UNIVERSIDAD DE LA REPÚBLICA Facultad de Ingeniería RESOURCE ALLOCATION IN NETWORKS FROM A CONNECTION-LEVEL PERSPECTIVE

Transcription

1 UNIVERSIDAD DE LA REPÚBLICA Facultad de Ingeneía Tess paa opta al Título de Docto en Ingeneía Eléctca RESOURCE ALLOCATION IN NETWORKS FROM A CONNECTION-LEVEL PERSPECTIVE (ASIGNACIÓN DE RECURSOS EN REDES DESDE LA PERSPECTIVA DE LAS CONEXIONES) Auto: Andés Feagut Decto de Tess: D. Fenando Pagann c Deechos de auto esevados (all ghts eseved) Montevdeo, Uuguay 2011

2 II UNIVERSIDAD DE LA REPÚBLICA ORIENTAL DEL URUGUAY INSTITUTO DE INGENIERíA ELÉCTRICA Los abajo fmantes cetfcamos que hemos leído el pesente tabajo ttulado Resouce allocaton n netwoks fom a connecton-level pespectve hecho po Andés Feagut y encontamos que el msmo satsface los equementos cuculaes que la Facultad de Ingeneía exge paa la tess del título de Docto en Ingeneía Eléctca. Fecha: 30 de agosto de 2011 Decto Académco: D. Pablo Monzón Decto de Tess: D. Fenando Pagann Tbunal examnado: D. Pablo Belzaena D. Mattheu Jonckheee D. Enesto Modeck

3 III ISSN: XXXX-XXXX (pnted veson) ISSN: XXXX-XXXX (electonc veson) Andés Feagut Tess de Doctoado en Ingeneía Eléctca Facultad de Ingeneía Unvesdad de la Repúblca Montevdeo, Uuguay, 2011.

4 IV UNIVERSIDAD DE LA REPÚBLICA ORIENTAL DEL URUGUAY Fecha: 30 de agosto de 2011 Auto: Ttulo: Insttuto: Gado: Andés Feagut Resouce allocaton n netwoks fom a connecton-level pespectve Insttuto de Ingeneía Eléctca Docto en Ingeneía Eléctca (D. Ing.) Se autoza a tavés de la pesente a la Unvesdad de la Repúblca Oental del Uuguay a hace ccula y copa esta tess con popóstos no comecales po equementos de ndvduos o nsttucones. Fma del auto El auto se eseva otos deechos de publcacón o utlzacón de la tess y/o de extactos de la msma sn su autozacón escta. El auto declaa que obtuvo pemso explícto paa el uso de todo mateal con deecho de auto que apaece en esta tess, excepto extactos o mencones de tabajos académcos con autozacón smla a la actual, cuyo uso es expesamente dentfcado.

5 Contents 1 Intoducton Outlne of ths wok and man contbutons Assocated publcatons Resouce allocaton and congeston contol n netwoks Resouce allocaton as an optmzaton poblem Rate allocaton n communcaton netwoks Congeston contol as a decentalzed optmzaton algothm Congeston contol algothms n the Intenet Netwok utlty maxmzaton beyond congeston contol I Connecton level esouce allocaton and use-centc faness 28 3 Connecton level models and use ncentves Intoducton Use ncentves Stochastc demands Use-centc faness Defnton Contollng the aggegate ates Contollng the numbe of flows: the sngle-path case Contollng the numbe of flows: the mult-path case Packet level smulaton Utlty based admsson contol Admsson contol n the sngle path case Stochastc model and ts stablty Flud lmt devaton fo the sngle path case

6 VI 5.4 Equlbum and stablty of the flud lmt Admsson contol n the mult-path case Smulaton examples Connecton level outng Connecton level outng polces A necessay condton fo stablty A decentalzed outng polcy Combnng admsson contol and outng Conclusons of Pat I 85 II Resouce allocaton and connecton level models fo multate weless netwoks 86 8 Resouce allocaton n multate weless envonments Backgound and pevous wok TCP esouce allocaton n a multate weless envonment Effcency and faness n multate weless netwoks The sngle cell case Multate andom ealy detecton Extenson to weless access netwoks Connecton level analyss of weless netwoks Model descpton Connecton level pefomance of cuent TCP-multate envonments Connecton level pefomance of the unbased allocaton Analyss of IEEE Calculatng the effectve ates: the mpact of oveheads Multate RED mplementaton Smulaton examples Conclusons of Pat II Geneal conclusons and futue lnes of wok 128

7 VII A Mathematcal pelmnaes 130 A.1 Convex optmzaton A.2 Lyapunov stablty A.3 Passve dynamcal systems A.4 Makov chans

8 Abstact In ths thess, we analyze seveal esouce allocaton poblems asng n the study of telecommuncaton systems. In patcula, we focus on data netwoks, of whch the most mpotant example s the global Intenet. In such netwoks, the scace esouce to be allocated s the amount of bandwdth assgned to each ongong connecton. Ths allocaton s pefomed n eal tme by the undelyng potocols, spltted acoss seveal logcal layes. Fom ths pont of vew, the netwok can be thought as a lage scale contol system, whee ndvdual enttes must follow gven contol laws n ode to fnd a sutable esouce allocaton. Snce the semnal wok of[kelly et al., 1998], ths poblem has been expessed n economc tems, though the Netwok Utlty Maxmzaton famewok. Ths fomulaton poved to be a valuable tool to analyze exstng mechansms and desgn new potocols that enhance the netwok behavo, and povded a cucal lnk between the tadtonal laye analyss of netwok potocols and convex optmzaton technques, leadng to what s called the coss-laye desgn of netwoks. In ths wok we focus on the analyss of the netwok fom a connecton-level pespectve. In patcula, we study the connecton-level pefomance, effcency and faness of seveal models of netwok esouce allocaton. We do so n seveal settngs, both sngle and multpath, and both wed and weless scenaos. We analyze n detal two mpotant poblems: on one hand, the esouce allocaton povded by congeston contol potocols wheneve multple connectons pe use ae allowed. We dentfy poblems wth the cuent paadgm of flow-ate faness, and we popose a new noton of use-centc faness, developng along the way decentalzed algothms that can be mplemented at the edge of the netwok, and dve the system to a sutable global optmum. The second mpotant poblem analyzed hee s the esouce allocaton povded by congeston contol algothms ove a physcal laye that allows multple tansmsson ates, such as weless netwoks. We show that the typcal algothms n use lead to mpotant neffcences fom the connecton-level pespectve, and we popose mechansms to ovecome these neffcences and enhance the esouce allocaton povded by such netwoks. Thoughout ths wok, seveal mathematcal tools ae appled, such as convex optmza-

9 IX ton, contol theoy and stochastc pocesses. By means of these tools, a model of the system s constucted, and seveal contol laws and algothms ae developed to each the desed pefomance objectve. As a fnal step, these algothms wee tested va packet-level smulatons of the netwoks nvolved, povdng valdaton fo the theoy, and evdence that they can be mplemented n pactce.

10 Resumen En esta tess, se analzan vaos poblemas de asgnacón de ecusos que sugen en el estudo de los sstemas de telecomuncacones. En patcula, nos centamos en las edes de datos, de los cuales el ejemplo más mpotante es la Intenet global. En este tpo de edes, el ecuso escaso que debe se asgnado es la cantdad de ancho de banda de cada conexón en cuso. Esta asgnacón se ealza en tempo eal po los potocolos subyacentes, que típcamente se encuentan dvddos en vaos nveles lógcos o capas. Desde este punto de vsta, la ed puede se pensada como un sstema de contol a gan escala, donde cada entdad debe segu un conjunto dado de leyes de contol, a fn de enconta una asgnacón adecuada de ecusos. Desde el nfluyente tabajo de[kelly et al., 1998], este poblema se ha expesado en témnos económcos, dando luga a la teoía conocda como Netwok Utlty Maxmzaton (maxmzacón de utldad en edes). Este maco ha demostado se una heamenta valosa paa analza los mecansmos exstentes y dseña potocolos nuevos que mejoan el compotamento de la ed. Popocona además un vínculo cucal ente el tadconal análss po capas de los potocolos de ed y las técncas de optmzacón convexa, dando luga a lo que se denomna análss mult-capa de las edes. En este tabajo nos centamos en el análss de la ed desde una pespectva a nvel de conexones. En patcula, se estuda el desempeño, efcenca y justca en la escala de conexones de vaos modelos de asgnacón de ecusos en la ed. Este estudo se ealza en vaos escenaos: tanto sngle-path como mult-path (edes con un únco o múltples camnos) así como escenaos cableados e nalámbcos. Se analzan en detalle dos poblemas mpotantes: po un lado, la asgnacón de los ecusos ealzada po los potocolos de contol de congestón cuando se pemten vaas conexones po usuao. Se dentfcan algunos poblemas del paadgma actual, y se popone un nuevo concepto de equdad centada en el usuao, desaollando a su vez algotmos descentalzados que se pueden aplca en los extemos de la ed, y que conducen al sstema a un óptmo global adecuado. El segundo poblema mpotante analzado aquí es la asgnacón de los ecusos ealzada po los algotmos de contol de congestón cuando tabajan sobe una capa físca que pemte

11 XI múltples velocdades de tansmsón, como es el caso en las edes nalámbcas. Se demuesta que los algotmos usuales conducen a nefcencas mpotantes desde el punto de vsta de las conexones, y se poponen mecansmos paa supea estas nefcencas y mejoa la asgnacón de los ecusos pestados po dchas edes. A lo lago de este tabajo, se aplcan vaas heamentas matemátcas, tales como la optmzacón convexa, la teoía de contol y los pocesos estocástcos. Po medo de estas heamentas, se constuye un modelo del sstema, y se desaollan leyes de contol y algotmos paa loga el objetvo de desempeño deseado. Como paso fnal, estos algotmos fueon pobados a tavés de smulacones a nvel de paquetes de las edes nvolucadas, popoconando la valdacón de la teoía y la evdenca de que pueden aplcase en la páctca.

12 Agadecmentos Como todo tabajo de lago alento, y vaya s esta Tess lo fue, cabe agadece a muchas pesonas que contbuyeon y puseon de su pate paa que este esfuezo llegue a buen pueto a lo lago de estos años. En pme luga, queo agadece a los membos del tbunal de Tess, po habe dedcado tempo y esfuezo a ecoe estas págnas y contbu con sus comentaos a mejoa la vesón fnal de este documento. En patcula, queo agadece a Enesto Modeck, quen hace ya bastante tempo me do el empujón ncal paa que oente m tabajo haca la nvestgacón, combnando las heamentas matemátcas con la ngeneía. Queo agadece tambén a ms compañeos de tabajo a lo lago de todos estos años. Empezando po el núcleo de nvestgacón que fomamos y bautzamos ARTES hace ya vaos años, y donde ntentamos planta la semlla de la nvestgacón matemátca aplcada a edes. Muy especalmente, queo agadece a Pablo Belzaena po su apoyo constante en su doble ol de oentado y tambén de jefe, como decto del IIE, po habe facltado tantas cosas. Oto agadecmento muy especal paa Laua Aspot, con quen las dscusones (académcas o no tanto), y las tades de tabajo, mate de po medo, sempe fueon enquecedoas. Más en geneal, a ms compañeos de la baa del IIE y del IMERL, que sempe estuveon pesentes a lo lago del camno. En patcula a Daío Buschazzo, poque con su constante Samu, cuándo te vas a doctoa? empujó sempe paa llega a este punto. Oto agadecmento especal a ms compañeos del Gupo MATE: Enque Mallada, Matín López, Macos Cadozo, Dego Feje, Juan Pablo Sabene, José Gacía, Fabán Kozynsk y Matín Zubeldía. Todos ellos, en mayo o meno medda, contbuyeon con este tabajo. Con ellos he compatdo el tabajo dao duante estos años y sus popuestas, deas y cítcas ceaon sempe un entono especal donde lleva adelante la nvestgacón. Tambén a Patca Cobo, que confó en mí paa segu desaollando tabajo de nvestgacón en el Uuguay. Un agadecmento especal paa Pablo Monzón, m Decto Académco, po sempe esta atento y dsponble, facltando muchas cuestones fomales, y sugendo y apotando sempe. Oto agadecmento muy especal paa m Decto de Tess: Fenando Pagann. No alcanzan las palabas paa descb el ol que Fenando tuvo en este tabajo. No fue un o-

13 XIII entado más, sno que se aemangó y tabajó constantemente paa que este poyecto llegue a buen pueto. Gacas, Fenando, po dame la opotundad, y poque con tu tabajo y soldadad pemanente apendí que hace nvestgacón de caldad es posble desde cualque luga. Solo hace falta (como s fuea poco) peseveanca, guosdad, mucho estudo y espítu cítco. Todo esto lo apendí compatendo las hoas de tabajo duante estos años. Apendí además, que todo esto puede hacese sn pede la sonsa, n la humldad. Un agadecmento tambén muy especal a m famla, ms pades Ael y Alca, ms hemanos Pablo, Gabel, Jave y Matías, po su apoyo desde sempe. Y a m famla amplada, Cstna, Pablo y Matías que sempe estuveon pesentes dándome fueza y consejos. Y po últmo, el agadecmento más mpotante es paa la coautoa esptual de esta tess: Natala, nunca había llegado hasta acá sn vos. Nunca seía lo msmo sn vos. Sempe ceíste en mí aún en los momentos más dfícles de este camno. Me empujaste a pesegu ms metas, y me enseñaste a dsfutalas, y sobe todo a dsfutalas juntos. Me enseñaste que el camno mpota, y que el camno es más lndo ecoelo juntos, y con una sonsa. Ojalá que las hoas (o días, o meses) sepaados po esta empesa, no empañen la constuccón pemanente que día a día hacemos de nuesta vda juntos, sno que sva paa hacela más fuete, y sobe todo, más lnda de ecoe ente los dos.

14 1 Intoducton The Intenet has evolved fom a loose fedeaton of academc netwoks to a global entty spannng mllons of end uses, equpments and nteconnectng technologes. The exponental gowth of avalable communcaton devces and applcatons have evolutonzed human communcaton, commece and computng. Ealy n ths evoluton, t was ecognzed that unestcted access to netwok esouces esulted n poo pefomance, manly because of hgh packet loss ates and the esultng low netwok utlzaton. Uses tansfeng data ove the netwok n an open loop fashon wll nevtably lead, as the netwok gows, to what s called congeston collapse. Ths phenomenon, fst obseved n the md 1980s, bought fowad the need of studyng the esouce allocaton that the netwok shall povde to ts uses. The man challenge s to allow data flows ove the netwok to egulate themselves n ode to fnd an effcent and fa opeatng pont. The soluton calls fo feedback mechansms that allow the end uses to egulate the demands based on some congeston sgnals povded by the netwok. Moeove, and due to the scale of the Intenet, t s necessay to have decentalzed contol algothms, that s, we cannot assume that a cental dspatche wll povde the esouce allocaton on the fly, based on cuent demand. Such an appoach would eque an unaffodable amount of computng and sgnallng esouces. The fst step n ths decton came wth the wok of[jacobson, 1988], n whch the autho poposed to add a congeston contol mechansm to the Tansmsson Contol Potocol (TCP)

15 CHAPTER 1. INTRODUCTION 2 whch was and s used on the Intenet to tansfe data fles. The algothm was based on the sldng wndow mechansm that was aleady n use fo flow contol, but wth the possblty to adapt the wndow sze based on netwok congeston. The man assumpton s that packet losses n a gven communcaton wee caused by some esouces of the netwok becomng congested, and thus upon detecton of a lost segment n the data tansfe, the sende must estan fom sendng too much data, by educng ts tansmsson wndow. The poposed algothm, whch we wll descbe late, as well as ts successve updates have been undoubtedly successful n steeng the taffc of the Intenet though ts global expanson. The mpotance of the congeston contol poblem has attacted a lage eseach communty. The man beakthough was the fomulaton n the semnal wok of[kelly et al., 1998]. Kelly poses the congeston contol poblem as a sutable convex optmzaton poblem, whee connectons ae epesented as economc actos n a bandwdth maket. The netwok congeston sgnals ae ntepeted as pces, and the maket equlbum becomes the esouce allocaton obtaned though the decentalzed mechansms. The esultng fomulaton has become known n the eseach communty as the Netwok Utlty Maxmzaton (NUM) poblem. Thoughout ths wok, we shall apply the NUM famewok to analyze seveal mpotant poblems of netwok esouce allocaton. One of the man ssues wth economc models fo netwok congeston contol s the fact that the utlty a connecton deves n the afoementoned bandwdth maket s assocated wth the potocol behavo. It s thus only a metaphocal descpton, not elated wth eal use wllngness fo bandwdth. In geneal, typcal uses ae not awae of ths, because the congeston contol algothms ae deep nsde the opeatng systems of the hosts. Howeve, seveal applcatons stated to cheat the bandwdth allocaton by openng multple connectons. If a use o entty s allowed to open seveal connectons, though the same o multple paths, the esouce allocaton poblem becomes moe complcated. We shall show n patcula that the end use has ncentves to open multple connectons, thus leadng to potental neffcences. Ths poblem motvates the fst pat of ths wok, whch focuses on fndng a sutable NUM poblem whee utltes eflect the eal use pefeences, and whee we develop decentalzed algothms that allow the system to opeate n a fa and effcent opeatng pont. We developed two knd of algothms, coopeatve and non coopeatve. The fst ones ae based on end systems havng fne-ganed contol of the numbe of connectons, and coopeatng among them to each the common goal of maxmzng netwok total utlty. The second class of algothms focuses on the case whee uses may not be coopeatve, and the netwok must potect tself fom geedy uses by pefomng admsson contol. In ou analyss, both sngle-path and mult-path stuatons ae consdeed, thus modellng dffeent types

16 CHAPTER 1. INTRODUCTION 3 of use connectons acoss the netwok. Anothe ecent development of netwoks s the ongong gowth of weless local aea netwoks (WLAN). In the last decade, the technology to povde weless Intenet connectvty has become ubqutous. In patcula, WLANs ae often the fnal hop n netwok communcaton, povdng connectvty to home and offce uses, college campuses, and even eceatonal aeas. Ths gowth of weless netwoks has been accompaned by eseach on the esouce allocaton povded by such systems. In patcula, the NUM famewok has been appled often to model dffeent types of weless Medum Access Contol (MAC) layes. Moeove, extensve eseach has been devoted to the analyss of collsons and ts nteacton wth the above tanspot potocols and congeston contol. In the second pat of the thess, we focus on the esouce allocaton of weless netwoks, wth a specal emphass on the effect of havng multple tansmsson ates, whch s a common featue n the technologes nvolved. The fact that multple tansmsson ates coexst n the same cell has been often ovelooked, wth the man focus beng placed on collsons. In ou wok, we analyze the mpact of havng multple ates, showng that t can lead to mpotant neffcences, not explaned by collsons alone. We extend the NUM theoy to nclude the multple ates found n WLANs and analyze ts nteacton wth the uppe laye potocols, n patcula congeston contol, and how the esouce allocaton affects the connecton level pefomance. We popose mechansms to enhance the cuent esouce allocaton n ode to mpove the effcency of these netwoks, though a smple and decentalzed packet level algothm. We also analyze the pefomance of the algothms n sngle cell and mxed wedweless scenaos, whch ae common deployments n pactce. 1.1 Outlne of ths wok and man contbutons The document s oganzed as follows. In Chapte 2 we ntoduce the eade to congeston contol n the Intenet, and the NUM famewok used to model t. Ths chapte pesents all the man esults n the lteatue elated to congeston contol and optmzaton. In patcula, the dffeent appoaches to the esouce allocaton poblem ae pesented, and we also descbe the algothms used n pactce to pefom congeston contol, and how they elate to optmzaton algothms. The thess s then splt nto two man pats. Pat I ntoduces the use-centc noton of faness we popose to enhance netwok esouce allocaton. In Chapte 3 we descbe the models used fo congeston contol when multple connectons ae nvolved, and we pove that the end-use has ncentves to ncease ts numbe of connectons to get moe bandwdth. These esults motvate the analyss of Chapte 4, whee ou noton of use-centc faness s defned, n ode to ovecome ths lmtaton of cuent potocols. We also descbe how to acheve

17 CHAPTER 1. INTRODUCTION 4 ths noton of faness by sutable modfcatons of standad congeston contol algothms, povded that the end use s capable of contollng the aggegate ate of ts connectons. We also analyze how to acheve the poposed esouce allocaton by usng the numbe of ongong connectons as a contol vaable, whch s clealy moe pactcal. Ths analyss assumes coopeaton between uses. In Chapte 5 we lft ths hypothess by movng the functon of connecton-level contol to the netwok. We develop decentalzed algothms fo connecton admsson contol that dve the system to the desed allocaton. The analyss of ths chapte s manly based on flud lmts fo the undelyng stochastc pocesses. In Chapte 6 we nclude the possblty of outng at the connecton-level tmescale, and analyze the achevable stablty egon fo these knd of algothms, genealzng pevous esults. Thoughout these chaptes, flud model and packet level smulatons ae pesented to llustate the behavo of the algothms n pactce. Conclusons of ths pat ae gven n Chapte 7. In Pat II, we analyze the esouce allocaton poblem n weless local aea netwoks, focusng specfcally on the multate capabltes of these technologes. In Chapte 8, we develop a model fo standad congeston contol on top of a multate weless netwok, showng that the esultng allocaton can be descbed though a NUM poblem. Howeve, ths allocaton s hghly based aganst the uses wth bette ates, thus leadng to mpotant neffcences. Ths motvates the analyss of Chapte 9, whee a new NUM poblem s poposed, moe suted to the patcula natue of these netwoks. Also, packet level algothms ae developed n ode to dve the system to ths moe effcent allocaton. These algothms ae decentalzed and can be mplemented wthn cuent technologes. We also analyze the possblty of extendng the noton of faness of ths NUM poblem to the case of multple hop weless and wed netwoks, n patcula povdng an mpotant genealzaton of the NUM famewok to nclude both cases smultaneously. In Chapte 10 we tun ou attenton to the connecton level tmescale, and develop models that enable us to detemne the stablty egon and pefomance of these systems when both the congeston contol and the lowe medum access contol laye ae taken smultaneously nto account. Stablty esults and pefomance metcs ae detemned fo both the cuent allocaton and the poposed enhancements. In Chapte 11, we apply the esults to the mpotant case of IEEE netwoks, commonly known as WF. These netwoks, whch ae now pesent n most deployments aound the wold, povde example of the neffcences descbed. We show how the developed algothms can be appled n such a settng, and povde smulaton examples of these algothms n pactce. Conclusons of ths pat ae gven n Chapte 12. Fnally, n Chapte 13 we pesent the man conclusons of ths wok and descbe the futue lnes of eseach. Appendx A evews some of the mathematcal pelmnaes used n the developments of ths thess.

18 CHAPTER 1. INTRODUCTION Assocated publcatons Below, we lst the publcatons assocated wth ths thess: A. Feagut and F. Pagann, Achevng netwok stablty and use faness though admsson contol of TCP connectons, n Poceedngs of the 42nd Annual Confeence on Infomaton Scences and Systems (CISS 2008), Pnceton, NJ, USA, Mach A. Feagut and F. Pagann, Utlty-based admsson contol: a flud lmt analyss, n Stochastc Netwoks Confeence, Pas, Fance, June A. Feagut and F. Pagann, A connecton level model fo IEEE cells, n Poceedngs of the 5th. IFIP/ACM Latn Amecan Netwokng Confeence (LANC 09), Pelotas, Bazl, Septembe A. Feagut and F. Pagann, Use-centc netwok faness though connecton-level contol, n Poceedngs of the 29th IEEE Infocom Confeence, San Dego, USA, Mach A. Feagut, J. Gacía and F. Pagann, Netwok utlty maxmzaton fo ovecomng neffcency n multate weless netwoks, n Poceedngs of the 8th Intl. Symposum on Modelng and Optmzaton n Moble, Ad Hoc, and Weless Netwoks (WOpt 10), Avgnon, Fance, May-June A. Feagut and F. Pagann, Connecton-level dynamcs n netwoks: stablty and contol, n Poceedngs of the 49th IEEE Confeence on Decson and Contol, Atlanta, USA, Decembe A. Feagut and F. Pagann, Resouce allocaton ove multate weless netwoks: a Netwok Utlty Maxmzaton pespectve, n Compute Netwoks, 55 pp , A. Feagut and F. Pagann, Netwok esouce allocaton fo uses wth multple connectons: faness and stablty. Submtted to IEEE/ACM Tansactons on Netwokng.

19 2 Resouce allocaton and congeston contol n netwoks In ths chapte, we wll pesent the man deas behnd the Netwok Utlty Maxmzaton (NUM) famewok, whch seve as a bass fo the development of the contbutons of the thess. The NUM theoy has evolved snce the wok of[kelly et al., 1998] and s nowadays used to model dffeent poblems n netwokng. Below we ntoduce the eade to the man esults and we establsh some mpotant defntons and notatons used thoughout ths wok. A man efeence fo NUM theoy s[skant, 2004]. 2.1 Resouce allocaton as an optmzaton poblem We begn by a smple example that llustates the elatonshp between esouce allocaton and optmzaton, and povdes ntuton about some of the concepts nvolved late. The basc esults on convex optmzaton theoy that we shall use ae summazed n Appendx A.1. Consde a set of enttes, whch we may call uses and we ndex by. These uses want to shae some scace esouce. Assume that the total avalable quantty of the esouce s lmted by some amount c> 0. If a gven use s allocated a gven amount x of the total esouce, t deves a utlty U (x ). Fom now on, we make the followng assumpton on utlty functons:

20 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 7 Assumpton 2.1. The utlty functon U ( ) s a contnuously dffeentable, non deceasng, stctly concave functon U :R + R. The utlty functon expesses the beneft a use obtans fom the esouces. The non deceasng hypothess states smply that the use s happe wheneve moe esouces ae allocated to t. The concave assumpton s standad, and expesses the fact the amount of magnal utlty povded by exta esouces s deceasng. A esouce allocaton s a vecto x=(x ), wth x 0, that ndcates the amount of esouces gven to each one of the uses. Snce the total amount s lmted by c, a esouce allocaton would be feasble f and only f: x c. (2.1) Clealy, thee ae seveal possble esouce allocatons. We ae nteested n fndng a esouce allocaton that gves an optmal tade-off between all the patcpatng enttes. A easonable choce s to maxmze the socal welfae, whch s gven by the followng convex optmzaton poblem: Poblem 2.1. subject to the constant: max x U (x ) x c. The above poblem s the optmzaton of a concave functon, due to Assumpton 2.1, subject to the lnea (and theefoe convex) nequalty constant (2.1). Applyng the technques of Appendx A.1, we wte the Lagangan of the poblem, whch s: (x,p)= U (x ) p x c, whee p denotes the Lagange multple assocated wth the constant. The Kaush-Kuhn-Tucke optmalty condtons fo ths poblem state that any soluton (x,p ) must satsfy: x U x=x = (x ) p = 0, p x c =0, whee p 0, and x satsfes the constant (2.1). We can dscuss two cases. In the fst case, optmalty s acheved wth stct nequalty n the constant,.e. x < c. In ths case, the second condton above mples that p = 0.

21 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 8 U (x ) U (x ) f (p) max x Demand cuve px x x p FIGURE 2.1: USER UTILITY AND DEMAND CURVE. Ths happens when thee s a pont n the nteo of the egon defned by (2.1) whee all uses acheve maxmal ndvdual satsfacton. In ths case, we say that the constant s not actve and the assocated Lagange multple wll be 0. A moe nteestng case s when all uses cannot be fully satsfed, and some tade-off must occu. Ths s the case when all the utltes ae nceasng and stctly concave, as n Assumpton 2.1. In ths stuaton we wll have x = c and p > 0, wth the esultng allocaton satsfyng: U (x )=p. (2.2) The above condton can be ntepeted n economc tems, as we shall see below. Consde fo the moment that some extenal entty s avalable, that has the powe to fx a pce p fo the usage of the scace esouce. In such a stuaton, a use allocated an amount x of the esouce wll peceve a utlty U (x ), but wll have to pay px fo ts usage. A atonal use would choose to ndvdually make the followng optmzaton: max x U (x ) px. Note that, snce U s dffeentable, the maxmum above s attaned by choosng x such that U (x )=p, o equvalently: x = f (p), whee f (p)=(u ) 1 (p) s called the use demand cuve. Fo a gven pce p, f (p) s the optmal amount of esouce a atonal use s wllng to puchase. Ths ntepetaton s depcted n Fgue 2.1. The demand cuve f (p) s a deceasng functon of the pce, as expected. The economc ntepetaton of equaton (2.2) s now clea: assume that, gven a fxed pce fo the shaed esouce, evey use puchases ts own optmal amount at ths pce. Then, the KKT optmalty condtons ensue that thee exsts an optmal pce p > 0, whch s exactly

22 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 9 the Lagange multple fo the system, such that the oveall allocaton wll be optmal fo the ognal esouce allocaton Poblem 2.1. By choosng ths optmal pce, we can decentalze the soluton of the poblem, snce each use wll puchase the amount of esouces dctated by ts demand cuve, but the oveall allocaton wll satsfy the KKT optmalty condtons of the esouce allocaton Poblem 2.1, and theefoe we acheve maxmal total welfae. The key to solvng the above poblem n a decentalzed way s theefoe fndng the ght pce fo the esouces. If the utlty functons of each use ae known, we can eadly fnd the optmum of Poblem 2.1, but n that case thee would not be any eal advantage fom the decentalzaton popety. A smple way to acheve ths s to adapt the pce by the followng dynamcs: ṗ= x c, wth adequate povsons to pevent the pce fom becomng negatve. Note that the above equaton smply states that the pce must be nceased wheneve the offeed esouce cannot satsfy the cuent total demand, and loweed when thee s an excess of esouces. Ths algothm s called the dual algothm and we shall exploe t late n moe geneal settngs. The decentalzaton popety apples also to the extenal entty choosng the pce, whch does not have to know the ndvdual use utltes, nethe the amount of esouce puchased by each one of them. It just needs to know the total demand fo the esouce n ode to adapt the pce accodngly. The smple example pesented hee llustates the man concepts and ntuton behnd the economc ntepetaton of esouce allocaton. We now move on to ate allocaton n telecommuncaton netwoks, whee the man deas eman the same, but the constants nvolvng the offeed esouces become moe elaboate. 2.2 Rate allocaton n communcaton netwoks Consde now a netwok composed of shaed esouces whch ae the netwok lnks. We ndex these lnks by l and assocate wth them a capacty c l, namely the amount of data they can cay n bts pe second (bps ). Ove ths netwok, we want to establsh connectons that can use seveal lnks along a path o oute denoted by. Fo the moment, we consde that each connecton uses a sngle set of esouces o oute, so we can also use to dentfy the connecton. We shall use the notaton l f lnk l belongs to oute. The esouce allocaton poblem conssts n detemnng a set of ates x at whch each connecton can tansmt data, takng nto account the capacty constants of the netwok, that s, the total amount of taffc caed by each lnk must not exceed the lnk capacty c l. Howeve, the allocaton should be effcent,.e. t should cay as much taffc as possble acoss

23 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 10 Route 3 c 1 c 2 Route 1 Route 2 FIGURE 2.2: PARKING LOT NETWORK OF EXAMPLE 2.1. the netwok, wthout volatng the capacty constants. Moeove, we want ths allocaton to be fa,.e. t does not stave some uses n beneft of some othe ones. Fo a gven netwok, thee ae multple possble esouce allocatons. It tuns out that the two objectves stated befoe can be conflctng. To llustate ths we consde the followng example, whch wll be used many tmes along ths wok. Example 2.1 (Lnea pakng-lot netwok). Consde a netwok wth two lnks of capactes c 1 and c 2 and thee outes. Connectons 1 and 2 tavel though a sngle lnk, whle connecton 3 uses both esouces. Ths netwok s depcted n Fgue 2.2. A sutable esouce allocaton x=(x 1,x 2,x 3 ) R 3 + must satsfy the netwok capacty constants, whch ae: x 1 + x 3 c 1 x 2 + x 3 c 2 Among the multple possble allocatons, we want to leave out those that unde-utlze esouces. We have the followng defnton: Defnton 2.1 (Paeto effcent allocaton). Fo a gven netwok, a feasble esouce allocaton x R + n s called Paeto effcent f and only f fo any othe feasble esouce allocaton y x we have: If y > x fo some s : y s < x s. (2.3) The atonale behnd the above defnton s that, we cannot ncease the ate allocated to some oute wthout deceasng the ate allocated to anothe one. An altenatve chaactezaton of Paeto effcency n the case of netwoks s gven by the followng condton:, l : x s = c l, (2.4) s :l s that s, each oute taveses a satuated lnk. We now pesent two possble allocatons fo the netwok of Example 2.1. Example 2.2 (Max-thoughput allocaton). In the context of Example 2.1, consde the allocaton x 1 = c 1, x 2 = c 2, x 3 = 0. Ths allocaton s the most effcent n tems of total netwok thoughput, howeve t s extemely unfa, stavng the long oute completely.

24 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 11 Example 2.3 (Max-mn fa allocaton). Consde the same netwok and assume that c 1 c 2. A possble allocaton s: x 1 = x 3 = c 1 2, x 2 = c 2 c 1 2. Ths allocaton s fa n a sense we wll defne clealy below. Note that the most congested esouce s equally splt between ts two outes, whle oute 2 s allocated the emanng capacty n the second lnk. Note also that the total thoughput obtaned by ths allocaton s stctly less that the one n Example 2.2. The above example llustates the concept of max-mn faness, whch s a noton boowed fom the economc theoy, and ntoduced by[betsekas and Gallage, 1991] n the context of netwoks. Below s the fomal defnton. Defnton 2.2 (Max-mn faness). A esouce allocaton x=(x ) s max-mn fa f, fo any othe allocaton y that satsfy the capacty constants of the netwok, the followng s tue: f y > x fo some oute, then thee exsts anothe oute s such that x s x and y s < x s. In othe wods, when the allocaton s max-mn fa, any othe allocaton that nceases the ate gven to some oute ( ) must also stctly decease the ate allocated to some othe oute s that was aleady n a pooe condton to begn wth. An altenatve chaactezaton of max-mn faness s the followng poposton, whose poof can be found n[betsekas and Gallage, 1991, Secton 6.5] Poposton 2.1. An allocaton x s max-mn fa f and only f fo evey thee exsts a lnk l such that: x s = c l and x x s s : l s, s :l s.e., each oute taveses a satuated lnk, and n that lnk t s among the ones that use most of the esouces. It s easy to see that n Example 2.3, the above condton s vefed. In ode to genealze the above allocatons,[kelly, 1997] ntoduced the followng famewok, whch s called Netwok Utlty Maxmzaton (NUM), and we wll use t as a base fo the analyss n the est of ou wok. Consde that each connecton n the netwok deves a utlty o beneft fom the ate allocated to t, whch we chaacteze by a functon U (x ). In the est of ou wok, unless othewse stated, we assume that utltes satsfy Assumpton 2.1. In ode to summaze the netwok capacty constants fo the geneal case, we make use of the followng:

25 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 12 Defnton 2.3 (Routng matx). We call the outng matx the matx R=(R l ) whose entes satsfy: 1 f l R l = 0 othewse. Wth ths notaton, the total ate y l that goes though lnk l vefes: y l = x = R l x, :l whch can be wtten n matx notaton as: whee y=(y l ) s the vecto of lnk ates. y= Rx, We ae now n poston to state the Netwok Utlty Maxmzaton poblem: Poblem 2.2 (Netwok Poblem). Gven a netwok composed of lnks of capacty c=(c l ), a outng matx R and whee each souce has a utlty functon U ( ) allocate the esouces such that the followng optmum s attaned: subject to: max x 0 U (x ) Rx c, whee the last nequalty s ntepeted componentwse. The esultng allocaton s unque due to Assumpton 2.1 and the fact that the capacty constants ae lnea. The soluton wll be such that the total utlty o socal welfae of the connectons s maxmzed wthout volatng the capacty constants. Note that unde the above assumptons, the esultng allocaton wll be automatcally Paeto-effcent, snce f t s not the case, we can ncease total utlty by nceasng the ate of the flow that s not satuated n condton (2.4). By choosng dffeent utlty functons, we can model dffeent esouce allocatons. An nteestng paametc famly of utlty functons, poposed by[mo and Waland, 2000], s the followng: Defnton 2.4 (α famly of utlty functons). Letα 0 be a paamete, and w > 0 a weght, the followng functon s calledα fa utlty: U (x )= w x 1 α 1 α α 1, w log(x ) α = 1.

26 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 13 Note that the above functons satsfy U (x )=w x α fo allα 0. When all connectons shae the sameα =α, we call the allocaton esultng fom Poblem 2.2 the (weghted)α fa allocaton. Note also that, foα=0, the esultng allocaton f the one that maxmzes netwok total thoughput, as n Example 2.2. Howeve, ths choce of utlty functon s not stctly concave, as equed by Assumpton 2.1. Foα we have the followng: Poposton 2.2 ([Mo and Waland, 2000]). If w = w> 0, theα fa esouce allocaton tends to the max-mn fa allocaton asα. If the weghts ae dffeent fo each oute, the lmt allocaton s called weghted max-mn faness. The ole of the paamete α s to enable us to consde dffeent ntemedate stuatons between the maxmum effcency (α 0) and maxmum faness (α ). An mpotant case sα=1 whch s called (weghted) popotonal faness by[kelly et al., 1998]. To futhe analyze the popetes of the esouce allocaton gven n Poblem 2.2, we wll make use of Lagangan dualty (see Appendx A.1). Let p=(p l ) denote the Lagange multples fo the lnk capacty constants, then the Lagangan of Poblem 2.2 s: (x,p)= U (x ) p l l R l x c l. The Kaush-Kuhn-Tucke (KKT) condtons fo optmalty n ths poblem ae: = U x (x ) R l p l = 0 l, p l R l x c l =0 l, whee as usual we assume that x s feasble and p l 0. Wth the economc ntepetaton descbed n Secton 2.1 n mnd, we shall call p l the lnk pce. It s also convenent to ewte the pecedng equatons n tems of lnk ates and oute pces. Recall that y l = R l x s the lnk ate. Also we defne: q = R l p l l as the oute pce fo oute. Due to the defnton of the outng matx, t s smply the sum of the Lagange multples of the lnk the oute taveses. In matx notaton we have: q= R T p, whee T denotes matx tanspose.

27 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 14 Wth these notatons, the KKT condtons can be ewtten as: U (x ) q = 0, (2.5a) p l (y l c l )=0 l. (2.5b) Equaton (2.5a) can be ntepeted as a demand cuve. The bandwdth allocated to oute must be such that the magnal utlty of the oute matches ths oute pce. We shall also wte ths equaton as x = f (q ) beng f ( )= U 1 ( ), wth f beng a deceasng functon due to Assumpton 2.1. Equaton (2.5b) s the complementay slackness condton that tells that ethe esouce l s satuated, o ts pce must be 0. Snce the objectve functon s concave and the constants ae convex, the condtons (2.5) ae necessay and suffcent fo optmalty (c.f. Appendx A.1), and thus chaacteze the esouce allocaton. To llustate the dffeent esouce allocatons defned up to now, we show a numecal example based on the netwok of Example 2.1. Example 2.4 (α faness n a pakng lot netwok). Consde the settng of Example 2.1, and take c 1 = 10 and c 2 = 5. Suppose the utltes come fom theα famly wth the sameα,.e. U (x )=x α. The KKT condtons fo ths poblem ae: x α 1 = q 1 = p 1, x α 2 = q 2 = p 2, x α 3 = q 3 = p 1 + p 2, p 1 (x 1 + x 3 c 1 )=0, p 2 (x 2 + x 3 c 2 )=0. Snce x α > 0 we must have p 1,p 2 > 0 and thus the lnks wll be satuated. We can educe the above equatons to: p 1/α 1 +(p 1 + p 2 ) 1/α = c 1, p 1/α 2 +(p 1 + p 2 ) 1/α = c 2. Solvng the above equatons we fnd the lnk pces fo dffeent values of α. Then, settng x = q 1/α we can fnd the esouce allocaton. In Table 2.1 we llustate the esults fo dffeent values ofα. As we mentoned befoe, the casesα 0andα coespond to the maxthoughput and max-mn allocatons espectvely. So fa, we have seen how dffeent esouce allocatons schemes can be acheved wth dffeent choces of utlty functons. If a global entty has access to the utlty functons of all the uses and knows the capactes of all the lnks, the esouce allocaton could be calculated by solvng the optmzaton poblem pesented above. Howeve, ths s clealy uneasonable snce ths eques a centalzed knowledge of all the paametes of the netwok. A moe easonable set of assumptons s the followng: Each connecton knows ts own utlty functon.

28 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 15 α x 1 x 2 x 3 p 1 p undef. undef. TABLE 2.1:α FAIR ALLOCATIONS FOR THE NETWORK OF EXAMPLE 2.1 FOR DIFFERENT VALUES OFα. Each netwok lnk knows the total nput ate. Thee s a potocol that allows the netwok to convey some nfomaton about the esouce congeston on a gven oute to the connectons along ths path. In the next Secton we wll pesent seveal algothms that solve the esouce allocaton poblem pecsely, and vefy the assumptons above. The key dea n all of these algothms s that each lnk computes the Lagange multples nvolved n ts capacty constant, an the souce eacts by adaptng ts tansmsson ate to the sum of these lnk pces. 2.3 Congeston contol as a decentalzed optmzaton algothm The dscusson of the pecedng Secton emphaszes the need of decentalzaton. As we shall see, the patcula stuctue of the netwok Poblem 2.2 enables us to use some smple algothms to solve t. Below we dscuss thee dffeent appoaches fom a theoetcal pespectve and then we poceed to elate them to cuent congeston contol algothms used n pactce The pmal algothm Consde that each lnk n the netwok s able to measue ts nput taffc y l and can geneate a congeston pce accodng to some statc map y l f l (y l ). We assume that f l ( ) s a nondeceasng functon such that the followng condton holds: y 0 f l (u)d u as y. The functons f l ( ) play the ole of a penalty functon. Intutvely, the value of f l ( ) must gow quckly when the lnk capacty constant s volated,.e. as soon as y l > c l. Consde now the followng unconstaned optmzaton poblem n the postve othant:

29 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 16 Poblem 2.3. whee as befoe y= Rx. max x 0 U (x ) l yl 0 f l (u)d u, Unde the assumptons fo U and f l, the objectve functon s stctly concave and thus Poblem 2.3 s a convex optmzaton poblem. Unde some mld exta assumptons on the penalty functons, the optmum of Poblem 2.3 les n the nteo of the postve othant. The fst ode optmalty condtons of Poblem 2.3 ae: U (x ) R l f l (y l )=0 l Identfyng p l = f l (y l ) we have that, as n the pevous Secton, the optmum s attaned when the magnal utlty of all uses matches the sum of the lnk pces ove those lnks tavesed by the oute. Note that these lnk pces ae detemned by the penalty functon, and do not necessaly concde wth the Lagange multples defned befoe. It s howeve convenent to oveload the notaton n ths way, snce they act as a pce vaable. As befoe, wth centalzed nfomaton of all the utltes and lnk pce functons of the netwok, a global entty could be capable of solvng the optmal allocaton.. Howeve, we would lke to defne an adaptaton algothm that enables the souces to fnd the optmum wthout the need of centalzed nfomaton. Consde now the followng dynamcs fo the souce ates: whee as befoe q = l R l p l ẋ = k U (x ) q, and k > 0 s a constant (step sze). Ths s a gadent seach algothm, whee the souces ty to follow the decton of the gadent (wth espect to x ) of the objectve functon. The complete dynamcs ae: ẋ = k U (x ) q, (2.6a) y= Rx, (2.6b) p l = f l (y l ) l, (2.6c) q= R T p. (2.6d) Equatons (2.6) defne what s called the pmal algothm. Its name comes fom the fact that adaptaton s pefomed on the pmal vaables of Poblem 2.2. Note that the equlbum of (2.6) satsfes the optmalty condtons of Poblem 2.3. Moeove, we have the followng: Poposton 2.3 ([Kelly et al., 1998]). The equlbum of (2.6) s the optmum of Poblem 2.3. Ths equlbum s globally asymptotcally stable.

30 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 17 Note that ths algothm s decentalzed: each souce must adapt ts ate followng ts own utlty functon and wth the knowledge of the pces along ts oute. Lnks geneate pces accodng to some statc map of ts own utlzaton, espectve of whch outes pass though. Theefoe, the algothm satsfes the hypotheses stated at the end of the pevous Secton. The man dawback s that, snce pces ae statcally geneated, the allocaton attaned s the soluton of Poblem 2.3, whch s only an appoxmaton of Poblem 2.2. In Secton 2.4 we shall show that the typcal TCP congeston contol algothm can be modelled as an mplementaton of ths pmal algothm. In the next Secton we shall show anothe algothm that allows us to solve exactly Poblem The dual algothm In ode to solve Poblem 2.2 exactly we must adapt not only the souce ates but also the lnk pces. To deve a sutable algothm, we tun to esults on convex optmzaton theoy. Consde agan the Lagangan of Poblem 2.2: (x,p)= U (x ) p l l R l x c l. (2.7) The dual functon of Poblem 2.2 conssts on maxmzng (x,p) ove x fo fxed p. It s convenent to ewte (2.7) as: (x,p)= U (x ) q x + p l c l. (2.8) It s now clea that, to maxmze wth espect to x, the outes must choose x n ode to maxmze ts suplus: x : max x U (x ) q x. Unde Assumpton 2.1, ths leads to the choce x = U 1 (q ). The coespondng suplus s then: and the dual functon s: S (q )= U (U 1 (q )) q U 1 (q ), l (p)= S (q )+ p l c l. (2.9) The optmum of Poblem 2.2 can be calculated by mnmzng (p) ove p 0 and choosng x accodngly. Snce the poblem s convex, thee wll be no dualty gap (c.f. Appendx A.1). To fnd a sutable algothm, we can make a gadent descent n the dual functon. Consde the devatve: p l = l S(q ) p l + c l. (2.10)

31 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 18 By usng the nvese functon theoem we can compute: S (q ) q = x, and thus: Substtutng n (2.10) we get: S(q ) p l = x q p l = x R l. p l = c l R l x = c l y l. Theefoe, mnus the gadent of the dual functon measues the amount each capacty constant s volated. A gadent descent algothm wll follow the opposte decton of the gadent. The complete dynamcs ae as follows: x = U 1 (q ), (2.11a) y= Rx, (2.11b) ṗ l =γ (y l c l ) + p l, (2.11c) q= R T p, (2.11d) wheeγ > 0 s a constant (step sze). The functon( ) + p s called postve pojecton and s smply a way to pevent pces fom becomng negatve. It has the followng defnton: (x) + p = x f p> 0 o p= 0 and x> 0 0 othewse. (2.12) The algothm n (2.11) s appopately called the dual algothm and ts convegence was establshed n the followng esult: Poposton 2.4 ([Pagann, 2002]). Assume that the matx R has full ow ank,.e., gven q thee exsts a unque p such that q= R T p. Unde ths assumpton, the dual algothm (2.11) s globally asymptotcally stable, and ts equlbum s the unque optmum of Poblem 2.2. The dual algothm s also decentalzed n the sense dscussed at the end of the pevous Secton, and moeove solves Poblem 2.2 exactly. In Secton 2.4 we wll show that ths class of algothms model the behavo of delay-based congeston contol potocols n the Intenet The pmal-dual algothm We can combne the pevous deas nto a sngle algothm, whee the connectons adapt the ate to a cuent oute congeston pce and the lnks geneate the pces dynamcally.

32 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 19 The algothm s then called pmal-dual and t was ntoduced n the context of netwoks by [Alpcan and Basa, 2003, Wen and Acak, 2003]. The closed loop dynamcs n ths case ae: ẋ = k (U (x ) q ), (2.13a) y= Rx, (2.13b) ṗ l =γ l (y l c l ) + p l l, (2.13c) q= R T p. (2.13d) We have the analogous convegence esult: Poposton 2.5 ([Wen and Acak, 2003]). Assume that the matx R has full ow ank. Unde ths assumpton, the dynamcs of (2.13) ae globally asymptotcally stable, and ts equlbum s the unque optmum of Poblem 2.2. Remak 2.1. In the algothms pesented above, we assumed that k > 0 andγ l > 0 ae constants. The same esults ae vald f we take k (x )>0whee k ( ) s a contnuous functon. Analogously, we can takeγ l (p l )>0. All the pevous algothms can be summazed n the followng dagam, whch shows the feedback stuctue of the contol loop. FIGURE 2.3: CONTROL LOOP FOR THE DISTRIBUTED CONGESTION CONTROL ALGORITHMS. The dagonal blocks n the dagam epesent the decentalzed contol laws, followed by evey souce and evey lnk n the netwok. The dffeent souces and lnks ae nteconnected va the outng matx R, whch allows us to map connecton ates nto lnk ates, and lnk pces to oute pces. So fa, we have dscussed the methods n a puely theoetcal appoach. In the next Secton we descbe the elatonshp between the contol laws so fa pesented and the congeston contol algothms deployed n the Intenet.

33 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS Congeston contol algothms n the Intenet The tansmsson of data flows ove the Intenet s done manly usng the Tansmsson Contol Potocol (TCP)[RFC 793, 1981]. Ths potocol povdes a elable tanspot ove an unelable datagam netwok. The potocol takes the nput flow of data fom the applcaton laye and segments t nto packets, whch wll be delveed to the netwok laye below, n chage of elayng these packets to the destnaton. In ode to ecove fom possble losses n the communcaton, TCP adds a sequence numbe to each packet and the destnaton sends an acknowledgment (ACK) packet n the evese path to tell the souce the data has coectly aved. When TCP detects a mssng acknowledgment, t wll ntepet t as a lost packet and etansmt the data. When ths happens, the eceve must buffe the packets untl t flls all the gaps, so t can delve an odeed steam to the applcaton laye above at the destnaton. In ode to contol the numbe of n-flght packets (.e. packets sent but whose ACK has not yet been eceved), the sende sde mantans a tansmsson wndow W, whch s the maxmum allowed numbe of n-flght packets. The smplest case s W = 1 whch s the stop-andwat algothm. In that case, the sende wats fo the ACK of the pevous packet befoe poceedng wth the next. Ths choce of W s clealy neffcent when the Round Tp Tme (RTT) between souce and destnaton s lage, snce t can only send one packet pe RTT. A moe effcent stuaton can be acheved by enlagng the numbe of n flght packets as depcted n Fgue 2.4. In that case, the aveage ate obtaned by the souce wll be gven by: x= W RT T. Suppose now that the netwok s able to povde a maxmum amount of bandwdth c (n packets pe second). In that case, an optmal choce of W wll be equal to the bandwdth delay RTT Sende 1 2 W 1 2 W Data ACK t Receve 1 2 W 1 t FIGURE 2.4: SLIDING WINDOW PROTOCOL AND RATE CALCULATION.

34 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 21 poduct c d, wth d the ound-tp popagaton delay of the netwok. Settng W geate than ths value wll only ncease the total RT T due to buffeed packets that cannot be mmedately delveed, and the aveage ate wll stll be equal to c. Settng W lowe than ths value wll not make use of the full capacty of the netwok. The poblem s that, even f the souce can measue ts RT T, t does not know c a po, and most mpotantly, the value of ths allocated capacty may be tme vayng due to othe connectons competng fo the same esouces. In[Jacobson, 1988], the autho descbed a smple algothm to adapt the value of W. We now descbe ths algothm and show how t can be ntepeted as a pmal adaptaton n the language of Netwok Utlty Maxmzaton Loss based congeston contol: TCP Reno and ts vaants The congeston contol mechansm desgned by Jacobson, poceeds as follows. At the stat of the connecton, the wndow s set to W = 1 and the slow stat phase begns. In ths phase, wth each ACK eceved, the value of W s nceased by 1. Ths has the effect of effectvely duplcatng the tansmsson wndow each RT T, snce fo each W packets sent n a cycle, W ACKs wll be eceved (assumng no packet s lost) and n the next cycle, the wndow wll be 2W. Ths pocedue contnues untl a cetan theshold named ssthesh (fo slow stat theshold) s attaned. A typcal ntal value of ssthesh s 32 Kbytes, whch n typcal netwoks tanslate to W 22 packets. So, the ssthesh s attaned ealy n the data tansfe, povded thee ae no losses, wth the slow stat phase lastng typcally 5 RT Ts. At ths pont the congeston avodance phase begns. In ths new phase, wth each ACK eceved the wndow s nceased by 1/W. Ths has the net effect of nceasng almost lnealy the congeston wndow by 1 packet on each cycle, whch s a slowe gowth than the slow stat phase. Fo a modeately lage connecton, most of the data tansfe wll be acheved n ths phase. The ncease n W contnues untl the sende detects a loss. In the ognal poposal, losses wee detected when a cetan amount of tme (based on the measued RT T ) has passed wthout ecevng the ACK of a cetan packet, an event called a tmeout. At ths pont, the sende: Retuns the tansmsson wndow to W = 1. Sets the ssthesh to half the value of the wndow po to the loss. Restats tansmsson fom the lost packet onwads pefomng agan the slow stat phase. Soon afte ts ntoducton, t was clea that the tmeout mechansm to detect losses led to a dastc educton n the ate, due to the combnaton of the dle tmes and the etun of W

35 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS TCP Tahoe 100 TCP NewReno Congeston wndow sze (packets) Congeston wndow sze (packets) Tme (s) Tme (s) FIGURE 2.5: TYPICAL WINDOW EVOLUTION FOR TCP-TAHOE (LEFT) AND RENO (RIGHT) to 1. To solve ths ssue, the Fast Retansmt mechansm was ntoduced. The man dffeence wth the ognal poposal s the detecton of losses va duplcate ACKs. If a sngle packet s lost, the steam of packets contnue to ave to the destnaton, but thee s a gap n the sequence numbe. Ths poduces a numbe of ACKs ndcatng that a packet s mssng and equestng t. Once thee of these duplcate ACKs ae eceved, the sende assumes that a loss has occued. It etansmts the equested packet and esumes fom the slow stat phase. The algothm so fa descbed became known as TCP Tahoe, snce t was fst mplemented n the Tahoe veson of the BSD opeatng system. A typcal evoluton of W unde the TCP-Tahoe algothm s shown on the left n Fgue 2.5. The vaants of the algothm ntoduced late, known as TCP-Reno and TCP-NewReno (see[floyd and Hendeson, 1999] and efeences theen) ted to futhe mpove the aveage ate by elmnatng the slow stat phase n steady state. These congeston contol algothms poceed as follows: Retansmt mmedately the equested packet. Reduce the ssthesh to half the value of the wndow po to the loss. Set the wndow to ssthesh and esume fom the congeston avodance phase (they skp the slow stat altogethe). The esult s a moe egula behavo of the congeston wndow, as depcted on the ght n Fgue 2.5. Tmeouts ae stll obseved, and n ths case the algothm defaults to TCP-Tahoe behavo, but n the typcal scenaos, the connecton emans most of the tme n the congeston avodance phase. A fst analyss on the aveage ate obtaned by the TCP-Reno mechansm was pesented n[maths et al., 1997]. The authos establsh the followng elatonshp between the aveage

36 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 23 wndow W and the packet loss pobablty q: W= 3 2q. Takng nto account the elatonshp between W and the aveage ate we have the followng equaton, called Maths fomula: x= 1 3 RT T 2q, (2.14) whee x s gven packets pe second. Note that equaton (2.14) s n the fom of a demand cuve of equaton (2.5a), whee the pce s the packet loss pobablty and the connecton puchases less ate wheneve ths pce s hgh. A moe detaled analyss of the TCP-Reno was gven n[kelly et al., 1998]. Assume that the netwok dops packets fom connecton wth pobablty q. Moeove, let RT T be the ound tp tme of connecton, and assume that t s fxed (ths s the case when popagaton delays ae domnant). Then the ate at whch the netwok dops packets fom connecton s x q. If we assume that the wndow W nceases by 1/W when a packet s coectly eceved, and deceases by a facto 0<β< 1 when a packet s lost, a sutable flud model fo the wndow evoluton wll be: Ẇ = 1 W x (1 q ) β W x q, whch can be ewtten n tems of the ate x as: ẋ = 1 RT T 2 (1 q ) β x 2 q. (2.15) As fo the loss pobabltes, a sutable model fo netwoks wth small buffes s to consde that each lnk dops packets popotonal to ts excess nput ate, namely: yl c + l p l =, whee as usual(x) + = max{x,0}. Moeove, f lnk loss pobabltes ae small (whch s the typcal case n hgh bandwdth scenaos), and assumng that each lnk behaves ndependently, we can appoxmate to fst ode the oute loss pobablty as: q = p l. y l l :l If oute loss pobabltes ae small, we can also smplfy equaton (2.15) by makng the

37 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 24 appoxmaton(1 q ) 1 n the fst tem. The complete dynamcs follow: ẋ =β x 2 1 β RT T 2x 2 q, (2.16a) y= Rx, (2.16b) yl c + l p l = l, (2.16c) y l q= R T p. (2.16d) The pevous dynamcs ae n the fom of a pmal contolle of Secton wth penalty functon f l (y)=(1 c l /y) +. Theefoe, TCP-Reno can be seen as a pmal algothm that decentalzes the esouce allocaton poblem. The coespondng utlty functon s: U (x )= 1 β RT T 2 x whch belongs to theα famly wthα=2and w = 1/(β RT T 2 ). Note also that the equlbum of (2.16) satsfes: o equvalently: 1 β RT T 2 ˆx 2 ˆq = 0, ˆx = 1 1, RT T βˆq whch s smla to equaton (2.14) fom the statc analyss. Wth mno changes, the TCP-Reno algothm evolved nto TCP-NewReno, whch fxes some ssues n the Fast Recovey/Fast Retansmt algothm and povdes a moe stable behavo. Ths has been the mansteam potocol fo data tansfe n the Intenet, and ts success has allowed the Intenet to scale. Only vey ecently new poposals lke TCP-Cubc o TCP-Redmond ae n consdeaton, wth nethe one beng a clea eplacement fo the pevous algothms. The man beakthough of[kelly et al., 1998] was to elate the behavo of these congeston contol algothms to economc models., Delay based congeston contol Besdes the loss based potocols studed n the pevous secton, some TCP vaants poposed the use of queueng delay as a sgnal of congeston. In such a potocol, the souce measues the mnmum obseved RTT. Ths measue s s an estmaton of the ound tp latency n the communcaton,.e. when all the buffes ae empty. In ths context, ths s called the base RTT n ths context), and the souce eacts to an ncease of the RTT above ths estmated value. Vaants such as TCP-Vegas[Bakmo and Peteson, 1995] and TCP-Fast[Cheng et al., 2004] ae based on ths dea. We now show that usng queueng delay as the pce can be assmlated to pefomng the dual algothm descbed n Secton

38 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 25 To see ths, ecall the dynamcs of the pce vaable n the dual algothm (2.11): ṗ l =γ l (y l c l ) + p l. Note that the tem y l c l tacks the excess nput ate n the lnk. Integatng ṗ l we have that p l s popotonal to the amount of data accumulated n the buffe of lnk l. By choosng γ l = 1/c l we have that p l epesents the nstantaneous queueng delay n the lnk. Snce the oute pce s q = l :l p l, ths pce tacks the total queueng delay suffeed by oute packets along ts path. Theefoe, f the connecton s able to measue ths queueng delay and eact to ths pce wth some demand cuve f (q ), we have a dual algothm. If the oute nstead of nstantaneously adaptng to the cuent queueng delay, chooses to adapt ts ate n a pmal fashon, we have a pmal-dual algothm. In the case of TCP-Vegas and TCP-Fast, the demand cuve s chosen n a way compatble wth the utlty functon U (x )=w log(x ), theefoe povdng weghted popotonal faness among the ongong connectons Random Ealy Detecton As we dscussed above, typcal TCP congeston contol algothms eact to lost packets. Howeve, ths leads to an undesable behavo n the netwok buffes, whch ae wokng almost full most of the tme. Ths leads to nceased delays, whch can affect fo nstance non TCP taffc used n eal tme netwok applcatons. To solve ths ssue, Floyd et. al. poposed n[floyd and Jacobson, 1993] to use a Random Ealy Detecton (RED) algothm. The man dea s to dscad packets andomly wth nceasng pobablty as the buffe stats to fll. Ths leads to an ealy detecton of congeston by the souces, whch coopeate loweng the ates. When appopately desgned, ths also helps to educe the synchonzaton between packet losses of dffeent souces, enablng a bette use of netwok esouces. A smple model of the RED algothm s the followng: consde that the packet loss pobablty s popotonal to the buffe occupancy, namely: p l =κ l b l. Fom the dscusson n the pevous secton, we know that the buffe occupancy obeys the followng equaton: ḃ l =(y l c l ) + b l. Puttng togethe both equatons we have: ṗ l =κ l (y l c l ) + p l,

39 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 26 so the packet loss pobablty now follows the dual pce adaptaton, wthγ l =κ l. If we assume that the loss pobabltes ae low, we have that the oute loss pobablty s agan q = l :l p l. Combnng all these equatons wth the model fo TCP-Reno (2.15) we have that TCP-Reno wth RED follows a pmal-dual dynamcs. In the case of TCP-Reno, ths wll solve exactly Poblem 2.2 wth utltes of theα famly wthα= Netwok utlty maxmzaton beyond congeston contol So fa, we have descbe how the NUM famewok ntoduced n[kelly et al., 1998] povded the mathematcal tools to analyze the behavo of typcal congeston contol algothms. Ths success led to a paadgm shft n netwok potocol desgn, based on convex optmzaton methods. In patcula, the NUM famewok has been appled extensvely n the lteatue to povde solutons to seveal esouce allocaton poblems, beyond tadtonal congeston contol. Fo nstance, seveal genealzatons of the above methods wee developed to tackle the multpath esouce allocaton poblem. Thee, connectons may be splt up along seveal paths acoss the netwok. Two man fomulatons have been gven: one n whch the paths ae pedefned (c.f. [Han et al., 2006, Voce, 2007]) and the tanspot laye s n chage of managng the taffc on each oute. Anothe fomulaton decomposes the mult-path poblem n a congeston contol laye, smla to the one dscussed above, and a outng poblem that can be solved by the ntemedate nodes n a decentalzed way, by usng smultaneously all of the esouces n the netwok[pagann and Mallada, 2009]. Ths s acheved by applyng a dual decomposton of the NUM poblem, leadng to sepaate optmzaton poblems to be solved n the dffeent netwok layes, and communcatng though sutably geneated netwok pces. Anothe poblem whee the NUM famewok poved useful s n the weless settng, whee seveal lnks may ntefee wth each othe. Hee, besdes the typcal esouce allocaton to each entty n the lnks, the addtonal schedulng poblem ases,.e. whch lnks may be actve at a gven tme, n ode to communcate effcently and wthout ntefeence. By means of the same dual decomposton technques, a sutable soluton s acheved, whee the poblem decomposes n the congeston contol and schedulng poblems, though the last can be dffcult to decentalze. A good efeence hee s[ln et al., 2006]. These examples motvate the study of esouce allocaton n netwoks followng the NUM famewok. The geneal technque can be summazed as follows: state the NUM poblem, that s, to maxmze a cetan utlty noton whch nvolves all the enttes n the netwok, and let the popetes of the netwok act as constants n the gven optmzaton poblem. If these constants ae convex, apply a dual decomposton analyss to tanslate the above poblem to sutable subpoblems that must be solved by the dffeent layes of the netwok, possbly

40 CHAPTER 2. RESOURCE ALLOCATION AND CONGESTION CONTROL IN NETWORKS 27 communcatng between them though well defned pces. If ths decomposton s possble, seveal algothms analogous to the pmal, dual, and pmal-dual algothms pesented n ths chapte can be appled n ode to dve the system to the optmal allocaton. A good efeence fo ths coss-laye analyss of netwoks s[chang et al., 2007]. In the followng, we shall use ths pocedue to tackle seveal poblems asng n netwok esouce allocaton.

41 Pat I Connecton level esouce allocaton and use-centc faness 28

42 3 Connecton level models and use ncentves The models and dscusson of Chapte 2 analyze the esouce allocaton n netwoks teatng each flow o connecton as an ndvdual entty, whch s pemanently pesent n the netwok. Howeve, n eal netwoks thee ae two mpotant phenomena that must be consdeed. In the fst place, netwok uses may open multple smultaneous connectons, maybe wth dffeent destnatons and along multple outes. On the othe hand, the numbe of establshed connectons o flows may be tme vayng, subject to the use demands and the fact that they may cay dffeent amounts of wok. In ths chapte, we evew the models avalable to descbe such stuatons. In Secton 3.1 we evew the elevant NUM famewok fo the connecton level pespectve as pesented by[skant, 2004]. A fst contbuton of ths thess s pesented n Secton 3.2, whee t s shown that unde sutable assumptons a use has ncentves to open moe connectons, n ode to get moe bandwdth. Ths s an ntutve esult, t has been used befoe[bscoe, 2006] as an agument aganst the flow-level noton of faness enfoced by cuent potocols, and we povde the fst goous poof. Fnally, n Secton 3.3 we pesent the standad model fo dealng wth the tme vayng aspect of connectons, and dscuss ts mplcatons.

43 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES Intoducton The models and algothms descbed n Chapte 2 assume that each connecton shang esouces n the netwok has an assocated nceasng and concave utlty functon that values ts dese fo esouces. Ths economc ntepetaton, summazed n Poblem 2.2 and the undelyng algothms s, howeve, only a metaphocal descpton. TCP-Reno and ts vaants, fo nstance, have been desgned n a heustc manne, and only late these algothms wee ntepeted n tems of economc deas. In patcula, the utlty functon that descbes TCP-Reno s detemned by the potocol, as we saw n Chapte 2. The end-use has lttle contol ove ths, snce t s mplemented n the kenel of the opeatng system, and thus changng t may be dffcult. As a consequence, the esouce allocaton povded by the cuently pedomnatng Intenet potocols does not eally eflect the use valuaton of bandwdth. Howeve, as we shall see below, uses can cheat on the esouce allocaton n a smple way: they can open moe connectons, and ths s ndeed a mechansm that has been used n the Intenet fo a long tme. The emegence of pee-to-pee fle exchange softwae, whch opens seveal paallel connectons, has deepened ts mplcatons nowadays. In ode to model ths stuaton, we shall focus on the esouce allocaton povded by the netwok n the pesence of multple connectons. The model below was ntoduced by [de Vecana et al., 1999, Bonald and Massoulé, 2001] and s summazed n[skant, 2004]. We consde, as n Chapte 2 a netwok composed of lnks, ndexed by l, wth capacty c l, and a set of paths o outes, ndexed by. End-to-end connectons (flows) tavel though a sngle path, specfed by the outng matx R. x denotes the ate of a sngle connecton on oute, and n wll denote the numbe of such connectons. We defne: ϕ = n x, whch s the aggegate ate on oute, obtaned by the whole set of ongong connectons on ths oute. The aggegate ate n lnk l fom all the outes s: y l = R l ϕ = R l n x. (3.1) Connectons pesent n the netwok egulate the ate though some TCP congeston contol algothm. Assume each connecton on oute can be modelled by some utlty functon U TC P, and connectons along the same path have the same utlty. Then, the esouce allocaton of the Netwok Poblem 2.2 educes to:

44 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES 31 Poblem 3.1 (TCP Congeston contol). Fo fxed{n }, n > 0, max ϕ ϕ n U T C P, n subject to capacty constants: R l ϕ c l fo each l. l The above optmzaton poblem gves a noton of flow-ate faness, whee the utlty U TC P s assgned to the ate x = ϕ of each TCP flow, modellng potocol behavo. Fo futue n convenence, we have chosen to expess the poblem n tems of theϕ vaables. TCP utltes ae assumed nceasng and stctly concave, and we wll manly focus on theα fa famly of [Mo and Waland, 2000]. The Kaush-Kuhn-Tucke (KKT) condtons fo Poblem 3.1 nclude U T C P whch ae equvalent to the demand cuve: ϕ n = q, x = ϕ n = f T C P (q ), (3.2) wth f T C P =[U T C P ] 1. In patcula, foα fa utltes we have f T C P (q )=(q /w ) 1/α. Theefoe, TCP congeston contol algothms behave as decentalzed ways of achevng the optmum of Poblem 3.1, whee the utlty functon U T C P models the potocol behavo. Ths defnes a mappngφ:n ϕ; that s, gven the numbe of connectons n=(n ) n each oute, the esouce allocatonϕ=(ϕ ) s gven as the soluton of Poblem 3.1. In the next Secton we deve an mpotant popety of ths mappng. 3.2 Use ncentves Fom the use-level pespectve, fo a gven numbe of connectons n each oute, the esouce allocaton s detemned by the flow-ate faness mposed by the netwok though the soluton of Poblem 3.1. Howeve, a use vyng fo moe esouces may challenge ths by openng moe connectons along the same oute. Ou fst esult states that ths ndeed allows the use to get a lage shae of bandwdth. Theoem 3.1. Assume R has full ow ank. Then the mapϕ=φ(n) s such that: ϕ n 0,n > 0.

45 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES 32 Poof. Consde the mapφ:n ϕ defned by Poblem 3.1. Ths map s contnuous when n > 0[Kelly and Wllams, 2004]. We wll also assume that n a neghbohood of the soluton of Poblem 3.1, all lnks ae satuated (f thee ae locally non-satuated lnks, they can be easly emoved fom the analyss). In ths case, the KKT condtons of Poblem 3.1 mply: U T C P ϕ n = q,, R l ϕ = c l l. Fom the fst goup of equatons we have that: ϕ = n f T C P (q ), (3.3) and substtutng n the lnk constants we have that the optmal lnk pces must satsfy: F(n,p)= Rdag(n)f T C P (R T p) c= 0. Hee, dag(n) denotes a dagonal matx wth the entes of n, c=(c l ) s the vecto of lnk ates and f T C P (R T p) s the vecto of flow ates detemned by the demand cuve n each oute. By usng the Implct Functon Theoem we have that: Defne now the followng matces: p F 1 F n =. p n N= dag(n), F= dag(f T C P (R T p)), F = dag(f T C P (R T p)). Note that the dagonal entes of N and F ae stctly postve, and the dagonal entes of F ae negatve, snce we assume the lnks ae satuated and theefoe the pces nvolved ae postve. Afte some calculatons we ave to: and thus: F p = RN F R T, F n = R F, p n = (RN F R T ) 1 (R F). Note that the fst matx s nvetble snce R has full ow ank and the dagonal matx has defnte sgn.

46 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES 33 We now tun to calculatng ϕ. Fom equaton (3.3) we have: n and theefoe: whee I s the dentty matx. ϕ n = F+ N F R T p n, ϕ n = I N F R T (RN F R T ) 1 R F, We would lke to pove that the dagonal tems of ths matx ae non-negatve. Snce F s a dagonal matx wth postve entes, we can educe the poblem to povng that the followng matx has postve dagonal entes: M= I DR T (RDR T ) 1 R, whee D= N F s also a dagonal matx wth stctly postve entes. We can thus defne D= dag( d ) and wte: D 1 M D= I DR T (RDR T ) 1 R D, whch s of the fom I A T (AA T ) 1 A. Note that ths last matx s symmetc and vefes(i A T (AA T ) 1 A) 2 = I A T (AA T ) 1 A. Thus, t s a self-adjont pojecton matx, and theefoe s postve semdefnte. Fom ths we conclude that the dagonal entes of D 1 M D ae non-negatve, and snce the dagonal entes of M ae not alteed by ths tansfomaton, M has non negatve dagonal entes, whch concludes the poof. The above esult states that fo a geneal netwok, ϕ R / n s non-negatve, and n fact examples can be constucted whee equalty holds. Howeve, n netwok scenaos whee thee s eal competton fo the esouces, typcally the nequalty above s stct. Theoem 3.1 theefoe mples that a geedy use has ncentves to ncease the numbe of connectons along a oute n ode to gane moe bandwdth. To fomalze ths futhe, consde that use has an assocated nceasng and concave utlty functon U (ϕ ) whee ϕ s the total ate t gets fom the netwok. We can defne a game between uses n whch the stategy space s the numbe of connectons each use opens, and the payoff functon s U (ϕ (n)). As a consequence of Theoem 3.1 we have the followng coollay: Coollay 3.1. The domnant stategy fo the connecton-level game s to ncease the numbe of connectons. Ths fomalzes aguments of[bscoe, 2006] egadng the lmtatons of flow-ate faness. We conclude that, f a subset of uses behave n ths way, the numbe of ongong connectons n each oute wll gow wthout bounds, whch s an undesable stuaton snce the

47 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES 34 oveheads nvolved would lead to an mpotant waste of esouces. Ths knd of stuatons ae known as the tagedy of the commons[hadn, 1968], n whch each ndvdual actng selfshly has ncentves to enlage ts use of the esouces, but when the goup behaves n ths way the common good s depleted, o at least neffcently used. We shall addess ths ssue futhe n Chapte 4 whee we develop algothms and admsson contol mechansms to contol ths selfsh behavo and dve the system to a sutable socal optmum. 3.3 Stochastc demands Anothe way of takng uses nto account s though a stochastc model fo demand. Hee, uses ae not stategc, nstead they open connectons on oute at tmes govened by a Posson pocess of ntenstyλ. Each connecton bngs an exponentally dstbuted wokload wth mean 1/µ. These knd of models wee ntoduced by[de Vecana et al., 1999] and [Bonald and Massoulé, 2001]. Fo each netwok state n=(n ), the ate at whch each connecton s seved s assgned accodng to Poblem 3.1. A tme scale sepaaton assumpton s often used to smplfy the analyss, namely, we assume that congeston contol potocols dve the system to the optmum of Poblem 3.1 fo a gven n nstantaneously;.e. the congeston contol opeates at a faste (packet-level) tmescale than the connecton aval/depatue pocess. We can defne the aveage load on oute asρ =λ /µ. Usng the tme-scale sepaaton assumpton between congeston contol and connecton-level dynamcs, the pocess n(t)=(n (t)) becomes a contnuous tme Makov chan (c.f. Appendx A.4), wth the followng tanston ates: n n+ e wth ate λ, (3.4a) n n e wth ate µ ϕ (n). (3.4b) Hee e denotes the vecto wth a one n coodnate and 0 elsewhee.ϕ (n) s the ate assgned to oute by the mapφdefned eale. The pocess defned by equatons (3.4) s a multdmensonal bth and death pocess wth coupled death ates. Ths pocess s not evesble and the global balance equatons cannot be solved explctly except n some specal cases. Howeve, ts stablty (n the sense of postve ecuence) can be analyzed. The man esult s: Theoem 3.2 ([de Vecana et al., 1999, Bonald and Massoulé, 2001]). Fo the weghtedα fa esouce allocaton wthα>0, the Makov model defned by (3.4) s stable (.e. postve ecuent) f: R l ρ < c l l. (3.5)

48 CHAPTER 3. CONNECTION LEVEL MODELS AND USER INCENTIVES 35 Moeove, f R lρ > c l, fo some l, the pocess s tansent. Note that ths theoem states that netwok stablty s guaanteed wheneve evey lnk load ρ l = R lρ does not exceeds lnk capacty. Ths esult may seem tval at fst sght, snce condton (3.5) s the natual condton asng n queueng theoy, wheneve each lnk s consdeed a sepaate queue. Note howeve that n fact, each oute behaves as a connecton level queue, seved though a complex polcy defned by the esouce allocaton ϕ(n). The fact that the stablty egon of the system concdes wth the nteo of the capacty egon,.e. the nteo of the set defned by the capacty constants, s due to the fact that theα fa polcy foα>0can be seen as a sutable genealzaton of the pocesso shang sevce dscplne to ths coupled system. We can also constuct smple examples foα=0 whee Theoem 3.2 does not hold. Theoem 3.2 also assumes that flow szes follow a exponental dstbuton. Ths s a estctve hypothess, snce t s known that typcal Intenet flows do not follow ths dstbuton. Nevetheless, the esult has also been extended n dffeent ways n[ln and Shoff, 2006], [Massoulé, 2007] and[pagann et al., 2009], showng that the stablty condton s the same unde moe geneal hypotheses fo the aval and wokload pocesses, howeve the theoy becomes moe nvolved n that case. The stochastc stablty egon s theefoe chaactezed. Howeve, faness between uses s not addessed by ths knd of esult: the connecton level dynamcs ae such that ethe the loads can be stablzed and uses ae satsfed, o the netwok s unstable. In the latte case, the numbe of connectons gow wthout bounds, up to a pont whee use maybe mpatence comes nto play and the connectons ae dopped. Ths undesable behavo s ndependent of the flow ate faness mposed by TCP (.e. the value ofα). Some authos[massoulé and Robets, 1999] ague that the above stuaton eques admsson contol of connectons. Whle admsson contol may ovecome nstablty, we beleve that t should be caed out n a way that faness between uses s taken nto account. To tackle ths ssue, n the followng Chapte we shall defne a sutable noton of use-centc faness, whee a use may epesent a bundle of connectons and outes ove the netwok. Applyng ths noton, we wll constuct dffeent decentalzed mechansms to dve the netwok to a fa opeatng pont.

49 4 Use-centc faness The models pesented n Chapte 2 make use of an economc ntepetaton of the esouce allocaton poblem to fomulate a sutable NUM poblem. On the othe hand, the dscussons of Chapte 3 show that ths economc language s metaphocal n the sense that use valuaton of bandwdth s not taken nto account, the undelyng packet-level dynamcs beng govened by some undelyng potocol and the flow-ate faness t mposes. Howeve, the economc language ntoduced suggests the possblty of moe closely elatng netwok esouce allocaton wth the eal ncentves of Intenet uses. A sgnfcant gap n the lteatue has emaned between these two ponts of vew. Among the many easons fo ths dsconnecton, we focus hee on one: uses do not cae about TCP connectons, but about the aggegate ate of sevce they eceve fom the netwok. Applcatons can, and often do, open multple connectons to ve fo a lage shae of the bandwdth pe, and n Theoem 3.1 we showed that ths s ndeed a atonal stategy. What we want s to defne a sutable noton of use-centc faness, whee esouce allocaton takes place n tems of the aggegate ate of connectons sevng a common entty, whch can be dentfed as a cetan use. These aggegatons can be qute geneal and may nvolve sngle-path o mult-path settngs: fo nstance, emegng pee-to-pee (p2p) systems dynamcally manage multple connectons wth multple stes, all sevng the common goal of a sngle download. Faness n ths context apples natually to the oveall download ate of flows gong though dffeent outes.

50 CHAPTER 4. USER-CENTRIC FAIRNESS 37 In ths Chapte we poceed to defne the noton of use centc faness by a sutable NUM poblem, whch can be solved va the standad technques of Lagangan pmal-dual decomposton pesented n Chapte 2. In Secton 4.1 we defne ths noton n the geneal case. Then, n Secton 4.2 we deve sutable decentalzed contol algothms fo the aggegate ate that dve the system to the desed esouce allocaton. Snce the aggegate ate s often dffcult to contol dectly by the use, n Secton 4.3 we poceed to analyze algothms based on contollng the numbe of connectons n the sngle path case. In Secton 4.4 we genealze these algothms to the mult-path settng. All the algothms of ths Chapte ae deved assumng some fom of coopeaton between uses. We postpone the dscusson on how the netwok may potect tself of non-coopeatve uses to Chapte 5. Fnally, n Secton 4.5 we dscuss a packet level mplementaton of the poposed algothms and show an example of the mechansm n acton. 4.1 Defnton Let us assume, as befoe, that the netwok s composed by a set of esouces o lnks of capacty c l. Routes, ndexed by ae establshed along these lnks, and we have, as befoe, a outng matx R l whch stoes the ncdence of outes on lnks. Above ths netwok, we have new enttes whch we call uses, and we ndex them by. Uses open one o moe end to end connectons though the netwok, ove a set of outes. If a gven use opens connectons ove oute we wte. Note that the same oute can be n use by moe than one use. In that case we duplcate the ndex, ndexng outes of dffeent uses wth dffeent labels. These connectons obtan fom the netwok an aggegate ateϕ on oute, and we defne the total ate a gven use obtans fom the netwok though all of ts outes as: ϕ = ϕ. Let U be an nceasng and concave utlty functon that models use pefeences o dese fo bandwdth, ndependent of the potocols n place n the netwok. We popose the followng noton of use-centc faness, defned va the followng NUM poblem: Poblem 4.1 (Use Welfae). subject to: max ϕ U (ϕ ), R l ϕ c l l.

51 CHAPTER 4. USER-CENTRIC FAIRNESS 38 The above Poblem amounts to fndng a sutable esouce allocaton fo evey oute n the netwok so that the total use welfae s maxmzed. By usng dffeent utlty functons (e.g. theα famly of Defnton 2.4) we can model dffeent notons of faness between uses. Note also that the famewok s vey geneal: a use s defned by a set of outes and a utlty functon. Ths can model fo nstance uses downloadng data fom seveal locatons, multple paallel paths, uses uploadng data to seveal ponts n the netwok, and of couse, the sngle path stuatons dscussed eale. As compaed to the ognal esouce allocaton Poblem 2.2 fom[kelly et al., 1998], the above defnton takes nto account not only the behavo of ndvdual connectons along a oute, contolled by the tanspot potocol TCP, but also on the numbe of actve connectons assocated wth each use. Ou man goal n ths pat of the thess s to fnd sutable decentalzed algothms that, takng the undelyng TCP congeston contol as gven, dve the system to the esouce allocaton of Poblem 4.1. A fst step n ou study s to assume that uses coopeate n ode to maxmze ts socal welfae. Even n ths case, Poblem 4.1 s moe dffcult that Poblem 2.2 snce the new objectve functon s not necessaly stctly concave, even when the utltes satsfy Assumpton 2.1. To see ths note that the objectve functon can be ewtten as: U (ϕ )= U ϕ. Theefoe, swtchng some esouces fom a oute to anothe oute, whle mantanng the value ofϕ does not change the value of the objectve functon. Thus any convex combnatonθϕ +(1 θ)ϕ poduces the same total welfae. Of couse, n the sngle-path case whee each use s confned to a sngle oute, Poblem 4.1 and Poblem 2.2 concde. In the followng Secton we wll show how to addess ths ssue, assumng that uses can dectly contol the ateϕ fo each. Then we wll poceed to fnd sutable laws fo n, the numbe of connectons n each oute, that emulate the desed behavo foϕ. 4.2 Contollng the aggegate ates In ths Secton, we assume that uses have dect contol of the total ates pe outeϕ,, and we would lke to popose a decentalzed dynamc contol law fo the vaablesϕ such that the netwok s dven to the soluton of Poblem 4.1. We wll esot agan to the Lagangan dualty famewok. We denoteϕ=(ϕ ) the ate vecto, p=(p l ) the pce vecto and, as befoe, y=(y l ) s the vecto of lnk ates and q=(q ) the vecto of oute pces.

52 CHAPTER 4. USER-CENTRIC FAIRNESS 39 The Lagangan of Poblem 4.1 s: (ϕ,p)= U (ϕ ) p l (y l c l ) l = U ϕ q ϕ + p l c l. (4.1) The KKT condtons that chaacteze the optmal oute ates can be obtaned by maxmzng oveϕ wth fxed pces. The followng condtons must be satsfed: Ethe o In patcula, equatons (4.2) mply that: U ϕ U ϕ l = q, (4.2a) < q andϕ = 0. (4.2b) U (ϕ, )=q := mnq. (4.3) Theefoe, n the optmal allocaton, use only sends taffc though the paths wth mnmum pce. The total ate of each use,ϕ, s detemned n the optmum by condton (4.3). Howeve, n geneal the optmal oute atesϕ need not be unque, snce t may be possble to have seveal allocatonsϕ wth ϕ =ϕ fo each, and poducng the same pces n the netwok. The Use Welfae Poblem 4.1 s smla to the congeston contol poblem descbed n Chapte 2, so a numbe of dstbuted appoaches ae avalable to dveϕ to the desed allocaton. Some dffcultes appea due to the lack of stct concavty n the objectve of Poblem 4.1. Fo nstance, n one of the vey fst woks on the subject,[aow et al., 1958] showed that the pmal-dual dynamcs may lead to oscllatoy behavo of the dynamcs. We llustate ths by a smple example: Example 4.1 (Pmal-dual dynamcs fo the Use Welfae poblem). We consde a sngle use whch has two outes avalable, as n Fgue 4.1. Fo smplcty, we choose fo the use the utlty functon U(ϕ) = log(ϕ), though the behavo wll be the same wth any utlty. Lnk capactes ae taken as c 1 = c 2 = 1 and snce thee s a oute though each lnk, R s the 2 2 dentty matx. In ths settng, Poblem 4.1 becomes: max ϕ log(ϕ 1+ϕ 2 ), subject to: ϕ 1 1, ϕ 2 1.

53 CHAPTER 4. USER-CENTRIC FAIRNESS 40 C 1 = 1 Use Seve C 2 = 1 FIGURE 4.1: NETWORK OF TWO PARALLEL LINKS AND ONE USER. 2 2 φ 1 p φ p 2 φ 1 p Tme Tme FIGURE 4.2: EVOLUTION OF THE PRIMAL DUAL DYNAMICS FOR EXAMPLE 4.1 WITH INITIAL CONDITION ϕ 1 = 0.5,ϕ 2 = 1.5 AND PRICES p 1 = p 2 = 0. It s clea that the soluton sϕ1 =ϕ 2 = 1 snce U s nceasng, and the equlbum pces ae p1 = p 2 = 1/2= U (ϕ1 +ϕ 2 ). Thus n ths case the soluton s unque. The behavo of the system fo the contol law ϕ = U (ϕ) q was smulated and t s shown on Fgue 4.2. We can see that even f the soluton s unque, the system may oscllate aound the equlbum value. To coect ths poblem, we wll popose now a vaant of pmal-dual dynamcs, wth an addtonal dampng tem, that enables us to obtan global convegence to the equlbum. Consde the followng contol law foϕ : + ϕ = k U ϕ q ν q, (4.4a) ϕ y= Rx, (4.4b) ṗ l =γ l (y l c l ) + p l, (4.4c) q= R T p. (4.4d) whee k > 0,γ l > 0 andν> 0. Recall that( ) + ϕ s the postve pojecton defned n equaton

54 CHAPTER 4. USER-CENTRIC FAIRNESS 41 (2.12), and gven by: (x) + y = x f y> 0 o y= 0 and x> 0 0 othewse. Note that the dampng temν q does not affect the equlbum of the dynamcs. We can see that, due to the pojecton( ) + ϕ the equlbum of (4.4) satsfes the KKT condtons (4.2): n patcula f U ϕ, < q thenϕ must be zeo. The man dea behnd the algothm n (4.4) s that uses, nstead of eactng to the pce q, they must eact to the pedcted oute pce q +ν q, thus antcpatng possble changes. Ths dea fst appeaed n[pagann and Mallada, 2009] n the context of combned multpath congeston contol and outng. We have the followng esult concenng the stablty of the algothm: Theoem 4.1. Unde the contol law gven n equatons (4.4) all tajectoes convege to a soluton of Poblem 4.1. Befoe poceedng wth the poof, we shall need the followng smple esult on the postve pojecton, whch s used as a standad agument n many congeston contol theoems. Lemma 4.1. Gven u 0, u 0 0 and v R, the followng nequalty holds: (u u 0 )(v) + u (u u 0)v. (4.5) Poof. If u> 0 the left and ght hand sdes of (4.5) concde. If u= 0, then the left hand sde s u 0 max{v,0}. If v> 0 ths tem s equal to u 0 v, so both sdes also concde. If nstead we have v 0, then the left hand sde t s 0 u 0 v, so the nequalty holds n all cases. Poof of Theoem 4.1. Consde the followng Lyapunov functon fo the system: 1 V(ϕ,p)= (ϕ ϕ 2k )2 + l 1 2γ l (p l p l )2 + whee(ϕ,p ) s an equlbum of the system, and y = Rϕ. l ν(c l y l )p l, (4.6) Fst note that V n the expesson (4.6) satsfes V 0 fo eveyϕ 0 and p 0. The fst two tems ae quadatc tems. The last tem s non negatve snce y l c l due to the poblem constants, and p l 0. Note that ths last tem vanshes at any equlbum due to equaton (4.4c), whch s n tun mposng the complementay slackness condton fo the optmum. When dffeentatng along tajectoes we obtan: + V= (ϕ ϕ ) U ϕ q ν q + (p l p l )(y l c l ) + p l + ϕ l + ν(c l y l )(y l c l ) + p l. (4.7) l

55 CHAPTER 4. USER-CENTRIC FAIRNESS 42 Notng thatϕ,ϕ, p l and p l ae all non negatve, we can apply Lemma 4.1 to get d of the postve pojecton by nsetng a sgn n the fst two tems. We have: V (ϕ ϕ ) l U ϕ q ν q + (p l p l )(y l c l )+ + ν(c l y l )(y l c l ) + p l. (4.8) By nsetng the values at equlbum appopately we can decompose (4.8) n the followng tems: We stat by notng that: V (ϕ ϕ U ) ϕ q + (ϕ ϕ )(q q ) (I I) ν(ϕ ϕ ) q (I I I) + (p l p l )(y l y l ) (I V) l + (p l p l )(y l c l ) l + ν(c l y l )(y l c l ) + p l l l (I) (V) (V I). (I I)+(I V)= (ϕ ϕ ) T (q q )+(y y ) T (p p ) = (ϕ ϕ ) T R T (p p )+(ϕ ϕ ) T R T (p p )=0. The complementay slackness condton n tun mples that(v) 0snce ethe y l = c l and the tem vanshes, o y l < c l and p l = 0, so each tem n the sum s p l(yl c l ) 0. The bound fo the fst tem deves fom the equlbum condton (4.2). We assocate the tems n(i) on each use : (ϕ ϕ U ) ϕ q =(ϕ ϕ, ) U ϕ U (ϕ, ) + (ϕ ϕ )(U (ϕ, ) q ). The fst tem of the ght hand sde s 0 due to the fact that U s a deceasng functon. The second tem s 0snce ethe U (ϕ. )= q f s a oute wth mnmum pce, o U (ϕ. )<q andϕ = 0. Summng ove we conclude that(i) 0.

56 CHAPTER 4. USER-CENTRIC FAIRNESS 43 The emanng tems can be gouped and vefy: (I I I)+(V I)= ν(ϕ ϕ ) T q+ν(c y ) T ṗ = ν(ϕ ϕ ) T R T ṗ+ν(c y ) T ṗ = ν(y y ) T ṗ+ν(c y ) T ṗ =ν(c y)ṗ =ν (c l y l )(y l c l ) + p l 0, l whee the last nequalty follows fom obsevng that, due to the pojecton, each summand of the last tem s 0 f p l = 0 and y l < c l o (y l c l ) 2 0 othewse. We conclude that the functon V s deceasng along the tajectoes. Stablty now follows fom the LaSalle Invaance Pncple (c.f. Appendx A.2). Assume that V 0. In patcula, the tems(i) and(i I I) +(V I) must be dentcally 0 snce they ae negatve semdefnte. Imposng(I) 0we conclude thatϕ =ϕ, fo all andϕ = 0 fo all outes whch do not have mnmum pce. Moeove, ϕ = We also have that: ϕ = 0 snceϕ must be n equlbum. (I I I)+(V I)=ν (c l y l )(y l c l ) + p l, l and snce each tem has defnte sgn, mposng(i I I)+(V I) 0eques that ethe p l = 0 o y l = c l at all tmes. Theefoe, ṗ= 0 and p must be n equlbum. It follows that q s n equlbum, and theefoe etunng to the dynamcs we must have ϕ = K a constant. If ϕ = K > 0, t would mean thatϕ mplyng that y l c l s volated at some lnk. Theefoe, K 0 and snce ϕ = 0, we must have K = 0. Theefoeϕ s n equlbum. We conclude that n ode to have V 0 the system must be n a pont that satsfes the KKT condtons (4.2), and theefoe the system wll convege to an optmal allocaton fo Poblem 4.1. Moeove, snce V s adally unbounded, ths convegence wll hold globally. Befoe poceedng wth the analyss, let us compae the behavo of the dynamcs (4.4) n the paallel lnks case gven n Example 4.1. Example 4.2 (Pmal dual dynamcs wth dampng). Consde agan the settng of Example 4.1 depcted n Fgue 4.1. Recall that the optmal allocaton fo Poblem 4.1 s gven byϕ 1 =ϕ 2 = 1, and the equlbum pces ae p 1 = p 2 = 1/2= U (ϕ 1 +ϕ 2 ). The behavo of the system fo the contol law (4.4) was smulated and t s shown on Fgue 4.3. We can see that n ths case the system conveges to the equlbum allocaton, and the pces fnd the ght equlbum pce, thus coectng the oscllatng behavo of Example 4.1. As a esult of Theoem 4.1, f uses could contol the aggegate nput ate ove each oute followng the dynamcs (4.4), the system wll convege to the optmal allocaton of Poblem

57 CHAPTER 4. USER-CENTRIC FAIRNESS φ 1 p φ p 2 φ 1 p Tme Tme FIGURE 4.3: EVOLUTION OF THE PRIMAL DUAL DYNAMICS FOR EXAMPLE 4.1 WITH INITIAL CONDITION ϕ 1 = 0.5,ϕ 2 = 1.5 AND PRICES p 1 = p 2 = 0 AND DAMPING FACTORν= Note that the poposed algothm assumes that use can contol the ate on each of ts outes, and the only knowledge t needs s the oute pce q and the pedctve tem q. Theefoe the algothm s decentalzed, snce uses only need nfomaton about the outes they ae usng to pefom the contol. Howeve, n eal netwoks t s not easonable to assume that the use has fne tuned contol ove the aggegate ate. As we dscussed n Secton 2.4, the tanspot laye gets n the way. Uses may open one o seveal connectons on each oute, but the ate at whch connecton s seved s detemned by the congeston contol algothms n place,.e. the TCP potocol. As we dscussed n Secton 3.1, the esouce allocaton pefomed by TCP can be modelled by Poblem 3.1. In concluson, the end use may contolϕ but only ndectly, though the numbe of ongong connectons on each oute. The emanng sectons n ths chapte addess ths ssue. 4.3 Contollng the numbe of flows: the sngle-path case Based on the pecedng dscussons, we would lke to fnd a sutable contol law fo the numbe of ongong flows n each oute such that the system conveges to the esouce allocaton defned by Poblem 4.1. In ths secton, we wll focus on the smple stuaton n whch each use opens connectons ove only one oute n the netwok. Theefoe we can dentfy a use and ts oute, and the aggegate ate obtaned by the use sϕ =ϕ. We wll assume also that the netwok opeates wth some sutable congeston contol algothm such that, fo a fxed numbe of connectons n=(n ), the netwok seeks to optmze the TCP Congeston Contol Poblem 3.1. Fo ease of exposton we wll assume that ths congeston contol s a dual algothm, but we dscuss the altenatves below.

58 CHAPTER 4. USER-CENTRIC FAIRNESS 45 Unde these assumptons, the netwok pat of the dynamcs wll be gven by: ṗ l =γ l (y l c l ) + p l l, q= R T p, x = f T C P (q ), ϕ = n x, y= Rϕ. Hee x epesents the ndvdual ate of each connecton on oute. Gven a cuent oute pce q, the ate s detemned by the condton: U TC P (x )=q, whee U T C P, models the TCP behavo. Theefoe, f T C P, s the demand cuve fo the utlty functon U T C P, o U 1 (q ). In the case ofα faness, whee U TC P (x )=w x α cuve s gven by: f T C P (q )= and t wll be a deceasng functon of the pce. q w 1/α, ths demand We popose to add, on top of ths laye, a contol law fo n such that n equlbum the ateϕ = n x s detemned by the use level utlty functon U nstead of the netwok laye utlty U T C P. The man ntuton s to use the cuent lowe laye congeston pce as a feedback sgnal fom the netwok. If unde the cuent condtons the esouce allocaton s such that use wth oute gets a ateϕ whch does not fulfll the use demand, the ateϕ must be nceased. In the lght of Theoem 3.1, we can do so by nceasng n. The contol law s as follows: ṅ = k (U 1 (q ) ϕ ). (4.9) Combnng t wth the pevous equatons we ave at the followng dynamcs fo the system: ṅ = k (U 1 (q ) ϕ ), (4.10a) x = f T C P (q ), (4.10b) ϕ = n x, (4.10c) y= Rϕ, (4.10d) ṗ l =γ l (y l c l ) + p l l, (4.10e) q= R T p. (4.10f) We would lke to analyze the asymptotc popetes of these dynamcs. We begn by chaactezng ts equlbum:

59 CHAPTER 4. USER-CENTRIC FAIRNESS 46 ϕ n x Connecton Level Contol TCP Congeston Contol Netwok q FIGURE 4.4: COMBINED CONNECTION LEVEL AND USER LEVEL CONTROL LOOP Poposton 4.1. The equlbum of (4.10),(n,p ), s such that the esultngϕ s the soluton of Poblem 4.1 and x s the soluton of Poblem 3.1 fo the gven n. Poof. Any equlbum pont(n,p ) of the dynamcs (4.10) must vefy: U 1 (q ) ϕ = 0, p T (y c)=0, x = f T C P(q )= U 1 T C P (q ), wthϕ = n x, y = Rϕ and q = R T p. The fst two equatons mply thatϕ,p satsfy the KKT condtons (4.2) of Poblem 4.1. Moeove, the last two equatons mply that(x,p ) must satsfy also the KKT condtons of Poblem 3.1. Snce n ths case both poblems have unque solutons due to the stct concavty of the utlty functons nvolved, the dynamcs (4.10) have a sngle equlbum whch s the smultaneous soluton of both poblems. The above agument justfes why we can use the same oute congeston pce q to contol both the ate of each connecton and also the numbe of connectons. The poposed contol loop s best explaned though Fgue 4.4, whee we can see that the connecton level contol s an oute contol loop added to the cuent netwok congeston contol algothm Local stablty n the netwok case We would lke to deve stablty esults fo the system. We begn by local stablty fo a geneal netwok. To pove ths esult we shall use the passvty appoach. A system wth nput u, state s and output v s called passve f thee exsts some stoage functon of the state V(s) (smla to a Lyapunov functon) such that: V u T v.

60 CHAPTER 4. USER-CENTRIC FAIRNESS 47 If the nequalty s stct, the system s called stctly passve. Fo some backgound on passvty of dynamcal systems we efe the eade to Appendx A.3. We note n patcula that the negatve feedback nteconnecton of two passve systems s stable. The passvty appoach was ntoduced n congeston contol by[wen and Acak, 2003]. In the pape, the authos pove that the system(ϕ ϕ ) (q q ) wth dual dynamcs n the lnks (equatons (4.10d), (4.10e), (4.10f)) s passve wth stoage functon: 1 V n e t (p)= (p l p l 2γ )2. l l Theefoe, we have to check passvty of the map(q q ) (ϕ ϕ ) wth the new contol law (4.9). Due to the decentalzed natue of dynamcs (4.10) we can check that the use system (q q ) (ϕ ϕ ) s tself passve fo each snce n that case thee would be a stoage functon V (n ) such that: V (n ) (q q )(ϕ ϕ ), and theefoe: V(n,p)=V n e t (p)+ V (n ) wll be a Lyapunov functon fo the system. Moeove, f the use system s stctly passve, we get asymptotc stablty. To obtan a local stablty esult, we shall check that the lneazed dynamcs aound the equlbum of the use pat of the system s passve, by usng the fequency doman chaactezaton of stct passvty pesented n Theoem A.5. We have to check that the nput-output tansfe functon H(s) of a lnea system satsfes Re(H(jω)) 0 ω R. If moeove the nequalty s stct, the system s stctly passve. We now calculate the tansfe functon of the lneazed use pat of the system and establsh ts stct passvty. Lemma 4.2. The lneazaton of the system(q q ) (ϕ ϕ ) gven by equatons (4.10a), (4.10b), (4.10c) has a eal postve tansfe functon and theefoe s stctly passve. Poof. We dop the subscpt fo smplcty. Let us denote byδ the devaton of the vaables of the equlbum values. Fom equatons (4.10b) and (4.10c) we have: δx= f T C P (q )δq= aδq, δϕ=δ(nx)=n δx+ x δn, whee a> 0 snce f T C P s deceasng n q. Combnng the above equatons we have: x δn=δϕ+ a n δq. (4.11)

61 CHAPTER 4. USER-CENTRIC FAIRNESS 48 Passng equaton (4.10a) nto the Laplace doman and lneazng we have: sδn= k[(u 1 ) (q )δq δϕ]= k bδq kδϕ, whee b> 0 snce U 1 s also deceasng n q. Multplyng both sdes by x and substutng (4.11) we get: whch can be ewtten as: s(δϕ+ a n δq)= k bx δq k x δϕ, δϕ δq = H(s)= a n s+ k bx s+ k x. So the lneazed dynamcs fomδq δϕ behaves as a lead-lag system H(s). It s easy to see that the cuve Re(H(jω)) s a ccle n the ght half plane though the ponts H(0)= b> 0 and H( )=a n > 0 and thus the tansfe functon s postve eal. We conclude that the use system s stctly passve. Combnng Theoem A.5 wth Lemma 4.2 and the esults fom[wen and Acak, 2003] we ave at the man esult of ths secton: Theoem 4.2. The equlbum of the system (4.10) s locally asymptotcally stable. Remak 4.1. Wth the same deas, f we eplace the potocol demand cuve f T C P n equaton (4.10b) by a pmal adaptaton algothm gven by: ẋ =β (U T C P (x ) q ), then t can be shown that the analog of Lemma 4.2 holds wth H(s) gven by: H(s)= (β n + k x a)s+ kβ x ab (s+β a)(s+ k x. ) Ths second ode system has one zeo and two poles. It can be seen that the zeo s always to the left of the pole k x and thus the phase of the system wll always be n( π,π), theefoe t s also postve eal. We conclude that the system (4.10) s also locally asymptotcally stable f a pmal-type congeston contol s n use Global stablty n the sngle bottleneck case Although the esult deved n the pevous subsecton s only local, as we shall see, smulaton esults of the system dynamcs ndcate that stablty holds globally as well. Howeve, t s not easy to deve a global esult snce the local passvty appoach does not gve an appopate Lyapunov functon fo the non-lnea system. To deve a global stablty esult we fst pefom a tme scale sepaaton analyss of the system. Typcally, we can assume that the congeston contol potocol opeates on a faste

62 CHAPTER 4. USER-CENTRIC FAIRNESS 49 tmescale than the connecton level contol of n. Snce congeston contol s known to be stable, t can be assumed that uses wll contol n by eactng to the equlbum pceˆq (n ) fom Poblem 3.1 eached by congeston contol. Below we fomalze ths appoach n tems of the sngula petubaton theoy (c.f. [Khall, 1996]) and deve a global stablty esult fo the sngle bottleneck case, whee the map n ˆq can be calculated explctly. Let us ewte the dynamcs (4.10) as a sngulaly petubed system: ṅ = k (U 1 (q ) ϕ ), (4.12a) εṗ l =γ l (y l c l ) + p l l, (4.12b) whee as usual x = f T C P (q ),ϕ = n x, y= Rϕ and q= R T p. Asε 0we can decompose the system n a bounday laye model epesentng the fast dynamcs fo p and a educed system whee n evolves as f p has eached equlbum. The bounday laye system s smply, fo fxed n: εṗ l =γ l (y l c l ) + p l l, x = f T C P (q ), ϕ = n x, y= Rϕ, q= R T p, whch s a dual congeston contol algothm smla to (2.11) studed n Chapte 2. It s known to be asymptotcally stable by Poposton 2.4. Its equlbum ˆx(n), ˆϕ(n), ˆp(n) and ˆq(n) s the soluton of Poblem 3.1. In patcula the map n ˆϕ(n) s the mapφwe studed n Secton 3.2. The educed system s autonomous n the state n and ts dynamcs s: ṅ = k(u 1 (ˆq (n)) ˆϕ(n)). (4.13) The man advantage of the sngula petubaton analyss s the fact that, f both the bounday laye system and the educed model ae asymptotcally stable, we can conclude that the complete system (4.12) s asymptotcally stable fo small values of the petubaton paamete ε. We now pesent an example showng both the complete model and the educed dynamcs fo a netwok stuaton, llustatng the stablty of the system. Example 4.3 (Contollng the numbe of connectons n a pakng-lot netwok). We consde a pakng lot topology, wth capactes and ound tp delays as depcted n Fgue 4.5. Each use opens TCP connectons, whch we model as havng the demand cuve: f T C P (q)= K q, whee K s chosen as n Maths fomula (2.14). In patcula, connectons n the the long oute wll have a lage RTT than the shote ones, and thus K wll be smalle.

63 CHAPTER 4. USER-CENTRIC FAIRNESS 50 Route 2 C 1 = 10 C 2 = 10 Route 0 Route 1 FIGURE 4.5: PARKING LOT TOPOLOGY. If each use opens only one connecton, the esultng allocaton would be the soluton of Poblem 3.1 wth n = 1 n each oute. Fo ths poblem ths allocaton tuns out to be x1 = 2.6, x2 = x 3 = 7.4. We would lke to acheve popotonal faness, so we choose U (ϕ )=logϕ fo each use/oute n ths topology. The soluton of the use welfae Poblem 4.1 n ths case s: ϕ 1 = 3.33, ϕ 2 =ϕ 3 = In Fgue 4.6 we show the esults, statng fom an ntal condton of n(0)=(1,2,3). We plot the tme evoluton of the system n n, p andϕ. In sold lnes, we plot the complete dynamcs (4.10). In dashed lnes we plot the educed dynamcs (4.13) wth tme scale sepaaton. Note that both dynamcs convege to the desed equlbum, and show smla behavo. Extensve testng wth seveal ntal condtons show that the system ndeed behaves n a stable manne n all cases. In a netwok stuaton, howeve, the educed system (4.13) s dffcult to analyze because ˆq and ˆϕ ae dffcult to expess n tems of n, snce they ae only chaactezed by Poblem 3.1. We now pesent a stablty esult fo a sngle-bottleneck case, whee the allocaton can be calculated explctly. We analyze ths system unde the followng assumptons: We consde a netwok wth a sngle bottleneck of capacty c whch s shaed by N uses, each wth use utlty functon U, = 1,...,N. We assume that the congeston contol algothm of all uses follows the same demand cuve f T C P. Fo nstance, ths could coespond to eveyone usng TCP-Reno at the tanspot laye, wth the same ound tp tmes. Unde these hypotheses, thee wll be a sngle pce and, snce thee s a common TCP demand cuve, the ndvdual flow ates wll satsfy: c ˆx (n)= N =1 n :=ˆx(n) (4.14) Theefoe, ˆq(n)= U T C P (ˆx(n)) and ˆϕ (n)=n ˆx(n), whch substtuted n (4.13) gve an autonomous system n the state n. We have the followng:

64 CHAPTER 4. USER-CENTRIC FAIRNESS n 1 20 n 2 n p 1 p 2 φ 1 φ 2 20 φ FIGURE 4.6: RESULTS FOR THE PARKING LOT TOPOLOGY. IN SOLID LINES, THE COMPLETE DYNAMICS, IN DOTTED LINES, THE REDUCED MODEL. Theoem 4.3. The solutons of the system (4.13) complemented wth (4.14) convege to an equlbum pont n such thatϕ = ˆϕ(n ) s a soluton of Poblem 4.1,.e. maxmzes the use welfae. Poof. Fst note that an equlbum of (4.13) togethe wth (4.14) and the lowe laye equlbum automatcally ensues the KKT condtons (4.2). We thus focus on the stablty. Denote by n = N =1 n the total numbe of connectons. Ths wayˆx(n)=c/ n. Addng equatons (4.13) n we obtan: n =k N =1 U 1 (ˆq(n)) N ˆϕ(n). =1

65 CHAPTER 4. USER-CENTRIC FAIRNESS 52 Note that, wheneve n 0 we have N =1ˆϕ(n)=c, and that: ˆq(n)= U T C P (ˆx(n))= U TC P (C/ n ). So the dynamcs of n vefy: n =k N =1 c g n c, (4.15) whee g (x)= U 1 (U T C P (x)). The functon g s nceasng snce t s the composton of two deceasng functons. Note that (4.15) s an autonomous dffeental equaton n the scala vaable n. Denotng by n the equlbum of (4.15), defne the followng canddate Lyapunov functon: By dffeentatng along the tajectoes we obtan: V 1 ( n )= 1 2k ( n n ) 2. (4.16) N c V 1 =( n n ) g c n =1 N c c =( n n ) g g n n, =1 whee the last step uses the equlbum condton. Notng that each g s deceasng n n, we have V 1 0 wth the nequalty beng stct f n n. Ths shows that n conveges to equlbum, and n patcula the ate of each connectonˆx(n) x = c/ n. We etun now to equaton (4.13). By defnngδn = n n we can ewte t as: δn = k[g (ˆx(n)) n ˆx(n)] = kˆx(n)δn + k[g (ˆx(n)) n ˆx(n)]. The tem b (t):= g (ˆx(n)) n ˆx(n) vanshes as t snceˆx(n) x and g (x )=ϕ = n x. Now fo each take the Lyapunov functon V 2 (δn )= 1 2k (δn ) 2, ts devatve s: V 2 = ˆx(n)(δn ) 2 + b (t)δn (4.17) Consde nowǫ> 0 abtay, and choose t 0 such thatˆx(n(t))> x 2 and b (t) < x 4 ǫ wheneve t> t 0. Then, n the egon whee δn ǫ and fo t> t 0 we have that: b (t)δn < x 4 ǫ δn x 4 (δn ) 2.

66 CHAPTER 4. USER-CENTRIC FAIRNESS 53 We can now bound V 2 as follows: V 2 = ˆx(n)(δn ) 2 + b (t)δn < x 2 (δn ) 2 + x 4 (δn ) 2 = x 4 (δn ) 2 x 4 ǫ2. The above nequalty poves that V 2 s stctly deceasng along the tajectoes, and thus eventually,δn wll each the set δn <ǫ n fnte tme. Snce the above s tue fo anyǫ> 0. we have thatδn 0 asymptotcally. Extendng the global stablty esults fo moe geneal topologes poves to be a dffcult task. In the next secton, we shall follow a dffeent appoach to deve a sutable dynamcs fo n, whch would enable us to pove global stablty esults. 4.4 Contollng the numbe of flows: the mult-path case So fa, we have studed a mechansm to contol the aggegate atesϕ dectly n ode to each the optmum of the Use Welfae Poblem 4.1 n a mult-path settng, whch led us to the dynamcs (4.4). We also analyzed the possblty of contollng n as a means to acheve the optmum of Poblem 4.1 by ndectly contollngϕ. We would lke to combne both deas n a sngle contol algothm fo n whch can be used n a mult-path settng. We begn by notng the followng smple elatonshp between n andϕ : ϕ = (n x )=x ṅ + n ẋ. (4.18) We would lke to contol n such thatϕ follows a contol law of the type dscussed n Secton 4.2. Consde the followng law fo n : ṅ = n k (U (ϕ ) q ν q ) ẋ. (4.19) x Wth ths choce of ṅ, vald wheneve x > 0, we can substtute n (4.18) to get: ϕ = x ṅ + n ẋ = n x k (U (ϕ ) q ν q ) ẋ + n ẋ x = k ϕ U (ϕ ) q ν q, whch s on the fom of the contol foϕ analyzed n (4.4), wth k eplaced by k ϕ.

67 CHAPTER 4. USER-CENTRIC FAIRNESS 54 Fo pactcal puposes, t s bette to expess the dynamcs above n tems of the oute pce q. We wll do so n the case of U TC P beng n theα famly. We have: x = f T C P (q )= q 1/α, w q 1/α 1 q, and theefoe: ẋ = f T C P (q ) q = 1 α w ẋ = w q. x α q The complete dynamcs fo the system now follows: ṅ = n k (U (ϕ ) q ν q )+ w q αq ϕ = n f T C P (q ),, (4.20a) (4.20b) y= Rϕ, (4.20c) ṗ l =(y l c l ) + p l l, (4.20d) q= R T p. (4.20e) We want to deve a global stablty esult fo these dynamcs. We begn by extendng the esult of Theoem 4.1 to the case whee the contolle gan s k ϕ. The esult s the followng: Theoem 4.4. Consde the dynamcs (4.4) wth the fst equaton eplaced by: ϕ = k ϕ U (ϕ ) q ν q. (4.21) Then, fo any ntal condtonϕ(0) such thatϕ (0)>0, the system conveges to a soluton of the Use Welfae Poblem 4.1. Poof. The man ssue wth equaton (4.21) s that fϕ = 0 at some tme t, thenϕ emans 0 n the futue, even f the desed allocaton s not 0 fo that oute. We show that ths ndeed cannot happen f we stat at an ntal condton wthϕ > 0. Consde the followng modfcaton of the Lyapunov functon V(ϕ,p) defned n (4.6): V(ϕ,p)= ϕ ϕ (σ ϕ ) k σ dσ+ l 1 (p l p l 2γ )2 + l l ν(c l y l )p l, (4.22) whch s well defned at least whenϕ > 0. Note that, when dffeentatng along the tajectoes, V satsfes equaton (4.7) and so t wll be deceasng along tajectoes by the same aguments of Theoem 4.1. s: Assume thatϕ s such thatϕ > 0, then the coespondng tem n the Lyapunov functon V (ϕ )= ϕ ϕ σ ϕ k σ dσ= 1 (ϕ ϕ k )+ϕ ϕ log. k ϕ

68 CHAPTER 4. USER-CENTRIC FAIRNESS 55 Theefoe, asϕ 0, V (ϕ ). Also fϕ, V (ϕ ). If nowϕ = 0 we have: V (ϕ )= ϕ ϕ σ ϕ k σ dσ= 1 k ϕ, so ths tem s well defned whenϕ 0 and we can extend V(ϕ,p) contnuously n ths case. Note also that V (ϕ ) whenϕ. We conclude that the set{v K} s compact fo any K> 0. Moeove, these sets do not contan the egonϕ = 0 fo those outes such thatϕ > 0. Theefoe, fo any ntal condton(ϕ 0,p 0 ) wthϕ 0 n the postve othant, the set{v V(ϕ 0,p 0 )} s compact, and fowad nvaant due to V deceasng along tajectoes. The esult now follows by LaSalle nvaance pncple: the system wll convege to the set whee V= 0 contaned n{v V(ϕ 0,p 0 )}. Snce we analyzed n the poof of Theoem 4.1 that ths can only happen wthϕ=ϕ, a soluton of the Poblem 4.1, ths concludes the poof. We now state the man esult of ths secton: Theoem 4.5. The dynamcs (4.20), statng fom any ntal condton wth n > 0 and q > 0, wll convege to an equlbum pont whch acheves the optmum of Poblem 4.1. Poof. It s easy to see that, f n > 0 and q > 0 thenϕ > 0, so the ntal condton foϕ vefes the hypotheses of Theoem 4.4. As dscussed befoe, (4.20) mply that ϕ must vefy (4.21) and due to Theoem 4.4,ϕ wll convege to a soluton of Poblem 4.1. Theefoe, n,p l wll convege to a pont such that use welfae s maxmzed. Obseve that the contol law fo n n equaton (4.20a) s smla to to the poposed contol law n the sngle path stuaton, but wth a devatve acton. Howeve, the devatve tems play opposng oles, and ths suggests that the smple contol law: ṅ = k n (U (ϕ ) q ) (4.23) should be also a sutable way to dve the system to the use welfae equlbum. In fact, when tanslated toϕ, the contol law (4.23) yelds vey smla dynamcs to (4.4) wth a dampng paameteν = w αk q. Howeve, ths dampng paamete s tme vayng and oute dependent, whch s not compatble wth ou eale stablty agument. One can also ty to pefom a tme-scale sepaaton analyss of (4.20) based on sngula petubaton, as n Secton 4.3. Howeve, the devatve acton hee does not allow to teat the n vaable as slowly vayng. The tme scale sepaaton analyss can be pefomed on (4.23),

69 CHAPTER 4. USER-CENTRIC FAIRNESS 56 Use 1 C 1 = 4 Mbps C 2 = 10 Mbps C 3 = 6 Mbps Use 2 FIGURE 4.7: DOWNLINK PEER-TO-PEER EXAMPLE NETWORK but n ths case global stablty esults ae hade to obtan, due to the fact that the map n ϕ s dffcult to analyze. So we have a choce between a moe complcated law (4.20a) wth guaanteed stablty popetes, and the smple one (4.23). Below we show a smulaton example llustatng that the smple dynamcs (4.23) s suffcent n some cases: Example 4.4 (Pee-to-pee downloadng example). The algothms deved n ths chapte can be appled to a pee-to-pee fle exchange envonment. In such a system, uses may open multple connectons to seveal pees to download shaed data fles. It s natual to consde the utlty of a use based on ts total download ate, athe on the ate of ndvdual connectons. We thus have a mult-path nstance of the Use Welfae Poblem 4.1, whee the set of flows pe use shae a common destnaton but wth dffeent souces. It s aguable that n a pee-to-pee envonment, a cetan level of coopeaton can be obtaned, at least among uses that access the system wth a common applcaton. Ths applcaton could be pe-pogammed wth a connecton-level contolle, that opens and shuts connectons accodng to the congeston pce t measues on each oute. Thee s also pee pessue to espect ths contol. The algothm n ths case would wok as follows: use measues the total download ate ϕ and the congeston pce q on each of ts outes. Then t apples the contol law (4.20a) (o the smple contol (4.23)) to egulate the numbe of actve connectons pe oute. Each connecton wll be ndvdually contolled by standad TCP, equng no change n cuent potocols. Fo nstance, f a loss-based TCP s used, the pce q would be the packet loss facton of the oute, whch can be deduced fom the numbe of etansmssons. As an example, consde the netwok depcted n Fgue 4.7, whee two uses shae a total

70 CHAPTER 4. USER-CENTRIC FAIRNESS Numbe of actve connectons 12 Use aggegate ate Numbe of connectons Use 1, Route 1 2 Use 1, Route 2 Use 2, Route 1 Use 2, Route Tme (s) Mbps φ 1 φ Tme (s) FIGURE 4.8: NUMBER OF CONNECTIONS ON EACH ROUTE AND AGGREGATE RATE (ϕ ) FOR THE PEER- TO-PEER DOWNLINK EXAMPLE. of fou lnks wth dffeent capactes, n ode to each the seves. We choose the use utlty U (ϕ )=log(ϕ ) to mpose popotonal faness and we suppose that dffeent outes have TCP-Reno lke demand cuves, wthα=2but dffeent weghts, n ode to epesent dffeent ound tp tmes on each oute. It s easy to see that the soluton of Poblem 4.1 s ths case satsfesϕ 1, =ϕ 2, = 10Mbps. In Fgue 4.8 we plot the esults, statng wth an ntal condton of 1 flow on each oute. As we can see, the algothm of (4.23) fnds the appopate numbe of flows pe oute such that the esouce allocaton s optmal. In the followng secton, a packet level mplementaton of the contol law (4.23) s dscussed, and we compae ts pefomance wth the above example. 4.5 Packet level smulaton We mplemented a packet level smulaton of the contol law (4.23) usng the netwok smulato ns-2[mccanne and Floyd, 2000]. Wth ths tool, we can eplcate the behavo of congeston contol potocols ove a netwok at the packet level, whch takes nto account the full TCP dynamcs. We smulated the topology of Fgue 4.7, wth each use begnnng wth a sngle TCP connecton pe oute. The congeston contol algothm s TCP-Neweno, whch s the latest veson of the standad TCP congeston contol. Wth ths choce of TCP, the oute pce s the packet loss ate on the oute. We pogammed a use nstance, that measues the loss ate on each of ts outes by countng packet etansmssons at the tanspot laye. It also keeps tack of the total ate pe outeϕ, and theefoe

71 CHAPTER 4. USER-CENTRIC FAIRNESS 58 s able to calculateϕ. Each use then mantans a vaableø Ö ØÆ Öµfo each oute, whch s the taget numbe of connectons. Ths vaable s updated peodcally by ntegatng the contol law (4.23). As n the pevous example, we choose logathmc utltes to mpose popotonal faness n the netwok. Each second, the use chooses whethe to open o close a connecton o oute based on the cuent value ofø Ö ØÆ Öµ, ounded to he neaest ntege. Results ae shown n Fgue 4.9, whch shows that the numbe of connectons appoxmately tack the equlbum values found n the pevous example. The aggegate ate of each use also conveges towads the equlbum allocaton ofϕ 1, =ϕ 2, = 10Mbps. 14 Numbe of admtted flows (n) Use 1,Route 1 Use 1,Route 2 Use 2, Route 1 Use 2, Route Tme (sec) 12 Use aggegate ate (ρ) 10 8 Mbps Use 1 Use Tme (sec) FIGURE 4.9: NUMBER OF CONNECTIONS ON EACH ROUTE AND AGGREGATE RATE (ϕ ) IN AN NS-2 SIM- ULATION FOR THE PEER-TO-PEER DOWNLINK EXAMPLE.

72 5 Utlty based admsson contol The analyss of Chapte 4 apples to the case of netwoks whee uses coopeate by contollng the numbe of ndvdual TCP connectons n ode to mantan some noton of faness between the total ates. Howeve, ths knd of use contol s often not possble. An applcaton may not be capable of tunng the numbe of connectons n such a fne way. In patcula, t may not be pactcal to temnate actve connectons, as may be equed by the poposed contol algothms, whch may need to educe n. Thee ae also nfomaton equements fo the end systems such as beng awae of the use aggegate ateϕ, whch may be dffcult to monto. Also, and moe categocally, a geedy use may smply efuse to coopeate, and ncement ts numbe of ongong connectons n ode to gab a lage shae of bandwdth, as we dscussed n Theoem 3.1. In these stuatons, the netwok should be able to espond and potect tself by mposng some admsson contol on the ncomng connectons. Pevous wok on admsson contol (see e.g. [Massoulé and Robets, 1999]) has been manly motvated by the equement of mantanng netwok stablty unde stochastc models of load, such as those pesented on Secton 3.3. Howeve, any admsson contol ule that lmts the maxmum numbe of ongong connectons n each oute wll guaantee stablty. The emanng queston s whethe thee s an admsson contol ule that, n addton to guaanteeng stablty, s able to dve an oveloaded system nto a fa opeatng pont, fom a use centc pespectve.

73 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 60 In ths chapte we focus on ths task, based on the noton of use centc faness ntoduced n Chapte 4, and usng the knowledge we acqued fom the poposed dynamcs fo n to dve the system to the optmal use welfae allocaton. The chapte s oganzed as follows: In Secton 5.1 we ntoduce the poposed admsson contol ule n the sngle path scenao. We poceed to analyze ths ule unde stochastc loads by usng the technque of flud lmts, and chaacteze the equlbum of these dynamcs, as well as showng that ths equlbum s locally asymptotcally stable. In Secton 5.5 we extend the poposed admsson contol ule to the mult path settng and also chaacteze the equlbum attaned. Fnally, n Secton 5.6 we pesent seveal scenaos of packet-level smulaton of the algothms, showng that they can be mplemented n eal netwoks. 5.1 Admsson contol n the sngle path case The models n Chapte 4 ae sutable fo a netwok whee uses may open o close connectons at wll. Howeve, a moe ealstc settng s one n whch uses open new connectons at andom aval tmes, and each connecton bngs a andom amount of wokload. Ths type of models whee ntoduced n[massoulé and Robets, 1999], and wee evewed n Secton 3.3. Also n[massoulé and Robets, 1999], the dea of admsson contol was advocated. In such a system, the netwok tself may decde whethe each ncomng connecton s allowed nto the netwok. Admsson contol allows the system to be stablzed even when the extenal load on each lnk s ove the lnk capacty. What we would lke to deve s a decentalzed admsson contol ule that can be enfoced at the netwok edge, such that n case of oveload the system allocates esouces accodng to the Use Welfae Poblem. We concentate hee on the sngle path case, so each use can be dentfed wth a oute. Each use s theefoe assgned a utlty functon U. Fom the analyss of Secton 4.3, we see that n ode to acheve faness, each use must ncease ts numbe of connectons wheneve U 1 (q )>ϕ, o equvalently, U (ϕ )>q. If on the othe hand, the nequalty eveses, the numbe of connectons must be deceased. It s theefoe natual to popose the followng admsson contol ule: wheneve a new connecton aves on oute, f U (ϕ )>q admt connecton, f U (ϕ ) q dscad connecton. (5.1) We call the pecedng ule Utlty Based Admsson Contol. Ths ule mposes a lmt on the numbe of connectons a gven use s allowed nto the netwok. A stategc use wll not get a lage shae of bandwdth by smply nceasng the

74 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 61 numbe of connectons, as dscussed n Theoem 3.1, because eventually the admsson condton (5.1) wll not be met, due to the nceasng pce on the oute. In a scenao whee all uses ae geedy, the netwok wll opeate n the egon whee U (ϕ )= q, whch ae the KKT condtons of Poblem 4.1, and thus the esultng allocaton wll be fa. Below we fomalze these aguments though a stochastc model of the system. 5.2 Stochastc model and ts stablty We wll make a stochastc model fo ths system, smla to the one pesented n Secton 3.3. Assume that each use on oute opens connectons though the netwok. Fesh connectons ave as a Posson pocess of ntenstyλ. Each connecton bngs an exponentally dstbuted wokload of mean 1/µ. Connecton aval and job szes ae ndependent, and also ndependent between uses. We also make the same tme scale sepaaton assumpton,.e. congeston contol opeates on a faste tmescale that the connecton level pocesses, and theefoe the ate at whch each connecton s seved s detemned by the soluton of the TCP congeston contol Poblem 3.1. Fo a gven state n=(n ) of connectons on each oute, x(n)=(x (n)) wll denote the soluton of Poblem 3.1. We denote byϕ (n)=n x (n) the aggegate ate on oute, and q (n) the oute pce. We assume that n Poblem 3.1, the utltes U T C P come fom theα fa famly wthα>0. Wth each use, we assocate an nceasng and concave utlty functon U (hee denotes the oute and the use), that epesents the use valuaton of bandwdth. The admsson contol ule (5.1) amounts to compang U (ϕ (n)) wth q (n) n each state, and only admt connectons wheneve the admsson condton s satsfed. Wth the above assumptons, the pocess n(t) s a contnuous tme Makov chan wth state spacen N, wth N beng the numbe of uses, and tanston ates: n n+ e wth ateλ 1 {U (ϕ )>q }, n n e wth ateµ ϕ, (5.2) whee e s the vecto wth a 1 n coodnate and 0 elsewhee, and 1 A s the ndcato functon. The model (5.2) s smla to the one pesented n Secton 3.3, but wth the admsson ule (5.1) ncopoated. Recall that, wthout the admsson condton, the Makov chan s stable (postve ecuent) only f the lnk loadsρ l = R l λ /µ < c l. Admsson contol should stablze the system n any stuaton. We now pove that ths s ndeed the case, fo the nontval admsson ule (5.1). We need the followng Lemmas on the map n ϕ foα fa esouce allocatons, poved n[kelly and Wllams, 2004]:

75 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 62 Lemma 5.1 ([Kelly and Wllams, 2004]). The functon n ϕ (n) s contnuous at each n whee n > 0. Lemma 5.2 ([Kelly and Wllams, 2004]). Fo any scale paamete L > 0, ϕ(ln) = ϕ(n). We have the followng: Poposton 5.1. The Makov chan (5.2) s stable. Poof. We shall pove that thee s a fnte set that the pocess n(t) defned by (5.2) cannot leave. It follows that the educble set contanng n= 0 s fnte and statng wth any ntal condton n that set (n patcula, an empty netwok), the pocess wll each a stable state. Consde the box set S={n : n n 0 }, wth n 0 to be chosen late. We would lke to pove that the pocess cannot leave S. Snce (5.2) defnes a multdmensonal bth and death pocess, we have to pove that, fo n such that n = n 0 the pocess cannot leave the box. To do so, consde the facet: Now fo each we defne: Snceϕ S ={n : n = n 0, n = n 0 }. ϕ = max{ϕ (n) : n = 1, n = 1}, ϕ = mn{ϕ (n) : n = 1, n = 1}. s contnuous by Lemma 5.1 n the compact set of the ght, these values ae well defned. Moeove,ϕ > 0 snce n > 0. If n S, usng Lemma 5.2, we have: snce U s deceasng. n ϕ (n)=ϕ ϕ U n 0 (ϕ (n)) U (ϕ ), (5.3) We also have fo any n S : n q (n) 1/α ϕ ϕ =ϕ (n)=n f T C P (q (n))=n 0, n 0 w and thus: q (n) w n α 0 ϕ α. (5.4) Now choose n 0 such that fo evey oute the followng s satsfed: w ϕ α n α 0 > U (ϕ ). (5.5) Then, fo that choce of n 0 and evey n S we can combne (5.3), (5.4) and (5.5) to yeld: U (ϕ (n)) U (ϕ )< kα ϕ α n α 0 q (n), and theefoe the admsson condton s not satsfed. The tanston n n+e s not allowed n S and thus the system must eman n the set S.

76 CHAPTER 5. UTILITY BASED ADMISSION CONTROL µ 1 ϕ 1 (n) λ 1 µ 1 ϕ 1 (n) 10 8 µ 2 ϕ 2 (n) µ 2 ϕ 2 (n) n 2 6 λ 2 λ 2 4 µ 1 ϕ 1 (n) µ 1 ϕ 1 (n) λ 1 2 µ 2 ϕ 2 (n) µ 2 ϕ 2 (n) n 1 FIGURE 5.1: STATE SPACE, ALLOWED TRANSITIONS AND ADMISSION CONTROL CURVES FOR A SINGLE BOTTLENECK LINK WITH TWO USERS.α=2 AND U = log(ϕ ). As an example, n Fgue 5.1 we plot the state space and admsson contol cuves of the model defned by (5.2) fo the case of a sngle bottleneck lnk wth two uses,α=2 and use utltes U (ϕ )=log(ϕ ). 5.3 Flud lmt devaton fo the sngle path case Now that stablty s assued, we poceed to analyze the faness of the admsson contol polcy. We wll do so by devng a sutable flud model fo the system (5.2). The model s based n what we call lage netwok asymptotc. The man dea s to scale the netwok sze appopately, by enlagng the capacty of the lnks and the aval ate of flows, such that a law of lage numbe scalng occus. An mpotant emak s that, fo the scalng to wok appopately, we also have to scale the use pefeences wth the sze of the netwok. Moe fomally, we take a scalng paamete L > 0 and consde a netwok wth lnk capactes scaled by L,.e. c L l = Lc l. We also assgn each use a utlty U L(ϕ)=LU (ϕ/l). Note that

77 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 64 wth ths choce, the utlty functons vefy the followng scalng popety: (U L ) (Lϕ)= U (ϕ). That s, the use magnal utlty fo obtanng L tmes bandwdth n the scaled netwok s the same that the magnal utlty fo the ognal amount n the ognal netwok. We denote byϕ L (n) and q L (n) the ate allocaton and oute pces n the scaled netwok, and as befoe ϕ(n) and q(n) denote the ognal values,.e. L = 1. The followng elatonshps hold: Lemma 5.3. Fo any L> 0, the esouce allocaton and oute pces satsfy: ϕ L (Ln)= Lϕ (n), q L (Ln)= q (n). Poof. We ecall thatϕ (n) and q (n) satsfy the KKT condtons fo Poblem 3.1: U T C P n ϕ (n) = q (n) wth n > 0, p l (n)(y l (n) c l )=0 l. We now want to vefy that the choceϕ L (Ln)=Lϕ (n) and p L l (Ln)=p l(n) satsfy the KKT condtons of the scaled veson of Poblem 3.1,.e. wth c l eplaced by Lc l and n by Ln. Notng that wth the above choce y L l (Ln)= Ly l(n) and usng the above equatons we have: ϕ U L (n) T C P = U ϕ (n) T C P = q (n)= q L Ln n (Ln) wth Ln > 0, p L l (Ln)(y L l(ln) Lc l )=p l (n)(ly l (n) Lc l )= Lp l (n)(y l (n) c l )=0 l. Theefoe,ϕ L and p L satsfy the KKT condtons of the scaled veson of Poblem 3.1, and theefoe ae the esouce allocaton and oute pces of the scaled netwok. Usng the above elatonshps, we now deve the flud model of the system. To avod techncaltes, we shall eplace the ndcato functon n (5.2) by a smooth appoxmaton f ε, such that f ε (x)=1f x>ε, and f ε (x)=0f x< ε. The ognal model can theefoe be appoxmated by: n n+ e wth ateλ f ε (U (ϕ ) q ), n n e wth ateµ ϕ. (5.6) Note that, asε 0 the above model appoxmates (5.2). We have the followng esult:

78 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 65 Theoem 5.1. Consde a sequence of pocesses n L (t) govened by equatons (5.6) wthλ L = Lλ, c L l = Lc l,µ L =µ, and utlty functons that satsfy U L (ϕ)= LU wth L> 0 a scalng paamete. Consde also a sequence of ntal condtons n L (0) such that ϕ n L (0)/L has a lmt n(0). Then, the sequence of scaled pocesses: L n L (t)= n L (t) L s tght and any weak lmt pont of the sequence whee n > 0 conveges as L to a soluton of the followng dffeental equaton:, ṅ =λ f ε (U (ϕ ) q ) µ ϕ, (5.7) whee ϕ(n) and q(n) ae the allocaton maps fo L = 1. Poof. The poof s vey smla to the flud lmt esult fom[kelly and Wllams, 2004], but wth addtonal consdeatons fo the admsson contol tem. We shall use the followng stochastc epesentaton of the pocess n L (t), n tems of standad Posson pocesses wth a tme scalng. Consde{E (t)} and{s (t)} to be a famly of ndependent Posson pocesses of ntenstesλ andµ espectvely. Consde also the followng pocesses: τ L (t)= t 0 T L (t)= t 0 f ε (U L (ϕl (n L (s))) q L (n L (s)))d s, ϕ L (n L (s))d s Heeτ tacks the amount of tme the admsson condton s satsfed, and T tacks the amount of sevce povded to oute. The Makov chan evoluton (5.6) of n L (t) statng at n L (0) can be wtten as: n L (t)=n L (0)+ E (Lτ L (t)) S (T L (t)), whee the tem Lτ L (t) comes fom the fact that the aval ate of n L s Lλ. Defne T L = T L /L. Now fo each and t 0: T L (t)= 1 L = t 0 t 0 ϕ L (n L (s))d s= 1 L ϕ ( n L (s))d s, whee we have used Lemma 5.3 foϕ L. t 0 ϕ L (L n L (s))d s= 1 L t 0 Lϕ ( n L (s))d s

79 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 66 We also have: τ L (t)= t = = = 0 t 0 t 0 t 0 f ε (U L (ϕl (n L (s))) q L (n L (s)))d s f ε (U L (ϕl (L n L (s))) q L (L n L (s)))d s f ε (U L (Lϕ ( n L (s))) q ( n L (s)))d s f ε (U (ϕ ( n L (s))) q ( n L (s)))d s. Theefoe, the pocess n L satsfes the followng: n L (t)= n L(0) L t T L (t)= L E (Lτ L (t)) 1 L S (L T L (t)), ϕ ( n L (s))d s, t τ L (t)= f ε (U (ϕ ( n L (s))) q ( n L (s)))d s. 0 The concluson now follows fom the same aguments n[kelly and Wllams, 2004] applyng the hypothess fo n L (0) and the functonal law of lage numbes fo the pocesses E and S, namely, 1 L E (Lu) λ u and 1 L S (Lu) µ u. Note that the functons T ae Lpschtz snce they ae the ntegal of a bounded functon (ϕ s bounded by the maxmum capacty n the netwok). Also,τ L s a Lpschtz functon because f ǫ s bounded by 1. So any lmt pont of n L must satsfy: t t n= n(0)+λ f ε (U (ϕ ( n(s))) q ( n(s)))d s µ ϕ ( n(s))d s. 0 Dffeentatng the above equaton at any egula pont of the lmt, we ave at: 0 n=λ f ε (U (ϕ ( n(t))) q ( n(t))) µ ϕ ( n(s)), whch s the desed esult. 5.4 Equlbum and stablty of the flud lmt We now analyze the system n the flud lmt. We would lke to chaacteze ts equlbum and pove stablty esults. Fom now on we focus on the flud dynamcs: ṅ =λ f ε (U (ϕ ) q ) µ ϕ, (5.8)

80 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 67 wheeϕ=ϕ(n) and q= q(n) ae the allocaton maps defned though the TCP Congeston Contol Poblem 3.1, and n les n the postve othant. We wll assume that the use aveage loadsρ =λ /µ do not vefy the netwok capacty constants,.e. the netwok wthout admsson contol s unstable. If t s stable, admsson contol s not needed and the flud lmt wll convege to n= 0 by the same aguments of [de Vecana et al., 1999, Bonald and Massoulé, 2001]. The equlbum condton fo (5.8) s: wheeρ s use aveage load. ϕ =λ µ f ε (U (ϕ ) q )=ρ f ε (U (ϕ ) q ), Snce f ε s bounded above by 1, the only possbltes ae: ϕ <ρ & U (ϕ ) q <ε o ϕ =ρ & U (ϕ )>q +ε. Asε 0 the above tanslate to: ϕ <ρ & U (ϕ )= q o (5.9a) ϕ =ρ & U (ϕ )> q. (5.9b) The ntepetaton of the above condtons s the followng: ethe the equlbum allocaton fo use s less than ts demand, and the system s n the bounday of the admsson condton, o the use s allocated ts full aveage demand and admsson contol s not appled. We would lke to elate ths to the Use Welfae Poblem 4.1 defned befoe. Consdeed the followng modfcaton of Poblem 4.1: Poblem 5.1 (Satuated Use Welfae). max U (ϕ ), subject to: Rϕ c, ϕ ρ. Poblem 5.1 has the followng ntepetaton: allocate esouces to the dffeent uses accodng to the utlty functons, but do not allocate a use moe than ts aveage demand. It amounts to satuatng the uses to a maxmum possble demand, gven by the valueρ. We have the followng:

81 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 68 Poposton 5.2. The equlbum ponts of of (5.8) asε 0 convege to the optmum of Poblem 5.1. Poof. Asε 0, the equlbum of (5.7) vefes equatons (5.9). We wll show that these ae equvalent to the KKT condtons of Poblem 5.1. The Lagangan of Poblem 5.1 s: (ϕ,p,η)= U (ϕ ) p T (Rϕ c) η T (ϕ ρ), whee p as befoe ae the lnk pces,ηs the vecto of multples fo the addtonal constant, andρ=(ρ ). The KKT condtons ae: U (ϕ )=q +η, (5.10a) p l η (ϕ ρ )=0, (5.10b) R l ϕ c l = 0 l. (5.10c) Now consde an equlbum allocaton ˆϕ that vefes (5.9). The last condton above s always satsfed by ˆϕ snce t s one of the KKT condtons of the TCP congeston contol poblem that defnes n ϕ. We can takeˆη = U (ˆϕ ) ˆq. Wth that choce, the tplet(ˆϕ,ˆq,ˆη) satsfes the KKT condtons (5.10). The fst condton s satsfed automatcally. Moeove, condton (5.9a) mplesˆη=0 and then the second KKT condton s vald n that case. Also, (5.9b) mplesˆη > 0 but at the same tmeˆϕ=ρ, so the second KKT condton s also vald n ths case. Theefoe, the equlbum allocaton unde admsson contol n an oveloaded netwok s a soluton of Poblem 5.1. Note that f the taffc demands ae vey lage (ρ ), Poblem 5.1 becomes the ognal Use Welfae Poblem 4.1, and admsson contol s mposng the desed noton of faness. Moeove, f some uses demand less than the fa shae accodng to Poblem 4.1, the esultng allocaton potects them fom the oveload by allocatng these uses the mean demand, and shang the est accodng to the use utltes. In Secton 5.6 we shall gve a numecal example of ths behavo. We now tun ou attenton to the stablty of the equlbum. We shall pove a local stablty esult by usng the same passvty aguments analyzed n Secton 4.3. To do so, we wll lft the tme scale sepaaton assumpton, but nevetheless assume that the dynamcs (5.8) s stll a good appoxmaton of netwok behavo. The esult s the followng: Poposton 5.3. Assume that utlty based admsson contol can be modelled by the dynamcs (5.8), and the lnks pefom the dual dynamcs pce adaptaton (2.11). Then the equlbum of the system s locally asymptotcally stable.

82 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 69 Poof. The dea of the poof s to show that the system (q q ) (ϕ ϕ ) lneazed aound the equlbum s stctly passve, snce the emanng half of the contol loop(ϕ ϕ ) (q q ) s passve, as shown by[wen and Acak, 2003], and dscussed n Secton 4.3. To do so, obseve that n the sngle path case, the use pat of the system s dagonal. We focus on one use and dop the subscpt accodngly. The Laplace doman veson of (5.8) aound equlbum s: sδn=λa( bδϕ δq) µδϕ = (abλ+µ)δϕ λaδq (5.11) whee a= f ε (U (ϕ ) q ) 0 and b= U (ϕ )>0 ae constants. Note also that: δϕ= n δx+ x δn= d n δq+ x δn (5.12) whee we have used thatδx= dδq wth d= f T C P (q )>0. Combnng (5.11) and (5.12) we obtan the tansfe functon fo the system: δϕ δq = n d s+λax s+ x (abλ+µ) = H(s) Fo smplcty we focus on the case whee a> 0, whch coesponds to the stuaton whee admsson contol s opeatng, H(s) s stctly postve eal and thus stctly passve (c.f. Theoem A.5). Snce each block n the dagonal s stctly passve we can fnd a sutable stoage functon fo the complete system by addng the stoage functon of each use subsystem, and we can constuct a stctly deceasng Lyapunov functon fo the feedback loop. Ths shows that the equlbum s locally asymptotcally stable. The case whee a = 0 (.e. when the equlbum fully satsfes use demand) can be teated n a smla way by usng moe detaled passvty esults fo the netwok sde (c.f. [Wen and Acak, 2003, Theoem 2]). As n the case of coopeatve contol studed n Secton 4.3, global stablty esults ae hade to obtan. 5.5 Admsson contol n the mult-path case We now analyze how to extend the esults on admsson contol to a stuaton whee the use opens connectons on seveal paths, and gets an utlty fo the aggegate. We assume that connecton avals on each path ae ndependent, followng a Posson pocess of ntensty λ, and wth exponentally dstbuted wokloads of mean 1/µ. Thus, the lowe layes of the netwok allocate esouces as n the sngle path stuaton, and n patcula each oute has an

83 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 70 aveage loadρ =λ /µ. Howeve, we shall contol the numbe of connectons n each path takng nto account the aggegate ate each use peceves though all ts paths. The logcal extenson of the admsson contol ule (5.1) to the case whee uses may open connectons along multple outes s the followng: If U (ϕ )> q admt connecton on oute. If U (ϕ ) q dscad connecton on oute. (5.13) wheeϕ = ϕ, as befoe. The analyss of the above polcy s smla to the sngle path case. Pefomng a flud lmt scalng of the undelyng Makov pocess, the flud lmt dynamcs of the system become: ṅ =λ f ε (U (ϕ ) q ) µ ϕ. (5.14) And we have the analogue of Poposton 5.2. Consde the followng poblem: Poblem 5.2 (Satuated Use Welfae (mult-path)). max U (ϕ ) subject to: Rϕ c, ϕ ρ We have the followng: Poposton 5.4. The equlbum ponts of (5.14) convege, as ε 0, to the optmum of Poblem 5.2. Stablty esults of ths equlbum, howeve, poved hade to obtan. We shall exploe the behavo of ths polcy though packet level smulaton n the followng secton. 5.6 Smulaton examples In ths secton we dscuss pactcal mplementaton ssues and exploe the pefomance of the polces developed though ns2 smulatons. We pesent two scenaos of admsson contol, and show how the desed level of faness s attaned. We mplemented n ns2 seveal mechansms: fst we mplemented a Use class that feeds the netwok wth a Posson pocess of avng connectons. Moeove, each connecton sze s an ndependent exponental andom vaable, as n the model of Secton 5.1. Each connecton s establshed n an ndependent way by usng a new nstance of TCP-Neweno.

84 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 71 At the netwok edge, the aggegate ate of each use s measued, as well as the loss pobablty, whch s the oute pce. The loss pobablty s estmated fom the numbe of etansmtted packets, and aveaged between all the connectons shang the same oute. Ths s done n ths case by snffng the connectons, and appoach that we undestand can be dffcult on eal netwoks. Howeve, ecent wok on explct congeston notfcaton (c.f. Re-ECN [Bscoe, 2008]) can also be used to communcate congeston pces to the edge oute, and can be a possble soluton to ths poblem. Admsson contol s pefomed by the ule (5.13), whee the utlty functon can be chosen fom theα famly Sngle path scenao Ou fst example conssts of a lnea pakng lot netwok, wth thee sngle path uses. We choose the capactes as n Fgue 5.2, and admsson contol s pefomed wthα=5, to emulate the max-mn fa allocaton, whch fo ths netwok s gven byϕ,1 = 5Mbps,ϕ,2 =ϕ,3 = 3Mbps. Use 3 C1=8Mbps C2=6Mbps Use 1 Use 2 FIGURE 5.2: PARKING LOT TOPOLOGY. In ou fst smulaton, the use loads ae such that the netwok s oveloaded, wth each use load geate that ts fa shae. The esults show that the uses ae admtted n 1 5, n 2 8 and n 3 15 smultaneous connectons n equlbum, wth total ates shown n Fgue 5.3. Thee we see that the max-mn allocaton s appoxmately acheved. Obseve that, despte havng the same equlbum ate as use 2, use 3 s allowed moe connectons nto the netwok because ts RT T s hghe, and thus ts connectons slowe. We smulated a second stuaton n whch we educed the aveage load of use 2 to below ts fa shae accodng to Poblem 4.1. In ths case,ρ 2 = 1Mbps and s theefoe satuated n the sense of Poblem 5.1. The new aveage ates fo ths stuaton ae shown n fgue 5.4. Note that admsson contol potects use 2 by allowng ts shae of 1Mbps nto the netwok. The emanng capacty s allocated as n Poblem 5.1, whose optmum sϕ,1 =ϕ,3 = 4Mbps, andϕ,2 = 1Mbps.

85 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 72 6 Use aggegate ate 5 4 Mbps Use 1 Use 2 Use Tme (sec) FIGURE 5.3: ADMISSION CONTROL RESULTS FOR THE OVERLOAD CASE. 5 Use aggegate ate 4 Mbps 3 2 Use 1 Use 2 Use Tme (sec) FIGURE 5.4: ADMISSION CONTROL RESULTS WHEN USER 2 IS BELOW ITS FAIR SHARE Mult path scenao In ths case, we etake the settng of a p2p downloadng system analyzed n Secton 4.5. The topology s the same, shown agan n Fgue 5.5, but now we ely on admsson contol to mpose faness, nstead of use coopeaton and contol. As befoe, uses geneate connectons as a Posson pocess n each oute and the loads ae chosen such that the netwok s n oveload. Admsson contol s pefomed at the edge of the netwok, takng nto account the total ate of each use, wth utltes chosen to obtan popotonal faness. The esults ae plotted n Fgue 5.6, whee we can see the numbe of ongong connectons and the esultng use aggegate ates. Note that the numbe of connectons s much nose that n the coopeatve example, due to the nheent andomness of connecton aval and depatues. Howeve, admsson contol manages to dve the aveage use ates n equlbum aound the fa shae of the Use Welfae Poblem 4.1, whch s agan n ths case ϕ 1, =ϕ 2, = 10Mbps. Note also that n the absence of admsson contol, the numbe of connectons would gow wthout bounds.

86 CHAPTER 5. UTILITY BASED ADMISSION CONTROL 73 Use 1 C 1 = 4 Mbps C 2 = 10 Mbps C 3 = 6 Mbps Use 2 FIGURE 5.5: DOWNLINK P2P EXAMPLE TOPOLOGY Use 1,Route 1 Use 1,Route 2 Use 2, Route 1 Use 2, Route 2 Numbe of admtted flows (n) 10 5 Mbps Tme (sec) Use aggegate ate Use 1 Use Tme (sec) FIGURE 5.6: DOWNLINK P2P EXAMPLE RESULTS: NUMBER OF CONNECTIONS (ABOVE) AND USER AGGRE- GATE RATES (BELOW.

87 6 Connecton level outng The analyss of the pecedng chapte assumes that each use establshes connectons though some set of pedefned outes, possbly wth multple destnatons. The use manages smultaneously seveal connectons ove these outes and deves a utlty fom the aggegate ate. Moeove, the use has an ndependent aval ateλ fo each oute. In ths chapte we focus on a slghtly dffeent stuaton: hee, each use has a set of outes avalable to communcate wth a gven destnaton n the netwok. These outes ae ndffeent fo the use, because all of them seve the same pupose. Each use bngs connectons nto the netwok, and at each connecton aval, the use o the edge oute may decde ove whch oute send the data. Ths s a typcal nstance of the mult-path load balancng poblem, but at the connecton level tmescale. The poblem of balancng the load between seveal possble outes has been analyzed befoe n seveal ways. In[Pagann and Mallada, 2009], the combned poblem of congeston contol and mult-path outng was studed, whee the end pont enttes and the netwok shae esponsblty on the ate and pce adaptaton mechansms n ode to fnd a sutable optmum. Howeve, n that wok, the numbe of ogn-destnaton pas s assumed fxed, and wth only one connecton. Fom the connecton level pespectve, n [Han et al., 2006] an algothm called Mult-path TCP was ntoduced, whch assumes that each connecton s seved by a congeston contol potocol that uses all all the possble paths smultaneously. Ths algothm enables to make an effcent use of the avalable capacty, but

88 CHAPTER 6. CONNECTION LEVEL ROUTING 75 t has seveal mplementaton poblems, due to the fact that dffeent packets of the same flow tavel though multple outes, causng eodeng poblems, so exta contol taffc s needed fo the system to wok. Fnally, an nteestng class of load balancng technques was studed n[jonckheee, 2006], called balanced outng, whch fnds an optmal andom splttng of the taffc usng the state of the netwok. Howeve, ts applcablty s educed because t assumes that the netwok esouce allocaton belongs to the class of balanced faness, whch s not n use n eal netwoks snce t s dffcult to decentalze. What we popose s to use the oute pces as a sutable ndcato of congeston ove the outes, and use them to deve a smple decentalzed algothm to assgn connectons to outes. We wll ely then on standad TCP congeston contol algothms to contol the ate of each connecton, mantanng each ndvdual connecton along a sngle path. Ths chapte s oganzed as follows: n Secton 6.1 we dscuss a stochastc model and ts flud veson fo outng polces n a netwok. We then show n Secton 6.2 a necessay condton fo stablty, genealzng the stablty condton dscussed n Secton 3.3 to the mult-path case. In Secton 6.3 we popose a decentalzed outng polcy based on the oute pces and analyze ts stablty n some topologes. Fnally, n Secton 6.4 we combne the outng polcy wth the admsson contol ules ntoduced n Chapte 5 to gve a combned utlty based outng and admsson contol polcy fo the system. 6.1 Connecton level outng polces We wll make a model of the system based on the stochastc model fo connectons descbed n Secton 3.3. We assume that uses, ndexed by, have multple outes avalable to seve the jobs. Uses geneate ncomng connectons as a Posson pocess of ntenstyλ, and exponental fle szes wth mean 1/µ. Thusρ =λ /µ epesents the use aveage load. Note that hee we do not dstngush between the outes, snce each use may be seved by a set of possble outes. As fo congeston contol, we assume that the TCP laye behaves as descbed n 3.1, that s, fo each numbe of ongong connectons, the TCP laye allocates esouces by optmzng Poblem 3.1. The use o the netwok may now choose, at the stat of the connecton, to whch oute send the data fom the set, but note that each connecton behaves ndependently afte that. In patcula each connecton follows a sngle specfed path thoughout ts sevce. Nevetheless, by appopately choosng the oute, the load may be dstbuted acoss multple paths. We fomalze a outng polcy n the followng way: gven the cuent state of the netwok, chaactezed by the vecto n=(n ) of ongong connectons pe oute, a outng polcy s a selecton fo a new connecton. We denote by A N N the set of netwok states such that

89 CHAPTER 6. CONNECTION LEVEL ROUTING 76 connectons avng fom use ae outed though oute. If the same physcal oute s possble fo many uses, we duplcate ts ndex accodngly, and N s the total numbe of outes. The only geneal equement fo the outng polcy s that the sets A ae a patton of the space fo each,.e.: 1 A 1. (6.1) Let us make a flud model of ths system, unde the same deas of Chapte 5. In a flud lmt, the dynamcs of the numbe of connectons s gven by: ṅ = + λ 1 A µ ϕ n. (6.2) Hee,ϕ =ϕ (n) as befoe denotes the total ate assgned to the flows on oute dependng on netwok state. The satuaton( ) + n s needed n ths case because some outes may not be used, and thus the numbe of flows must eman at 0. Remak 6.1. Note that we could have also consdeed moe geneal outng polces, n whch each outng decson s assgned a pobablty p (n) fo each netwok state. The outng polcy constant (6.1) n that case wll be the same. Howeve, n the followng we wll only focus on detemnstc outng polces. Note also that the sets A may be qute geneal. Howeve, fo pactcal mplementaton, t s necessay that the outng polcy s decentalzed,.e. the decson of outng a flow of use ove oute should be pefomed on local nfomaton. We defe ths dscusson to Secton 6.4 and focus now on necessay condtons fo stablty of the system. 6.2 A necessay condton fo stablty Ou goal s to chaacteze the stablty egon of a outng polcy, wth dynamcs gven by (6.2). Moe fomally, we would lke to know fo whch values ofρ the flud model goes to zeo n fnte tme. We ecall that fnte tme convegence of the flud model s elated (c.f.[robet, 2003, Chapte 9]) to the stochastc stablty of the coespondng Makov chan models. We wll fst deve a necessay condton fo stablty, whch genealzes the stablty condton of (3.5) to the outng case. Fo ths pupose, ntoduce fo each use the smplex of possble taffc factons among avalable outes: = α =(α ) :α 0, α = 1. The followng s the man esult of ths secton:

90 CHAPTER 6. CONNECTION LEVEL ROUTING 77 Theoem 6.1. A necessay condton fo the exstence of a polcy{a } that stablzes the dynamcs (6.2) s the exstence fo each use of a spltα such that: R l α ρ c l l. (6.3) l Remak 6.2. Note that condton (6.3) s the non-stct veson of (3.5) fo the spltted taffc loadsρ =α ρ. Thus, equaton (6.3) means that fo a outng polcy to exst, t s necessay that the netwok s stablzable, n the sense that thee s a patton of the use loads such that the undelyng sngle path netwok s stable. Of couse, f each use has only one possblty, then ={1} and we ecove the sngle path stablty condton. Condton (6.3) was obtaned n[han et al., 2006] fo stochastc stablty n the case of Mult-path TCP. In that case, howeve, the TCP laye must be modfed to make full smultaneous use of the avalable outes. Hee each oute emans sngle-path, wth standad TCP congeston contol, and the outng polcy s used to acheve the same stablty egon, wheneve possble. Poof of Theoem 6.1. Consde the convex and compact subset ofr L, wth L the total numbe of lnks: = z R L : z l = R l α ρ c l,α The set epesents the excess of taffc n each lnk fo each possble splt. If (6.3) s not feasble fo a gven load vectoρ, then the set s dsjont wth the closed negatve othantr L. By convexty, thee exsts a stctly sepaatng hypeplane (c.f.[boyd and Vandebeghe, 2004, Secton 2.5]),.e. a fxedˆp R L wthˆp 0 and such that: ˆp T z a+ε z, ˆp T z a z R L. The second condton mples n patcula thatˆp has nonnegatve entes, snce fˆp l < 0, takng z l we haveˆp l z l + and theefoe the nequalty s volated fo any a. Also, snce z= 0 R L, we have that a 0. Defne nowˆq= R Tˆp and consde the followng state functon: 1 V(n)= n. µ ˆq Note thatˆq 0 andˆq 0. Dffeentatng V along the tajectoes of (6.2) we get: 1 1 V= ṅ = [λ 1 A µ µ µ ϕ ] ˆq ˆq + n 1 λ 1 A µ µ ϕ, ˆq

91 CHAPTER 6. CONNECTION LEVEL ROUTING 78 whee n the last step we used the fact that the tem nsde the pojecton s negatve wheneve the pojecton s actve. Reodeng the tems we ave at: V ˆq l ρ 1 A ϕ = R l ˆp l ρ 1 A R l ˆp l ϕ ˆp l R l ρ 1 A ˆp l c l. l In the last step, we used the fact that R l ϕ c l due to the esouce allocaton beng feasble. Regoupng the tems we ave at: V l ˆp l R l ρ 1 A c l a+ε>0, snce z l = R l ρ 1 A c l by the defnton of outng polcy. Theefoe, V s stctly nceasng along the tajectoes, and beng a lnea functon of the state, f the condton (6.3) s not met the numbe of connectons gows wthout bounds. Thus (6.3) s necessay fo stablty. Remak 6.3. Note that the above Theoem emans vald f we change the polcy 1 A andom splttng polcy, possbly dependent on the state p (n), snce p (n) must also be n the set fo all n. We theefoe have shown that f taffc cannot be splt among the outes such that each lnk s below ts capacty on aveage, then the system cannot be stablzed unde any outng polcy. The analogue to the suffcent stablty condton (3.5) n ths case s: l l by a, α : R l α ρ < c l l, (6.4) l whch s the stct veson of (6.3). The followng poposton s dect fom the sngle-path stablty Theoem 3.2: Poposton 6.1. If (6.4) holds, then thee exsts a outng polcy that stablzes the system. Poof. If such a choce ofα exsts, then the andom splttng polcy that sends an ncomng connecton on oute wth pobabltyα stablzes the system. Ths s because the system s equvalent to a sngle path pocess wth aval atesλ α due to the Posson thnnng popety, and condton (6.4) s the stablty condton of Theoem 3.2 n that case.

92 CHAPTER 6. CONNECTION LEVEL ROUTING 79 The above shows that the stablty egon of ths system s chaactezed completely. Howeve, the andom splttng polcy mentoned n the poof of Poposton 6.1 s not useful n a netwok envonment, snce t s not decentalzed. Each use must know n advance the aveage loads of the whole system n ode to select the weghtsα accodngly, n ode to fulfll (6.4). We would lke to ntoduce a sutable feedback polcy that takes nfomaton fom the netwok to decde on whch oute send an ncomng connecton. We popose such a polcy n the next secton, and analyze ts stablty unde condton (6.4). 6.3 A decentalzed outng polcy In a mult-path settng n whch each use may choose among a set of outes, t s natual to ty to balance the loads by usng the netwok congeston pces measued on each oute. A smple feedback polcy fo outng s, at each aval, choose the oute wth the cheapest pce. In ou pevous notaton ths amounts to take: A = n : q (n)= mn q (n). (6.5) Implct n the above equaton s some ule fo beakng tes when thee ae multple outes wth mnmum pce. Snce congeston pce s nvesely elated wth connecton ate, ths s equvalent to outng new flows to the path wth the best cuent ate fo ndvdual connectons. Note also that, n the case of equal paallel lnks wth smla TCP utltes, t can be seen as a genealzaton of the Jont the Shotest Queue polcy fo pocesso shang queues. We shall nvestgate the stablty of ths polcy unde condton (6.4). We need the followng Poposton: Poposton 6.2. Gven n=(n ) R N +, letϕ (n) and q (n) be the coespondng ate allocaton and oute pces fom the TCP Congeston Contol Poblem 3.1. Choose alsoα that satsfes (6.4). Then thee exsts δ > 0 such that: q (n)α ρ :n >0 q (n)ϕ (n) δ q (n). (6.6) To pove the above Poposton, we must deal wth the quanttes q (n)ϕ (n) when n 0. We have the followng: Lemma 6.1. Assume that n s such that n = 0. Consde n ε = n+εe Then q (n ε )ϕ (n ε ) 0 whenε 0. Poof. Fom the devaton of Theoem 3.1 we know that fo n> 0: q n = R T (RDR) 1 R F= B F,

93 CHAPTER 6. CONNECTION LEVEL ROUTING 80 wth D, F> 0 and dagonal. Obseve that B 0 and theefoe b 0. We conclude that n the egon n> 0: q n 0 and thus q (n ε ) deceases asε 0, theefoe t has a lmt q 0. If q 0 = 0 we ae done snceϕ (n) s bounded by the maxmum lnk capacty. If q 0 > 0 then x (n ε )= f T C P (q (n ε )) emans bounded and theefoeϕ (n ε )=n ε x (n ε ) 0. Poof of Poposton 6.2. Snce the nequalty n (6.4) s stct, we can chooseδ>0 such that: satsfes the lnk capacty constants Rψ c. ψ =α ρ +δ Consde now n > 0. Snce ϕ(n) s the optmal allocaton, and ψ s anothe feasble allocaton fo the convex Poblem 3.1, we have that the nne poduct: ϕ n U TC P (ϕ /n ) (ψ ϕ) 0. In fact, f t wee postve, we can mpove the solutonϕ by movng t n the decton ofψ. Applyng the above condton toψ =α ρ +δ and notng that: n U TC P (ϕ /n ) = U ϕ T C P = q (n), ϕ n we conclude that: whch can be ewtten as: q (n) α ρ +δ ϕ (n) 0, q (n)α ρ q (n)ϕ (n) δ q (n) whch poves the esult fo n> 0 componentwse. esult. If now n s such that n = 0 fo some, we can take lmts and nvoke Lemma 6.1 to get the The pevous bound s smla to the one used n[bonald and Massoulé, 2001] to pove stablty n the sngle path scenao, but wth the gadent evaluated at the optmum, nstead of anothe feasble allocaton. We now apply ths bound to obtan the followng chaactezaton of the outng polcy: Theoem 6.2. Suppose (6.4) holds. Then unde the dynamcs (6.2) wth the outng polcy gven by (6.5) we have: q ṅ δ q (6.7) µ

94 CHAPTER 6. CONNECTION LEVEL ROUTING 81 Poof. We have that: q ṅ = q ρ + 1 A µ ϕ q n ρ 1 A :n >0 q ϕ, whee the nequalty s tval f the pojecton s not actve. If the pojecton s actve, n = 0 and thus q ϕ = 0 so the coespondng negatve tem vanshes. Regoupng the above n each use we get: q ṅ ρ q 1 A µ q ϕ :n >0 = ρ mnq q ϕ :n >0 ρ q α q ϕ, :n >0 whee we have used the defnton of the outng polcy and the fact that the mnmum pce can be uppe bounded by any convex combnaton of the pces. Applyng Poposton 6.2 we complete the poof. It emans to see f we can use the nequalty (6.7) to establsh asymptotc stablty of the flud dynamcs though a Lyapunov agument. Note that, although the Lyapunov functon used n the poof of Theoem 6.1 has a smla shape, and now the nequalty has been evesed, thee s an mpotant dffeence: the q facto of (6.7) s a functon of the state. Ths mples that the left hand sde of (6.7) may not be ntegated easly to get a Lyapunov functon fo the state space whose devatve along the tajectoes yelds the desed esult. We now focus on a specal case whee we can gve an affmatve answe. Assume that the netwok s composed by a set of paallel bottleneck lnks as depcted n Fgue 6.1. Each use n ths netwok has a set of outes establshed n any subset of the lnks. Moeove, assume that all uses have dentcal α fa utltes denoted by U( ) and fle szes ae equal fo each use, so we can take wthout loss of genealtyµ = 1. In such a netwok, the esouce allocaton of Poblem 3.1 can be explctly computed as a functon of the cuent numbe of flows n. In patcula, all flows though bottleneck l face the same congeston pce q = p l, and as they have the same utlty, they wll get the same ate, gven by: By usng that q (n)= U (x (n)) we have: wth l such that l. x (n)= q (n)= x (n) α = c l l n. l n c l α,

95 CHAPTER 6. CONNECTION LEVEL ROUTING 82 Use 1 Lnk 1 Routng Use n Routng Lnk L FIGURE 6.1: PARALLEL LINKS NETWORK WITH ROUTING COVERED IN THE ANALYSIS. Snce q = p l f l the lnk pces should have the fom: p l (n)= : l n c l α. We now state the stablty esult fo ths type of netwok: Poposton 6.3. Fo the netwok of paallel lnks unde consdeaton, let the system be gven by (6.2) wth the outng polcy (6.5). Unde the stablty condton (6.4), the state n conveges to 0 n fnte tme. Poof. Consde the canddate Lyapunov functon: V(n)= 1 α+1 l 1 c α l α+1 n. (6.8) : l The above functon s contnuous and non-negatve n the state space, adally unbounded and s only 0 at the equlbum n= 0. Its devatve along the tajectoes vefes: V= l : l n α c α l ṅ = p l ṅ = q ṅ. (6.9) : l l : l Invokng Theoem 6.2 we conclude that V 0 n the state space, and t s only 0 when all pces ae 0 whch can only happen at the ogn. Ths mples asymptotc stablty of the flud dynamcs.

96 CHAPTER 6. CONNECTION LEVEL ROUTING 83 To obtan fnte tme convegence note that V vefes: V(n)= 1 α+1 whee K> 0 s an appopate constant. We can obtan the followng bound: and thus: V α l α+1 α pl c l (K max p l ) α+1 α, l α+1 K max p l K p l K q, l δ q δ K V α α+1. Combnng ths wth the esult fom Theoem 6.2 we get: V δ K V α α+1. l Integatng the above nequalty yelds: V(t) 1 1+α V(0) 1 1+α δ K(1+α) t, and we conclude that V, and theefoe n, each 0 n fnte tme, popotonal to V(0) 1 1+α. Remak 6.4. Note that n the above poof the tme of convegence to 0 statng fom an ntal condton n(0) scales lnealy wth n snce V(0) s of ode n(0) α+1. We can theefoe conclude that the stochastc system wll also be stable, va the stablty of the flud lmt. 6.4 Combnng admsson contol and outng To end ths chapte, let us analyze the possblty of extendng the admsson contol ules deved n Chapte 5 to the connecton outng settng. Recall that, n ode to pefom admsson contol, we assocate wth each use an utlty functon U (ϕ ) wthϕ beng the total ate the use gets fom the netwok. Admsson contol ove a oute was pefomed by compang the magnal utlty wth the oute pce. In the new settng, the end use may choose among seveal outes, and thus the natual way to mege the esults of 5 wth the connecton level outng s the followng combned law: Admt new connecton f mn q < U (ϕ ) If admtted: oute connecton though the cheapest path The combned flud dynamcs fo admsson contol and outng can then be wtten as: + ṅ = λ 1 A 1 {U (ϕ )>q } µ ϕ. (6.10) n

97 CHAPTER 6. CONNECTION LEVEL ROUTING 84 The above dynamcs should convege to 0 wheneve the stablty condton (6.4) s met. The nteest of admsson contol s n the case whee (6.3) s volated,.e. the netwok s oveloaded. In that case, we have the analogous of Poposton 5.4. Poposton 6.4. The equlbum ponts of (6.10) ae optmal solutons of the followng poblem: subject to: max ϕ U (ϕ ) Rϕ c, ϕ ρ fo each. The poof s smla to 5.4. The above optmzaton poblem s smla to the Satuated Use Welfae Poblem 5.1 but the constantϕ ρ s mposed on the aggegate ate of each use and ts total aveage load, nstead of the aveage load pe oute. Stablty condtons howeve, poved hade to obtan n ths case, and the poblem s open fo futue wok.

98 7 Conclusons of Pat I In ths pat of the thess, we poposed a new paadgm fo esouce allocaton n netwoks, whch ntends to bdge the gap between classcal NUM appled to congeston contol and a use centc pespectve. New notons of faness appea, as use utltes ae evaluated n aggegates of taffc, whch can model dffeent nteestng stuatons. We showed how the contol of the numbe of connectons can be used to mpose these new notons of faness, and how the uses can coopeate n ode to dve the netwok to a fa equlbum. Moeove, we showed how admsson contol and outng based on typcal congeston pces can be used to potect the netwok n oveload, and smultaneously mpose faness between ts uses. Fnally, we showed pactcal mplementatons of the mechansms deved n ou wok, and smulatons based on these mplementatons show that the poposals accomplsh the goals. Seveal lnes of wok eman open, both theoetcal and techncal. Fo nstance, genealzng the stablty esults fo multpath admsson contol, and the stablty egon of the outng polcy poposed ae two mpotant theoetcal questons. In pactcal tems, we plan to exploe new netwok mplementatons based on cuent congeston notfcaton potocols, that wll help make these decentalzed admsson contol mechansms scalable to lage netwoks

99 Pat II Resouce allocaton and connecton level models fo multate weless netwoks 86

100 8 Resouce allocaton n multate weless envonments Weless local aea netwoks, n patcula those based on the IEEE standad (commonly known as WF)[IEEE , 2007], ae becomng nceasngly pevalent to delve Intenet access. Moe and moe use equpments have WF capablty, and thee s an nceasng demand fo bandwdth and communcaton speed n these netwoks. The IEEE standad has been vey successful, standadzng the behavo of the physcal (PHY) and medum access contol (MAC) layes of communcaton. These layes defne how to send the data ove the ado channel, and moe mpotantly, how to shae the common access medum between the dffeent uses n a decentalzed way. The appoach, based on andom access potocols[betsekas and Gallage, 1991], poved successful n wed netwoks such as Ethenet, and s pesent agan n One mpotant featue these netwoks ntoduce s the capablty of usng dffeent tansmsson ates n the physcal laye, n ode to communcate wth statons unde dffeent ado channel condtons. Ths enables a staton to connect to ts access pont (AP) even n bad channel condtons, by educng ts modulaton speed and tansmttng data moe slowly. A cucal pefomance ssue howeve emans to be analyzed: how ths featue nteacts wth hghe laye potocols such as TCP, and how ths mpacts the esouce allocaton among competng uses. The laye was desgned ndependently of the uppe layes of the net-

101 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 88 wok, and thus a coss-laye appoach fo the analyss s needed to examne the effcency and faness of the whole system. The man pupose of ths pat of the Thess s thus to analyze the effcency and faness of cuent potocols unde the Netwok Utlty Maxmzaton pespectve, dentfyng possble poblems, and analyzng some possble solutons. We begn n ths chapte by studyng the esouce allocaton povded by TCP ove a smplfed multate laye. The chapte s oganzed as follows: In Secton 8.1 we dscuss the pevous wok on the subject, and then n Secton 8.2 we pesent ou model fo TCP esouce allocaton n multate netwoks and deve some conclusons. 8.1 Backgound and pevous wok Weless local aea netwoks (WLANs) ae nowadays pesent n most netwokng deployments aound the wold. These hotspots povde Intenet connectvty to neaby uses va the use of the ado fequency (RF) medum. The man standad used nowadays s the IEEE [IEEE , 2007], whch s commonly known as WF. Among othe thngs, the standad defnes the way multple uses access the shaed medum though the use of Medum Access Contol (MAC) potocols. In patcula, the IEEE standad defnes the Dstbuted Coodnaton Functon (DCF) MAC potocol, whch s the most wdely deployed of the possble choces n the standad. Ths potocol defnes a andom access polcy, based on Cae Sense Multple Access wth collson avodance (CSMA/CA). A detaled explanaton of how DCF woks wll be pesented n Chapte 11. Howeve, we wll explan the basc deas hee n ode to motvate the poblem. In a andom MAC potocol such as CSMA/CA, each use tes to access the medum wheneve t senses the medum s not beng used by anothe staton (cae sense). If upon a packet aval, the staton fnds the medum busy, t wll wat untl the end of the tansmsson. When the medum becomes fee, many statons may be watng to tansmt, so the followng dstbuted mechansm s pefomed: each staton wats a andom numbe of tme slots befoe attemptng tansmsson. Ths s done by decementng a backoff counte whch was pevously chosen at andom n some nteval, ndependently by each staton. Ths mechansm allows to desynchonze statons afte the medum becomes fee. When some staton decdes to tansmt, the est wll sense the medum busy and wat fo the next oppotunty. It may also happen that two o moe statons attempt tansmsson at the same tme. In ths case, the ecovey of ndvdual tansmtted fames becomes mpossble: ths s called a collson and the data s lost. Afte each packet tansmsson, an acknowledgment packet s sent f the data aved coectly. A mssng acknowledgment s ntepeted as collson by the statons and the packet s etansmtted at the followng attempt.

102 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 89 In a typcal scenao, the total numbe of statons s not known to each ndvdual staton. It s clea that the backoff counte must be adapted to the numbe of statons, n ode to lowe the access pobablty wheneve the netwok s congested. Ths s pefomed n DCF by an exponental backoff mechansm, n whch the backoff nteval s doubled afte each unsuccessful tansmsson, and eset to a cetan pedefned value wheneve the tansmsson s successful. As we mentoned befoe, afte some staton gets access to the medum, t may tansmt at dffeent data ates, to ensue ecepton at the destnaton. Snce the packet length n Intenet data tansfes s typcally fxed (wth 1500 bytes beng the domnant value), tansmttng at dffeent data ates tanslates nto usng the medum a dffeent amount of tme. Ths, howeve, s neve taken nto account by the DCF mechansm, and as we shall see ths can lead to unfaness n the system. The poneeng wok of[banch, 2000] analyzed the pefomance of DCF n though a caeful modellng of collsons and the backoff esponse pocess stpulated by the standad. Ths led to accuate means of pedctng effectve long tem ates of a set of statons shang the medum. The analyss of[banch, 2000] s howeve lmted, snce t s assumed that all statons have the same data ate (the only one avalable at the tme). Moe ecently n[kuma et al., 2007] the analyss was extended to consde multple physcal data ates pesent n a sngle cell. The analyss povdes an accuate model of the MAC laye unde satuaton assumptons,.e. the tansmttng statons always have packets to send. Ths s, howeve, a vey dealzed model of the uppe laye potocols. Some smple ways to account fo the tanspot laye wee suggested n[kuma et al., 2007, Lebeugle and Poutee, 2005], but thee s no full consdeaton of the TCP congeston contol mechansms that nteact wth the MAC laye to detemne use level pefomance. As we ntoduced n Chapte 2, congeston contol n the uppe layes s commonly modelled though the Netwok Utlty Maxmzaton (NUM) appoach of[kelly et al., 1998]. Ths appoach has been extended n ecent yeas to coss-laye optmzaton n weless netwoks (see fo example[ln et al., 2006, Chang et al., 2007] and efeences theen). A sgnfcant poton of ths lteatue s howeve based on scheduled MAC layes. In these netwoks, coodnaton among tansmttng statons s needed n ode to choose the appopate set of lnks to tun on at each tme, n ode to avod ntefeence. The soluton of ths poblem can be tackled va dual decomposton, and the schedulng component of the poblem can be solved, though t poses dffcultes fo decentalzaton. Moeove, a sepaate queue pe destnaton s needed at each node of the netwok. In the case of andom MAC potocols, most of the wok has been theoetcal, wth an deal-

103 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 90 zed model of the access potocol and patculaly of collsons. In[Lee et al., 2006] a patcula model (based on ALOHA potocols) s analyzed, whle a moe ecent lne of wok developed n [Jang and Waland, 2008, Poutèe et al., 2008] shows how a fne-ganed contol of medum access pobabltes allows n pncple to appoach the pefomance of the scheduled case. Whle ths eseach has deep theoetcal mplcatons, t has lmted mpact n cuent pactce snce t does not epesent potocols such as IEEE Addtonally, we ague that fo weless LAN technologes, the emphass on collsons s msplaced. In ecent vesons of the IEEE standad, the loss of pefomance due to collsons s not as mpotant as one mght expect, due to two man easons: on one hand, most of the taffc s typcally downlnk, wth an Access Pont (AP) povdng connectvty, and sendng taffc to ts attached statons. Ths access pont does not collde wth tself. Secondly, wth the default paametes, collson pobabltes and esoluton tme ae low when compaed to data tansmsson, wheneve the numbe of statons s not too lage. Fa moe elevant to use pefomance s the nteacton between congeston contol potocols lke TCP and the multple PHY data ates. In the followng secton we pesent a fst contbuton towads undestandng ths elatonshp, usng a NUM appoach. Recall that typcal TCP potocols ely on losses to detect congeston. Howeve, n the access pont the buffe s seved at dffeent data ates dependng on the destnaton, and so a efned model of the buffe AP s needed. We wll show that couplng the TCP behavo wth these multple data ates can lead to neffcences n the system, wth uses able to tansmt at hghe data ates beng seveely penalzed. 8.2 TCP esouce allocaton n a multate weless envonment We begn by consdeng a sngle cell scenao whee N uses, ndexed by, ae downloadng data fom a sngle access pont. Each staton establshes a downlnk TCP connecton, and we denote by x the sendng ate. These ates wll be contolled by TCP n esponse to congeston. The packets of each connecton wll be stoed n the nteface queue of the AP, befoe beng put nto the shaed medum to each ts destnaton. The combned PHY and MAC layes offes each destnaton a dffeent modulaton ate, whch combned wth othe factos such as potocol oveheads esults n an effectve ate C offeed to the connecton packets. We assume fo now these ates as gven, and postpone to Chapte 11 the dscusson on how to calculate them n netwoks. The fst step n ou analyss s to detemne the aveage sevce ate y attaned by each staton n such a queue. As we mentoned befoe, we assume all packets havng a fxed length L, and thus the tme needed to tansmt a sngle packet to staton s T = L/C. Typcally the AP buffe s seved wth a FIFO dscplne. Let p HO L, denote the pobablty that the Head of

104 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 91 Lne packet s fo staton. We make the followng assumpton: p HO L, = x meanng that the pobablty of havng a packet fo the staton as HOL s popotonal to the nput ates n the queue. The aveage tme between packet tansmssons s T= j p HO L,j T j. On aveage, staton wll send a packet of length L wth pobablty p HO L, on each packet tansmsson, so the aveage ate fo staton can be calculated as: y = p HO L, L T = j x j, p HO L, L x j p = HO L,j L/C j j x. (8.1) j/c j The above equaton s smla to equaton (5) n[kuma et al., 2007], whch s establshed though smla aguments but consdeng collsons. In (8.1) we omtted the collson tems wnce we ae focusng n the downlnk case. To complete the loop, we now model the TCP behavo that detemnes the nput ates x. Recall fom Chapte 2 that TCP congeston contol algothms can be modelled as pefomng the followng adaptaton: ẋ = k(x )(U (x ) p ), whee U (x) s an nceasng and stctly concave utlty functon, p s the congeston pce (nomally the loss pobablty), and k(x )>0 a scalng facto. In ou analyss we shall estct ouselves to theα famly of utlty functons of Defnton 2.4. A smple model fo the lnk loss ate fo use s the ato between excess nput ate and total ate, namely: + x y + 1 p = = 1 x j x = p, j/c j whee we have used (8.1) fo y, and( ) + = max(,0) as usual. Note that wth ths model, loss pobabltes fo all uses ae equalzed by the buffe. The complete closed loop dynamcs now follows: ẋ = k(x )(U (x ) p), + 1 p= 1 j x. j/c j (8.2a) (8.2b) We would lke to chaacteze the equlbum of these dynamcs n tem of a NUM poblem. Fo ths pupose, consde the followng functonφ:r N + Rgven by: x x Φ(x)= 1 log, C wheneve x > 1 and 0 othewse. C C

105 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 92 Lemma 8.1. Φ s a convex functon of x. Poof. Note thatφ(x) can be wtten as the composton mapφ(x)= g(f(x)) whee f(x)= x /C s a lnea functon of x and g(u)=(u 1 log(u))1 {u>1}, wth 1 A beng the ndcato functon of the set A. It s easy to see that fo u> 0, g (u)=(1 1/u) + whch s a nonnegatve and nceasng functon of u. Theefoe, g s nceasng an convex and thusφs convex [Boyd and Vandebeghe, 2004, Secton 3.2.4]. Consde now the followng convex optmzaton poblem: Poblem 8.1 (TCP Multate Pmal Poblem). max x 0 1 C U (x ) Φ(x) (8.3) The above poblem s n the fom of the pmal netwok Poblem 2.3, the objectve functon beng a total utlty mnus a netwok cost, captued n the functonφ. We have the followng: Theoem 8.1. The equlbum of the dynamcs (8.2) s the unque optmum of Poblem 8.1. Moeove ths equlbum s globally asymptotcally stable. In ode to pove the Theoem, let V(x) denote the objectve functon of Poblem 8.1. The followng popetes of V ae needed: Lemma 8.2. The functon V has compact uppe level sets{x R N + unbounded,.e. lm x V(x)=. : V(x) γ}, and s adally Poof. We splt the poof n seveal cases. Fst we analyze the case 0<α<1, whee we have the followng bound: 1 U (x )= C K K w x 1 α C (1 α) = max x 1 α C x C 1 α. w C α 1 α 1 α x w C Hee, K= N max α and we have used the fact that x 1 α 1 α s nceasng. Now consde y = x /C, and assume that y > 1 so the functonφdoes not vansh. Usng the pevous bound we have: C V(x) K y 1 α y+ 1+log(y)=h(y). (8.4)

106 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 93 Snceα>0, t s clea that h(y) when y. We conclude that V(x) when y, and snce y s smply a weghted L 1 nom ove the postve othant, the same concluson wll hold when x fo any (equvalent) nom. We also have: {x : V(x) γ} {x : y< 1} {x : h(y) γ}, and the above sets ae both compact, the second due to the fact that h s contnuous and h(y) when y. Fo the caseα=1 we use the bound x C y wth y as befoe, and theefoe: 1 C w log x 1 C w logc y K 1 + K 2 log(y), whch gves a bound analogous to (8.4) wth h(y)= K 1 +(K 2 + 1)log(y) y+ 1 whch also goes to fo y, and the est follows. The thd case s even smple snce foα>1 the utlty s bounded above by 0, so we have V(x) y+ 1+log(y) and the esult follows. We now pove the man esult: Poof of Theoem 8.1. By Lemma 8.1 and the stct concavty of the utltes we have that the objectve functon s stctly concave, and so Poblem 8.1 has a unque optmum, that must satsfy the optmalty condtons: 1 U C (x ) Φ(x)=0. x By substtutng the expesson foφand dffeentatng, the above s equvalent to: + 1 U C (x 1 ) 1 j x =0. j/c j Identfyng the last tem as p= p(x) n (8.2) the optmalty condtons become: U (x ) p= 0, whch s the equlbum condton of (8.2). We now consde V(x) as a Lyapunov functon of the system. Dffeentatng along the tajectoes we get: V= V ẋ= k(x ) C (U (x ) p) 2 0, and so V s nceasng along the tajectoes. Moeove, V= 0 only when x= x, the soluton of Poblem 8.1. Theefoe the equlbum s asymptotcally stable. Snce by Lemma 8.2 V s adally unbounded, stablty holds globally.

107 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 94 The functonφn Poblem 8.1 plays the ole of a penalty functon. It nceases to nfnty wheneve x /C > 1. Note that ths last nequalty can be ntepeted n tems of the tme shae allocated to uses. Ths suggests that the above poblem can be ntepeted as an appoxmaton of the followng Netwok Utlty Maxmzaton: Poblem 8.2 (Scaled multate netwok poblem). subject to: 1 max U (x ), x C x C 1. Ths NUM poblem has two vaatons wth espect to the standad Netwok Poblem 2.2. The fst s that the constant s ewtten n tems of x /C, whch as mentoned above can be ntepeted as the tme popoton the shaed medum s used by connecton. The sum of the allocated tme popotons less than one s a natual way to genealze the wed capacty constants to a shaed multate substate. The second mpotant dffeence s the scalng facto 1/C fo the use utlty. Ths s not a natual featue of the poblem, nstead t s a consequence of the congeston contol and esouce shang mechansms actually n use n these netwoks. The man consequence of ths fact s a bas aganst uses of hgh physcal ates, whch ae gven less weght n the net utlty. The effect of ths bas s a adcal equalzaton of ates domnated by the slowest statons, as shown by the followng example. Example 8.1. Assume 3 uses ae downloadng data fom a sngle AP, and they have equal utltes U(x)= 1 whch models TCP/Reno esponse wthτbeng the connecton ound tp τ 2 x tme. Assume the lowe laye effectve ates ae C = 10. In ths case the soluton of Poblem 8.2 s x = fo all thee uses. Now f fo nstance use 3 changes ts ado condtons and obtans a lowe value of effectve ate such as C 3 = 1, the new allocaton s: x 1 = x 2 = x 3 = Note that despte the fact that only one use wosened ts ado condtons, all thee uses ae downgaded, and n patcula the fastest ones ae moe heavly penalzed due to the use 3 neffcency. Ths poblem has been obseved n pactce n envonments, and we wll exhbt t by smulaton n Chapte 11. Note that f nstead of solvng Poblem 8.2 we plug the values nto 8.1, the soluton would be x = 0.89 fo all thus showng that the bae functon appoxmaton s vey close. The key n the above example s that all uses shae the same utlty functon. In that case t s easy to show the followng esult:

108 CHAPTER 8. RESOURCE ALLOCATION IN MULTIRATE WIRELESS ENVIRONMENTS 95 Poposton 8.1. In the case whee all uses shae a common utlty functon U = U, the soluton of Poblem 8.2 s: ates. x = 1 j 1/C. (8.5) j The allocated ates ae theefoe equalzed to the hamonc mean of the effectve data Ths s n accodance wth esults obtaned n[kuma et al., 2007] fo multple ate netwoks, whee collsons ae consdeed. In fact, n that wok the ate n equaton (8.5) appeas as an uppe bound on the ealstc ate of pemanent connectons. Howeve, we emak that n[kuma et al., 2007] the TCP laye s not modelled; athe, t s assumed that the AP has equal pobablty of sevng all uses. Hee we have modelled TCP, and we fnd that (8.5) only holds unde the assumpton of equal utltes, but ndependent of the TCP flavo (the value of α). Howeve, f uses have dffeent utlty functons, as s the case when TCP-Reno connectons have dffeent RTTs, ths soluton s no longe vald, and the allocaton must be calculated by solvng Poblem 8.2. The pecedng dscussons suggest that ways of emovng the atfcal bas fom Poblem 8.2 should be exploed. Ths s the man pupose of the followng chapte.

109 9 Effcency and faness n multate weless netwoks Up to ths pont, we focused on analyzng the esouce allocaton establshed by cuent standad potocols. Ths led us to establsh that n ths case esouces ae allocated as n the Scaled multate netwok Poblem 8.2, whch ntoduces a bas aganst the uses o statons wth moe effectve usage of the shaed medum. We would lke to devse mechansms to dve the system to a NUM nsped esouce allocaton, but wthout the pecedng bas, leadng to a moe effectve, yet fa, esouce shang. Moeove, we would lke these mechansms to be decentalzed, and compatble wth cuent pactces, n patcula wth typcal congeston contol potocols, as well as weless standads, n ode fo them to be deployable. In ths chapte we focus on ths task. Fst, n Secton 9.1 we analyze the case of a sngle cell fom a theoetcal pespectve. Then n Secton 9.2 we descbe a sutable packet level algothm to dve the system to the desed equlbum. Fnally, n Secton 9.3 we dscuss moe geneal topologes, n patcula mxed wed-weless ones, and dscuss how to extend the esults to ths case.

110 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS The sngle cell case The natual optmzaton poblem to solve n the sngle cell scenao s smla to Poblem 8.2 but wthout the tems 1/C that scale the dffeent utltes. We fomulate t as follows: Poblem 9.1 (Weless Multate Netwok Poblem). max U (x ), x subject to the constant: x C 1 (9.1) As fomulated, the above NUM poblem becomes a specal case of those n the lteatue of scheduled weless netwoks (see e.g[ln et al., 2006] and efeences theen), whee the set of feasble ates s taken to be the convex hull of ates achevable by ndependent (nonntefeng) sets of lnks. In the cuent scenao, only ndvdual lnks ae schedulable wthout ntefeence, at ate C. The convex hull then becomes (9.1). Howeve, n contast wth these efeences, we wll seek a soluton to Poblem 9.1 that does not use a complcated schedulng mechansm n the AP, whch would mply a sgnfcant depatue fom cuent netwoks. In ode to do so, let us wte the Lagangan of Poblem 9.1, whch s: x (x,p)= U (x ) p 1. C The KKT condtons fo ths poblem ae: U (x ) p = 0, C x p 1 = 0. C The fst equaton s of patcula nteest: t tells us that even f thee s a sngle pce assocated wth the constant (9.1), each ndvdual connecton must be chaged wth a scaled veson of ths pce, wth the scalng facto 1/C. That s, f we call p = p/c, the pce seen by souce, ths pce wll be hghe fo connectons wth lowe ates, theeby chagng the souces accodng to the own channel neffcency. To see the dffeence wth Poblem 8.2 moe clealy, let us consde agan the case whee all uses shae the same utlty functon U (x)= U(x) fom theα fa famly wth equal weghts w = 1. In ths case, the KKT condtons become: U (x )=x α = p, C x p 1 = 0. C

111 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 98 Snce p> 0, mposng equalty n the second equatons yelds: p 1 α fom whee we obtan the optmal ates: 1 α C 1 = 1, x = C 1 α 1 j C α 1 j. (9.2) We see that, n contast to (8.5), ates ae no longe equalzed n the soluton, and uses wth lage C wll eceve a lage shae of esouces. We analyze now some mpotant cases. Fst we obseve that, when α, we ecove the allocaton (8.5) whch s also the max-mn fa allocaton fo the system. As noted befoe, ths can be vey neffcent, and moeove can heavly penalze the hghe ate uses. Whenα 0we ecove the max-thoughput allocaton, whch s smply to allocate all esouces to the uses wth the hghe effectve ates C. Ths howeve staves all the slowe statons. The caseα=1, whch s popotonal faness, has the followng nce popety, whch s vefed dectly fom equaton (9.2): Poposton 9.1. In the case whee all connectons shae the same utlty U(x)=wlog(x) (α= 1), the equlbum of Poblem 9.1 s: x = C N = 1,...,N. In patcula, the allocated ate fo use depends only on ts own effectve ate and the total numbe of uses n the cell. Equvalently: unde popotonal faness tme s shaed equally among uses, but those wth a moe effcent use of tme (.e. hghe C ) can obtan a popotonally geate ate. Ths potects fast uses fom the lowe ate ones. x If we etun to Example 8.1, we see that when all uses have C = 10, the allocaton s = 3.333, but when use 3 changes ts ate to C 3 = 1, the allocated ates x1 and x 2 do not change, and x3 = 0.333, so only use 3 s penalzed by the change. When utltes ae chosen to epesent TCP-Reno connectons, that sα=2, we do not have complete potecton, but the stuaton s nevetheless mpoved fom the max-mn case. The followng s a numecal example. Example 9.1. Consde agan the same stuaton of Example 8.1, and utltes chosen fom the α fa famly wthα=2 and w = 1. When all uses have C = 10, the optmal allocaton fom

112 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 99 Poblem 9.1 s x allocaton s: = as befoe. When use 3 changes ts ado condtons to C = 1 the x 1 = x 2 = 1.93, x 3 = The total netwok thoughput nceases by 80% wth espect to Example 8.1, and the fastest uses ae not as heavly penalzed. Havng agued that Poblem 9.1 s ndeed a easonable way to allocate esouces, we concentate now n fndng a decentalzed algothm that dves the system to the desed allocaton. 2: Consde the followng pmal-dual algothm, smla to the ones ntoduced n Chapte ẋ = k(x ) ṗ= U (x ) pc x C 1 whee as usual, k(x )>0 and( ) + p s the postve pojecton. + p, (9.3a). (9.3b) The followng Theoem s a dect consequence of the esults n[aow et al., 1958] and [Feje and Pagann, 2009]: Theoem 9.1. The equlbum of the dynamcs (9.3) s the unque optmum of Poblem 9.1. Moeove, ths equlbum s globally asymptotcally stable. Theefoe, congeston contol theoy povdes us wth a satsfactoy algothm to dve the system to the equlbum. Note that the AP must ntegate equaton (9.3b) and poduce a pce p, wheeas each connecton should eact accodng to equaton (9.3a); n patcula each connecton should eact to a scaled veson of the pce p/c, dependng on ts effectve ate. If we want to mplement such an algothm n pactce, the man ssue s how to communcate the C, whch ae dependent on the ado condtons and known only by the AP, to the end ponts of the connectons. Ths s what we study n the followng secton. 9.2 Multate andom ealy detecton The pupose of ths secton s to develop a packet-level algothm that mmcs the behavo of the dynamcs (9.3). The fst step towads developng such an algothm s to fnd an ntepetaton of the magntudes nvolved. We begn wth the pce p. In the wed case, typcal dual algothms have ntepeted the pce vaable as the queueng delay[low and Lapsley, 1999, Low et al., 2002]. Ths s also the case hee, n ths modfed veson. By ntegatng ṗ n equaton (9.3b) we see that p tacks the

113 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 100 amount of tme the shaed medum s not capable of copng wth the demands, and thus the ncomng wokload s accumulated n the queue geneatng delay. Moe fomally, let b denote the amount of data of connecton n the AP buffe (assume t s non empty). Then: ḃ = x y = x x j x j/c j, (9.4) whee we have used equaton (8.1) fo the danage ate y. The delay n the queue fo an ncomng packet wll be the tme needed to seve all the data befoe ts aval. Snce the data b s seved at an effectve ate C we have that: b d=. C By dffeentatng and substtutng the expessons fo ḃ we ave at: ḃ d= = C x C 1. Moeove, when the buffe s empty, d = 0 and t can only ncease (wheneve whch s exactly what s allowed by the postve pojecton. x C > 1), Theefoe, the pce p n equaton (9.3b) can be ntepeted as the queueng delay n the AP. If n patcula all the effectve ates ae equal (C = C ), we ecove the delay based model of[low and Lapsley, 1999, Low et al., 2002] fo wed netwoks. The poblem wth usng the queueng delay as the pce vaable s that thee s no smple way to scale t n ode to tansmt the scaled veson p = p/c to the souces. The queueng delay s unfom acoss the souces, and theefoe t s the TCP laye that must know the C n ode to eact accodngly. Howeve, n ode to do ths, the souce must be awae of ts MAC level ate, whch s nfeasble snce souces may be fa away n the netwok and may not even be awae that they ae congested at a weless lnk. Moeove, typcal TCP mplementatons eact to packet losses. We now dscuss a pactcal method to ovecome these lmtatons, wthout esotng to a complcated schedulng mechansm. In ode to dve the system to the optmum of Poblem 9.1, we popose to use a smple Actve Queue Management polcy, whch we call the Multate Random Ealy Detecton (MRED) algothm. Instead of usng queueng delay as the pce, we popose to use as a poxy the buffe length b. To geneate the pce, the AP dscads packets fo connecton andomly wth pobablty p popotonal to b/c. Ths gves a lnea Random Ealy Detecton (RED) algothm [Floyd and Jacobson, 1993], but wth pobabltes elated to the effectve data ates C. Note that ths mechansm can be mplemented n the AP, esotng only to local nfomaton. The AP must know the destnaton addess fo the ncomng packet, the effectve data

114 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 101 ate wth that destnaton and the buffe contents n ode to decde whethe to dscad the packet. We now make a model fo the system. Let p =κb/c be the loss pobablty of connecton, wthκ>0the popotonalty constant of MRED. The TCP souce eacts to ths loss pobablty, and t can be modelled by a pmal contolle (equaton (2.6)). A smple model fo the total buffe content s: + ḃ= x y, b.e. the ate of change of the content n the buffe s the dffeence between total nput ate and total output ate. Substtutng the expesson fo y n equaton (8.1) we get the followng dynamcs fo the system: ẋ = k(x ) ḃ= y U (x ) κb, (9.5a) C x C 1 + b. (9.5b) Note that the second equaton n (9.5) s smla to the equaton fo p n the pmal-dual algothm, wth an added state dependent gan y > 0. In patcula, the equlbum values of (9.5) x and p =κb satsfy the KKT condtons of Poblem 9.1. As fo stablty, global stablty esults fo these dynamcs ae hade to obtan, due to the vayng gan tem. Howeve, we ague now that locally aound the equlbum the system above behaves exactly as the dual dynamcs (9.3b). Theefoe, the system wll be locally asymptotcally stable. Consde equaton (9.5b), and denote by δb, δx, δy the system vaables aound equlbum. We have that, nea the equlbum, the system vefes: δb= y δx 1 + δy C x C 1 and the second tem s 0 due to the equlbum condton of the ognal dynamcs. Theefoe: δx δb=γ 1, C whch s exactly the lnea behavo of the dual dynamcs (9.3b) wth a fxed ganγ>0. Snce the pmal-dual dynamcs ae asymptotcally stable, the new system (9.5) wll be locally asymptotcally stable. We have thus poved that: Poposton 9.2. The dynamcs (9.5) ae locally asymptotcally stable and ts equlbum s the soluton of Poblem 9.1.,

115 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 102 Note that the algothm (9.5) s ease to mplement at the packet level than the ognal pmal dual dynamcs, whch nvolve measung the buffeng delay. Snce only local stablty esults wee obtaned, n Chapte 11 we wll exploe ts behavo by smulaton. 9.3 Extenson to weless access netwoks The analyss of Secton 9.1 s vald n a sngle cell settng. We would lke to genealze t to moe complcated netwoks. In patcula, we ae nteested n the case of weless access systems,.e. a wed backbone whch has non-ntefeng weless cells as stub netwoks. Ths s a common settng n pactce fo weless access n offce buldngs and college campuses. In such netwoks, one obtans a combnaton of classcal capacty constants fo wed lnks (that do not ntefee wth any othe lnks), and constants of the type (9.1) fo lnks of the same stub cell ntefeng wth one anothe, potentally wth dffeent effectve data ates. Fo such netwoks, we would lke to develop a pce scalng method that enables decentalzed uses to allocate esouces to maxmze utlty. Of couse, one could futhe consde moe geneal ntefeence models, such as dffeent weless cells ntefeng wth each othe, o moe abtay ntefeence pattens as has been consdeed n the schedulng lteatue[ln et al., 2006]. These, n addton to the complexty of schedulng, lead to optmzaton poblems that ae dffcult to decentalze. Fo ths eason, we chose to focus on a naowe settng whch nevetheless coves scenaos of pactcal mpotance, and whch can be addessed though a smalle depatue fom cuent pactce, n patcula usng cuently deployed MAC layes. Consde then a netwok composed of lnks l= 1,..., L. These lnks can be wed o weless, and each one has an effectve tansmsson ate C l. In the case of wed lnks, C l s the lnk capacty. In the weless case, t s the effectve data ate dscussed befoe. Let = 1,...,n epesent the connectons, wth ate x, and R the classcal outng matx,.e. R l = 1 f connecton taveses lnk l. To epesent the contenton nheent to the netwoks unde analyss, we goup the lnks l nto contenton sets: two lnks belong to the same contenton set f they cannot be tansmttng smultaneously, and we defne a matx gven by G k l = 1 f lnk l belong to contenton set k, and 0 othewse. We call G the contenton matx. If a lnk s wed, the contenton set s a sngleton, snce t does not ntefee wth any othe lnk. In the case of a weless cell, the contenton set s composed of all lnks that depend on the same access pont. We shall assume that each lnk belongs to only one contenton set: ths s the estcton we mpose on the ntefeence model n ode to obtan tactable solutons. The netwok capacty constants, followng the dea of expessng constants as tme

116 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 103 popotons of allocated medum of (9.1), can be wtten as: Hx 1, H= G C 1 R, wth R and G as defned above, C= dag(c l ) and 1 a column vecto of ones. Note that each constant s assocated wth a contenton set, and each ow of the above equaton coesponds to the tme popotons assocated wth each use n the contenton set summng less than 1. It s ease to see how ths famewok enables us to model dffeent stuatons va some examples, whch we pesent below. Example 9.2 (Wed netwok). If all lnks ae wed, the contenton matx G s the dentty matx. By takng R and C as befoe, we ecove the classcal wed constants: R l x C l. Example 9.3 (Sngle weless cell). If thee s only one weless AP wth N uses n the cell, we can take R as the dentty matx, C as the dagonal matx wth weless effectve capactes and G= 1 T, snce thee s only one contenton egon contanng all the lnks. We then ecove the constant: dscussed n Secton 9.1. x C 1, Example 9.4 (Weless dstbuton system). To see a moe complete example, consde the netwok composed of wed and weless lnks shown n Fgue 9.1. Ths topology appeas n outdoo weless dstbuton scenaos, whee the backhaul nodes dstbutes the access to seveal hotspots (two n ths example) usng one channel o fequency ange of the weless medum. The local hotspots communcate on anothe fequency wth the hosts, thus non ntefeng wth the backhaul communcaton. takng: We can model the capacty constants of ths netwok wthn the above famewok by G=, R= , C= dag(c,c AP1,C AP2,C 1,C 2,C 3,C 4 ).

117 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 104 C1 AP1 C2 Access (C) CAP1 CAP2 Backhaul C3 AP2 C4 FIGURE 9.1: TOPOLOGY OF A MIXED WIRED-WIRELESS DISTRIBUTION SYSTEM WITH 4 END-USERS. Wthn the above famewok, we can now pose the geneal Netwok Utlty Maxmzaton poblem fo ths class of netwoks, whch s: Poblem 9.2 (Geneal Wed-Weless Netwok Poblem). max x U (x ) subject to: Hx 1. The pevous poblem seeks an optmal allocaton wthn the natual constants of the netwok, expessed n tems of allocated tme popotons. These constants ae smla to the ones used n the schedulng lteatue. We wll show that fo the specal stuctue unde consdeaton, a decentalzed soluton can be obtaned, that nvolves smple buffe management solutons and standad congeston contol, povded a sutable pce scalng as the one ntoduced n Secton 9.1 s appled. Consde now the Lagangan of Poblem 9.2: (x,p)= U (x ) p T (Hx 1), (9.6) whee p=(p 1,...,p K ) T s the vecto of pces. We see theefoe that we have one pce fo each contenton set.

118 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 105 By denotng q= H T p, the KKT condtons of Poblem 9.2 ae: whee q s gven by: p k U (x )= q, l G k l R l C l x 1 q = l : l k :l k = 0, p k C l. (9.7) Theefoe, the connecton must eact to a pce whch s the sum of the pces of the contenton sets t taveses, dvded by the lnk capactes t uses wthn ths set. Agan, to solve Poblem 9.2 we can use a pmal-dual algothm wth the followng dynamcs: ẋ = k(x )(U (x ) q ), ṗ=(hx 1) + p, (9.8a) (9.8b) q= H T p. (9.8c) These dynamcs, ae globally asymptotcally stable due to the esults n[aow et al., 1958] and[feje and Pagann, 2009]. Its equlbum s exactly the soluton of the Geneal Wed- Weless Netwok Poblem 9.2. The key emak hee s that, f each contenton set n the matx H s assocated wth only one AP, as t was n the pecedng examples, the pce dynamcs of (9.8) tacks the queueng delay at ths AP. Theefoe, a decentalzed mechansm can be acheved, snce the contenton pce that must be tansmtted to the souce wll be the queung delay at each AP tavesed by the souce, scaled by the effectve capacty of the lnk assocated wth ths AP. A packet doppng mechansm such as MRED can be used to scale and tanslate ths queueng delay nto a sutable loss pobablty, whch aggegated along the outes wll poduce the coect q fo each souce. If lnks ae allowed to ntefee acoss APs, the stuaton s moe complcated, snce the queueng delays become coelated, and souces must eact to congeston pces whch ae not on the oute, thus decentalzaton s not possble wthout message passng between the APs. As mentoned befoe, we wll leave out ths case and analyze how we can decentalze the algothm n some examples of pactcal mpotance. Example 9.5 (Mxed wed-weless access netwok). A typcal confguaton fo weless access coveage s to dstbute access ponts n non ovelappng channels acoss the egon to cove, and we them to the Intenet access node. Ths poduces a tee lke topology as Fgue 9.2. Thee, the dffeent APs ae connected to a cental swtch, whch s also connected to the

119 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 106 AP1 Intenet Dstbuton (Cdst) Access (Caccess) Dstbuton (Cdst) APn FIGURE 9.2: TOPOLOGY OF A MIXED WIRED-WIRELESS ACCESS NETWORK. oute handlng the Intenet connecton. End uses ae then connected to the APs va fo example. In ths case, each use taveses thee contenton sets: the contenton n ts own AP and the two wed lnks. Assumng the lnk capactes and use dstbutons shown n Fgue 9.2, the coespondng pce fo connecton s calculated accodng to equaton (9.7) as: whee p a c c e s s and p d s t j q = p a c c e s s + p d s t j + p AP j C a c c e s s C d s t j C ae the queueng delays of the wed lnks tavesed by packets of connecton, wheeas p APj s the queueng delay of the FIFO queue at the coespondng AP. All pces ae scaled by the coespondng lnk capactes, whch ae locally known. By usng fo nstance the MRED algothm of Secton 9.2, we can tansmt ths pce to the souce, and mpose the noton of faness of Poblem 9.2 by emulatng the dynamcs (9.8). Example 9.6 (Weless dstbuton system). A vaaton of the above example occus when the aea to cove s lage, as s the case n lage outdoo deployments. In ths case, the dstbuton lnks that connect each AP wth the wed netwok ae eplaced by a weless cell that handles the backhaul, ecoveng the stuaton of Example 9.4, shown n Fgue 9.1. In ths case, the coespondng pce fo connecton can agan be calculated accodng to (9.7) as: q = p a c c e s s C a c c e s s + p BH C APj + p AP j C The man dffeence wth the above example s that p BH s common to all dstbuton lnks, t eflects the FIFO queueng delay at the backhaul node, and C APj s the capacty the backhaul

120 CHAPTER 9. EFFICIENCY AND FAIRNESS IN MULTIRATE WIRELESS NETWORKS 107 AP uses to communcate wth the AP of connecton. Agan, by usng Multate RED n ths tee topology we can mpose the noton of faness of Poblem 9.2. In Chapte 11 we wll analyze the behavo of these algothms n some of ths examples though packet level smulatons. To end ths chapte, we would lke to pont out whch s the common chaactestc all these netwoks shae that enables a smple dstbuted soluton. The essence s that each lnk belongs to a sngle contenton set, and each such set s seved by a common queue. Unde ths assumpton, the nne sum n (9.7) conssts of a sngle tem,.e. we can wte: 1 q = p k(l), C l l : l whee k(l) s the contenton set assocated wth lnk l, and p k(l) the coespondng pce. By pefomng the coect capacty scalng, ths pce can be appopately elayed to the souces.

121 10 Connecton level analyss of weless netwoks In ths chapte we tun to the analyss of multate weless netwoks at the connecton level, focusng on the case of a sngle cell. We develop a stochastc model fo the evoluton of the numbe of connectons pesent n the cell, that tes to captue the tme and spatal behavo of a connecton avals, as well as the esouce allocaton that the lowe layes (TCP, multate MAC) mpose on the ate of these connectons. Based on the models of pevous chaptes, we constuct a model fo andom avals of connectons n the coveage egon of the cell. Ou appoach genealzes the connecton level stochastc model descbed n Secton 3.3, n ode to take nto account that avals n dffeent egons of the cell lead to dffeent ado condtons, and thus dstnct effectve data ates. The man pupose s to deve stablty condtons fo ths stochastc pocess, as well as some pefomance measues, n patcula the connecton level thoughput acheved by dffeent esouce allocatons methods. The oganzaton of ths chapte s as follows. In Secton 10.1 we descbe ou stochastc model fo the connectons n the cell. Then, n Secton 10.2 we analyze the model assumng that the undelyng allocaton s done by standad TCP and MAC level algothms, as studed n Chapte 8. Fnally, n Secton 10.3 we deve esults fo the connecton level tmescale assumng that the netwok opeates unde the moe effcent esouce allocatons of Chapte 9.

122 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS y x FIGURE 10.1: A TYPICAL CELL WITH GEOMETRICALLY RANDOM CONNECTION DEMANDS AND CIRCULAR RATE REGIONS Model descpton In ths secton we descbe the undelyng model and assumptons that wll be used n ths chapte. We would lke to model a sngle cell scenao whee downlnk connectons ave andomly, wth andom wokload szes. Snce the effectve data ates that connectons get at the MAC laye ae detemned by ado condtons, we would lke ou model to captue ths stuaton by epesentng connecton avals assocated to a locaton n the cell. Ths locaton wll detemne the effectve data ate. Assume that the AP s located at the ogn of the planer 2. Let R j R 2 be the egon of the plane whee uses can acheve a tansmsson ate C j. As an example, n an outdoo settng the R j could be concentc dscs aound the ogn, wth deceasng physcal laye ates as depcted n Fgue Ou model, howeve, eques no assumptons on the shape of the dffeent ate egons. We descbe fst the aval of connectons. Statons can be anywhee n the cell and connectons may ave andomly n tme. We model ths by assumng that connectons follow a spatal bth and death pocess[baccell and Zuyev, 1997]. New connectons appea n the cell as a Posson pocess of ntenstyλ(x),x R 2 whch epesents the fequency pe unt of aea. In patcula, uses ave to egon R j wth ntensty: Λ j = λ(x)d x. R j Each connecton demands some andom amount of wokload whch we assume exponental wth mean 1/µ. Fo smplcty we assume that these wokloads have the same mean n all

123 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS 110 egons. Ths s a easonable assumpton snce the connecton szes ae geneally ndependent of the locaton. The use of exponental dstbutons fo connecton sze s questonable, we do ths n ode to obtan a tactable model. We efe to[bonald and Massoulé, 2001, Pagann et al., 2009] fo a dscusson. We fst note that the aval pocess to each egon j s a Posson pocess wth total ntenstyλ j, espectvely of whethe new connectons ae geneated by dffeent statons o they belong to the same staton (n the same poston n space). Both stuatons can be captued by the total aval ateλ j. To complete the system, and thus deve stablty condtons and pefomance metcs at the connecton level tmescale, we must specfy the sevce ate of each connecton. In ths egad, we wll analyze both the based esouce allocaton dscussed n Chapte 8, and the altenatve poposed n Chapte Connecton level pefomance of cuent TCP-multate envonments. Suppose that ates ae allocated to ongong connectons followng the TCP esouce allocaton of Poblem 8.2. As agued n Secton 8.2, ths appoxmately models the behavo of the pevalng loss-based TCP congeston contol when connectons shae a FIFO queue, sevced wth multple lowe-laye ates. Fo smplcty, assume that all connectons shae the same utlty functon, fo whch the soluton of Poblem 8.2 s gven by equaton (8.5), that s, the hamonc mean of the effectve data ates of ongong connectons. In ths case, gven that at a cetan moment of tme thee ae n j connectons of ate C j, the ate fo evey connecton s gven by x(n)= 1 j n j/c j ; (10.1) hee we denote by n the vecto of n j. Puttng togethe all the pevous consdeatons we have the followng Makov chan model fo the vecto n(t). n n+ e j wth ntenstyλ j, (10.2a) n n e j wth ntenstyµn j x(n), (10.2b) whee e j denotes the vecto wth a 1 n the j coodnate and 0 elsewhee. The fst queston of nteest fo ths model s the stochastc stablty egon,.e. the egon of aval ntenstes that poduce an egodc Makov pocess, and hence a statonay dstbu-

124 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS 111 ton. We ae also nteested n calculatng the thoughput of connectons. Both ssues wll be studed by dentfyng the above model wth a well known queue. as Substtutng equaton (10.1) n (10.2) we can ewte the death ates of the Makov pocess whee g j := 1 C j. µn j x(n)=µc j g j n j k g k n k, Wth ths notaton, we can dentfy the tanston ates of (10.1) wth those of a Dscmnatoy Pocesso Shang (DPS) queue[altman et al., 2006], wth total capacty 1, and whee fo each class j the aval ate sλ j, the mean job szeν j =µc j and the DPS weght s g j. Wth ths dentfcaton, the followng esult follows dectly fom the stablty condton of the DPS queue: Poposton The Makov pocess descbng the numbe of connectons s egodc f and only f: Λ j = < 1. (10.3) µc j j It s woth specalzng equaton (10.3) to the mpotant case n whch connectons ave unfomly dstbuted n the cell,.e. λ(x) = λ, a constant. Ths can epesent a stuaton whee uses do not know whee the AP s. In that case, f A j s the aea of the egon R j,λ j =λa j and the stablty condton becomes λ µ < 1 A j j C j. (10.4) Ths s of the fomρ< C wheeρ s the taffc ntensty n b t s/(s m 2 ) and C can be thought as a cell taffc capacty, whch captues the geomety, and s a weghted hamonc mean of the effectve ates. We have thus poved the followng: Poposton Fo a sngle cell wth unfom andom avals, and whee each effectve ate C j coves a egon of aea A j, the maxmum cell capacty (n ate pe aea unt) s gven by: C = 1. A j j C j The second ssue we ae nteested n s pefomance, whch we measue by the connecton level thoughput, wheneve the system s stable (.e. < 1). Ths can be evaluated by calculatng the expected tme n the system fo jobs n a DPS queue. The expected watng tme n a DPS queue was analyzed n[fayolle et al., 1980], and extended by[havv and van de Wal, 2008]. Letτ j be the tme n the system fo a job of class j.

125 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS 112 Connecton level thoughput (Mbps) System load γ 1 γ 2 FIGURE 10.2: CONNECTION LEVEL THROUGHPUTS, 2 CLASSES, C 1 = 10, C 2 = 1 MBPS. ARRIVAL RATES ARE PROPORTIONAL TO COVERAGE AREAS. Fo the case of two classes thee s an explct fomula: 1 E(τ 1 )= 1+ Λ 2(C 1 C 2 ) µc 1 (1 ) µc 2 2(2 ), 1 E(τ 2 )= 1+ Λ 1(C 2 C 1 ) µc 2 (1 ) µc 1 2(2 ). whee s the system load as n equaton (10.3). Obsevng that a use can send an aveage 1/µ bts dung tmeτ j connecton level thoughput as: 1 γ j = µe(τ j ) we can measue the As an example, we calculate now the connecton level thoughput fo a cell wth two allowed ates as n Example 8.1, namely C 1 = 10 and C 2 = 1 Mbps, by applyng the pevous fomula. Example We assume that connectons ave unfomly n space, so the popoton of slow connectons s geate. Results ae shown n Fgue 10.2, that shows connecton level thoughput unde vayng cell load. Note that when the load nceases, the connecton level thoughput of both classes equalzes to the detment of faste uses, wth no appecable gan fo the slowe ones; ths s a consequence of the allocaton that equalzes pe-connecton ates. Fo the geneal case wth multple classes smple explct fomulas fo theτ j ae not avalable. Rathe, these values can be obtaned by solvng a system of lnea equatons: m E(τ j )= B j 0 + E(τ k )Λ k B j k (10.5) k=1 whee the B j k depend on the system paametes (we efe the eade to[altman et al., 2006] fo the expessons). Theefoe, we can calculate the connecton level thoughputsγ j by nu-

126 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS 113 mecally solvng ths system fo the effectve data ates of the netwok. A case study fo the IEEE standad wll be gven n Chapte 11. Results smla to the above example wee gven n[bonald and Poutèe, 2006], whee a connecton level model s also analyzed. Once agan, howeve, n ths wok the lowe layes ae not modeled, and a smple tme-shang mechansm s assumed fo the medum. Hee we have found that the downwad" equalzaton of connecton-level thoughputs wll occu wth any α fa congeston contol, n patcula cuent mplementatons, povded utltes ae the same fo all souces Connecton level pefomance of the unbased allocaton We now analyze the connecton level pefomance of the esouce allocaton of Poblem 9.1, whch emoves the bas n utlty aganst fast uses, and as dscussed n Secton 9.2 can be mplemented though a pce scalng mechansm. Agan, we wll assume that all uses shae the same utlty functon U(x)=x 1 α /(1 α) of theα fa famly. In ths case the soluton of Poblem 9.1 s gven by (9.2); gven that at a cetan moment of tme thee ae n j connectons of ate C j, we obtan the connecton ates x (n)= C 1/α j n j C 1/α 1 j. (10.6) The connecton level pocess behaves agan as a DPS queue, wth aval atesλ j, job szes µc j and weghts dependng onαas: g j = C 1/α 1 j. Theefoe, fo anyα, the system wll be stable when the loads vefy the stablty condton (10.3), and the expected job sevce tmes and connecton level thoughput can be calculated by the same method we descbed above. It s woth notng that asα, we ecove the weghts of the cuent allocaton analyzed n the pevous secton, whch s max-mn fa but can be hghly neffcent, penalzng the hghest ates whch suffe the most slowdown. Fo the case of α = 1 coespondng to popotonal faness, the weghts assocated to each class become equal and the pefomance metcs can be explctly calculated as: E(τ j )= 1 µc j (1 ), and: γ j = C j (1 ).

127 CHAPTER 10. CONNECTION LEVEL ANALYSIS OF WIRELESS NETWORKS 114 Connecton level thoughput (Mbps) γ 1 wth α= γ 1 wth α=2 γ 1 wth α=1 γ System load FIGURE 10.3: CONNECTION LEVEL THROUGHPUTS, 2 CLASSES, C 1 = 10, C 2 = 1 MBPS. FOR DIFFER- ENT FAIRNESS NOTIONS UNDER PRICE SCALING. CLASS 2 THROUGHPUT IS ONLY PLOTTED ONCE SINCE RESULTS ARE SIMILAR IN ALL THREE CASES. Ths allocaton has the nce popety that connecton level thoughputs become popotonal to the lowe laye offeed ate, the popotonalty constant beng the slowdown of the pocesso shang queue, whch only depends on the cell total load. In patcula, the slowdown of each class wth espect to ts own effectve data ate s homogeneous among classes, whch suggests ths s a bette way of allocatng esouces. The case of cuent TCP Reno-lke algothms wll be an ntemedate one, coespondng toα=2. In Fgue 10.3 we compae the connecton level thoughputs of a cell wth two allowed ates as n Example 8.1, namely C 1 = 10 and C 2 = 1 Mbps, n the case of max-mn, popotonal fa and Reno-lke allocatons. In Chapte 11 we wll specalze these esults to evaluate the connecton level thoughputs obtaned by TCP ove an IEEE cell.

128 11 Analyss of IEEE In ths chapte, we apply the pevous models to quantfy the behavo of TCP ove an IEEE MAC laye. We begn by calculatng the effectve data ates povdes to TCP connectons, and then we poceed to analyze though smulaton seveal examples Calculatng the effectve ates: the mpact of oveheads. In the models deved n the pecedng sectons, an mpotant quantty s C, the effectve data ate at whch TCP packets fom a gven use ae seved by the undelyng layes. When the AP wants to tansmt a packet to use of length L at a physcal laye modulaton ate PHY usng , t must comply wth a sees of backoff and watng tmes, as well as heades ncluded by the PHY laye. Ths means that the MAC laye offes a sevce to the uppe laye consstng of a tansmsson ate C PHY. Ths has been analyzed befoe[banch, 2000, Kuma et al., 2007] and we wll ecall and extend ths analyss hee. Evey tme a packet s ntended to be tansmtted n , the DCF algothm fo medum shang comes nto play. In patcula, evey tansmsson conssts of a cae sense phase, whee the medum s sensed to detemne whethe othe statons ae tansmttng. Ths s called the DI FS tme. Afte that, the staton pefoms a backoff phase consstng of a andom numbe of tme slots, of duatonσ. The numbe of tme slots s unfomly chosen between

129 CHAPTER 11. ANALYSIS OF IEEE and C W, the contenton wndow paamete. Afte that, a physcal laye heade s added, known as the PLCP heade, whch specfes the physcal laye modulaton ate to be used subsequently. Then the data s put nto the medum at ate PHY. Afte tansmsson, the staton wats fo a SI FS tme to eceve the coespondng acknowledgment, whch n tun needs a M AC _AC K tme to be tansmtted. Consdeng all of the above phases, the tme t takes to send ths packet has a fxed component gven by T 0 := DI FS+ H+ L + SI FS+ M AC _AC K, (11.1) PHY that ncludes the tme n the a and all oveheads, plus a andom numbe of tme slots Kσ, whee K U{0,...,C W}. In Table 11.1 we show typcal values of these paametes fo g. Paamete Value Slot tmeσ 9µs SI FS 10µs DI FS 28µs PLCP Heade H 28µs PHY C W m n 6Mbps... 54Mbps 15 slots M AC _AC K 50µs TABLE 11.1: IEEE G PARAMETERS We ae nteested n the aveage ate obtaned by a staton to study the uppe laye effectve ate. Obsevng that each packet s teated ndependently, the tansmsson tmes of successve packets fom a enewal pocess, and the enewal ewad theoem[felle, 1965] tells us that n the long ange the aveage ate s: C 0 = L E Kσ+T 0 = C W m n 2 L σ+t 0, (11.2) whee we substtuted K fo ts mean. We also took C W= C W m n snce we ae modelng downlnk taffc fom the AP, whch does not collde wth tself. We also assume the appopate PHY has been used so that one can neglect packet tansmsson eos. The denomnato of the pecedng expesson (mean total tme) s denoted by T. In Table 11.2 we show the coespondng MAC level ates C 0 fo the dffeent PHY ates allowed n g wth paametes as n Table Note the mpact of oveheads n the hghest modulaton ates: ths s due manly to the fact that physcal and MAC laye oveheads ae fxed n tme, ndependent of the modulaton ate PHY chosen fo the data. Ths mples that

130 CHAPTER 11. ANALYSIS OF IEEE hghe modulaton ates can fnsh the data pat of the packet moe quckly, but they stll have to send the fxed length heades and wat fo the backoff slots. When TCP connectons ae taken nto account, anothe ovehead must be consdeed: the TCP ACK packet. These packets wee desgned to have low mpact on the evese path, by havng a length of 40 bytes. Howeve, due to the oveheads added by the MAC laye, the TCP ACK becomes non neglgble, specally at hgh modulaton speeds. We assume that one TCP ACK s sent n the uplnk decton fo evey TCP packet sent downlnk. We wll also assume that collson pobabltes ae low between downlnk packets and the TCP ACKs. Unde these assumptons, the TCP ACK packet ntoduces anothe ovehead tme n the system. The effectve data ate then becomes: C = L T + TC P_AC K (11.3) whee T C P_AC K s the aveage tme to tansmt a TC P_AC K packet and s gven by: TC P_AC K := DI FS+ C W m n σ+h+ L a c k + SI FS+ M AC _AC K (11.4) 2 PHY whee L a c k s typcally 40 bytes. These effectve data ates C ae also shown n Table Agan, note the stong mpact of the TCP ACKs n the pefomance of the potocol at hgh modulaton ates, due to the fact that the lowe laye potocol oveheads ae fxed n tme. PHY ates MAC ate (C 0 ) Eff. ate (C ) Measued ate TABLE 11.2: MAC AND EFFECTIVE DATA RATES FOR THE CORRESPONDING PHY RATES OF G, WITH A PACKET SIZE OF L= 1500 BYTES. VALUES ARE IN MBPS. THE MEASURED RATES ARE ESTIMATED WITHIN 0.1Mbps OF ERROR. To valdate the above expessons, we smulated n ns-2[mccanne and Floyd, 2000] seveal ndependent eplcatons of a long TCP connecton n a sngle weless hop scenao. The aveage thoughput fo each PHY ate s epoted n the last column of Table 11.2, showng good ft wth the pedcted values. In the followng, we shall not ty to modfy the mpact of oveheads and consde them gven, snce they ae ncluded n the standads. Fo the pupose of modellng, we wll use the

131 CHAPTER 11. ANALYSIS OF IEEE C values of Table 11.2, as the effectve data ates povded by to TCP connectons, and use ths values fo the algothm mplementatons Multate RED mplementaton As we dscussed n Chapte 9, the pce to whch a TCP connecton should eact n ode to attan the equlbum of Poblem 9.1 s the queueng delay. Howeve, ths pce should be scaled by the effectve data ate C the connecton expements n each lnk. Clealy, ths s dffcult to mplement wthout esotng to schedulng. Moeove, typcal TCP connectons use loss based congeston contol mechansms, such as TCP-Reno. Theefoe, we popose to use the MRED algothm developed n Secton 9.2 at each node to attan the optmum of Poblem 9.1. To test the poposal n a eal envonment, we mplemented ths algothm n the Netwok SmulatoÒ ¹¾. Ou mplementaton s based on the lbay ¼¾½½ÑÖ[Baldo et al., 2007]. Two mpotant extensons wee made to the lbay: the exstng ARF mechansm was updated to cope wth the possblty of a sngle node havng dffeent modulaton ates fo dffeent destnatons, whch eflects the eal behavo of cuent APs. The second modfcaton was to mplement the Multate RED (MRED) queue, whee the descbed ealy packet dscad takes place. Note that the coss-laye nfomaton needed fo mplementaton of the mechansm s mnmal: wheneve a packet fo next-hop j s eceved, t s dscaded wth pobablty p j = κb/c j wheeκacts as a scalng paamete, b s the cuent queue length, and C j s the coespondng effectve ate fo the cuent modulaton ate the AP mantans wth destnaton j (as n Table 11.2). In the case of wed lnks, the lnk capacty s used to scale ths dop pobablty. The non-dopped packets ae seved then on a FIFO bass Smulaton examples We now pesent seveal smulaton scenaos to llustate the behavo of the poposed algothm. Fst we compae the dffeent allocatons n a sngle-cell settng, and show the mpovement n effcency obtaned by ntoducng the pce scalng algothm. Then we analyze the esouce allocaton n a settng whee connectons have dffeent RTTs, and thus dffeent utlty functons, n ode to show that the models ntoduced n Chaptes 8 and 9 also captue the behavo n ths stuaton. Then we valdate and compae the connecton level thoughputs obtaned unde the dffeent allocatons. Fnally, we pesent a case whee the netwok s moe complex, and the technques ntoduced n Secton 9.3 ae valdated.

132 CHAPTER 11. ANALYSIS OF IEEE AP Bad lnk Coveage Aea FIGURE 11.1: TOPOLOGY OF A SINGLE-CELL SCENARIO Sngle-cell scenao We smulate the topology shown n Fgue 11.1, whch conssts of a sngle cell g scenao n whch 3 uses ae connected wth a modulaton ate PHY = 54Mbps, and some tme late, a fouth use s added at the lowest possble modulaton PHY 4 = 6Mbps. All fou connectons use TCP-Neweno and shae equal Round Tp Tmes (RTTs), then havng smla utlty functons. Results ae shown n Fgue We smulated 50 ndependent eplcatons of the expement, and the aveage thoughput fo each connecton type as well as 95% confdence ntevals ae shown. Fo these modulaton ates, the effectve data ates accodng to Table 11.2 ae C = 19.5Mbps, = 1,2,3 and C 4 = 4.74Mbps. In the fst gaph of Fgue 11.2, we see that ntally the fast uses ae shang effcently the total thoughput. Afte the ntoducton of the slow use, all connectons convege to the same thoughput, whch s appoxmately x = 2.74Mbps, the hamonc mean dscussed n Poposton 8.1. In the second gaph, we show the behavo of the system unde the MRED algothm. In ths case, afte the ntoducton of the slow use, the allocaton conveges appoxmately to x = 4.2Mbps, = 1,2,3 and x 4 = 2.1Mbps, whch s the exact soluton of Poblem 9.1. Note that the total thoughput n the netwok s nceased by moe than 30% Dffeent RTTs scenao The pupose of ths example s to show that Poblem 8.2 captues the behavo of the system when the TCP connectons have dffeent RTTs, and thus dffeent utltes, and to show how

133 CHAPTER 11. ANALYSIS OF IEEE Connecton thoughput n a sngle cell scenao wth DopTal 54Mbps connectons 6Mbps connectons Pedcted equlbum 7 6 Connecton thoughput n a sngle cell scenao wth MRED 54Mbps connectons 6Mbps connectons Pedcted equlbum 5 5 Thoughput (Mbps) 4 3 Thoughput (Mbps) Tme (secs) Tme (secs) FIGURE 11.2: COMPARISON BETWEEN THROUGHPUTS: WITHOUT MRED (ABOVE), WITH MRED (BE- LOW). TCP1 TCP2 AP FIGURE 11.3: WIRED-WIRELESS TOPOLOGY. effcency can also be mpoved n ths case wth the MRED algothm. We consde the topology of Fgue 11.3, whee two connectons wth dffeent RTTs shae a weless bottleneck lnk. In ths example, connecton 1 has a longe RTT than that of connecton 2, and ts staton s close to the AP, havng a modulaton ate PHY 1 = 54Mbps. The second connecton has a modulaton ate of PHY 2 = 6Mbps. Both connectons use TCP- Neweno, whch we model by the utlty functon U(x)= 1/(τ 2 x) wthτthe connecton RTT. Pluggng these values nto Poblem 9.1 usng the effectve data ates of Table 11.2, the allocaton esults ae x1 = 2.43Mbps and x 2 = 4.14Mbps. In the fst gaph of Fgue 11.4 we show the esults of 50 ndependent eplcatons of the expement, whch shows that ndeed the connecton thoughputs convege appoxmately to the values pedcted by Poblem 8.2. By usng MRED n the AP we can change the allocaton to the one poposed n Poblem 9.1, emovng the bas of Poblem 8.2. The esultng allocaton s x1 = 4.38 and x 2 = In the second gaph of Fgue 11.4 we show the coespondng smulaton esults. We see that the

134 CHAPTER 11. ANALYSIS OF IEEE Connecton thoughput n a dffeent RTT scenao wth DopTal TCP 1 TCP 2 Pedcted equlbum 7 6 Connecton thoughput n a dffeent RTT scenao wth MRED TCP 1 TCP 2 Pedcted equlbum 5 5 Thoughput (Mbps) 4 3 Thoughput (Mbps) Tme (secs) Tme (secs) FIGURE 11.4: WIRED-WIRELESS TOPOLOGY SIMULATION. ABOVE: ORIGINAL ALLOCATION. BELOW: MRED ALGORITHM. MRED algothm appoxmately dves the system to the new equlbum. Note that ths new equlbum s also moe effcent IEEE connecton level thoughputs We now apply the esults of Chapte 10 to evaluate the TCP connecton level pefomance when wokng wth a andom wokload and an undelyng IEEE MAC laye. As befoe, we focus on a downlnk scenao whee all TCP connectons ave at the coveage zone at a andom pont, wth total aval ateλ j fo the ate C j. These ates ae chosen as n Table 11.2, whch ae vald fo g. As a fst example, we consde a sngle cell scenao wth two PHY ates. Uses nea the cell establsh connectons at the hghest possble modulaton ate of 54M b p s and the emanng uses use the lowest possble modulaton of 6Mbps. We smulated the andom aval of connectons n ns-2 and measued the connecton level thoughput fo dffeent values of the cell load. To take nto account the fact that the low modulaton ate has a geate coveage aea, the aval ntenstes wee chosen popotonal to the sze of the coveage aeas. In Fgue 11.5 we plot the connecton level thoughputs obtaned by smulaton and the pedcted thoughputs usng the esults of Chapte 10 fo dffeent values of the total cell load. The fst gaph shows the connecton level thoughputs when the poposed Multate RED s not n use, and the second one shows the esults fo a cell usng Multate RED. In both cases esults show a good ft wth the model. In each case, the connecton level thoughputs stat at the effectve data ate fo each class, C, coespondng to the case whee each connecton aves to an empty netwok, and thus s

135 CHAPTER 11. ANALYSIS OF IEEE Connecton level thoughputs fo a sngle cell wth DopTal γ 1 γ 2 16 Connecton level thoughput (Mbps) System load Connecton level thoughputs fo a sngle cell wth DopTal γ 1 γ 2 16 Connecton level thoughput (Mbps) System load FIGURE 11.5: CONNECTION LEVEL THROUGHPUTS FOR AN IEEE G CELL WITH TWO MODULATION RATES. ABOVE: WITHOUT MRED, BELOW: WITH MRED IN USE. able to obtan ts full ate. When the offeed load begns to gow the thoughputs go to zeo, as expected. We obseve that n the case whee MRED s not n use, the hgh modulaton ate uses ae moe heavly slowed down, and thoughputs tend to equalze. When all the data ates ae allowed, the connecton level thoughputs can be calculated by solvng the lnea system of equatons dscussed n Secton In Fgue 11.6 we plot the pedcted connecton level thoughputs fo such a settng, wth nceasng cell load. Agan, the aval ates fo each class ae chosen popotonal to the estmated coveage aeas. In the fst gaph, we show the esults fo cuent g cells, whee pce scalng s not used, and thus

136 CHAPTER 11. ANALYSIS OF IEEE Connecton level thoughputs (Mbps) Connecton level thoughputs (Mbps) Connecton level thoughputs wthout MRED System load Connecton level thoughputs wth MRED System load FIGURE 11.6: CONNECTION LEVEL THROUGHPUTS FOR AN IEEE G CELL. ABOVE: WITHOUT PRICE SCALING, BELOW: WITH MRED. EACH LINE CORRESPONDS TO A DIFFERENT PH Y RATE IN DE- CREASING ORDER. penalzng the hghe ates. In the second gaph we show the connecton level thoughputs unde the pce scalng mechansm, emovng the bas. We can see that the hghe ates get bette thoughput n all cell loads, whle the lowe ones ae mostly unaffected by the change. To evaluate the dffeence between the two mechansms, n Fgue 11.7 we plot the ato between the hghe and lowe thoughputs fo thee dffeent stuatons: wthout MRED, wth MRED n use, and the theoetcal popotonal fa stuaton. We can see that applyng MRED wth cuent TCP mplementatons gves an ntemedate stuaton, mpovng on the bas aganst hghe thoughputs wth espect to the cuent stuaton An applcaton to a mxed wed-weless tee topology In ths example, we smulate the topology of Example 9.5. Ths topology s llustated n Fgue 11.8 wth two dstbuton APs and two uses n each AP. The access lnk capacty s c a c c e s s = 20M b p s epesentng a typcal access capacty (e.g. a DSL lne). The dstbuton lnks have c d s t = 100Mbps and thus ae ovepovsoned. The weless cells ae dentcal and have two uses each, wth modulaton ates PHY 1 = PHY 3 = 54Mbps and PHY 2 = PHY 4 = 6Mbps.

137 CHAPTER 11. ANALYSIS OF IEEE Rato of best vs. wost thoughput Wthout MRED TCP + MRED Popotonal Fa Cell load ρ FIGURE 11.7: RATIO OF BEST VS. WORST THROUGHPUT FOR AN IEEE G CELL. AP1 Intenet Dstbuton (Cdst) Access (Caccess) Dstbuton (Cdst) APn FIGURE 11.8: TOPOLOGY OF A MIXED WIRED-WIRELESS ACCESS NETWORK. Each use has a sngle TCP connecton and all connectons have equal RTTs. Pluggng these values nto Poblem 9.2 gves the followng allocaton: x 1 = x 3 = 6.4Mbps x 2 = x 4 = 3.2Mbps Note n patcula that both the access lnk and the weless cells ae satuated n the esultng allocaton. Ths s a dffeence wth typcal wed-only models wth tee topologes. In patcula, n ths case, thee s a postve pce (queueng delay) both at the APs and the wed access lnk. By usng the MRED algothm as dscussed n Secton 9.2, we can dve the system to ths allocaton. Results ae shown n Fgue 11.9, whee agan 50 ndependent eplcatons wee pefomed, and the aveage thoughputs as well as 95% confdence ntevals ae shown. We

138 CHAPTER 11. ANALYSIS OF IEEE TCP thoughputs fo a weless access scenao wth MRED n use TCP 1 TCP 2 TCP 3 TCP 4 7 Thoughput (Mbps) Tme (sec) FIGURE 11.9: THROUGHPUTS OF TCP CONNECTIONS FOR A WIRELESS ACCESS SCENARIO WITH 4 USERS. MRED IS IN USE. see that the thoughputs appoxmately convege to the above equlbum. Note also that, f we choose not to use the MRED algothm, the allocaton would be gven by the soluton of Poblem 8.2, whch s x 3.8Mbps fo each use. In that case, the full access capacty would not be used.

139 12 Conclusons of Pat II In ths pat of the thess, we appled the Netwok Utlty Maxmzaton famewok to chaacteze the coss-laye nteacton between the TCP tanspot potocol wth an undelyng MAC whee multple modulaton ates coexst. Ths stuaton s pesent n typcal IEEE deployment scenaos. We descbed the esouce allocaton mposed by cuent weless netwoks n ths famewok, showng that a bas s mposed aganst uses of hgh modulaton ates. We then poposed an altenatve esouce allocaton that genealzes the faness and effcency notons of TCP n wed netwoks to ths context, and ovecomes the neffcences of cuent potocols. We developed a smple mechansm to mpose these moe effcent equlba n sngle cell scenaos and genealzatons of ths pocedue to moe complex topologes. We also showed how the connecton level dynamcs can be analyzed though a Makov pocess, whch can be dentfed n some cases wth a well known queue. Ths enables us to chaacteze the stablty egons and connecton-level thoughput obtaned by the cuent esouce allocaton and ou poposed altenatve. Fnally we appled the pevous esults to the IEEE MAC laye, establshng the effectve data ates and valdatng the esults by smulatons. Seveal lnes of wok eman open. An mpotant one s how to extend the poposed mechansms to the new addtons, whee these ssues become moe mpotant due to the hghe data ates nvolved, and the use of packet aggegaton. We also would lke to study

140 CHAPTER 12. CONCLUSIONS OF PART II 127 the pefomance of a system whee the uplnk taffc s not neglgble, as s the case n seveal usage models of today, and collsons have to be taken nto account.

141 13 Geneal conclusons and futue lnes of wok Thoughout ths thess, we have analyzed seveal poblems of netwok esouce allocaton, unde the famewok of Netwok Utlty Maxmzaton. The wok manly focused on analyzng the connecton level pefomance, effcency and faness of seveal models of netwok esouce allocaton, n seveal settngs, both sngle and mult-path, and both wed and weless scenaos. We have analyzed n detal two mpotant poblems: on one hand the esouce allocaton povded by congeston contol potocols wheneve multple connectons pe use ae allowed. We dd so n the context of wed netwoks, and we dentfed poblems wth the cuent esouce allocaton algothms. In patcula, we showed that ndvdual geedy uses can cheat on the esouce allocaton by openng multple connectons. We theefoe ntoduced a new paadgm of esouce allocaton focused on the aggegate ates allocated to each use, whch we called use-centc faness. We then showed how smple and decentalzed algothms can be ntoduced nto the netwok n ode to dve the system to a sutable effcent and fa opeatng pont. The algothms ely on the contol of the numbe of connectons usng the congeston pces aleady povded by the netwok. Ths contol can be ethe done by the use, o as an admsson contol by the netwok. We modelled the two stuaton and povded stablty esults fo the esultng systems. Also, the algothms developed wee tested wth packet level smulatons, valdatng ou esults, and povdng evdence that eal netwok mplementatons ae possble.

142 CHAPTER 13. GENERAL CONCLUSIONS AND FUTURE LINES OF WORK 129 The second mpotant poblem analyzed hee s the esouce allocaton povded by congeston contol algothms ove a physcal laye that allows multple tansmsson ates, whch s a common stuaton when weless netwoks ae nvolved. We showed that the typcal algothms ntoduce undesed neffcences. We poposed a new esouce allocaton model, and the coespondng algothms to mpose ths allocaton n a sngle weless cell scenao. We also analyzed how to extend these algothms to moe complcated topologes, n patcula mxed wed-weless netwok topologes common n pactce. We showed stablty esults fo the poposed algothms, and appled them to the patcula case of the IEEE standad, whch s elevant snce t s one of the most wdely used weless technologes. Agan, packet level smulatons wee povded, showng that the algothms can be appled n pactce. The eseach eled on seveal mathematcal tools. The most mpotant wee convex optmzaton, non-lnea and lnea contol technques, Lyapunov theoy, Makov chan modellng and flud lmts. We have ted to keep balance between the mathematcal models and mplementaton. In all of the algothms, the development was always wth a pactcal vew n mnd. sometmes dscadng moe complex algothms fo the sake of mplementaton. Howeve, we have analyzed the pefomance of all the contol algothms poposed, n patcula we povded goous poofs of the local o global asymptotc stablty. In the case whee we wee unable to obtan these esults, we have tested them va smulatons. Seveal lnes of futue eseach eman open. Fo the esults on the fst pat of the thess, the man poblems ae elated to mplementaton ssues. We have povded a theoetcal analyss of the algothms, and though packet level smulatons, a poof of concept that these algothms can be mplemented n eal netwoks. In futue eseach, developng these algothms n ode to put them n poducton-level netwoks s moe than a tval task. In patcula, developng sutable mechansms to communcate pces between connectons and the outes, n ode to ease mplementaton of admsson contol algothms. Fo the weless netwoks analyzed n the second pat, t would be nce to extend the esults pesented hee to the upcomng weless standads, whee new featues appea. In patcula, packet aggegaton algothms ae ntoduced to lowe the mpact of havng multple ates n the cell. Howeve, the esults of ths wok povde evdence that, f not done popely, neffcences can nevetheless appea. Futhe study on ths subject s equed, whch can lead to new algothms that enhance pefomance n these netwoks.

143 A Mathematcal pelmnaes In ths Appendx, we povde a bef evew of some of the mathematcal concepts that ae needed to develop the esults pesented n ths Thess. A.1 Convex optmzaton We pesent below the man concepts of convex optmzaton. A good ntoducton to ths subject s[boyd and Vandebeghe, 2004]. We stat by some defntons. Defnton A.1 (Convex set). A set C R n s called convex f the followng holds: αx+(1 α)y C x,y C, 0 α 1. The above defnton states that fo any gven two ponts n C, the lne segment jonng both ponts s entely contaned n the set. Convex sets fom the basc egons whee convex optmzaton s pefomed. Defnton A.2 (Convex functon). A functon f : C R, whee C R n s a convex set, s sad to be convex f and only f: f(αx+(1 α)y) αf(x)+(1 α)f(y) x,y C, 0 α 1. The functon s sad to be stctly convex f the above nequalty s stct wheneve x y and α (0,1).

144 APPENDIX A. MATHEMATICAL PRELIMINARIES 131 Defnton A.3 (Concave functon). A functon f : C R, whee C R n s a convex set, s sad to be concave f and only f f s convex. Moeove, t s stctly concave f f s stctly convex. Convex functons can be vsualzed as a bowl shape: fo any two ponts(x, f(x)) and (y, f(y)) on the gaph of f, the lne segment jonng the two ponts stays above the gaph of the functon. Convesely, concave functons have the shape of an nveted bowl, wth the gaph of the functon beng above the lne segments jonng ts ponts. Lnea o affne functons ae lmt cases, and ae both concave and convex, though clealy not n the stct sense. In the development of the theoy, we shall fequently encounte convex optmzaton poblems, whee we want to maxmze a concave functon subject to convex constants,.e. fndng the optmum of a concave functon ove a convex set. The followng Lemma s useful: Lemma A.1. Consde a concave functon f defned ove a convex set C R n. Then the followng holds: If x s a local maxmum of f n C, then t s also a global maxmum ove C. If moeove f s stctly concave, then x s the unque global maxmum on C. If C s compact (closed and bounded), then a global maxmum exsts ove C. Theefoe, convex optmzaton poblems of ths fom ae well defned, and a soluton always exsts povded that the constant set s compact. The emanng queston s how we can fnd ths optmum. Fom now on, we shall focus on one class of convex optmzaton poblems that we shall fnd fequently n ou analyss. Consde a concave functon f, that we want to optmze, subject to lnea constants. The poblem s the followng: Poblem A.1. subject to the constants: max f(x) x Rx c. whee the nequalty above s ntepeted componentwse. We wll assume that f s dffeentable, and R s a matx of appopate dmensons. It s easy to see that the feasble set,.e. the ponts x that satsfy the constants fom a convex set. The fst step towads solvng such a poblem s to fom the Lagangan functon, gven by: (x,p)= f(x) p T (Rx c). (A.1) Hee, p=(p l ) s a vecto, whose entes ae called Lagange multples, wth the only constant beng p 0. The supescpt T denotes matx tanspose.

145 APPENDIX A. MATHEMATICAL PRELIMINARIES 132 One of the man esults on convex optmzaton, that we shall use fequently n ou wok, s the followng: Theoem A.1 (Kaush-Kuhn-Tucke condtons). Consde Poblem A.1, wth Lagangan gven by (A.1). Then, the optmum of the poblem must satsfy the Kaush-Kuhn-Tucke (KKT) condtons: x = f(x) R T p= 0, p T (Rx c)=0, Rx c, p 0. (A.2a) (A.2b) (A.2c) (A.2d) The poof of ths Theoem can be found n[boyd and Vandebeghe, 2004]. Fo a concave optmzaton poblem wth lnea constants, equatons (A.2) completely chaacteze the soluton. Note that equaton (A.2a) s a gadent condton on the Lagangan, whee the soluton must coespond to a statonay pont. The second condton, gven n (A.2b), s called the complementay slackness condton. It states that, f the soluton s acheved wth stct nequalty n a gven constant, the coespondng Lagange multple s 0. In ths case the constant s called nactve. If on the othe hand, the soluton s acheved n wth equalty n the constant, the coespondng Lagange multple may be stctly postve. An ntepetaton of ths condton s gven n Secton 2.1 n tems of pces. In the text, we efe often to the KKT condtons, and typcally we mpose (A.2a) and (A.2b), always assumng the emanng condtons as mplct n the statement. The Lagangan and the KKT condtons ae pat of a moe extensve theoy of convex optmzaton, whch s Lagangan dualty. In ths theoy, convex optmzaton poblems ae analyzed though the dual functon gven by: D(p)= max (x,p), x C whee C s the set defned by the constants of Poblem A.1. It s easy to see that, f x s a soluton of A.1, then t must satsfy: f(x ) nf p 0 D(p). To see ths, note that fo any x C and p 0 we have: f(x) f(x) p T (Rx C), and thus: max x C f(x) max x C f(x) p T (Rx C)=max (x,p)= D(p). x C

146 APPENDIX A. MATHEMATICAL PRELIMINARIES 133 Snce the left hand sde does not depend on p, we conclude that: max x C f(x) nf D(p). p 0 The left hand sde s called the pmal optmzaton poblem. The ght hand sde defnes what s called the dual optmzaton poblem. Note that the dual poblem s always convex and pefomed ove the Lagange multples. The above s called the weak dualty theoem, that states that a soluton of the dual poblem s always an uppe bound of the pmal optmum. Wth addtonal hypotheses, called Slate constant qualfcaton condtons, the above nequalty becomes an equalty (equvalently, t s sad that thee s zeo dualty-gap). Slate qualfcaton condtons ae always satsfed when the objectve functon s concave and the constants ae lnea. Thus, fo Poblem A.1, we wll always have zeo dualty gap. A.2 Lyapunov stablty We now pesent some esults on Lyapunov stablty theoy that ae used n seveal pats of ths Thess. We wll summaze only the basc esults, and efe the eade to[khall, 1996] fo futhe developments. Consde a dynamcal system defned by the followng ntal value poblem of a dffeental equaton: ẋ=f(x), x(t 0 )=x 0. (A.3a) (A.3b) Hee, f : R n R n defnes the dynamcs of the system, and we assume that f satsfes the Lpschtz condton: fo any gven x R n thee exsts a constant K x such that: f(y) f(x) K x y x y R n. Hee, epesents a sutable nom nr n. Wth the above condton, the ntal value poblem defned by (A.3) has a well defned soluton ove an nteval ofrcontanng t 0. We ae nteested n the equlbum solutons of ths system,.e. solutons of the fom: x(t) x 0 t. Fo the system (A.3) to be n equlbum, t s necessay that x s a soluton of: f(x)=0. The exstence of equlbum tajectoes s of lttle pactcal mpotance, snce ths equlbum cannot be obseved n pactce f they ae not stable, n the sense that small petubatons

147 APPENDIX A. MATHEMATICAL PRELIMINARIES 134 of the system n equlbum do not poduce dastc depatues fom the equlbum behavo. Below we defne seveal mpotant notons of stablty. Wthout loss of genealty, we shall assume that the coodnate system s chosen such that the soluton of f(x) = 0 unde study s at x= 0. Defnton A.4 (Stablty). The equlbum pont s sad to be stable f, gven anyε>0, thee exsts δ > 0, such that x(t) <ε, t t 0, wheneve: x 0 <δ. Defnton A.5 (Asymptotc stablty). The equlbum pont s sad to be (locally) asymptotcally stable f t s stable and thee exsts aδ>0, such that wheneve: lm x(t) = 0, t x 0 <δ. Defnton A.6 (Global asymptotc stablty). The equlbum pont s sad to be globally asymptotcally stable f the lmt: holds fo any ntal condton x 0. lm x(t) = 0, t Intutvely, stablty mples that f the ntal condton s petubed a small amount fom the equlbum pont, the tajectoy stll stays nea ths equlbum pont n the futue. The asymptotc stablty condton s stonge, and mples that when the petubaton s small, the tajectoy wll convege to the equlbum pont. Thus the equlbum pont acts as a local attacto of the system. The global stablty condton states that the equlbum pont s eached eventually statng fom any ntal condton fo the system. The stablty analyss of a gven dynamcal system can be pefomed n seveal ways. The fst one s though lneazaton. Consde a lnea dynamcal system of the fom (A.3). In ths case, the dynamcs can be ewtten as: ẋ= Ax, (A.4a) x(t 0 )=x 0, (A.4b) whee A s a n n matx. In ths case, the soluton s gven by: x(t)=e A(t t 0) x 0 t R n, whee e M s the exponental matx. A dect analyss of ths matx leads to the followng poposton:

148 APPENDIX A. MATHEMATICAL PRELIMINARIES 135 Poposton A.1. If all the egenvalues of A have negatve eal pats, then the ogn s a globally asymptotcally stable equlbum fo the system (A.4). A matx A that satsfes the above condton,.e. ts egenvalues have stctly negatve eal pats, s called a Huwtz matx. When the system s non-lnea, but t can be sutably appoxmated by a lnea system aound the equlbum, local asymptotc stablty can stll be obtaned: Poposton A.2. Consde a dynamcal system of the fom (A.3), whee f s contnuously dffeentable at the equlbum x= 0. If the Jacoban matx: A= f x s Huwtz, then the ogn s locally asymptotcally stable. The man dea s that, locally, the system can be appoxmated by the lnea system gven by ẋ= Ax, wth A the jacoban matx. Stonge stablty esults fo non lnea systems ae often acheved va Lyapunov functons. The Lyapunov theoy of stablty povdes us wth a suffcent condton based on a sutable chosen functon that acts as an enegy functon fo the system. The man esult s the followng: Theoem A.2 (Lyapunov, 1892). Consde a contnuously dffeentable postve defnte functon V(x),.e. such that: and V(0)=0. V(x)> 0, f x 0, Defne the devatve along the tajectoes of the system as: V(x)= d V(x(t))= V f(x). d t Then the followng condtons fo stablty hold: 1. If V(x) 0, then the ogn s a stable equlbum pont fo the system. 2. If n addton, V(x) < 0 x 0, then the equlbum pont s asymptotcally stable. 3. If n addton to 1. and 2., V s adally unbounded,.e.: V(x) wheneve x, then the ogn s globally asymptotcally stable.

149 APPENDIX A. MATHEMATICAL PRELIMINARIES 136 The dea behnd Lyapunov functons s that, once a soluton entes the set{x : V(x) c}, t cannot leave, snce V deceases along tajectoes. We say that the set{x : V(x) c} s fowad nvaant fo the dynamcs. Snce V s contnuous and V(0)=0, we can choose a set nea x= 0 such that the soluton neve leaves ths egon. If moeove V< 0, we can enclose the soluton n a sequence of such sets thus povng asymptotc stablty. If moeove V s adally unbounded, fo any ntal condton the set{x : V(x) V(x 0 )} s compact, and thus the soluton cannot leave the set and must convege to the equlbum. An mpotant efnement of Lyapunov s esult along the pevous deas s due to LaSalle. We need the followng: Defnton A.7. A set M s sad to be nvaant wth espect to the dynamcs (A.3) f: x(t 0 ) M x(t) M t R. We say that a set M s fowad nvaant f: x(t 0 ) M x(t) M t t 0. Equlbum ponts ae the most basc nvaant sets. Howeve, the soluton of the dynamcs can have lmt cycles o even moe complcated behavo. The elatonshp between nvaant sets and Lyapunov functons s the followng esult: Theoem A.3 (LaSalle Invaance Pncple). LetΩ R n be a compact set that s fowad nvaant fo the dynamcs (A.3). Let V : Ω R be a contnuously dffeentable functon such that V(x) 0 nω. Let E={x : V(x)=0}, and M be the lagest nvaant set n E. Then, evey soluton of (A.3) wth x 0 Ω appoaches the set M as t. The noton of convegence to a set s smply that d(x(t),m) 0 when t, whee d(x,m)=nf y M d(x,y). Note that the above esult does not eque V to be postve defnte. It meely states that, f we can fnd a Lyapunov functon fo the system on a compact and fowad nvaant set, the soluton must convege to the set whee V = 0, and moeove, t must convege to an nvaant set n ths egon. Note also that the constucton ofωmay not be ted to V. Howeve, a typcal choce s{x : V(x) c} fo an appopate value of c. Ths leads to a coollay of the above Theoem, ognally due to Kasovsk: Coollay A.1. Let x= 0 be an equlbum pont of (A.3), and V : R n R a contnuously dffeentable and adally unbounded postve defnte functon such that V(x) 0fo all x R n. Let S={x : V(x)=0}, and note that 0 S. If the only soluton of (A.3) that can stay dentcally n S s x = 0, then the ogn s globally asymptotcally stable. Note that when V s negatve defnte, S={0} and ths s a estatement of Lyapunov s esult. Howeve, f V s only negatve semdefnte, we have a stonge esult, povded that x(t) 0s the only admssble soluton wth V 0.

150 APPENDIX A. MATHEMATICAL PRELIMINARIES 137 A.3 Passve dynamcal systems A useful appoach n analyzng the stablty of nteconnected dynamcal systems, elated to the Lyapunov theoy, s the passvty appoach. Consde a dynamcal system wth nput u R m, output y R m and state x R n, gven by: ẋ= f(x,u), (A.5a) y= h(x,u). (A.5b) Hee f : R n R m R n s locally Lpschtz and h : R n R m R m s contnuous, and we assume that the equlbum condtons ae f(0,0)= 0, h(0,0)=0. Note n patcula that we assume the system has the same numbe of nputs and outputs. Defnton A.8 (Passve system). The system (A.5) s called passve f thee exsts a postve semdefnte dffeentable functon V(x), whch only depends on the state, satsfyng: V(x)= V f(x) u T y (x,u) R n R m. V s called the stoage functon of the system. Addtonally, the system s called state stctly passve f: V(x) u T y ρψ(x) (x,u) R n R m. whee ρ > 0 s a constant and ψ(x) s a postve semdefnte functon such that: ψ(x(t)) 0 x(t) 0, fo all solutons of (A.5) wth abtay u(t). The tem ρψ(x) s called the dsspaton ate. Thee ae seveal othe notons of stct passvty, howeve, we shall estct ouselves to state stct passvty, and smply say that a system s stctly passve wheneve t s state stctly passve. A dect consequence of passvty s the stablty of the ogn when the nput s u 0. Lemma A.2. Assume that the system (A.5) s passve, and u 0. Then the ogn s a stable equlbum pont. If moeove the system s state stctly passve, the ogn wll be asymptotcally stable. If addtonally V s adally unbounded, the ogn s globally asymptotcally stable. The poof of the Lemma follows fom Lyapunov aguments usng V, the stoage functon, as a canddate Lyapunov functon fo the system. The full powe of the passvty appoach, howeve, becomes evdent when studyng nteconnected feedback systems. Let H 1 and H 2 be two systems of the fom (A.5), n a negatve feedback connecton, depcted n Fgue A.1. Note that the output of a system s the nput of the othe. We have the followng:

151 APPENDIX A. MATHEMATICAL PRELIMINARIES 138 u 1 y 1 H 1 y 2 H 2 u 2 FIGURE A.1: NEGATIVE FEEDBACK INTERCONNECTION OF SYSTEMS. Theoem A.4. Consde the feedback system of Fgue A.1, whee H s gven by: ẋ = f (x,u ), y = h(x,u ), fo = 1,2. Assume that the feedback system has a well defned state space model: ẋ= f(x,u), y= h(x,u). whee x=(x 1,x 2 ) T, u=(u 1,u 2 ) T and y=(y 1,y 2 ) T, f s locally Lpschtz, h s contnuous and f(0,0)=h(0,0)=0. If H 1 and H 2 ae passve, then the ogn s a stable pont of the dynamcs: ẋ= f(x,0) If moeove, H 1 and H 2 ae state stctly passve, then the ogn s asymptotcally stable. If addtonally the stoage functons V of each system ae adally unbounded, then the ogn s globally asymptotcally stable. The man dea fo the poof s to use V(x)=V 1 (x 1 )+V 2 (x 2 ) as a Lyapunov functon fo the closed loop system. Fne esults on asymptotc stablty can also be obtaned wth othe notons of passvty, o by means of the LaSalle nvaance pncple. Fo an applcaton of these concepts to Intenet congeston contol, we efe the eade to[wen and Acak, 2003]. In the case of lnea tme nvaant systems, a smple fequency doman condton can be obtaned that enables us to check whethe a system s (stctly) passve. Consde a lnea

152 APPENDIX A. MATHEMATICAL PRELIMINARIES 139 nput-output tme nvaant system, gven by: ẋ= Ax+ Bu, (A.6a) y= C x+ Du, (A.6b) whee A, B,C and D ae matces of appopate dmensons. In the Laplace doman, such a system wll have a tansfe functon matx gven by: G(s)=C(s I A) 1 B+ D We have the followng defnton: Defnton A.9 (Postve eal tansfe matx). A p p tansfe functon matx G(s) s called postve eal f: All elements of G(s) ae analytc n Re(s)>0. Any magnay pole of G(s) s smple and has a postve semdefnte esdue matx. Fo all ealωsuch that jω s not a pole of G, the matx G(jω)+ G T ( jω) s postve semdefnte. If moeove G(s ε) s postve eal fo some ε > 0, we say that G s stctly postve eal. Note that when p= 1 (sngle nput-sngle output system) and G s Huwtz,.e. t has no poles n Re(s) 0, the defnton above educes to checkng whethe the Nyqust plot G(jω) of the system satsfes Re(G(jω)) 0. The elatonshp between postve eal tansfe functons and passvty s establshed by usng the Kalman-Yakubovch-Popov Lemma (c.f. [Khall, 1996]), that elates the exstence of a sutable quadatc stoage functon V wth the eal postve condton. The esult s the followng: Theoem A.5. A lnea tme nvaant system wth a mnmal ealzaton tansfe functon G(s) s passve f and only f G(s) s postve eal. Moeove, t s state stctly passve f and only f G(s) s stctly postve eal. Theefoe, the (stct) passvty of a system can be checked though the fequency doman condton on ts tansfe functon. Note that fo sngle nput sngle output systems ths s extemely smple.

153 APPENDIX A. MATHEMATICAL PRELIMINARIES 140 A.4 Makov chans A useful mathematcal concept fo the modellng of telecommuncaton systems ae Makov chan stochastc pocesses. We pesent now the man defntons and esults of Makov Chan theoy. Let(Ω,,P) be a pobablty space. A contnuous tme stochastc pocess wth state space E s a famly of andom vaables X(t,ω), whee: X :R Ω E. Hee X(t, ) s a andom vaable n E and X(,ω) s a andom functon t X(t,ω) called the tajectoy of the pocess. The ndex paamete t Rs often ntepeted as tme, and we call the andom vaable X(t) = X(t, ) the state of the pocess at tme t. Fo smplcty, the explct dependence on the expementω s dopped. If the state space E s countable, an nteestng class of stochastc pocess ae Makov chans, whch ae defned by the followng popety: Defnton A.10. A contnuous tme stochastc pocess X(t) wth denumeable state space E s a Makov chan f and only f: P(X(t+ s)= j X(t)=,X(t 0 )= 0,...,X(t n )= n )=P(X(t+ s)= j X(t)=) (A.7) fo all, j E, 0 t 0 < t 1 <...< t n < t, k E and s 0. Equaton (A.7) s called the Makov popety, and t states that the behavo of the pocess afte the pesent tme t, gven the value of the state at ths tme, s ndependent of the pevous values of the state. If moeove the tanston pobabltes: P(X(t+ s)= j X(t)=) do not depend on t, we call the Makov chan homogeneous. The matx: P j (t)=p(x(t)=j X(0)= ) s called the tme t tanston matx, and{p(t) : t R + } the tanston semgoup. Note that each ow of P(t) must add up to 1, snce these ae the condtonal pobabltes of beng n some state, gven that at tme 0 we ae n state. By applyng the Makov popety (A.7) fo ntemedate states, the followng elatonshp between tanston matces can be establshed: P(t+ s)= P(t)P(s),

154 APPENDIX A. MATHEMATICAL PRELIMINARIES 141 whch s called the Chapman-Kolmogoov equaton. Defnng a Makov chan model va ts tanston semgoup can be had, snce we ae dealng wth a famly of matces. We can smplfy ths by notng that, f the devatve P (s) s well defned at s = 0, then the Chapman-Kolmogoov equatons togethe wth the condton P(0)=I mply: P (t)=p(t)q, whee Q= P (0). Theefoe: P(t)=e Qt. Thus, the matx Q= P (0) completely detemnes the tanston pobabltes, and t s called the nfntesmal geneato of the chan. The entes of the matx Q ae ntepeted as tanston ates. Note that n patcula q = j q j. Moe fomally, the tanston pobabltes of the Makov chan satsfy the followng popety: Poposton A.3. Let X(t) be a Makov chan wth nfntesmal geneato Q. Then, fo any j E we have: P(X(t+ h)= j X(t)=)=q j h+ o(h). We ae nteested n the lmtng behavo of a Makov chan pocess. An mpotant defnton s the followng: Defnton A.11 (Ieducble Makov chan). A Makov chan s called educble f fo evey, j E thee exsts s 0 such that P j (s)>0. The ntuton s that we can each evey state fom any othe state n fnte tme wth postve pobablty. We now tun ou attenton to the dstbuton of the andom vaable X(t). Note that: P(X(t)=j)= P(X(t)= j X(0)= )P(X(0)=)= P(X(0)= )P j (t), o n vecto notaton, fν s a ow vecto wthν (t)=p(x(t)=), we can wte: ν(t)=ν(0)p(t). An nvaant dstbuton fo a Makov chan s a pobablty dstbuton π on E such that, when ntepeted as a ow vecto, vefes: π=πp(t) t 0. Note that fν(0)=π, then the dstbuton of X(t) emans the same fo all t. It s not dffcult to see that n ths case, the Makov chan pocess s statonay, and we say that the

155 APPENDIX A. MATHEMATICAL PRELIMINARIES 142 pocess s n steady state. Note that the above condton s equvalent to fndng a vecto π wthπ 0 and such that: πq= 0, π1=1, whee Q s the nfntesmal geneato and 1 s a column vecto of ones. Not evey Makov chan has an nvaant dstbuton. The exstence of nvaant dstbutons s lnked to how often the states ae vsted. An mpotant defnton s the followng: Defnton A.12 (Postve ecuent Makov chan). Let E and consde the followng stoppng tmes: τ = nf{t 0 : X(t) }, τ = nf{t τ : X(t)=}. τ s the ext tme fom andτ s the etun tme to. A state s called postve ecuent f and only f: E(τ X(0)=)<. If all of the states ae postve ecuent, we say that the chan s postve ecuent. In patcula, fo educble chans, ethe all states ae postve ecuent o not. Note that f the chan s postve ecuent, we expect to vst all of the states fequently, so the chan can effectvely be found afte some tme n any of the states. Ths s lnked to the asymptotc behavo of the chan, whch we summaze as follows: Theoem A.6 (Egodc theoem fo Makov chans). If a Makov chan s educble and postve ecuent, thee exsts a unque nvaant pobablty dstbuton π gven by: whch also satsfesπ > 0. π = E(τ X(0)= ) E(τ X(0)= ), Moeove, fo any ntal dstbutonν(0), the dstbuton of the pocess at tme t,ν(t) conveges to π as t n dstbuton. Due to ths esult, we call educble and postve ecuent chans stable. Ths s because no matte what ntal dstbuton we choose fo the system, afte some tme, t wll exhbt a steady state egme chaactezed by the dstbutonπ.

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