data

### RESOURCE POOLING AND STAFFING IN CALL CENTERS WITH SKILL-BASED ROUTING

RESOURCE POOLING AND STAFFING IN CALL CENTERS WITH SKILL-BASED ROUTING by Rodney B. Wallace IBM and The George Washington University rodney.wallace@us.ibm.com Ward Whitt Columbia University ward.whitt@columbia.edu

### Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations

Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations Roy D. Williams, 1990 Presented by Chris Eldred Outline Summary Finite Element Solver Load Balancing Results Types Conclusions

### Analyzing Mission Critical Voice over IP Networks. Michael Todd Gardner

Analyzing Mission Critical Voice over IP Networks Michael Todd Gardner Organization What is Mission Critical Voice? Why Study Mission Critical Voice over IP? Approach to Analyze Mission Critical Voice

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### IBM Research Report. A Pricing Problem under Monge Property

RC23823 (W0512-010) December 2, 2005 Mathematics IBM Research Report A Pricing Problem under Monge Property Oktay Günlük IBM Research Division Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights,

### The Trip Scheduling Problem

The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

### Transportation. Transportation decisions. The role of transportation in the SC. A key decision area within the logistics mix

Transportation A key decision area within the logistics mix Chapter 14 Transportation in the Supply Chain Inventory Strategy Forecasting Storage decisions Inventory decisions Purchasing & supply planning

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Asymptotic Behaviour and Cyclic Properties of Tree-shift Operators

Asymptotic Behaviour and Cyclic Properties of Tree-shift Operators University of Szeged, Bolyai Institute 29th July 2013, Gothenburg Banach Algebras and Applications 1 Introduction, Motivation Directed