A DISCRETE-TIME MODEL OF TCP WITH ACTIVE QUEUE MANAGEMENT

A DISCRETE-TIME MODEL OF TCP WITH ACTIVE QUEUE MANAGEMENT by Hui Zhang B.Sc., Insiuion of Informaion and Communicaion, Dong Hua Universiy, P. R. China, 1 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Maser of Applied Science In he School of Engineering Science Hui Zhang 4 SIMON FRASER UNIVERSITY Augus 4 All righs reserved. This wor may no be reproduced in whole or in par, by phoocopy or oher means, wihou permission of he auhor.

APPROVAL Name: Degree: Tile of Thesis: Hui Zhang Maser of Applied Science A Discree-Time Model of TCP wih Acive Queue Managemen Examining Commiee: Chair: Dr. Glenn H. Chapman Professor, School of Engineering Science Dr. Ljiljana Trajović Senior Supervisor Professor, School of Engineering Science Dr. Uwe Gläesser Supervisor Associae Professor, School of Compuing Science Dr. Qianping Gu Examiner Professor, School of Compuing Science Dae Defended/Approved: ii

ABSTRACT The ineracion beween Transmission Conrol Proocol TCP) and Acive Queue Managemen AQM) can be capured using dynamical models. The communicaion newor is viewed as a discree-ime feedbac conrol sysem, where TCP adjuss is window size based on pace loss deecion during he previous Round Trip Time RTT) inerval. In his research, a discree-ime dynamical model is proposed for he ineracion beween TCP and Random Early Deecion RED), one of he mos widely used AQM schemes in oday s Inerne. The model, consruced using an ieraive map, capures he deailed dynamical behaviour of TCP/RED, including he slow sar and fas reransmi phases, as well as he imeou even in TCP. Model performance for various RED parameers is evaluaed using he ns- newor simulaor. The resuls are hen compared wih oher exising models. Based on evaluaions and comparisons of our resuls, we can conclude ha he proposed model capures he dynamical behaviour of he TCP/RED sysem. In addiion, he model provides reasonable predicions for sysem parameers such as RTT, pace sending rae, and pace drop rae. An exension of he proposed model is also presened, which is a dynamical model of TCP wih Adapive RED ARED), a modified AQM algorihm based on RED. iii

ACKNOWLEDGEMENTS I would lie o han my senior supervisor, Professor Lijljana Trajović, for her suppor, guidance, and nowledge. I grealy appreciae ha she gave me his precious chance o coninue my sudies and become a member of he Communicaion Newors Lab. In addiion, I han her for encouragemen, grea help and sincere care. A he same ime, I would lie o acnowledge he suppors of all of he members in our lab, especially Judy Liu, Kenny Shao, Niola Cacov, Vladimir Vuadinović, and Wan Zeng Gang. My special hans o Dr. Qianping Gu and Dr. Uwe Glaesser for heir valuable commens and for serving on my supervisory commiee. iv

TABLE OF CONTENTS Approval...ii Absrac...iii Acnowledgemens... iv Table of Conens... v Lis of Figures...vii Lis of Tables... ix 1 Chaper One: Moivaion... 1 Chaper Two: Bacground... 3.1 TCP algorihm... 3.1.1 TCP congesion conrol... 3.1. TCP implemenaions... 5. RED algorihm and evaluaion of is parameers... 7..1 Acive Queue Managemen... 7.. RED algorihms... 8..3 Evaluaion of RED parameers... 11..4 Recen sudies on AQM... 13.3 Simulaion ool: ns-... 14 3 Chaper Three: A survey of exising models... 15 3.1 Overview of modeling TCP Reno... 16 3. Overview of modeling TCP Reno wih RED... 4 Chaper Four: A discree approach for modeling TCP RENO wih Acive Queue Managemen... 6 4.1 TCP/RED model... 6 4.1.1 Case 1: No loss... 8 4.1. Case : One pace loss... 3 4.1.3 Case 3: A leas wo pace losses... 3 4. Properies of he S-RED Model... 33 4.3 Model validaion... 34 4.3.1 Validaion wih defaul RED parameers... 35 4.3. Validaion for various ueue weighs w... 41 4.3.3 Validaion for various drop probabiliies p max... 45 4.3.4 Validaion wih various hresholds min and max... 49 4.3.5 Validaion summary... 5 4.4 Comparison of S-RED model and M-model... 53 4.4.1 Comparison for defaul RED parameers... 54 v

4.4. Comparison for various ueue weighs w... 54 4.4.3 Comparison for various drop probabiliies p max... 57 4.4.4 Comparison for various hresholds min and max... 58 4.5 Model modificaion... 6 4.6 Model exension: TCP wih Adapive RED S-ARED model)... 64 4.6.1 S-ARED model validaion... 67 4.6. Model evaluaion... 7 5 Chaper Five: Conclusions and fuure wor... 7 Reference Lis...74 vi

LIST OF FIGURES Figure 1. Evoluion of he window size in TCP Reno, consising of slow sar, congesion avoidance, fas reransmi, and fas recovery phases.... 4 Figure. RED drop probabiliy as a funcion of he average ueue size... 1 Figure 3. RED drop probabiliy wih p max =... 1 Figure 4. RED drop probabiliy wih p max = 1... 1 Figure 5. TCP window size evoluion... 16 Figure 6. Topology of he modeled and simulaed newor.... 7 Figure 7. Evoluion of he window size in he S-RED model. The fas recovery phase has been simplified... 34 Figure 8. Topology of he simulaed newor... 35 Figure 9. Evoluion of he window size wih defaul RED parameers: a) S-RED model, and b) zoom-in, c) ns- simulaion resuls and d) zoom-in. Waveforms show good mach beween he S-RED model and he ns- simulaion resuls... 36 Figure 1. Evoluion of he average ueue size wih defaul RED parameers: a) S-RED model and b) zoom-in, c) ns- simulaion resul and d) zoomin. The average ueue size obained using he S-RED model is higher han he average ueue size obained using ns- simulaions.... 38 Figure 11. S-RED model: evoluion of he window size for w =.6... 41 Figure 1. ns-: evoluion of he window size for w =.6... 4 Figure 13. Comparison of he average ueue size for various w... 43 Figure 14. S-RED model: evoluion of he window size for p max =.5... 46 Figure 15. ns-: evoluion of he window size for p max =.5... 46 Figure 16. Comparison of he average ueue size for various p max... 47 Figure 17. S-RED model: evoluion of he window size for min = 1 paces.... 49 Figure 18. ns-: evoluion of he window size for min = 1 paces.... 5 Figure 19. Comparison of he average ueue size for various min and max... 5 Figure. M-model: evoluion of he average ueue size wih defaul RED parameers.... 54 Figure 1. Comparison of he average ueue size for various w.... 55 vii

Figure. Comparison of he average ueue size for various p max... 57 Figure 3. Comparison of he average ueue size for various min and max.... 59 Figure 4. S-RED model wih modificaion: a) evoluion of he window size and b) he average ueue size.... 61 Figure 5. Comparison of he average ueue size for various w... 6 Figure 6. Comparison of he average ueue size for various p max... 63 Figure 7. Comparison of average ueue size for various min and max... 63 Figure 8. Evoluion of he window size: a) S-ARED and b) ns- simulaion resuls... 68 Figure 9. Evoluion of he average ueue size: a) S-ARED and b) ns- simulaion resuls... 69 viii

LIST OF TABLES Table 1. Defaul RED parameers in ns-.... 35 Table. Saisic values of sae variables for S-RED model and ns-... 4 Table 3. Saisical analysis of window size for various w.... 43 Table 4. Saisical analysis of he average ueue size for various w.... 44 Table 5. Sysem variable for various w.... 45 Table 6. Saisical analysis of he window size for various p max... 47 Table 7. Saisic analysis of he average ueue size for various p max... 48 Table 8. Sysem parameers for various p max... 48 Table 9. Saisical analysis of he window size for various max and min... 51 Table 1. Saisical analysis of he average ueue size for various max and min.... 51 Table 11. Saisic analysis of he average ueue size for various w.... 5 Table 1. Sysem parameers.... 53 Table 13. Sysem variables for various w.... 56 Table 14. Sysem variables for various p max... 58 Table 15. Sysem parameers wih various min and max.... 6 Table 16. Defaul ARED parameers in ns-.... 67 Table 17. Average values of he window size, average ueue size, and sysem parameers.... 7 ix

1 CHAPTER ONE: MOTIVATION Today s Inerne applicaions, such as he World Wide Web, file ransfer, Usene news, and remoe login, are delivered via he Transmission Conrol Proocol TCP). Wih an increasing number and variey of Inerne applicaions, congesion conrol becomes a ey issue. Acive Queue Managemen AQM) ineracs wih TCP congesion conrol mechanisms, and plays an imporan role in meeing oday s increasing demand for ualiy of service QoS). Random Early Deecion RED), a widely employed AQM algorihm, is a gaeway-based congesion conrol mechanism. An accurae model of TCP wih RED can aid in he undersanding and predicion of he dynamical behaviour of he newor. In addiion, he model may help in analyzing he sysem s sabiliy margins, and providing he design guidelines for selecing newor parameers. These design guidelines are imporan for newor designers whose aim is o improve newor robusness. Therefore, modeling TCP wih RED is an imporan sep owards improving he newor efficiency and he service provided o Inerne users. Modeling TCP performance has received increasing aenion during he las few years due o he benefis i offers o he neworing communiy. Analyical TCP models enable researchers o closely examine he exising congesion conrol algorihms, address heir shorcomings, and propose mehods for heir improvemen. They may also be used o compare various TCP schemes and implemenaions, and o deermine heir performance under various operaing condiions. Moreover, hese models help examine he ineracions beween TCP and he ueuing algorihms implemened in rouers. Hence, hey aid in he improvemen of exising algorihms and in designing beer algorihms, such as AQM echniues. Finally, such models offer he possibiliy of defining TCP-friendly behaviour in erms of hroughpu for non-tcp flows ha coexis wih TCP connecions in he same newor. 1

In his hesis, our goal is o model TCP and o invesigae he nonlinear phenomena in a TCP/RED sysem. We are paricular ineresed in he bifurcaion and chaos phenomena ha are recenly observed in a TCP/IP newor [4]. We use an ieraive map o model he sysem. A second-order discree model is derived in order o capure he ineracions of he TCP congesion conrol algorihm wih he RED ueuing mechanism. We use he conceps proposed in [34] and consruc a nonlinear dynamical model of TCP/RED ha employs wo sae variables: he window size and he average ueue size. The change of he window size reflecs he dynamics of TCP congesion conrol, while he average ueue size capures he ueue dynamics in he RED gaeway. The novely of he proposed model is in capuring he deailed dynamical behaviour of TCP/RED. The proposed model, called S-RED model, considers he slow sar, he congesion avoidance, he fas reransmi, and pars of fas recovery phases. I also aes ino accoun imeou evens. In addiion, he ineracion of TCP wih Adapive RED ARED), a revised RED algorihm, is discussed. We also derive a discree-ime model for TCP wih ARED, named S-ARED model, based on S-RED model.

CHAPTER TWO: BACKGROUND.1 TCP algorihm TCP, or Transmission Conrol Proocol, is designed o provide a connecion-oriened, reliable, and bye-sream service [4]. Before communicaion can begin beween wo hoss, a connecion mus be esablished by employing a hree-way handshae algorihm. This algorihm also allows he sender and receiver o iniialize and synchronize imporan parameers, seuence numbers, adverised window size, and maximum segmen size. The basic concep of he TCP congesion conrol algorihm is window based flow conrol. The window size deermines he amoun of daa o be sen by he source. Is value will increase when all of he acnowledgemens are properly received; oherwise, he window size may decrease. There are numerous versions of TCP implemenaions. In his research, he primary focus is on TCP Reno [1], [], [1], as i is one of he mos widely used versions in oday s newor. Oher versions of TCP implemenaions are briefly overviewed in Secion.1..1.1 TCP congesion conrol To adjus he window size, he TCP Reno congesion conrol mechanism employs four algorihms: slow sar, congesion avoidance, fas reransmi, and fas recovery, as shown in Figure 1. They were inroduced by Jacobson [], [1] and are described in RFCs 11 [4] and 581 [1]. 3

Figure 1. Evoluion of he window size in TCP Reno, consising of slow sar, congesion avoidance, fas reransmi, and fas recovery phases. In order o avoid congesing he newor wih large daa burss, an esablished TCP connecion firs employs he slow sar algorihm o deec he available bandwidh in he newor. Typically, a TCP sender iniializes is congesion window cwnd) o one or wo segmens, depending on he TCP implemenaion. Upon receip of each acnowledgemen ACK) for new daa, TCP incremens cwnd by one segmen size. When cwnd exceeds a hreshold sshresh), he sender swiches from slow sar o he congesion avoidance phase. During congesion avoidance, cwnd is incremened by one segmen size per Round Trip Time RTT). Each ime a pace is sen by he sender, a imer is se. Pace loss is deeced by his imeou mechanism if he imer expires before receip of he pace has been acnowledged. When muliple paces are dropped from he same window or he TCP sender does no receive enough paces o rigger fas reransmi, he TCP sender may need o wai for a imeou o occur before he los paces ge reransmied. If a pace loss is deeced by he imeou mechanism, he TCP sender adjuss is sshresh and swiches bac o slow sar. The duraion of he imeou TO) is usually updaed wih he RTT. If anoher imeou occurrs before he los pace is successfully reransmied, he period of imeou TO) doubles o wice he value of he previous duraion TO). This duraion ime coninues o 4

increase exponenially for each unsuccessful reransmission unil he imeou period reaches 64 imes he firs imeou period 64TO). The imeou period hen remains consan a 64TO. The fas reransmi algorihm is used for recovery from losses deeced by riple duplicae ACKs; i.e. four consecuive ACKs acnowledging he same pace. When an ou-of-order segmen is received by a TCP receiver, a duplicae ACK is immediaely sen ou o inform he sender of he seuence number for he pace ha he receiver is expecing. The receip of riple duplicae ACKs is used as an indicaion of pace loss. The TCP sender reacs o he pace loss by reducing cwnd by half and by re-ransmiing he los pace wihou waiing for he reransmission imer o expire. The fas recovery algorihm is used o conrol daa ransmission afer fas reransmission of he los pace. During his phase, he TCP sender firs ses is sshresh o half of he curren cwnd. Then i increases is cwnd for each duplicae ACK received. The fas recovery algorihm recognizes each duplicae ACK as an indicaion ha one pace has lef he channel and has reached he desinaion. Since he number of ousanding paces has decreased by one, he TCP sender is allowed o incremen is cwnd. When a non-duplicae ACK is received, TCP sender ses cwnd eual o sshresh and swiches from fas recovery o he congesion avoidance phase..1. TCP implemenaions Older TCP implemenaions, released in he early 198s, employed a simple windowbased congesion conrol mechanism as specified in RFC 793 [38]. TCP Tahoe, released in he lae 198s, employed he slow sar, congesion avoidance, and fas reransmi algorihms. When pace loss is deeced, he sender reransmis he los pace, reduces he congesion window o one and eners slow sar. When he sender receives an ACK for he reransmied pace, i will coninue o send paces from he seuence number indicaed in he ACK even if some of hem are already sen. Thus, if any pace is los in he 5

meanime, i will be auomaically reransmied via his go-bac procedure. In oher words, TCP Tahoe will no have o wai for a imeou in he case where muliple paces are los from he same window. However, freuen losses will force he TCP sender o operae in slow sar phase for mos of he ime. TCP Reno, inroduced in he early 199s, added he fas recovery algorihm. This algorihm does no reuire slow sar for every pace loss. Once he hree duplicae ACKs are received, he TCP sender reransmis he los pace, reduces is congesion window by half, and eners he fas recovery phase insead of he slow sar algorihm. When he firs non-duplicae ACK is received, he sender exis fas recovery mode and eners he congesion avoidance phase. TCP Reno's fas recovery algorihm is opimized for he case when one pace is los from a window. When muliple paces are los from he same window, TCP Reno generally mus wai for imeou. TCP Reno s congesion conrol algorihm wors well when dealing wih congesioninduced loss. However i resuls in reduced hroughpu wihou providing any benefis. I is also inappropriae when faced wih loss due o corrupion raher han congesion, such as saellie newors. In addiion, using ns- simulaions, Fall and Floyd [11] demonsraed ha TCP Reno exhibis poor performance in erms of lin uilizaion whenever muliple paces are dropped from a single window of daa. To alleviae his problem, hey inroduced wo modificaions o TCP Reno: TCP New-Reno and TCP SACK [5]. A large number of Inerne Web servers sill use TCP Reno and is varians [35]. For example, in he TCP SACK implemenaion, TCP receivers can inform senders exacly which segmens have arrived, raher han replying on TCP s cumulaive acnowledgemens. This allows a TCP sender o efficienly recover from muliple los segmens wihou revering o using a cosly reransmission imeou o deermine which segmens need o be reransmied. I also hasens recovery and prevens he sender from becoming window limied, hus allowing he pipe o drain while waiing o learn abou los segmens. This abiliy is 6

of paricular benefi in eeping he pipe full and allowing ransmission o coninue while recovering from losses. Oher TCP implemenaions, such as TCP Vegas [5] and TCP Weswood [7], use various echniues o avoid congesion. They adjus he congesion window size based on esimaes of he hroughpu a he bolenec. For insance, TCP Vegas conrols he congesion window size by esimaing he RTT and by calculaing he difference beween he expeced flow rae and he acual flow rae. I linearly adjuss TCP s congesion window size upwards or downwards, so as o consume a consan amoun of buffer space in he newor. I deecs pace losses earlier han Reno and uses a slower muliple decrease han Reno. TCP Vegas eliminaes he need o une he receive window o serve as an upper bound on he size of he congesion window. I can avoid newor congesion wihou overdriving he lin o find he upper bound, even when operaing wih large windows. TCP Vegas increases is congesion window size more slowly han Reno by measuring he achieved hroughpu gain afer each increase o deec he available capaciy wihou incurring loss.. RED algorihm and evaluaion of is parameers..1 Acive Queue Managemen A radiional DropTail ueue managemen mechanism drops he paces ha arrive when he buffer is full. However, his mehod has wo drawbacs. Firs, i may allow a few connecions o monopolize he ueue space so ha oher flows are sarved. Second, DropTail allows ueues o be full for a long period of ime. During ha period, incoming paces are dropped in burss. This causes a severe reducion in hroughpu of he TCP flows. One soluion, recommended in RFC 39 [], is o employ acive ueue managemen AQM) algorihms. The purpose of AQM is o reac o incipien congesion before he buffer overflows. AQM allows responsive flows, such as 7

TCP flows, o reac imely and reduce heir sending raes in order o preven congesion and severe pace losses... RED algorihms The mos popular AQM algorihm is Random Early Deecion RED), proposed by Floyd and Jacobson [16]. The RED mechanism conains wo ey algorihms. One is used o calculae he exponenially weighed moving average of he ueue size, so as o deermine he bursiness ha is allowed in he gaeway ueue and o deec possible congesion. The second algorihm is for compuing he drop or maring probabiliy, which deermines how freuenly he gaeway drops or mars arrival paces. This algorihm can avoid global synchronizaion by dropping or maring paces a fairly evenly-spaced inervals. Furhermore, sufficienly dropping or maring paces, his algorihm can mainain a reasonable bound of he average delay, if he average ueue lengh is under conrol. The RED algorihm is given in Algorihm 1, where is he average ueue size and is he insananeous ueue size. Le w be he weigh facor and +1 be he new insananeous ueue size. A every pace arrival, he RED gaeway updaes he average ueue size as. 1) + 1 = 1 w ) + w + 1 During he period when he RED gaeway ueue is empy, he sysem will esimae he number of paces m ha migh have been ransmied by he rouer. Hence, he average ueue size is updaed as m + 1 = 1 w ), ) m = idle _ ime ransmission _ ime, 3) where idle_ime is he period ha he ueue is empy and ransmission_ime is he ypical ime ha a pace aes o be ransmied. 8

Algorihm 1: The pseudo-code of RED algorihm. Iniializaion: = ; coun = -1; for each pace arrival { calculae } if ueue empy) { m = idle_ime / ransmission_ime = 1- w ) m ; } else { = 1- w ) + w ; } if min < < max ) { coun = coun +1; calculae Pa: Pb = p max min ) / max min ); Pa = Pb / 1- coun Pb); Drop arriving pace wih Pa; if arrival pace is dropped, coun = ) } else if > max ) or = buffersize)) { } Drop arriving pace; coun = ; 9

The average ueue size is compared o wo parameers: he minimum ueue hreshold min, and he maximum ueue hreshold max. If he average ueue size is smaller han min, he pace is admied o he ueue. If i exceeds max, he pace is mared or dropped. If he average ueue size is beween min and max, he pace is dropped wih a drop probabiliy p b ha is a funcion of he average ueue size p b + 1 = 1 + 1 max min min p max if if + 1 + 1 oherwise min max, 4) where p max is he maximum pace drop probabiliy. The relaionship beween he drop probabiliy and average ueue size is shown in Figure. Figure. RED drop probabiliy as a funcion of he average ueue size. p b 1 p max min max The final drop probabiliy p a is given by E. 5). I is inroduced in order o avoid he severe increase of he drop rae shown in Figure. p p = b a coun p. 5) 1 b 1

Here, coun is he cumulaive number of he paces ha are no mared or dropped since he las mared or dropped pace. I is increased by one if he incoming pace is no mared or dropped. Therefore, as coun increases, he drop probabiliy increases. However, if he incoming pace is mared or dropped, coun is rese o...3 Evaluaion of RED parameers Unlie DropTail, he performance of he RED algorihm is no deermined only by he buffer size. Oher parameers, such as ueue weigh w ), maximum drop rae p max ), and he wo ueue hresholds min and max ), are also of grea imporance. The newor designer should selec hese appropriaely, so ha he sysem can achieve beer performance. w is an exponenially weighed average facor, which deermines he ime consan for he averaging of he ueue size. If w is oo low, hen he esimaed average ueue size is probably responding oo slowly o ransien congesion. If w is oo high, hen he esimaed average ueue size will rac he insananeous ueue size oo closely, and may resul in an unsable sysem. In [17], uaniaive guidelines are given for seing w in erms of he ransien burs size ha he ueue can accommodae wihou dropping any paces. The opimal seing for w is beween.1 and.5. The pace drop probabiliy is calculaed as a linear funcion of he average ueue size if he average ueue size is beween min and max. However, he maximum drop probabiliy deermines he oscillaion of he drop rae. If p max is se oo small, hen he newor behaves similar o he DropTail shown in Figure 3. However, if his value is se oo high Figure 4), he pace drop probabiliy increases, hus enforcing he sysem o oscillae severely and experience a decrease in hroughpu. I has been shown ha seady-sae pace drop raes of 5% o 1% would no be unusual a a rouer. There is, herefore, no need o se p max higher han.1 for real newor implemenaions [17]. 11

Figure 3. RED drop probabiliy wih p max =. p b 1 p max = min max Figure 4. RED drop probabiliy wih p max = 1. pb p = 1 max min max The opimal values for min and max depend on he desired average ueue size. If he ypical raffic paerns are fairly bursy, hen min mus be correspondingly large o allow he lin uilizaion o be mainained a an accepably high level. This value should also be associaed wih he buffer size of he newor. If min is se oo small, i leads o low bandwidh uilizaion. Conversely, if min is se oo high, i may resul in unfair compeiion for bandwidh among muliple lins, hereby cancelling ou he benefis of he RED algorihm. The opimal value for max depends in par on he maximum average delay ha can be allowed by he gaeway. The RED 1

gaeway funcions mos effecively when he difference beween max and min is larger han he ypical increase in he calculaed average ueue size, in one roundrip ime [16]. A useful rule of humb is o se max o a leas wice he size of min. If he difference beween max and min is se oo small, hen he compued average ueue size can regularly oscillae up o max. This behaviour is similar o he oscillaions of he ueue size up o he maximum ueue size experienced wih DropTail gaeways...4 Recen sudies on AQM Alhough RED is one of he mos prominen acive ueue managemen schemes, a number of research effors have focused on possible shorcomings of he original RED algorihm and have proposed modificaions and alernaives, such as Adapive RED ARED) [15], CHOKe [36], BLUE [13], Sabilized RED, Flow RED, and Balanced RED. For example, ARED, proposed in 1, aims o improve he robusness of he original RED wih only minimal changes. I aemps o achieve a sable average ueue size around a pre-se arge value. Is defaul seing is half way beween min and max. I has been shown [15] hrough various simulaion resuls ha ARED ends o be more sable and performs beer han he original RED. The ey modificaion of ARED is updaing he maximum drop rae for every inerval period so ha he pace drop probabiliy changes more slowly han in he original RED algorihm. The calculaion of he modified p max is given in Algorihm. Oher algorihms are aimed a solving problems ha he original RED could no address. For insance, CHOKe [36] is designed o achieve beer performance on fairness, while he BLUE algorihm performs ueue managemen direcly based on pace loss and lin uilizaion, raher han on he insananeous or average ueue lenghs. In addiion, he BLUE algorihm can reac faser o subsanial increases of raffic load han he RED algorihm. 13

Algorihm : The pseudo-code of ARED algorihm. For every inerval usually he inerval is se o.5 seconds): { If > arge ) && p max <=.5)) } where { p max = p max + α ; } Else if < arge) && p max >=.1)) { } p max = p max * β ; arge: [ min +.4 * max min ), min +.6 * max min )]; α : min.1, p max /4); β :.9..3 Simulaion ool: ns- Ns- [33] is a discree even newor simulaor argeed a neworing research, which is currenly suppored by he Defence Advanced Research Projec Agency DARPA) and he Naional Science Foundaion NSF). I provides subsanial suppor for simulaion of TCP, rouing, and mulicas proocols over wired and wireless local and saellie) newors. In his research, ns- was used o perform he newor simulaion and he simulaion resuls were used o evaluae he proposed S-RED model. 14

3 CHAPTER THREE: A SURVEY OF EXISTING MODELS Many papers have been published on differen models for TCP. In his chaper, we describe he differen models and implemenaions of TCP. From he viewpoin of flow characerisics, analyical TCP models can be classified in hree caegories based on he duraion of he TCP flows. This, in urn, deermines he TCP congesion conrol algorihms o be modeled and he aspecs of TCP performance ha can be capured by he model []. The firs caegory models shor-lived flows where TCP performance is srongly affeced by he connecion esablishmen and slow sar phases [3]. These models ypically approximae average laency or compleion ime, i.e., he ime i aes o ransfer a cerain amoun of daa. The second caegory models long-lived flows, which characerize he seady-sae performance of bul TCP ransfers during he congesion avoidance phase [8], [3], [34], [41]. These models approximae aspecs such as he window size, average hroughpu, sending rae, and goodpu goodpu is used o describe he daa ha has been properly received). The final caegory includes models for flows of arbirary duraion, i.e., hose ha can accommodae boh shor and long-lived flows [6], [8]. All hese models provide funcions o compue he average values of hroughpu, sending rae, goodpu, or laency of TCP flows. From he conrol heoreic poin of view, he developed models of TCP and TCP/RED [18], [19], [3], [4], [31], [39], [4], can be classified ino wo ypes: averaged models and ieraive map models. An averaged model is described by a se of coninuous differenial euaions. I neglecs he deailed dynamics and only capures he low freuency characerisics of he sysem. I can be used o analyze he seady-sae performance and o predic low freuency slow-scale bifurcaion behaviour, such as Hopf bifurcaions. Examples of such models are given in [18], [19], [3]. In conras, an ieraive map model has a discree-ime form and employs a se 15

of difference euaions. I provides relaively complee dynamical informaion. The discree-ime map is adeuae o explore nonlinear phenomena such as period-doubling and saddle-node bifurcaions, which may appear across a wide specrum of freuencies and cause he exisence of soluions in he high freuency range. Examples of ieraive maps are given in [4], [39], [4]. 3.1 Overview of modeling TCP Reno Several models have been proposed in order o analyze he performance of pace newors. Prior o 1997, research was primarily focused on modeling he TCP congesion avoidance phase and ignored slow sar, fas recovery, and imeou evens. In [7], he auhors assume random pace loss wih a consan probabiliy p, which means ha approximaely 1 / p paces are o be sen followed by one pace loss. Hence, he evoluion of he cwnd resuls a periodic sawooh shown in Figure 5. Figure 5. TCP window size evoluion. 16

The cwnd is halved when a pace is dropped. During his period, here are 1 W W 3W + W ) = paces o be delivered, where W is he maximum cwnd. Since, 8 1 3W p = 8, i follows ha cwnd is W = 8 3p. 6) As cwnd increases by one pace size every RTT, he ime reuired o send 3W 8 paces is W RTT. Hence, he sending rae becomes BW 3 M W = 8 W RTT M C = RTT p, 7) where consan C = 3 /. By selecing an appropriae value for he consan C, he model is able o esimae he TCP performance over a range of loss condiions and in environmens wih boh DropTail and RED ueuing [8]. However, he predicion of he model is no very accurae, because i simplifies he TCP congesion conrol algorihm. A more precise seady-sae model of TCP Reno, inroduced in [34], also employs he seady-sae sending rae as a funcion of loss rae and RTT of a bul daa ransfer TCP flow. The model provides a deailed explanaion of he TCP algorihm. In addiion o capuring he essence of TCP s fas reransmi and congesion avoidance phases, i also accouns for he effec of he 17

imeou mechanism, which is imporan from a modeling perspecive. The model also aes ino accoun delayed ACKs, which are no discussed in his research. However, his model ignores he influence of ueue managemen schemes. In [34], auhors ae several seps o analyze he sending rae by separaing he congesions caused by hree-duplicae ACKs and imeous. They also consider oher upper bound parameers, such as adverised window size. The model describes he sysem in erms of duraion of he RTT, called rounds. The model is buil based on he following assumpions: RTT is a consan and i is considered o be independen of he window size; he maximum window size is also consan; only long lived connecions are sudied, which implies ha he source buffer always has enough daa o send; any pace loss is considered as lin congesion; he pace loss is correlaed wihin one round, i.e., if a pace is los, hen he remaining paces ransmied in ha round are also los. However, he pace loss is independen in differen rounds. Slow sar and fas recovery phases are no considered in his model. The following wo siuaions describe he cases sudied in [34]. Case one: The imeou is no considered and congesion is only deeced by hree duplicae ACKs. The expecaion of he oal number of paces Y ha are o be ransmied before a pace loss, is given by 1 p E[ Y ] = + E[ W ] p 1 8 5 p = + p 3p, 8) where p is he drop probabiliy and W is he value of congesion window size. The ime reuired o send Y paces is noed by A. The expecaion of A is 18

RTT 8 5 p E[ A] = 3 + ). 9) 3p From Es 8) and 9), he sending rae BW can be derived as BW = = E[ Y ] E[ A] 1 8 5p + p 3p. 1) RTT 8 5p 3 + ) 3p Euaion 1) can be simplified o E. 11) for small values of p as BW 1 3 =. 11) RTT p Case Two: The loss indicaion is deeced by hree duplicae ACKs and imeous: The sending rae in E. 1) is modified o BW E[ Y ] + Q E[ R] =, 1) TO E[ A] + Q E[ Z ] where Q is he probabiliy ha a pace loss is deeced by a imeou, R is he oal number of paces ha are o be sen during he imeou inerval, and Z TO is he duraion of he imeou. The final simplified euaion for small values of p) for he sending rae is BW 1, 13) p 3p RTT + T min1,3 ) p1 + 3 p ) 3 8 where T is he iniial value of a imeou. Measuremens demonsraed ha he model was adeuae over a range of loss raes [34]. 19

The model developed in [6] is a furher exension of he model developed in [34], which aims for a more precise model of TCP laency. The model aes ino accoun he ime used o se up he connecion, which is nown as he hree-way handshae algorihm, and he slow sar phase. This informaion is very imporan for modeling shor-lived connecions. In addiion, he model considers deailed informaion abou daa ransmission including congesion avoidance, fas reransmi, fas recovery, and imeou evens. The oal laency of he newor is E T ] = E[ T ] + E[ T ] + E[ T ] + E[ T ], 14) [ ss loss ca delac where T ss is he duraion of daa ransfer wihin he slow sar and congesion avoidance phases, T loss is he cos for imeou or fas recovery, T ca is he ime reuired o ransmi he remaining paces before loss recovery, and T delac is he cos of he delayed ACKs. 3. Overview of modeling TCP Reno wih RED A simplified firs-order discree nonlinear dynamical model was developed for simplified TCP wih RED conrol in [4], [39], [4]. The exponenially weighed average ueue size was used as he sae variable. The model describes he sysem dynamics over large parameer variaions and samples he buffer occupancy a cerain ime insances. This dynamical model was used o invesigae he sabiliy, bifurcaion, and roues o chaos of a newor for various sysem parameers. Based on he developed model, he auhors demonsraed ha nonlinear phenomena, such as bifurcaion and chaos, migh occur if he sysem parameers were no properly seleced. However, his discree model neglecs he dynamics of TCP. The derived map is given by ave + 1 1 w) = 1 w) 1 w) ave ave ave + w B N K + w p C d ) M if if ave ave oherwise ave u ave l, 15)

where ave 1 + : average ueue size in round +1 ave : average ueue size in round ave u : upper bound of he average ueue size ave l : lower bound of he average ueue size w : ueue weigh in RED algorihm B : buffer size N : number of TCP connecions K : consan [1, 8 / 3 ] p : drop probabiliy in round C : lin capaciy d : round-rip propagaion delay M : pace size. In [31], a second-order nonlinear dynamical model was developed o examine he ineracions of a se of TCP flows wih RED. The model employs he fluid-flow and sochasic differenial euaions. Window size and average ueue lengh are used as sae variables. From a se of coupled ordinary differenial euaions, he auhors develop a numerical algorihm o obain he ransien average behaviour of ueue lengh, round rip ime, and hroughpu of TCP flows. This model is described by nonlinear differenial euaions ha employ he average values of ey newor variables, given by 1

C W R N R p R R R W W R W = = ) ) ) ) )) )) )) ) ) 1 ) D D, 16) where ) W : expeced TCP window size ) : expeced ueue lengh ) R : round rip ime ) N : load facor number of TCP sessions) ) p : probabiliy of pace mar/drop C : lin capaciy. A hird-order dynamical model ha describes he ineracion of TCP flows wih an RED conrolled ueue was developed in [3]. The sae variables of he model are average ueue lengh, insananeous ueue lengh, and hroughpu. The TCP sources are idealized o operae only during he congesion avoidance phase, when he congesion window size follows he rule of linear increase and muliplicaive decrease. This dynamical model is used o explore various parameer seings and observe ransien and euilibrium behaviour of he sysem. The validiy of he model is verified by comparison wih simulaion resuls. The ineracion beween TCP and RED is modeled as )) )) )) )) ) ) ) )) / 1 ) ) ) / ) ))) 1 ))) 1 ))) 1 ) / ) )) ) ) / ) s p s p P R R m R P R m R P d d s p R c d s R s d d K K L L L K π π λ λ λ λ λ π µ π λ β λ + = + = = =, 17)

where s ) : expecaion of he exponenially averaged ueue lengh ) : expecaion of he insananeous ueue lengh λ ) : expecaion of TCP hroughpu a he source P L ) : loss probabiliy in he ueue a ime π )) : seady-sae probabiliies for he ueue o be full K π )) : seady-sae probabiliies for he ueue o be empy p : drop probabiliy R : round rip ime β : ueue weigh in RED algorihm m : number of idenical TCP sources. In [18], [19], he auhors describe a second-order linear model for TCP and RED, by linearizing he fluid-based nonlinear TCP model. Window size and average ueue lengh are used as sae variables of he sysem. The auhors performed analysis of TCP ineracions wih RED from a conrol heoreic viewpoin. They presened design guidelines for choosing RED parameers ha led o local sabiliy of he sysem. In addiion, hey proposed wo alernaive conrollers o improve he ransien behaviour and sabiliy of he sysem: a proporional P) conroller ha possesses a good ransien response, bu suffers from seady-sae errors in ueue regulaion, and a classical proporional-inegral PI) conroller ha exhibis zero seady-sae regulaion error and accepable ransien behaviour. One imporan conribuion of his wor is providing a good example of how o use classical conrol heory o solve problems in complex communicaion sysems. The linearized model around he operaing poin is described as 3

4 ) 1 ) ) ) )) ) 1 )) ) ) R W R N C p N C R R C R R W W C R N W δ δ δ δ δ δ δ δ δ = + =, 18) where W W W = δ = δ p p p = δ ),, p W : he se of operaing poins W : expecaion of he TCP window size : expecaion of he ueue lengh R : round rip ime C : lin capaciy p T : propagaion delay N : load facor number of TCP sessions) p : probabiliy of pace mar/drop. A muli-lin muli-source model, developed in [7], was used o sudy he sabiliy of a general TCP/AQM sysem. A local sabiliy condiion in he case of a single lin wih heerogeneous sources and he sabiliy region of TCP/RED were derived. The sae variables of his model are window size, insananeous ueue lengh, and average ueue lengh. The auhors demonsraed ha TCP/RED becomes unsable when he lin capaciy increases. Finally, hey devised a new disribued congesion conrol algorihm ha mainains local sabiliy for arbirary

delay, capaciy, and raffic load. Preliminary simulaion resuls were provided in order o illusrae he model s behaviour. 5

4 CHAPTER FOUR: A DISCRETE APPROACH FOR MODELING TCP RENO WITH ACTIVE QUEUE MANAGEMENT 4.1 TCP/RED model The basic idea behind RED is o sense impending congesion before i happens and o ry o provide feedbac o senders by eiher dropping or maring paces. Hence, from he conrol heoreic poin of view, he newor may be considered as a complex feedbac conrol sysem. TCP adjuss is sending rae depending on wheher or no i has deeced a pace drop in he previous RTT inerval. The drop probabiliy of RED can be considered as he conrol law of he newor sysem. Is disconinuiy is he main reason for oscillaions and chaos in he sysem. Hence, i is naural o model he newor sysem as a discree model. In his hesis, we model he TCP/RED sysem using a sroboscopic map, which is he mos widely used ype of discree-ime maps for modeling power converers [3], [9], [1], [43]. This map is obained by periodically sampling he sysem sae. In our sudy, he sampling period is one RTT. Since window size and ueue size behave as sep funcions of RTT, one RTT is he sampling period ha capures heir changes [14]. Higher sampling raes would no significanly improve he accuracy of he model. Conversely, lower sampling raes would ignore he changes and affec he model accuracy. The sae variables employed in he S-RED model are window size and average ueue size. These sae variables capure he deailed behaviour of TCP/RED. Variaions of he window size reflec he dynamics of TCP congesion conrol. The window size increases exponenially and linearly during he slow sar and he congesion avoidance phases, respecively. I muliplicaively decreases when loss occurs. The average ueue size capures he ueue dynamics 6

in RED as i is updaed upon every pace arrival. In our sudy, we did no consider insananeous ueue size as an independen sae variable. The S-RED model is suied for invesigaing he nonlinear behavior of TCP/RED and analyzing he sysem s ransiions ino chaos. Hence, in his hesis, we consider a simple newor opology as shown in Figure 6. I consiss of one TCP source, wo rouers, and a desinaion. RED is employed in he firs rouer. A TCP Reno connecion is esablished beween he source and he desinaion. Daa paces are sen from he source o he desinaion, while raffic in he opposie direcion consiss of ACK paces only. We made several assumpions in order o consruc an approximae model. We assume ha ACK paces are never los. The connecion is long-lived and he source always has sufficien daa o send. Round rip propagaion delay d beween he source and desinaion is consan. Daa pace size M is consan. The lin ha connecs he wo rouers is he only bolenec in he newor. We also assume ha he imeou is only caused by pace loss and ha he duraion of he imeou period is 5 RTTs [14]. The sae variables of he sysem are sampled a he end of every RTT period. We assume ha he ueue size is consan during each sampling period. The model includes hree cases, depending on he number of paces los in he previous RTT period: no loss, single loss, and muliple pace losses. Figure 6. Topology of he modeled and simulaed newor. Bolenec Source Rouer 1 Rouer Desinaion 7

4.1.1 Case 1: No loss Le W,, and be he window size, insananeous ueue size, and average ueue size, respecively, a he end of he sampling period. If no pace is dropped during he las RTT period, TCP Reno increases is window size. The window size is increased exponenially in he slow sar phase and linearly in he congesion avoidance phase: W + 1 min W, sshresh) if W < sshresh =, 19) min W + 1, rwnd ) if W sshresh where rwnd is he receiver s adverised window size; i.e., he larges window size ha he receiver can accep in one round. Usually, rwnd is greaer han he window size. In his case, rwnd does no affec he variaions of he window size. In he even ha he window size increases linearly and reaches he value rwnd, he window size is ep a rwnd unil a loss occurs in he newor. The round rip ime in +1 round is calculaed by M RTT +1 = d +, ) C where RTT + 1 : round rip ime in round + 1 d : round rip propagaion delay C : lin capaciy M : pace size. In order o calculae he average ueue size given by E. 1), we need o now he ueue size a he sampling period +1. The insananeous ueue size depends on he ueue size in he previous period, he curren window size, and he number of paces ha have lef he ueue during he previous sampling period. Therefore, he insananeous ueue size is given by 8

9 M d C W C M d M C W M RTT C W = + + = + = + + + + + 1 1 1 1 1 ). 1) where 1 + : insananeous ueue size in round + 1 : insananeous ueue size in round +1 W : curren TCP window size in round + 1 Subsiuing +1 in E. 1) yields he average ueue size:,) max ) 1 1 1 M d C W w w + = + +. ) Because RED updaes he average ueue size a every pace arrival, is updaed W +1 imes during he curren sampling period. For he firs pace arrival, he average ueue size is updaed according o,) max ) 1 M d C W w w + =. 3) For he second pace arrival, he average ueue size is given by [ ] ), max ), max ) 1 ) 1 M d C W w M d C W w w w + + =. 4) Since we assume ha ueue size is consan during each period, and ha here are W +1 pace arrivals during his ime inerval, is updaed W +1 imes:,) max ) ) 1 1 ) 1 1 1 1 1 M d C W w w W W + = + + + +. 5)

In summary, if p W <.5, which implies ha no pace loss occurred in he previous sampling period, he sae variables of he model are W + 1 + 1 minw, sshresh) if = min W + 1, rwnd) if = 1 w ) W + 1 + 1 1 w W ) W W + 1 < sshresh sshresh ) max W + 1. 6) C d,) M 4.1. Case : One pace loss If.5 p W < 1.5, which implies ha one pace loss occurred in he previous RTT period, he congesion conrol mechanism of TCP Reno halves he window size in he curren sampling period: 1 W = W + 1. 7) The average ueue size is updaed in he same manner as in Case 1: W + 1 + 1 1 = W = 1 w ) W + 1 + 1 1 w ) W + 1 ) max W + 1. 8) C d,) M 4.1.3 Case 3: A leas wo pace losses In his case p W 1.5, which implies ha a leas wo paces are los in he previous RTT period. When muliple paces are los from he same window, TCP Reno may no be able o send a sufficien number of new paces in order o receive hree duplicae ACKs for each los pace. The TCP source will probably have o wai for a imeou o occur before reransmiing he los pace [11]. During he imeou period, he source does no send paces ino he newor. In our model, window size is euivalen o he number of paces ha are sen by he source during one RTT period. Hence, we assume ha he window size is zero during he imeou period. 3

The RED mechanism updaes he average ueue size for each pace arrival. However, when he ueue is empy, i calculaes he duraion of his period named idle ime. Then, he average ueue size is updaed based on esimaing he number of paces ha could have been sen during his ime. Our model does no ae ino accoun he idle ime. Therefore, during he imeou period, here are no pace arrivals. The average ueue size is no updaed and has he same value as in he previous RTT period. The TCP/RED sysem during he imeou period is described a W + 1 + 1 =. 9) = The pseudo-code describing he S-RED model is given in Algorihm 3. 31

Algorihm 3: The pseudo-code of he S-RED model. Iniializaion: ; ; p ; for every round { calculae he produc of if p W <. 5 p W { compare W wih sshresh if W < sshresh { } else { } W + 1 minw, sshresh) W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + W 1 min W 1, rwnd) + + W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + calculae he drop probabiliy using E. 4) } elseif.5 p W < 1. 5 { W 1 +1 W W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + calculae he drop probabiliy using E. 4) C d,) M C d,) M C d,) M 3

} else { W + 1 + 1. calculae he drop probabiliy using E. 4) } } 4. Properies of he S-RED Model The proposed second-order discree-ime model capures he ineracions beween he TCP Reno congesion conrol algorihm and he RED mechanism. I models he dynamics of he TCP/RED sysem. Unlie pas models [3], [4], [34], [39], [4], he S-RED model includes he slow sar phase. I also aes ino accoun imeou, common in TCP [34], ha he models in [3], [4], [39], [4] ignore. However, our model does no capure all he deails of he fas recovery phase. Congesion window size in he S-RED model is no increased for each duplicae ACK received afer reransmiing he los pace. Insead of inflaing he window size, we assume ha he TCP sender swiches o he congesion avoidance phase wihou performing slow sar. Evoluion of he window size in he S-RED model is shown in Figure 7. The S-RED model capures he mos imporan characerisics of he RED algorihm. The average ueue size is updaed afer each pace arrival; i.e., W +1 imes in he sampling period +1. In conras, models presened in [4], [39], [4] updae he average ueue size only once every RTT period. However, in deriving he new model, we have also made simplificaions o he RED algorihm. RED uses coun o modify he drop probabiliy according o E. 5), while we ignored he coun ha eeps rac of he number of pace arrivals since he las pace drop. We 33

have also ignored he idle ime period, since i has no significan impac on he dynamics of he sysem. Figure 7. Evoluion of he window size in he S-RED model. The fas recovery phase has been simplified. 4.3 Model validaion In order o evaluae he accuracy of he S-RED model, we compare is performance wih he ns- simulaion resuls. The opology of he simulaed newor is shown in Figure 8. I consiss of one source, one sin, and wo rouers: R1 and R. RED is employed in rouer R1. The lin beween R1 and R is he only bolenec in he newor. Is capaciy is 1.54 Mbps and is propagaion delay is 1 ms. The capaciy of he lins beween he source and R1, and beween R and he sin, is 1 Mbps. This is sufficien o guaranee no congesion in hese lins. Their propagaion delay is ms. We evaluaed he window size and he average ueue size in he S- RED model using MATLAB [9] and compared he resuls wih ns- simulaion resuls. 34

Figure 8. Topology of he simulaed newor. The model validaion is divided ino four sages. Firs, we used ns- defaul RED parameers. Second, we chose various ueue weighs w, while eeping oher sysem parameers consan. In he hird scenario, we varied he maximum drop probabiliy p max. Finally, we varied he minimum and maximum ueue hresholds min and max simulaneously, while eeping heir raio max / min = 3. In each simulaion scenario, we observed sysem behaviour and capured average RTT, sending rae, and drop probabiliy. 4.3.1 Validaion wih defaul RED parameers In order o validae ha he S-RED model can capure he deails of he sysem behaviour, we evaluaed he ime waveforms of he wo sae variables: window size and average ueue size. The ns- simulaion resuls are obained using defaul RED parameers lised in Table 1. Table 1. Defaul RED parameers in ns-. Pace size M byes) 5 Maximum drop probabiliy p max.1 Minimum ueue hreshold min paces) 5 Maximum ueue hreshold max paces) 15 Queue weigh w. The waveforms of he window size for various ime scales are shown in Figure 9. The proposed S-RED model and ns- simulaion resuls are uie similar, especially during he seadysae. 35

Figure 1 shows waveforms of he average ueue size for various ime scales. The average ueue size in he S-RED model is approximaely one pace larger han he average ueue size resuling from he ns- simulaion resuls. This difference is due o he simplificaions ha we inroduced in he S-RED model. Our S-RED model employs pace-maring/drop probabiliy calculaed by E. 4), while he RED algorihm adops a smooh drop probabiliy p a ha increases slowly as he coun increases: p a p = b, 3) 1 coun p b where p b is he drop probabiliy given by E. 4) and coun measures he number of paces ha have arrived since he las dropped pace. In he S-RED model, p b is used as he final drop probabiliy and coun is ignored. Since p b < p a, he average ueue size of he S-RED model is larger han ha obained via ns- simulaions. Figure 9. Evoluion of he window size wih defaul RED parameers: a) S-RED model, and b) zoom-in, c) ns- simulaion resuls and d) zoom-in. Waveforms show good mach beween he S-RED model and he ns- simulaion resuls. a) 35 Window size paces) 3 5 15 1 5 1 3 4 5 6 7 8 Time sec) 36

b) 35 Window size paces) 3 5 15 1 5 4 41 4 43 44 45 46 47 48 49 5 Time sec) c) 35 Window size paces) 3 5 15 1 5 1 3 4 5 6 7 8 Time sec) 37

d) 35 3 Window size paces) 5 15 1 5 4 41 4 43 44 45 46 47 48 49 5 Time sec) Figure 1. Evoluion of he average ueue size wih defaul RED parameers: a) S-RED model and b) zoom-in, c) ns- simulaion resul and d) zoom-in. The average ueue size obained using he S-RED model is higher han he average ueue size obained using ns- simulaions. a) Average ueue size paces) 1 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 Time sec) 38

b) 8 Average ueue size paces) 7.5 7 6.5 6 5.5 5 4 41 4 43 44 45 46 47 48 49 5 Time sec) c) Average ueue size paces) 1 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 Time sec) 39

d) 8 Average ueue size paces) 7.5 7 6.5 6 5.5 5 4 41 4 43 44 45 46 47 48 49 5 Time sec) Table summarizes he saisic values of he wo sae variables: window size and average ueue size. I also compares he resuls wih he ns- simulaion. We calculaed he average, maximum, and minimum values of he window size and average ueue size over he duraion of he simulaion. In addiion, he maximum and minimum values during he seady-sae are also compued. We observed ha he window size obained from he S-RED model is similar o ha obained via he ns- simulaion. The difference of he average ueue size is approximaely one pace. Table. Saisic values of sae variables for S-RED model and ns-. Window size paces) S-RED model ns- %) Average ueue size paces) S-RED model ns- %) Average 15.35 15.1.5 7.4 5.95 1.63 Max 3 31.38-1. 7.69 8.14-5.6 Min.1.5 77.45 Max seady-sae) 1 1.86-3.93 7.69 6.51 18.16 Min seady-sae) 1 9 11.11 7. 5.41 9.79 4

4.3. Validaion for various ueue weighs w In order o evaluae he S-RED model for various sysem parameers, we varied he ueue weigh w beween.1 and.1. The window size and he average ueue size obained from S- RED model are compared o hose from ns- simulaion. The window size of S-RED model and ns- are similar. For example, Figures 11 and 1 show he waveform of he window size, for w =.6, for S-RED model and ns-, respecively. Figure 11. S-RED model: evoluion of he window size for w =.6. 3 Window size pace) 5 15 1 5 1 3 4 5 6 7 8 Time sec) 41

Figure 1. ns-: evoluion of he window size for w =.6. 3 Window size paces) 5 15 1 5 1 3 4 5 6 7 8 Time sec) The waveforms in Figures 11 and 1 show ha S-RED model and ns- have similar resuls for he window size. However, ns- has coarser waves. We also compared he waveforms of he window size beween S-RED model and ns- for various w. The resuls are similar o hose presened in Figures 11 and 1. Therefore, he S-RED model can capure he dynamical behaviour of he congesion window size, an imporan variable of he TCP algorihm. Figure 13 shows he simulaion resuls of he average ueue size for various w during seady-sae. When w increases, he average ueue size slowly decreases. The difference in he average ueue size beween S-RED model and ns- simulaions is approximaely one pace for various w. Tables 3 and 4 summarize he saisical analysis of he window size and average ueue size for various w during seady-sae. 4

Figure 13. Comparison of he average ueue size for various w. Average ueue size paces) 1 8 6 4 S-RED model ns- 1 3 4 5 6 7 8 9 1 W x 1-3 Table 3. Saisical analysis of window size for various w. Parameers weigh w ) S-RED model Average window size paces) ns- %) n Variance: x x) n n 1) S-RED ns- model.1 15.39 15.9 1.99 11.6 1.74. 15.35 15.1.5 11.7 1.71.4 15.14 14.98 1.7 1.61 1.43.6 15. 14.84 1.1 1.1 1.67.8 15. 14.89.74 1.4 11.7.1 15. 13.83 8.45 1.4 9.11 43

Table 4. Saisical analysis of he average ueue size for various w. Parameers weigh w ) S-RED model Average ueue size paces) ns- %) n Variance: x x) n n 1) S-RED ns- model.1 7.38 5.98 3.41.7.1. 7.4 5.95 1.63.7.46.4 7.6 5.91 19.45.96.1.6 6.9 5.8 18.55.165.93.8 6.89 5.84 17.97.87.54.1 6.89 5.79 18.99.435.618 From Tables 3 and 4, we observed ha as w increases, he average ueue size and he window size slowly decrease. However, w has an obvious impac on he variance of he average ueue size. Large w implies large variance of he average ueue size. This may lead o an unsable sysem. For w beween.1 o.8, he window size of he S-RED model is in agreemen wih ha of ns-. Alhough he resuls of S-RED model and ns- have some difference, hey follow he same rend. We also evaluaed he S-RED model for oher sysem variables: average RTT, sending rae, and pace loss rae. The resuls are summarized in Table 5. Values obained using he S- RED model and ns- are uie similar. We observed ha small variaions in ueue weigh have significan influence on he average RTT. 44

Table 5. Sysem variable for various w. Parameers Average RTT msec) Sending rae paces/sec) Drop rae %) weigh w ) S-RED model ns- %) S-RED model ns- %) S-RED model ns- %).1 4.3 36.1 11.63 384.99 384.71.73.55.54 1.9. 39.9 36. 1.83 384.98 384.77.56.56.55.56.4 39.4 36. 8.8 385.11 384.79.83.59.56 6.1.6 39. 35.8 8.93 385.8 384.73.93.6.56 7.91.8 39. 35.8 8.9 385.1 384.68.19.61.55 11.11.1 38.9 35.7 8.96 385. 384.7.83.61.55 11.7 4.3.3 Validaion for various drop probabiliies p max In his simulaion scenario, we evaluaed he S-RED model for various p max. Oher parameers remain as lised in Table 1. When he maximum drop probabiliy is se o a very small value, he RED algorihm has only a small influence on he sysem behaviour. In his case, he sysem behaves similarly o TCP wih DropTail, which leads o bursy pace losses and longer ueuing delays. However, if he value of p max is close o one, a high dropping rae will cause he sysem o become unsable and decrease he hroughpu. We show he waveforms of he window size wih p max =.5. Figure 14 gives he resuls from he S-RED model and Figure 15 gives he resuls from he ns- simulaion. The resuls of he average ueue size are shown in Figure 16. The average ueue size decreases as he maximum drop rae increases. Resuls obained from he S-RED model and ns- simulaion resuls show he same rend. 45

Figure 14. S-RED model: evoluion of he window size for p max =.5. 3 Window size paces) 5 15 1 5 1 3 4 5 6 7 8 Time sec) Figure 15. ns-: evoluion of he window size for p max =.5. 3 Window size paces) 5 15 1 5 1 3 4 5 6 7 8 Time sec) 46

Figure 16. Comparison of he average ueue size for various p max. Average ueue size paces) 1 8 6 4 S-RED model ns-.1..3.4.5.6.7.8.9 1 Pmax Saisical analyses of he wo sae variables are lised in Tables 6 and 7. When p max increases, he average value of he window size and he average ueue size decrease. Table 6. Saisical analysis of he window size for various p max. Parameers Average window size paces) n Variance: x x) n n 1) p max S-RED model ns- %) S-RED model ns-.5 17. 15.67 8.61 14.3 1.61.1 15.35 15.1.5 11.7 1.71.5 14. 14.4 -.77 1.1 1.1.5 13.54 14.17-4.44 8.44 11.4.75 13.37 13.6-1.84 8.63 16.51 47

Table 7. Saisic analysis of he average ueue size for various p max. Parameers p max S-RED model Average ueue size paces) ns- %) n Variance: x x) n n 1) S-RED ns- model.5 8.5 6.66.87.46.8.1 7.4 5.95 1.63.7.46.5 5.94 5.36 1.8.16.9.5 5.39 5.1 3.45.13.44.75 5.5 5. 5..15.417 Validaion resuls for he average RTT, sending rae, and drop rae are lised in Table 8. The resuls show ha he sysem variables in our S-RED model change in a similar manner o he resuls obained from he ns- simulaions. As expeced, when he maximum drop probabiliy increases, he acual drop rae increases. I leads o a decrease of he sending rae. A he same ime, he average RTT decreases, which indicaes a lower ueuing delay. p max Table 8. Sysem parameers for various p max. Average RTT msec) Sending rae paces/sec) Drop rae %) S-RED model ns- %) S-RED model ns- %) S-RED model ns- %).5 44.3 38.1 16.7 385.13 384.7.11.45.51-11.76.1 39.9 36. 1.83 384.98 384.77.6.56.55.56.5 36.5 34.5 5.8 384.93 384.73.5.65.59 11.8.5 35.3 34. 3.8 384.98 379.37 1.48.73.61 19.9.75 34.8 35.1 -.85 384.63 357.55 7.6.74.65 14.37 48

4.3.4 Validaion wih various hresholds min and max In his validaion scenario, he model is evaluaed for various ueue hresholds. Values of min and max are changed simulaneously, while mainaining he raio max / min = 3, as recommended in []. Here, we show he waveform of he window size of he S-RED model Figure 17) and ns- Figure 18), wih min = 1 paces and max = 3 paces. The simulaion resuls for he average ueue size are shown in Figure 19. Figure 17. S-RED model: evoluion of he window size for min = 1 paces. 4 35 Window size paces) 3 5 15 1 5 1 3 4 5 6 7 8 Time sec) 49

Figure 18. ns-: evoluion of he window size for min = 1 paces. 4 35 window size paces) 3 5 15 1 5 1 3 4 5 6 7 8 Time sec) Figure 19. Comparison of he average ueue size for various min and max. Average ueue size paces) 5 15 1 5 S-RED model ns- 4 6 8 1 1 14 16 18 min paces) 9 and 1. The saisical analysis for he window size and he average ueue size are given in Tables 5

Table 9. Saisical analysis of he window size for various max and min. Parameers min paces) S-RED model Average window size paces) ns- %) n Variance: x x) n n 1) S-RED ns- model 3 1.75 13.3 -.14 8.44 8.53 5 15.35 15.1.5 11.7 1.71 1 1. 19.61 7.8 18.7 17.8 15 6. 4.8 7.8 3.34 31.35 3.5 9.6 5. 4.31 38.64 Table 1. Saisical analysis of he average ueue size for various max and min. Parameers min paces) S-RED model Average ueue size paces) ns- %) n Variance: x x) n n 1) S-RED ns- model 3 4.67 3.97 17.63.13.3 5 7.4 5.95 1.63.7.46 1 13.1 1.77 1.81.13.175 15 18.39 15.86 15.95.474 1.18 3.5.55 1.16 1.113 1.65 When max and min increase, which implies ha more paces can be admied o he ueue wihou being dropped and mared, he window size, insananeous ueue size, and average ueue size, ogeher wih heir variances, increase. As shown in Table 11, he average RTT increases significanly as a resul of he increase in insananeous ueue lengh in he buffer, when he hresholds max and min increase. A he same ime, he drop rae reduces and he sending rae increases slowly. Hence, if he buffer is 51

large enough, increasing he wo hresholds can help o increase he sending rae and decrease he drop rae; however, his is a he cos of long delays and bursy raffic. Table 11. Saisic analysis of he average ueue size for various w. Average RTT msec) Sending rae paces/sec) Drop rae %) min pace s) S- RED model ns- %) S-RED model ns- %) S-RED model ns- %) 3 33.4 31.1 7.4 383. 38.44..78.71 1.1 5 39.9 36. 1.83 384.98 384.77.6.56.55.56 1 54.7 48.1 13.7 385.1 384.85.6.31.33-6.34 15 67.7 6.3 1.7 385.6 384.83.6.. -1.71 79.1 73. 8.36 385.3 384.95.9.15.16-5.66 4.3.5 Validaion summary We evaluaed he S-RED model using MATLAB and compared is performance o he ns- simulaion resuls. Firs we evaluaed he waveforms of he wo sae variables: window size and average ueue size. While he window sizes mached uie well, here were differences in he average ueue sizes during he seady-sae. The difference is due o he simplificaions o he RED s dropping algorihm ha we made in he S-RED model. Neverheless, he average ueue size in he S-RED model and ns- resuls have he same rend when sysem parameers vary. We have also performed hree ses of ess based on various sysem parameers: w, p max, min and max. Sysem variables RTT, sending rae, and drop rae have been evaluaed. The comparisons wih he ns- simulaion resuls show good agreemen. 5

4.4 Comparison of S-RED model and M-model In his secion, we compare our S-RED model wih he Maryland model, ermed M-model in his hesis, which is also a discree nonlinear dynamical model of TCP Reno wih RED gaeway [4], [39], [4]. A brief descripion of he M-model is given in Secion 3.. We have compared S- RED model wih ns- resuls in Secion 4.3. In his secion, we compare S-RED model, M-model, and ns- simulaion resuls. S-RED model and M-model are evaluaed using MATLAB. The comparison sudy is also divided ino four sages: wih defaul RED parameers, wih various ueue weighs w, wih various drop probabiliies p max, and wih various hresholds min and max. The defaul RED parameers are lised in Table 1. Oher sysem parameers corresponding o S-RED model and M-model are summarized in Table 1. Table 1. Sysem parameers. S-RED model M-model Lin capaciy 1.54e+6 1.54e+6 Pace size 5 5 Round-rip delay.8.8 Buffer size _ 1 Slow sar _ Number of TCP 1 1 Consan: K _ 3 53

4.4.1 Comparison for defaul RED parameers The M-model [4], [39], [4] is a firs-order discree dynamical model whose sae variable is he average ueue size. Hence, we can only observe he ime waveform of he average ueue size Figure ). Figure. M-model: evoluion of he average ueue size wih defaul RED parameers. Average ueue size paces) 8 7 6 5 4 3 1 1 3 4 5 6 7 8 Time sec) The value of he average ueue size of he M-model closely maches he ns- resuls shown in Figures 13 c) and d). Afer he ransien sae, he average ueue size reaches a consan value 5.71 paces), while he average ueue size for he ns- simulaions varies around is fixed poin. Conversely, he proposed S-RED model capures he dynamical characerisics of he average ueue size. 4.4. Comparison for various ueue weighs w In his simulaion se, we varied he ueue weigh w from.1 o.1, while eeping he parameers lised in Table 1 and Table 1 fixed. The simulaion resuls of he average ueue size for various w are shown in Figure 1 wih he resuls of S-RED model and ns-. 54

Figure 1. Comparison of he average ueue size for various w. Average ueue size paces) 1 8 6 4 S-RED model M-model ns- 1 3 4 5 6 7 8 9 1 W x 1-3 The average values of he average ueue size of M-model are closer o ns- han hose of S-RED model wih various w. However, for each fixed w, he average ueue size of M-model eeps consan during seady sae and can no show he dynamical behaviour of he sysem. The M-model was hen evaluaed for oher sysem variables, namely he average RTT, sending rae and pace drop rae. Resuls are summarized in Table 13 wih he resuls obained from S-RED model and ns- simulaions. is he difference beween S-RED model wih ns- and beween M-model wih ns-. 55

Table 13. Sysem variables for various w. Parameers Average RTT msec) weigh w ) S-RED model %) M-model %) ns-.1 4.3 11.63 45.8 6.87 36.1. 39.9 1.83 41.3 14.7 36..4 39.4 8.8 39.4 8.84 36..6 39. 8.93 38.8 8.38 35.8.8 39. 8.9 38.5 7.54 35.8.1 38.9 8.96 38.3 7.8 35.7 Parameers Sending rae paces/sec) weigh w) S-RED model %) M-model %) ns-.1 384.99.7 35.89-15.9 384.71. 384.98.6 355.6-7.67 384.77.4 385.11.8 37.4-3.83 384.79.6 385.8.9 374.9 -.55 384.73.8 385.1.11 377.5-1.93 384.67.1 385..8 378.37-1.65 384.7 Parameers Drop rae %) weigh w ) S-RED model %) M-model %) ns-.1.55 1.9.67 3.39.54..56.56.7 8.1.55.4.59 6.1.71 7.7.56.6.6 7.91.71 7.7.56.8.61 11.11.71 9.33.55.1.61 11.7.71 3.4.55 As shown in Table 13, we also evaluaed he M-model for he same sysem variables: average RTT, sending rae, and pace drop rae. Excep for w =.1 and., when he S- RED model performs beer, he discrepancy in predicing he RTT for he wo models are similar. In all cases, he S-RED model more accuraely predics he sending and drop raes. 56

4.4.3 Comparison for various drop probabiliies p max In his comparison sudy, we compare he M-model, he S-RED model, and ns- simulaion resuls for various drop probabiliies p max. Oher parameers remain as lised in Tables 1 and 13. The average ueue size for various p max is shown in Figure. Figure. Comparison of he average ueue size for various p max. Average ueue size paces) 1 8 6 4 S-RED model M-model ns-.1..3.4.5.6.7.8.9 1 Pmax The average values of he average ueue size of M-model mach beer he ns- resuls han he of S-RED model for various p max. Similar o Figure, for each fixed p max, he average ueue size of M-model is consan during seady sae. I is no able o show he deail dynamics of he sysem. The average RTT, sending rae, and drop rae for he M-model are also summarized in Table 14. Wih various drop probabiliies p max, he S-RED model beer esimaes he average RTT, he sending rae, and he drop rae. 57

Parameers Table 14. Sysem variables for various p max. Average RTT msec) p max S-RED model %) M-model %) ns-.5 44.3 16.7 43.4 13.91 38.1.1 39.9 1.83 41.3 14.7 36..5 36.5 5.8 39.9 15.65 34.5.5 35.3 3.8 39.4 15.88 34..75 34.8 -.85 39. 11.68 35.1 Parameers Sending rae paces/sec) p max S-RED model %) M-model %) ns-.5 385.13.11 354.3-7.9 384.7.1 384.98.6 355.6-7.67 384.77.5 384.93.5 356. -7.46 384.73.5 384.98 1.48 356.7-6.9 379.37.75 384.63 7.6 356.33 -.34 357.55 Parameers Drop rae %) p max S-RED model %) M-model %) ns-.5.45-11.76.63 3.53.51.1.56.56.7 8.1.55.5.65 11.8.74 6.5.59.5.73 19.9.76 3.98.61.75.74 14.37.77 19.1.65 4.4.4 Comparison for various hresholds min and max In his subsecion, we invesigae he comparison of he S-RED model, M-model, and ns- resuls for various ueue hresholds. We change he values of he hresholds min and max simulaneously, eeping he raio max / min fixed a 3, as recommended in [], and seing he oher parameers o he values in Tables 1 and 1. The average ueue size for various min and max is shown in Figure 3. The sending rae, RTT, and drop rae are summarized in Table 15. 58

Figure 3. Comparison of he average ueue size for various min and max. Average ueue size paces) 5 15 1 5 S-RED model M-model ns- 4 6 8 1 1 14 16 18 min paces) A comparison wih he M-model suggess ha he proposed S-RED model is more accurae in predicing average RTT, sending rae, and drop rae. However, M-model has a beer predicion for he average ueue size during seady sae. Similar o Figure, for each fixed min and max, he average ueue size of M-model says consan. Therefore, S-RED model can capure more dynamical deails of he sysem han M-model. 59

Parameers min paces) Table 15. Sysem parameers wih various min and max. Average RTT msec) S-RED model %) M-model %) ns- 3 33.4 7.4 34.1 9.65 31.1 5 39.9 1.83 41.3 14.7 36. 1 54.7 13.7 6.8 6.4 48.1 15 67.7 1.7 83.1 37.81 6.3 79.1 8.36 19.1 49.45 73. Parameers min paces) S-RED model Sending rae paces/sec) %) M-model %) ns- 3 383.. 366.13-4.6 38.44 5 384.98.6 355.6-7.76 384.77 1 385.96.6 33.94-14.1 384.85 15 385.6.6 311.1-19.19 384.83 385.3.9 96.7-3.4 384.95 Parameers Drop rae %) min paces) S-RED model %) M-model %) ns- 3.78 1.1.96 35.4.71 5.56.56.7 8.1.55 1.31-6.34.37 11.78.33 15. -1.71. -1.79..15-5.66.14-11.95.16 4.5 Model modificaion The difference in he average ueue size beween he S-RED model and ns- is due o he simplificaions o he RED s pace discarding algorihm. The S-RED model employs he probabiliy p b E. 4) as he final drop probabiliy, while RED in ns- uses p a E. 5). If a modified drop probabiliy p a = α p b is used, he window size and he average ueue size would evolve as shown in Figure 4 a) and b), respecively wih he defaul RED parameer seings). 6

Comparisons show ha he average ueue size well maches he ns- simulaion resuls for modified drop probabiliies wih α = 1.8. Figure 4. S-RED model wih modificaion: a) evoluion of he window size and b) he average ueue size. a) 35 Window size paces) 3 5 15 1 5 1 3 4 5 6 7 8 Time sec) b) 8 Average ueue size paces) 7 6 5 4 3 1 1 3 4 5 6 7 8 Time sec) 61

For furher deail, we show a comparison of he average ueue size of he modified model wih he original S-RED model, modified S-RED model, and ns-, for various w, p max, and min in Figures 5, 6, and 7, respecively. From he following figures, we observed ha afer modifying he drop probabiliy, he modified model has beer agreemen wih ns- simulaion resuls. Figure 5. Comparison of he average ueue size for various w. Average ueue size paces) 1 8 6 4 S-RED model S-RED modified model ns- 1 3 4 5 6 7 8 9 1 w x 1-3 6

Figure 6. Comparison of he average ueue size for various p max. Average ueue size paces) 1 8 6 4 S-RED model S-RED modified model ns-.1..3.4.5.6.7.8.9 1 pmax Figure 7. Comparison of average ueue size for various min and max. Average ueue size paces) 5 15 1 5 S-RED model S-RED modified model ns- 4 6 8 1 1 14 16 18 min 63

4.6 Model exension: TCP wih Adapive RED S-ARED model) Adapive RED ARED) [15] is an exension o he RED algorihm proposed for improving RED s robusness. One of he main weanesses of RED is ha he average ueue size is oo sensiive o he parameer seings and he level of congesion. When he lin load is ligh or p max is high, he average ueue size is closer o min. On he oher hand, if he lin load is heavy or p max is low, hen he average ueue size is near or even greaer han max [15]. As ueuing delay is deermined by he average ueue size, he delay is also very sensiive o he parameers and is no easy o predic and conrol. Thus, he goal of ARED is o reduce he sensiiviy of he parameers ha affec RED performance and achieve a sable and predicable average ueue size based on he RED algorihm. The main idea of his approach is o adjus p max in response o dynamical changes of he average ueue size. The robusness of his algorihm depends on he freuency of adjusing p max. The pseudo-code of ARED algorihm was given in Secion..4. The ARED algorihm is an exension o he RED algorihm wih he modificaion of p max. We derived a TCP/ARED model based on our S-RED model discussed in Secion 4.1. We call i S-ARED model. The pseudo-code for he S-ARED model is described in Algorihm 4. We ep mos of he conceps and assumpions of he proposed S-RED model and incorporaed he modificaion of he sysem parameer p max. The value of he inerval, which refers o he freuency of updaing he value of p max, is a funcion of C RTT, where C is a consan. 64

Algorihm 4: The pseudo-code for he S-ARED model. Iniializaion: p ; ; ; sample_inerval C; par.4 max min ); α min.1, p max /4); β.9; for every round { if sample_inerval == { if > max - par && p max <=.5 } else { } calculae { p max p max + α } else if < min + par && p max >=.1 { p max p max β } sample_inerval = C sample_inerval = C-1; p calculae he produc of if p W <. 5 { p W 65

compare W wih sshresh if W < sshresh { } else { } W + 1 minw, sshresh) W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + W 1 min W 1, rwnd) + + W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + calculae he drop probabiliy using E. 4) } elseif.5 p W < 1. 5 { } W 1 +1 W W + 1 W + 1 1 w ) + 1 1 w ) ) max W 1 + 1 = + calculae he drop probabiliy using E. 4) C d,) M C d,) M C d,) M else { W + 1 + 1. calculae he drop probabiliy using E. 4) } } 66

4.6.1 S-ARED model validaion In order o validae he S-ARED model, we evaluaed he model using MATLAB and compared is performance wih he ns- simulaion resuls. For consisency, we used he same simulaion opology as shown in Figure 8. In order o evaluae how accuraely he S-ARED model can capure he dynamical behaviour of he sysem, we sudied he ime waveforms of he wo sae variables: window size and average ueue size. The ns- simulaion resuls are obained using defaul ARED parameers, lised in Table 16. Table 16. Defaul ARED parameers in ns-. Pace size M byes) 5 Maximum drop probabiliy p max Minimum ueue hreshold min paces) Maximum ueue hreshold max paces).1 5 15 Queue weigh w. Sample_inerval C 1 Figures 8 and 9 show he waveforms of he simulaion resuls for he window size and he average ueue size for he S-ARED model and ns-, respecively. 67

Figure 8. Evoluion of he window size: a) S-ARED and b) ns- simulaion resuls. a) 3 Window size paces) 5 15 1 5 4 6 8 Time sec) b) Window size paces) 35 3 5 15 1 5 4 6 8 Time sec) 68

Figure 9. Evoluion of he average ueue size: a) S-ARED and b) ns- simulaion resuls. a) 1 Average ueue size paces) 1 8 6 4 4 6 8 Time sec) b) 1 Average ueue size paces) 1 8 6 4 4 6 8 Time sec) 69

The waveforms for he window size and he average ueue size in ns- simulaions oscillae more severe han hose of S-ARED. A deailed comparison of simulaion resuls of S- ARED and ns-, for he wo sae variables and hree sysem parameers RTT, sending rae, and drop rae) are lised in Table 17. Table 17. Average values of he window size, average ueue size, and sysem parameers. S-ARED model ns- %) Window size paces) 17.66.1-1.13 Average ueue size paces) 9.69 8.69 11.5 RTT msec) 45.9 44. 3.84 Sending rae paces/sec) 385.3 377.41. Drop probabiliy %).44.6 69.3 The RTT and sending rae of he S-ARED model and ns- mach uie well. The difference in he average ueue size remains o be one pace. Due o he differen drop probabiliy, he difference in he average value of he window size is more han wo paces. The S-ARED model sill capures he dynamical behaviour of he sysem 4.6. Model evaluaion The purpose of he ARED algorihm is o mainain he average ueue size a a specified arge value which is approximaely max + min )/. From he resuls, he average ueue size in he S-ARED model is 9.69 paces. This is close o he arge value of 1 paces. Compared o he resuls of 7.4 paces from he S-RED model Table ), he average ueue size is less dependen on he parameer min in ARED algorihm han in RED algorihm. Also, from he evaluaion resuls, S-ARED model exhibis an improved sending rae and he drop probabiliy. However, S-ARED model is uie sensiive o he sampling inerval of p max. If he sampling 7

inerval is small, he TCP/ARED sysem can have a faser response o he newor. As a resul, i is less sable. Conversely, if he sampling inerval is oo long, he sysem behaves lie TCP/RED. 71

5 CHAPTER FIVE: CONCLUSIONS AND FUTURE WORK The TCP/RED sysem can be viewed as a complex feedbac conrol sysem, where TCP adjuss is sending rae depending on he pace loss probabiliy deermined by RED. In his hesis, we have inroduced a second-order discree-ime analyical model for he ineracion beween TCP Reno and RED algorihms. We used an ieraive map o consruc he discree-ime model of he sysem. The S-RED model capures he dynamical behaviour of TCP/RED and may be used o sudy is nonlinear behaviour. Unlie oher models, i aes ino accoun he TCP slow sar and imeou evens. We evaluaed he model by comparing is performance o ns- simulaions. Validaion of he S-RED model illusraes he performance of he model for various RED parameers. The comparison shows ha our S-RED model leads o similar resuls as ns-, paricularly for he window size, sending rae, RTT, and drop rae. However, here is difference in he average ueue size due o he model simplificaion. In Secion 4.5, afer modifying he S-RED model, boh sae variables, he window size and he average ueue size, mach well he ns- simulaions. We compared he S-RED model o he M-model, an exising firs-order discree-ime analyical TCP/RED model. Alhough he average value of he average ueue size of he M-model maches he ns- simulaion resuls beer han he S-RED model, he S-RED model capures more dynamical deails of he TCP and RED algorihms. Moreover, for he window size, sending rae, RTT, and drop rae, he S-RED model ouperforms he M-model. We also discussed a discree-ime model for TCP and ARED S-ARED) based on he proposed S-RED model. The S-ARED model can be considered as an exension o he S-RED model. The S-ARED model is validaed by comparing i o he ns- simulaion resuls. From he 7

S-ARED model, we find ha he ARED algorihm can help reduce some weanesses of he RED algorihm. The average ueue size is less dependen on he sysem parameer min, especially when he load of he newor is ligh. Thus, i is easier o predic he average ueue size as well as he lin delay. The S-ARED model can achieve a higher hroughpu and a lower drop rae. However, he sampling inerval of p max can affec he sabiliy of he sysem. Based on he comparisons of he resuls, we conclude ha he S-RED and S-ARED models can capure he dynamical behaviour of TCP/RED sysem and provide reasonable predicions of sysem parameers. There are wo avenues for fuure wor. Firsly, he new discree-ime model may be used o sudy he non-linear phenomena of he sysem, such as bifurcaion and chaos. Secondly, given he flexibiliy of he S-RED model, i can be used o evaluae TCP/RED based AQM schemes and develop new and improved algorihms. 73

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