Stability and Convergence of TCP-like Congestion Controllers in a Many-Flows Regime

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1 Stablty and Convergence of TCP-lke Congeston Controllers n a Many-Flows Regme Supratm Deb Coordnated Scence Lab Unversty of Illnos Urbana Illnos USA Emal: deb@uuc.edu Sanjay Shakkotta Department of Electrcal and Computer Engneerng Unversty of Texas Austn USA Emal: shakkott@ece.utexas.edu R. Srkant Coordnated Scence Lab Unversty of Illnos Urbana Illnos USA Emal: rsrkant@uuc.edu Abstract Wth the rapd growth of Internet parameter desgn and analyss for large-scale networks has become a topc of actve nterest. Snce smulaton of such large scale systems s not easy determnstc flud models have been wdely used for both qualtatve understandng of the behavor as well as parameter desgn for such networks. In ths paper we frst study a determnstc flud model for Internet congeston control when there are multple TCP-lke flows present. We provde condtons under whch such a system s globally asymptotcally stable n the presence of feedback delay. We then study the correspondng system wth the addton of web mce and other non-responsve flows modeled as stochastc dsturbances. We show that when there are a large number of flows choosng parameters based on the global stablty crteron for the determnstc system wth the nose replaced by ts mean value) ensures global stablty for the stochastc system as well. umercal examples and smulaton results wth some popular actve queue management mechansms valdate the parameter choces from analyss. The results ndcate that a system wth multple TCP-lke flows s globally stable as long as the bandwdthdelay product per flow s not very small. I. ITRODUCTIO The desgn phlosophy of the current Internet s based on the end-to-end paradgm wheren most of the ntellgence s at the end hosts. The network s task s to smply notfy the end systems whenever t detects congeston n the network. Congeston detecton s based on the aggregate flow behavor at the router and the end-hosts are notfed by smply droppng or by markng packets usng the explct congeston notfcaton EC) bt 4. The end-host reacts to ths nformaton by decreasng ts transmsson rate thus adaptng to network congeston. In ths manner end-to-end control s mantaned usng only mnmal network resources. Ths end-to-end desgn phlosophy has motvated a lot of work 7 0 usng a utlty functon maxmzaton framework leadng to a class of end-to-end rate control mechansms for Internet congeston control. Based on the choce of the utlty functon varous types of farness amongst users can be acheved. Further t has been shown n 0 that the congeston-avodance phase of TCP flow-control can be consdered a specal case of the above framework for approprately chosen utlty functons. Such determnstc rate based models of TCP have led to better understandng of TCP and ponts the way to mprove exstng congeston control mechansm used n the Internet. Determnstc rate update models whch explctly account for round-trp delay have been the focus of much study n the recent past. An mportant queston that has been addressed deals wth stablty of network controllers based on these determnstc rate adaptaton mechansms. In 8 a stablty condton for sngle proportonally far congeston controller wth delayed feedback was provded. Snce then ths result has been extended to networks n 6 7 and n 3 smlar results were shown for a dfferent class of controllers. We also refer the reader to 5 for other related analyss of congeston controllers wth delay. All of the above work dealt wth local stablty of the lnearzed controllers n the presence of round-trp delay. More recently suffcent condtons were derved n 3 for global exponental stablty for the case of a sngle flow accessng a lnk. However one may ask why we should consder determnstc models for Internet congeston control. In realstc systems there are two sources of randomness. Frst there can be flows that do not react to congeston control. For nstance these could be n the form of web-mce whch are short flows whch termnate before they can react to congeston control. Such uncontrolled flows can be modeled as stochastc dsturbances at the router. Second the markng decsons at the router could be probablstc. To see ths consder a partcular tme-nstance where the router decdes to mark 0% of the packets. Due to the constrant that the router can toggle the state of only a sngle bt n the packet header a possble strategy s to mark each packet ndependently and wth probablty 0.. In ths paper we concern ourselves wth randomness generated due to uncontrolled short flows n a many-flows regme. We refer the reader to for an analyss of TCP behavor n the context of probablstc markng at the router. In 6 the authors justfed usng flud determnstc model for studyng proportonally far congeston controllers. They showed that n the many-flows regme the trajectory of the average rate at the router converged to that of the determnstc model wth the nose replaced by ts mean /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

2 A. Man Contrbuton In ths paper we consder a system consstng of a sngle lnk accessed by a large number of TCP-lke flows each wth dentcal feedback delay but wth possbly) dfferent ntal condtons and also accessed by a large number of uncontrolled flows. By a TCP-lke mechansm we refer to the rate control model of the congeston avodance phase of TCP proposed n 0 whch s also closely related to the model n 5. We are nterested n relatng ths stochastc model to a determnstc model where the nose process s replaced by ts mean. We show that n the presence of uncontrolled flows modeled as stochastc nose the determnstc flud model wth nose replaced by ts mean value s accurate n the followng sense: The average rate of the flows behaves lke a sngle flow asymptotcally n the number of flows and tme. Thus unlke n the proportonally-far case studed n 6 where convergence was shown for each tme as opposed to asymptotcally n tme) here the trajectory of the stochastc system does not converge to that of the determnstc system n the many-flows regme. However f the number of flows s large enough the global stablty crteron for a sngle flow wth mnor modfcatons) s also a global stablty condton for the stochastc system wth multple flows. Thus the mplcaton s that parameter desgn can be carred out usng determnstc analyss based on the sngle flow model. Further for some standard markng functons used n lterature we show that TCP-lke sources wth standard TCP parameters satsfy the stablty crteron when the bandwdthdelay product per source s suffcently large. B. Organzaton of ths Paper We begn wth a descrpton of the model n Secton II. We consder multple TCP-lke flows along wth uncontrolled flows n the model. In Secton III we study the system by smply consderng the mean of the uncontrolled flow rate through the lnk. We present condtons on the congeston control gan for the global stablty of such a system. The next natural queston s whether such a choce of gan parameter s the rght one when the dsturbance process due to uncontrolled flows s not a constant but a tme-varyng possbly stochastc) process. We answer ths queston n Secton IV. We show that the choce of gan parameter wth uncontrolled flows modeled as a constant s approprate when the number of flows s large. In Secton V we study the condtons derved n the context of current TCP parameters and gve examples to llustrate the results. We provde some smulaton results n Secton VI before concludng n Secton VII. II. SYSTEM MODEL Our model s that of a sngle bottleneck lnk beng accessed by many TCP-lke flows. The delay n the forward and the reverse path s d/ so that the round-trp delay of each source s d. Such a model can be applcable n a scenaro when multple users behnd an ISP access a server through a common bottleneck lnk as n Fgure. The number of users SOURCES Fg.. d/ LIK d/ The system model n the system s whch s also the scalng parameter. We consder a sequence of such systems ndexed by. Inthe -th system there are sources accessng the lnk and the capacty of the lnk s scaled as c so that capacty per source s mantaned at c. Further n the -th system there are uncontrolled flows accessng the lnk. Before we descrbe the rate update mechansm of the sources we frst comment on the markng functon of the lnk. The lnk has a markng functon pλ C) whch denotes the fracton of packets marked when the total arrval rate nto the lnk s λ and the lnk capacty s C where C = c). The markng functon s assumed to satsfy the followng condtons. Assumpton : Markng Functon) ) The functon pλ C) s ncreasng n λ and s Lpschtz contnuous n λ. ) We further assume that pλ C) =pλ/c ). The frst assumpton s obvous snce pλ C) s the fracton of packets marked. The second assumpton says that the fracton of packets marked smply depends on the rato of the total arrval rate and the lnk capacty. To understand ths property n the context of our scalng suppose n the -th system the rate of the -th controlled flow s x and the rate of the -th uncontrolled flow s e for. Then the markng functon for the -th system s ) x + e pλ c) =p x + e ) c = p c ) where x and e are the average rate of the controlled and the uncontrolled flows respectvely. Thus under ths assumpton the markng functon n the -th system smply depends on the average flow rate through the lnk for some fxed capacty per flow. Two examples of markng functons whch have ths property are: ) B λ. pλ C) = C aλ. pλ C) = C a)λ. The frst markng functon has the nterpretaton of the queue sze beng B or larger n an M/M/ queue wth arrval /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

3 rate x. The second markng functon can be used a rate based model for REM for a sutable choce of a 8. We remark that any reasonable markng functon should satsfy the second assumpton. Ths ensures the scalablty of the markng functon n the number of flows. Thus from now on we wll nterpret the arguments x and c of the markng functon px c) as the average arrval rate and the capacty per flow respectvely. Further n the systems we consder from now on the capacty per flow c wll reman constant and the only tme-varyng parameter s the average rate x. Thus to avod unnecessary notaton we wll hde the dependence of p c) on c and let px) px c) =px/c ) In addton to the controlled flows we assume that the system s accessed by uncontrolled flows. These are flows whch do not react to congeston sgnals and are modeled as stochastc processes wth mean a. Inthe-th system there accessng the as..d and bounded stochastc t) s satsfy are uncontrolled flows {e ) t) + a} = lnk. We model {e ) t)} = processes wth mean 0. We assume that e ) lm sup t 0T ) = e ) t) =0a.s.. We now descrbe our model for the controlled flows. We consder a flud model for the rate update of the controlled flows. Denote by y ) t) the flow rate of the -th flow at tme t when there are such flows present n the system. Further denote by x ) t) the average flow rate of the controlled flows through the lnk at tme t and so x ) t) = = y ) t). Smlarly denote by e ) t)+a) the mean flow rate of the uncontrolled flows through the lnk. The fracton of packets marked by the lnk s px ) +a+e ) t)) where the average flow rate at the lnk conssts of the average flow rate of the controlled flows x ) t) and the average flow rate due to uncontrolled flows a + e ) t). The rate update of the -th flow s governed by ẏ ) t) =κw y ) t)y ) t d)px ) t d)+a + e ) t d/)) =... ) where κ > 0 s a constant and determnes stablty n the presence of delay. ote that the feedback delay s constant. Ths s reasonable f we employ early congeston notfcaton schemes usng vrtual queues whch lead to neglgble queueng delays at the router. We comment that choosng κ =/3 and κw = /d results n the rate control model of TCP 0. It can be shown that n the absence of delay and by smply takng the mean of the nose process contrbuted by the uncontrolled flows the above system converges to the unque equlbrum pont gven by the soluton of py + a) = w. y In the rest of the paper we fnd condtons under whch the system converges to the equlbrum pont n the presence of feedback delay wth and wthout stochastc dsturbance ntroduced by the uncontrolled flows. We also dscuss the mplcaton of our results n the context of TCP. III. MULTIPLE TCP-LIKE FLOWS WITH IDETICAL ROUD-TRIP DELAYS AD COSTAT OISE In ths secton we study the system when the nose process due to the uncontrolled flows s smply modeled by a constant process of rate a. For the purposes of ths secton we keep the number of flows fxed and so we drop the superscrpt on y ) t) and x ) t) and smply denote them by y t) and xt) respectvely. The rate update of the -th flow s ẏ t) = κw y t)y t d)pxt d)+a) =... and the trajectory of the average flow rate through the lnk xt) can be descrbed by ẋt) =κw y t)y t d) pxt d)+a). Our goal s to fnd sutable condtons on κ for the above system to converge to ts unque equlbrum pont. We frst provde condtons under whch the average flow rate xt) s bounded. The followng result whch s derved n 5 provdes such a condton. Lemma 3.: 5 Suppose κd < β. Fx δ > 0 where δ can be arbtrarly small. Then there exsts t 0 δ) < such that for all t>t 0 xt) M β where M β s the smallest postve number satsfyng MβpM β + a w) wβ ) w + δ M β For the rest of ths paper we wll assume that δ n the precedng theorem s a very small number fxed at say δ = Wthout loss of generalty we study the system evoluton from tme t 0 onwards. Thus by shftng the tmeaxs) throughout the paper we assume that xt) M β for all t 0. Suppose the ntal value of the average rate les n some compact set 0K n whch the equlbrum pont of the system s ncluded. Then clearly there exsts a β for whch M β K. We now assume that ntal condton for each flow satsfes Assumpton : Intal Condton) The ntal trajectory for any user {...} satsfes M β ɛ) y s) <M β + ɛ) s d /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

4 for some ɛ<. Essentally ths says that the ntal values of the ndvdual user rates are not too far away from each other. Snce the value of M β s larger than the equlbrum rate not allowng the ntal user rates to be more than twce the value of M β s a reasonable assumpton. We now ntroduce the followng notaton for every par of flows. r j t) =y t) y j t) j) {...} ) Our goal s to show that r j t) converges to zero for approprately chosen κ. Ths wll enable us to show that the system ndeed converges to the unque equlbrum pont under sutable condtons. Frst note that the dynamcs of r j t) can be descrbed by the followng. ṙ j t) = κpxt d)+a)y t)y t d) y j t)y j t d) = κpxt d)+a)y t)y t d) y t)y j t d) +y t)y j t d) y j t)y j t d) = κpxt d)+a)y t)r j t d)+ y j t d)r j t) Before we state our result on the convergence of r j t) we restate the followng result from 3 on general tmevaryng lnear delay dfferental equatons. Ths wll be useful n provng the convergence of r j t). Lemma 3.: 3 Consder the delay dfferental equaton gven by ẋt) =at)xt)+bt)xt d) wth some ntal condton xt) =φt) t d 0. If there exsts q> such that at) and bt) satsfy d at) q max as) + bs) ) < sgnbt)) t d s t bt) for all t t then V t) <qv0)e αt t t where V xt)) = sup t d s t x s) and α>0 q > are constants. We are now n a poston to state and prove the followng result on the convergence of r j t). Theorem 3.: If κd satsfes κd < mn β 6M β pm β + a) then lm sup r j t) =0. t j) {...} Remark: Before we go nto the detals of the proof we wll llustrate the key dea n the proof nformally. Frst we can show that as long as all the flow rates {y t)} are less than 3M the dfference between ther rates {r j t)} wll decrease. Ths wll follow from the sngle flow global stablty condton from 3M M M y t) j Tme y t) Fg.. Man proof dea n Theorem 3. r t) j Lemma 3.. Second recall from Lemma 3. that the average rate s upper-bounded as well.e. xt) <M see Fgure ). Suppose at some tme say t t happens that for a partcular flow l y l t ) = 3M and up to tme t we have all the ndvdual flow rates strctly less than 3M. As the average rate at ths tme xt ) <Mthere wll be some flow k whose rate y k t ) <M. On the other hand by assumpton the ntal value of {r j.)} s less than M. From the decreasng property of {r j.)} t follows that r lk t ) < M. Ths along wth the fact that y k t ) <M mples that y l t ) < 3M leadng to a contradcton. Therefore we must have that all the flow rates are strctly less than 3M for all tme and the requred result wll follow. We now formally prove ths result. Proof of Theorem 3.: Snce β s assumed to be fxed throughout the proof we drop the subscrpt n M β and smply use M n ths proof. Defne t =nf{t : max y t) 3M} 3) t>0 {...} where we nclude the possblty of t beng nfnty whch would mean max y t) < 3M for all t>0). Further defne the functon V j t) as V j t) = sup rjs). t d s t We dvde the proof nto two steps. In the frst step we show that under the condton on κd gven by the statement of the theorem V j t) <qv j 0) exp αt) for all t<t for every j) and for some constants q and α. In the second step we show that t = and use ths to conclude that r j t) converges to zero. Step : Consder the delay-dfferental equaton ṙ j t) = κpxt d)+a)y t)r j t d)+y j t d)r j t) for every par j). Suppose that κd max pxt d)+a)y t)+y j t d)) < t d s t + y jt d) y t) /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

5 Snce we are consderng t such that t<t wehave0 < y t) < 3M for all {...}. Further the average rate xt) <M by our assumpton. Thus a suffcent condton for the precedng nequalty to be satsfed s κd6mpm + a) <. Snce we also have κd < β for M to be an eventual upper bound on xt) thus f κd satsfes the condton gven by the statement of the theorem and q s chosen to satsfy then by Lemma 3. < q< 6MpM + a)κd V j t) <qv j 0) exp αt) t <t where α s a functon of q. In ths case choose q such that < ) q<mn ɛ 6MpM + a)κd where ɛ s such that the ntal rates of the ndvdual flows le n M ɛ)m + ɛ). We wll consder q n the above range for the rest of the proof. Step : We now show that t as defned by 3) s not fnte. We wll show t by contradcton. Suppose t <. Snce the trajectores of y t) s are contnuous n t wehave max y t ) = 3M. Suppose k = arg max y t ). {...} If κd satsfes the condtons gven n the statement of the theorem we further have from Step that V kj t) < qv kj 0) exp αt) for all t<t. Snce V kj 0) < Mɛ) for all j from our assumpton on the ntal condton we have V kj t) <qmɛ) exp αt) for all t<t.fromthe contnuty of V kj t) n t whch n turn mples whch further mples We also have V kj t ) <qmɛ) sup t d s t r kj t ) < Mɛ q y k t ) y j t ) < Mɛ q j. y t ) <M = snce xt) <M for all t>0. If the average of quanttes s less than M there must be at least one of them less than M. Let that element be ndexed by l so that y l t ) <M.ote that l k snce y k t )=3M. We thus have y k t ) y k t ) y l t ) + y l t ) < M qɛ + M<3M 4) where we have used the fact that qɛ <. Buty k t )=3M. Thus we have arrved at a contradcton and so for all t>0 y t) < 3M for all {...}. Ths along wth Step mples that V j t) <qv j 0) exp αt) t>0. Ths proves the convergence of r j t). To show that the convergence s unform n all pars j) we smply note that the exponent n the exponental convergence only depends on the choce of κ and not on any specfc flow. Thus we have shown that the trajectores of all the flows get coupled f ther round-trp delays are the same. Usng ths we can now study the stablty wth multple flows by usng stablty results from the sngle flow case. Frst note that the average flow rate through the lnk xt) can be wrtten as xt) = y t) = = y k t)+ y t) y k t) = = y k t)+δt) where y k t) s any partcular flow and δt) s a term whch goes to zero exponentally. ext we rewrte the update equaton for flow k as follows. ẏ k = κw y k t)y k t d)py k t d)+a + δt d)) = κw y k t)y k t d)py k t d)+a) + κy k t)y k t d)py k t d)+a) y k t)y k t d)py k t d)+a + δt d)) = κw y k t)y k t d)py k t d)+a) y k t)y k t d)p β k t)+a)δt d) ẏ k = κw y k t)y k t d)py k t d)+a) + ηt) 5) The second last step follows from the mean-value theorem and β k t) =y k t d)+fδt d)) for some f such that 0 <f<. ote that snce δt) 0 exponentally and all the other terms are bounded ηt) 0 exponentally. Thus we can vew the trajectory of the k-th flow as a sngle flow accessng the lnk except for an addtonal term whch s neglgble for large t. It s thus natural to beleve that the stablty crteron for u = κw ut)ut d)put d)+a) 6) s suffcent to guarantee the stablty of the system wth multple flows. We show ths n the next theorem whch s a smple extenson of the global stablty result wth sngle flow n 3. Theorem 3.: Suppose κd < mn β R 6M β pm β + a) where R>0 s such that κd < R s a suffcent condton for 6) to be globally stable. Then the system descrbed by 5) s globally asymptotcally stable /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

6 Before we prove the above result we state a useful result on functonal dfferental equaton 9. Lemma 3.3 page 79 9): Consder the retarded functonal dfferental equaton ẋt) =fx t ) x 0 = φ 7) where x t = {xt + θ) : d θ 0} CB d 0 and φ CB d 0. Assume f : CB d 0 R n s contnuous Lpschtz and f0) = 0. Then 7) s exponentally stable f and only f there exsts a functonal V t φ) such that c φ V t φ) c φ V c 3 x t c 3 V c V t φ) V t ξ) c 4 φ ξ where the norms of the functons are defned as φ =sup d θ 0 φθ) ξθ) and c are some postve constants. Proof of Theorem 3.: Snce ẏ k = κw y k t)y k t d)py k t d)+a) s globally exponentally stable 3 we have from Lemma 3.3 the exstence of a Lyapunov functon V k t y kt ) such that V k γv k and satsfyng the propertes gven n Lemma 3.3. ow apply the Lyapunov functonal V t) V t y t )= to the system gven by5). Snce t s easy to see that from whch t follows that V k t y kt ) 8) k= sup η k t) K exp αt) k V γv + K c 4 exp αt) V t) V 0) exp γt) K c 4 exp γt) exp αt). γ α The result thus follows snce all the ntal condtons are assumed to le n a compact set. We note that the exponent n the exponental stablty can be chosen as mnγα). We next extend the results to the case when the nose s modeled as a tme-varyng process but satsfes a strong law of large number type property as dscussed n Secton II. IV. MULTIPLE TCP-LIKE FLOWS WITH TIME-VARYIG OISE We now consder a varaton of the earler system. We consder a sequence of systems so that n the -th system the rate adaptaton of the -th flow s descrbed by ẏ ) = κw y ) t)y ) t d)px ) t d)+a + e ) t d/)) {...} where e ) t)+a) s the average nose due to uncontrolled flows and x ) t) s the average flow rate of TCP-lke flows. Let us assume lm sup t 0T e ) t) =0 a.s. 9) where e ) t) s the average nose due to uncontrolled flows. Condtons for 9) to hold are gven n 6. Before we extend our results to stochastc and bounded nose process satsfyng 9) n Subsecton IV-A we frst study a system wth the nose process satsfyng a slghtly relaxed assumpton as follows. lm sup e ) t) =0. 0) t 0 ) The case wth e ) t) satsfyng 9) s dscussed n Subsecton IV-B and the results follow from those derved n Subsecton IV-A. A. TCP-lke flows wth nose process satsfyng SLL type property over nfnte tme We assume that lm sup t 0 ) e ) t) =0. ote that the value of κ does not change wth the scalng. We also have an upper bound M on the average rate when the dsturbance due to uncontrolled flows s smply assumed to be a constant rate a. Snce the average dsturbance process becomes arbtrarly small as grows the followng Lemma s easy to show usng the upper bound on the average rate proved n 5. Lemma 4.: Suppose κd < β. Then gven ɛ > 0 there exsts and tɛ ) such that ) and t t) x ) t) M β + ɛ where M β s as gven n Lemma 3. As n the prevous secton defne r ) j t) =y ) t) y ) j t) j) {...} ) As before t can be shown that ṙ ) j t) = κpxt d)+a + e ) t d/)) t)r ) y ) j t d)+y ) j t d)r ) j t). ) Snce the addtonal nose term becomes arbtrarly small for large we can expect {r ) j } to become small for condtons /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

7 when κd gven by Theorem 3. hold. We remnd the reader that for all the ntal condtons are assumed to satsfy the condtons gven n Assumpton. Theorem 4.: If κd < mn β 6M β pm β + a) then gven ɛ > 0 tɛ ) ) such that >) sup r ) j t) <ɛ t t ) Further f κd satsfes the condton gven n Theorem 3. then gven ɛ > 0 ɛ ) such that > ) lm t y kt) y ɛ k {...} where y s the soluton of w = y py + a). Proof: We refer the reader to 4 for detals of the proof. B. TCP-lke flows wth nose process satsfyng SLL type property over fnte tme If the nose process s stochastc and bounded then t can be shown 6 that lm sup t 0T e ) t) =0 a.s. 3) where e ) t) s the average nose due to uncontrolled flows. It s thus natural to beleve that the results n the prevous subsecton hold for almost all realzatons of the nose process satsfyng the above condton. Theorem 4.: Suppose e ) t) satsfes 3). If κd satsfes the condton gven n Theorem 3. then gven ɛ > 0 ɛ ) such that > ) y ) k T) y ɛ k {...} a.s. where y s the soluton of w = y py). Proof: See 4 for the proof. Fnally we note that n practce the sources may update ther rates at dscrete tme slots. We can vew each tme slot as a measurement nterval over whch rates are measured n the system and control actons by the routers and flows are updated. Typcally ths measurement nterval s measured n terms of the number of packets that can be processed by a typcal router. For example the tme-step could be 00 packets long. By scalng both the tme-step and the capacty we mantan a constant tme-step as measured n packets 6. To ths end let each tme tme-step n the -th system be /. Thus the update of the -th system at the k +)-th tme-step can be descrbed by the followng. y ) k +=y ) k+ κ w y ) ky ) k dpx ) k d+ e ) k d/ + a) ote that snce the delay d as measured n seconds s fxed the delay n the -th system corresponds to d tme slots. The contnuous tme model can now be embedded as y ) t) =y ) t for t wth a straght lne approxmaton used between ntegers. Smlarly the average rate process x ) t) and the nose process e ) t) can be defned. For such a system of the contnuous tme processes y ) t) the results n Theorem 4. can be proved as well usng technques analogous to those used n 6. For brevty we skp the detals here. V. STABILITY CODITIOS WITH STADARD TCP In prevous sectons we have seen that that stablty condton wth a sngle TCP connecton accessng a lnk along wth some addtonal condton s enough to ensure the stablty and the convergence of multple TCP-lke flows n a many flows regme. In ths secton we use the earler results to derve stablty condtons wth the standard TCP parameters. Snce the global stablty crteron of a sngle TCP-flow accessng a bottleneck lnk plays an mportant role n the many flows regme we frst state the stablty condton wth a sngle TCP flow wth parameters mpled by standard TCP. Recall that the congeston avodance phase of TCP can be modeled as 0 ẋt) = d xt)xt d)pxt d)+a c) 4) 3 where x s n segments each segment may correspond to 5 bytes) per tme unt and a s the mean flow rate due to the uncontrolled flows. We defne wt) as wt) =dxt) The evoluton of the congeston-avodance phase of TCP can be re-wrtten as ẇt) = ) 3 wt d)+ad wt)wt d)p. 3d cd 5) Snce xt) s the rate n segments per unt tme and d s the round trp tme wt) can be nterpreted as the congeston wndow sze n segments. Further the quantty cd determnes the desred bandwdth-delay product per source at the equlbrum. Thus 5) s a contnuous tme verson of the wndow update algorthm of TCP algorthm n the congeston avodance phase. The followng lemma provdes suffcent condtons under whch 5) s globally exponentally stable. The condton follows from the results derved n 3. We state the stablty condton of 5) below whch follows from Theorem. n 3. For the the rest of ths secton we let w ) qw) =p cd. Lemma 5.: The controller gven by 5) s globally exponentally stable f 3 < l 3 + M R) M 3 R 3 + l R) /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

8 where R =sup l w M qw) +wq w)) M s the smallest postve number satsfyng M M ) p M +ad) 3 and l s the largest postve number satsfyng l l + ) 3 M qm + ad) q l + ) 3 M qm + ad) +ad 3 ow we consder multple TCP-flows. We further consder only the mean flow rate of the uncontrolled flows snce the stablty of such a system s suffcent to ensure the stablty of the system wth stochastc dsturbances when the number of flows s large. Contnung wth the earler model let there be flows wth the update equaton of the -th flow descrbed by w t) = 3 3d w t)w t d)qwt d)+ad) {...} 6) where wt) s the average wndow sze wt) = dxt) where xt) s the average flow rate) of the flows. Usng Theorem 3. we can easly derve stablty condtons for the system descrbed by 6). The followng result provdes such condtons. Corollary 5.: The system gven by 6) s globally stable f the followng condtons hold: ) We have 3 M qm +a) < 6 where M s the smallest postve number satsfyng M ) q M + ad ) ) 3 M. ) The condton gven by Lemma 5. s satsfed. We next present examples for two dfferent markng functons to demonstrate the usefulness of the prevous result. We are nterested n fndng the range of the equlbrum bandwdth-delay product to ensure stablty and convergence of multple TCP flows. We now demonstrate that for the examples consdered we can ensure global stablty f the bandwdth-delay product s large enough. So far we have assumed that the markng functon s a functon of the total flow rate through the lnk. However note that one could also consder queue based markng at the router such as RED or REM). Gven a queue based markng functon usng a reflected Brownan moton RBM) approxmaton a technque to fnd an equvalent rate based markng functon was suggested n 8. We consder two markng functons n the followng examples: frst a rate based markng functon and second a rate based markng functon equvalent to REM whch s a queue based markng mechansm. Example M/M/ type markng functon): We consder the markng functon x ) B w ) B px c) = = c cd. 7) Here x s the average flow rate of the sources through the lnk and c s a parameter whch can be adjusted for a desred bandwdth per source at the equlbrum. Such a markng functon has the nterpretaton of probablty of the buffer sze beng larger than B n an M/M/ queue wth arrval rate x. The equlbrum rate per source x n ths case s gven by ) 3 x B+ B d = cd) B+. We are nterested n fndng values of equlbrum bandwdthdelay product x d to guarantee global stablty of multple TCP flows. We show that the condtons gven by Corollary 5.) are satsfed as x d. To make our calculatons easy we assume that cd) B/B+) > 5 for reasons whch wll become obvous soon. In other words we seek values of x d n the range 53/) /B+) ) to ensure global stablty of multple dentcal TCP-lke flows. For the gven markng functon frst note that we have for M gven n Lemma 5. where M < +cd) B B+ 3/) B+ < cd) B B K = It can also be easly seen that ) B+ ) B+ M qm) <K B+.. = Kcd) B B+ Also for l gven n Lemma 5. we have l > +cd) B B+ 3/) B+ 3 M pm) 3 > cd) B B+ ) B+ 5 3 KB+ ). It follows from some algebrac manpulatons that Condton n Corollary 5. s satsfed when ) 3 M l + R < 3 whch s satsfed f cd) B B+ 3 KB+ +B)+ 3 ) B+ 5 K 3 KB+)). ext note that M the upper bound on the average rate n condton of Corollary 5. s such that ) 3 B+ B M < cd) B+ +<K cd) B B /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

9 where K = ) B+ One can check that a suffcent condton for Condton n Corollary 5. to be satsfed s cd) B B+ 4K B+. Snce x d = 3/) /B+) cd) b/b+) a suffcent global stablty crteron for multple TCP-lke flows s x d ) 3 B+ max 3 KB+ + B)+ K 3 ) B+ 5 3 KB+ )) 4KB+ 5. It follows that the system s globally stable as x d. Wth a more accurate numercal calculaton based on Corollary 5. t can be verfed that for B = 8 the system s globally stable for x d 6.66 whch corresponds to at least 8 packets per source at the equlbrum snce each packet approxmately conssts of segments. For B =5a suffcent condton s x d 3.43 correspondng to at least 7 packets per source at the equlbrum wth packet szes of segments. One can obtan smlar results for the markng functon x ) B c x )) c px) = ) x B+ c whch can vewed as the blockng probablty n an M/M//B queue wth arrval rate x and servce rate c. A suffcent condton wth B =8s and wth B =5 x d 4.97 x d.44 provdes a suffcent condton for global stablty. Example Random Early Markng or REM): REM marks a packet wth probablty exp θw)) f t arrves to fnd a workload of W already present n the vrtual queue. Usng a reflected Brownan moton approxmaton 8 ths can be vewed as a mechansm wth the followng markng functon: θσ x px c) = θσ 8) x +c x) Here σ denotes the varablty of the traffc and c can be tuned to obtan a desred rate allocaton at the equlbrum.. For smplcty we assume θσ =0.5. The equlbrum rate allocaton can be obtaned by solvng It can be verfed that 0.5x d) 3 0.5x d +cd x d) = 3. x d 6cd) 3. Frst we argue that the global stablty condton s satsfed as x d. Suppose we consder cd n the range cd 5 whch corresponds to x d.63. We are nterested n fndng x d n the range.37 ) to guarantee stablty. ote that the parameter M n Lemma 5. satsfes K cd) 3 M K cd) 3 for sutable constants K and K. Further t can be shown that l K 3 cd) 3 and R = sup qw)+wq w)) K 4 cd) 3 l w M for approprate postve constants K 3 and K 4.Itfollowsthat Condton n Corollary 5. s satsfed when ) 3 M l + R < 3 whch s satsfed f cd) 3 3 K ) + K 4 K 3 Smlarly condton s Corollary 5. can be expressed as cd) 3 K5. Snce x d 6cd) 3 a suffcent condton for global stablty n ths case s x d K for a sutable constant K. One can use numercal calculatons to obtan the a suffcent condton for global stablty wth multple TCP flows as x d Ths corresponds to at least 5 packets per source at the equlbrum wth packet szes of segments. The above examples clearly ndcate that: For reasonable markng functons and large enough target bandwdth-delay product per source multple TCP flows eventually behave lke a sngle flow and the system s globally asymptotcally stable /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

10 VI. SIMULATIO RESULTS In ths secton we present smulaton results to llustrate the effectveness of determnstc flud models n desgnng markng functon to meet a target equlbrum rate allocaton. The purpose of the smulaton s two-fold. Frst we show that f the target equlbrum wndow sze or the bandwdth-delay product s large n the range predcted by the global stablty crteron) then the equlbrum wndow sze n a packet based smulaton s close to the one predcted by the determnstc flud models. Second n our earler sectons we have derved condton for global stablty of a system wth TCP-lke flows. The natural queston s: Why s stablty so mportant? To llustrate the mportance of desgnng parameters to ensure stablty we compare the performance of TCP wth a modfed verson. We show that large oscllatons can happen due to nstablty n a badly desgned system. We smulate a sngle bottleneck lnk beng accessed by 00 persstent flows. All the long-lved flows are TCP-flows usng the dynamc wndow flow control mechansm of TCP. The smulaton s packet based and the lnk marks a fracton of the packets n the event of a congeston and the marks are echoed back to the source. There are also uncontrolled flows around 0% of the lnk bandwdth).e. flows that do not react to congeston. The lnk has an M/M/ type markng functon gven by the followng. ) B pˆx) =. ˆx C Here ˆx s the estmated arrval rate at the lnk and C s the vrtual capacty. Usng the flud determnstc model the value of C requred to meet a target per source equlbrum rate of y s gven by 3 C = y + a ) B+ B d ) where d s the round trp tme s the number of flows and a s the mean flow rate due to the unresponsve flows. In our smulatons we choose = 00 d =50ms and a = 0% of the lnk bandwdth. The parameter B n the markng functon s chosen as B =8. For the purposes of our smulaton we smply smulate the congeston avodance phase of TCP so that each successful transmsson ncreases the current wndow sze W by / W and each mark reduces the wndow sze by half. We also note that such a descrpton of the wndow update assumes the wndow sze to be n segments. In the followng we assume that each segment s equvalent to one packet. We frst choose C to ensure total equlbrum rate of 40.5 packets/ms whch s equvalent to an equlbrum wndow sze per flow of roughly 6 packets. The plots comparng the total rate nto the lnk and the target rate are shown n Fgure 3. The plot s representatve of smulatons done wth dfferent choces of parameters and for a large number of flows. The smulaton seems to ndcate that for moderately large target equlbrum wndow sze n the range predcted by the stablty crteron) determnstc flud models based parameter and markng functon desgn s approprate. Total rate at lnk packets/msec) tme msec) Fg. 3. Plot showng evoluton of average rate averaged over one round trp tme) of a typcal flow and the rate predcted by the flud model Total rate at the lnk packets/msec) Total rate at the lnk packets/msec) Gan = tme msec) Gan = tme msec) Fg. 4. Plot showng evoluton of average rate averaged over one round trp tme) of a typcal flow and the rate predcted by the flud model for two cases: the standard TCP and a modfed unstable TCP We next demonstrate the oscllatons that can happen n a badly desgned system. We consder a modfed verson of TCP n whch each successful transmsson ncreases the wndow sze W by 0/ W and each marked packet reduces the wndow sze by half as n standard TCP. In Fgure 4 we compare the average rate of ths modfed TCP wth that of a standard one for a target equlbrum rate of 8.33 packets/ms. Clearly the modfed verson of TCP has larger fluctuatons n the average rate. Ths may even lead to a lower throughput n a drop based system where packets are dropped at congeston nstead of beng marked) wth re-transmsson of dropped packets. The varance of total rate at the router n the case of ths modfed TCP s about 6 whereas the varance s 0.63 for the case of standard TCP. Ths underlnes the mportance of ensurng stablty /03/$7.00 C) 003 IEEE IEEE IFOCOM 003

11 VII. COCLUSIO We have studed a system consstng of a sngle lnk accessed by a large number of TCP-lke flows each wth dentcal delay access but wth possbly) dfferent ntal condton and also accessed by a large number of uncontrolled flows. The contrbutons of ths paper are: ) Our man result s that n the presence of uncontrolled flows stochastc nose) f the number of flows s large enough the global exponental stablty crteron for a sngle flow wth mnor modfcatons) s also a global exponental stablty condton for the stochastc system wth multple flows. Thus the mplcaton s that parameter desgn can be carred out usng determnstc analyss based on the sngle flow model. ) For the rate adaptaton model of TCP 0 we have shown that the stablty s ensured f the target equlbrum delay-bandwdth product wndow sze) per flow s large enough and we have derved upper bounds on ths quantty. Thus we have derved suffcent condtons for global stablty. umercal examples wth two popular markng functons ndcate that the target wndow sze per flow requred to ensure stablty s actually not very large. ) Usng smulatons we have verfed that the determnstc flud models that are stable can be used to desgn actve queue management mechansms. L. Massoule Stablty of dstrbuted congeston control wth heterogenous feedback delays IEEE Transactons on Automatc Control 476): June F. Pagann J. Doyle and S. Low Scalable Laws for Stable etwork Congeston Control Proceedngs of the IEEE Conference on Decson and Control Dec 00 4 S. Deb S. Shakkotta and R. Srkant Stablty and Convergence of TCP Flows n a Many-flows regme Techncal Report Coordnated Scence Lab UIUC Urbana IL 00 5 S. Shakkotta R. Srkant and S. Meyn Bounds on the Throughput of Congeston Controllers n the Presence of Feedback Delay Proceedngs of the IEEE Conference on Decson and Control Florda Dec 00 6 S. Shakkotta and R. Srkant Mean FDE models for Internet Congeston Control Techncal report. A shorter verson to appear n the Proceedngs of IEEE IFOCOM 00 7 G. Vnncombe On the stablty of end-to-end congeston control for the Internet Unversty of Cambrdge Techncal Report 00 ACKOWLEDGMET The authors would lke to acknowledge that the research was supported by AFOSR URI F and DARPA Grant F REFERECES S. Athuralya D. E. Lapsley and S. H. Low Random early markng for Internet congeston control Proceedngs of IEEE GLOBECOM 999. F. Baccell and D. Hong Interacton of TCP Flows as Bllards Techncal Report Aprl 00 IRIA Rocquencourt 3 S. Deb and R. Srkant Global Stablty of Congeston Controllers for the Internet Proceedngs of IEEE Conference on Decson and Control Las Vegas Dec S. Floyd TCP and explct congeston notfcaton ACM Computer Communcaton Revew 4:0-3 October C.V. Hollot V. Msra D. Towsley and W. Gong On desgnng mproved controllers for AQM routers supportng TCP flows Proceedngs of IFOCOM 00 Anchorage Alaska Aprl R. Johar and D. Tan End-to-End Congeston Control for the Internet: Delays and Stablty IEEE/ACM Transactons on etworkng 96):88-83 December F. P. Kelly A. Maulloo and D. Tan Rate control n communcaton networks: shadow prces proportonal farness and stablty Journal of the Operatonal Research Socety 49: F.P. Kelly Models for a self-managed Internet Phlosophcal Transactons of the Royal Socety A358: V. B. Kolmanovsk and V. R. osov Stablty of Functonal Dfferental Equatons Academc Press IC Orlando Florda USA S. Kunnyur and R. Srkant End-to-end congeston control: utlty functons random losses and EC marks Proceedngs of IFOCOM 000 Tel Avv Israel March 000. S. Kunnyur and R. Srkant Analyss and desgn of an adaptve vrtual queue algorthm for actve queue management Proceedngs of ACM Sgcomm San Dego CA August /03/$7.00 C) 003 IEEE IEEE IFOCOM 003