Which Protocol? Mutual Interaction of Heterogeneous Congestion Controllers. Vinod Ramaswamy, Diganto Choudhury and Srinivas Shakkottai Member, IEEE

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1 Whh Protool? Mutual Interaton of Heterogeneous Congeston Controllers Vnod Ramaswamy, Dganto Choudhury and Srnvas Shakkotta Member, IEEE Abstrat A large number of ongeston ontrol protools have been proposed n the last few years, wth all havng the same purpose to dvde avalable bandwdth resoures among dfferent flows n a far manner Eah protool operates on the paradgm of some onepton of lnk pre suh as paket losses or paket delays that determnes soure transmsson rates Reent work on network utlty maxmzaton has brought forth the dea that the fundamental pre or Lagrange multpler for a lnk s proportonal to the queue length at that lnk, and that dfferent ongeston metrs suh as delays or drops are essentally ways of nterpretng suh a Lagrange multpler We thus ask the followng queston: Suppose that eah flow has a number of ongeston ontrol protools to hoose from, whh one or ombnaton should t hoose? We ntrodue a framework wheren eah flow has a utlty that depends on throughput, and also has a dsutlty that s some funton of the queue lengths enountered along the route taken Flows must hoose a ombnaton of protools that would maxmze ther payoffs We study both the soally optmal, as well as the selfsh ases to determne the loss of system-wde value nurred through selfsh deson makng, so haraterzng the pre of heterogenety We also propose tollng shemes that nentvze flows to hoose one of several dfferent vrtual networks aterng to partular needs, and show that the total system value s greater, hene makng a ase for the adopton of suh vrtual networks Index Terms Congeston ontrollers, non-ooperatve games, TCP/IP, PMP I INTRODUCTION RECENT years have seen the desgn of a large number of ongeston ontrol protools for use on the Internet Ther desgns all revolve around the dea that lnk ongeston s ndated by some noton of pre, whh the soure an respond to Dfferent ongeston pre metrs nlude paket loss, paket marks, paket delays or some ombnaton thereof However, the relatve value of one protool versus another s not well understood For example, t mght be onjetured that a delay senstve applaton would onsder usng a protool that has a delay-based ongeston metr, and a throughput Manusrpt reeved Aprl 20, 202; revsed Deember 7, 202; aepted February 27, 203; approved by IEEE/ACM TRANSACTIONS ON NET- WORKING Edtor C Dovrols Ths work was supported n part by the NSF under Grants CNS , CNS , and CNS-49458; the Google Faulty Researh Awards program under Grants HDTRA and FA ; and the Google Faulty Researh Awards program V Ramaswamy and S Shakkota are wth Dept of ECE, Texas A&M Unversty, College Staton, Texas, USA e-mal: vnod83@tamuedu, sshakkot@tamuedu D Choudary was wth Dept of ECE, Texas A&M Unversty, College Staton, Texas, USA He s now wth Maxm Integrated, Dallas, Texas, USA e-mal: dhoudhury@tamuedu Color versons of one or more of the gures n ths paper are avalable onlne at Dgtal Objet Identer 009/TNET maxmzng applaton mght favor a loss-based metr How should applatons hoose the protool to use? An analytal framework for network resoure alloaton was developed n semnal work by Kelly et al [] If the flow has a rate x 0 and the utlty assoated wth suh a flow s represented by a onave, nreasng funton U x, the objetve s max U x N st y l l, l L 2 where N s the set of soures, L the set of lnks, l the apaty of lnk l L Also let R be the routng matrx wth R l = f the route assoated wth soure uses lnk l The load on lnk l s y l = r N R lrx r The problem an be solved usng deas based on Prmal-Dual system dynams [] [5] to yeld a set of ontrollers At the soure we have Soure: ẋ t = κ U x t + R l p l t, 3 l:l L x where k > 0,, and the notaton ϕ + ξ s used to denote the funton { ϕ + ϕ ξ > 0 ξ = 4 max{ϕ, 0} ξ = 0 4 ensures that x s non-negatve The ontroller n 3 has an attratve nterpretaton that the soure rate of flow responds to feedbak n the form of lnk pres p l t, wth the end-to-end pre beng alulated as the sum of pres on all lnks that the flow traverses somethng that s ommon to all ongeston ontrol protools Soure rate s always non-negatve, whh s enfored by the defnton of the funton n 4 The pre p l t at lnk l s alulated usng Lnk: ṗ l t = ρp l t R lj x j t l j N + p l t 5 5 ensures that the pre s non-negatve Eah lnk has a buffer n whh pakets are queued If the total load at a lnk l gven by j N R ljx j t s greater than the apaty l, the queue length nreases, whle f t s less than l, the queue length dereases as seen n 5 The queue length s always non-negatve, as enfored by the defnton n 4 The gan parameter ρp l s any postve funton Thus, the lnk-pre p l t an be dentfed wth the queue length at lnk l It has been shown [] [5] that the above ontrol sheme onverges to the optmal soluton to the problem n

2 2 Whle ths framework ndates that the fundamental pre of a lnk s proportonal to queue length, ongeston ontrol protools use several dfferent ongeston metrs For example, TCP Reno [6] uses paket drops or marks as ts pre metr, whle TCP Vegas uses end-to-end delay [2] Other protools nlude Salable TCP [7] that uses lossfeedbak, and allows salng of rate nreases/dereases based on network haratersts, FAST-TCP [8] that uses delayfeedbak, and s meant for hgh bandwdth envronments, and TCP-Illnos [9] that uses loss and delay sgnals to attan hgh throughput However, drops, marks, and delays are all funtons of the queue length Thus, a key dfferene between protools s ther way of nterpretng queue length nformaton A fall out of dfferent pre-nterpretatons s that when flows hoose dstnt ongeston ontrol protools, they do not obtan the same throughput on shared lnks For example, studes suh as [0] [3] study nter-protool as well as ntraprotool farness, whle [4] onsders a game of hoosng between protools, assumng that a ertan throughput would be guaranteed per ombnaton Throughput alone does not fully apture the performane of an applaton, sne t mght also be mpated by queueng effets suh as delay and paket loss We onsder applatons that mght have dfferent senstvtes to queueng Indeed, a large fraton of Internet traff onssts of fle transfers less delay senstve and buffered vdeo streams more delay senstve from data enters or ontent dstrbuton networks We model these flows as havng possbly dfferent utltes for throughput, and dsutltes for the queueng enountered on ther respetve paths We antpate for a future Internet arhteture where multple ongeston ontrollng shemes are avalable to ater the needs of dfferent serve lasses and the flows are allowed hoose the ones aordng to ther serve preferenes Hene, we assume that flows play far n that they hoose to follow the onstrants mposed by employng some form of ongeston ontrol Thus, the flows hoose from a set of reasonable ongeston ontrol mehansms, for example varants of TCP, so as to maxmze ther payoff that s utlty mnus dsutlty Our objetve s smlar to the proposal n [5], where a system desgn for vrtual lnks talored for flows that are rate senstve R and delay senstve D s presented The dea s that an R-flow would pk the vrtual lnk where t s guaranteed hgher rate, whereas a D-flow would pk one where t s guaranteed a lower delay However, unlke that work, we have two bas dfferenes Frst, we expltly model utlty for throughput and dsutlty for queung for all knds of flows, rather than assume that D-type flows would be wllng to lve wth smaller rate Ths enables us to explore the spae of multple lasses of serve wth tollng, sne t gves an objetve measure on the hoe made by the flow Seond, we allow a hoe between TCP flavors e, nterpretaton of queue length by ongeston ontrollers aordng to the applaton n queston However, n [5] the only way to redue delay s to have short buffers for the D serve lass, whh mght also result n more losses Our fndng s that f the number of flows n the system s large, the optmal strategy of a flow s to hoose a pre nterpretaton from among the spae of avalable ones that s most smlar to ts dsutlty funton Usng ths fndng, we an haraterze the total system value to all flows, and we show that the rato of ths value to the optmum value an be arbtrarly small Fnally, we onsder the stuaton n whh we reate multple vrtual networks wth tollng, wth eah flow havng a hoe between networks and between protools We show that we an fx the tolls suh that the overall system value an be nreased sgnfantly, n-spte of the toll We next present our model and summarze our man results II MODEL AND MAIN RESULTS We onsder a system n whh eah flow N has a so-alled far utlty funton [6], U x w x /, 6 wth, and a dsutlty that depends on the vetor of lnk pres p as Ũ x, p l L R l p l /τ x, 7 where > s a onstant The overall payoff s the dfferene of the two, gven by F x, p U x Ũp 8 The far utlty funton was proposed by Mo et al [6] as a method of apturng a large lass of farness measures based on the value of used For nstane, they showed that results n proportonal farness, whle results n max-mn farness The form of the dsutlty funton s suh that based on, the dsutlty an be almost lnear n queue length whh n turn s proportonal to delay, weghted by the parameter T, to gradually nreasng onvexty as rses, to a sharp utoff for large The threshold parameter τ n 7 models the flow s senstvty to queue length, wth a small value of τ ndatng hgh senstvty eg, delay senstve applatons need short queue lengths and a large value ndatng low senstvty eg, loss senstve applatons are affeted only by buffer overflow We defne a set of protools T, wth ardnalty T = T Eah protool z assoated wth a prenterpretaton funton m z p l p l /T z Note that these pre-nterpretaton funtons take the same form as dsutltes, and model the way n whh a partular protool z T nterprets lnk pres Agan, a loss-based protool would have a hgh value of T z, whle a dealy-based protool would have a low value Ths orresponds to the fat that n a protool that s modulated by buffer over flows suh as TCP Reno, the queue length has no mpat untl a maxmum threshold buffer sze s reahed, after whh the pre s very hgh T z = buffer sze here Smlarly, TCP Vegas approxmately dedes on whether the aheved throughput s too hgh or too low as ompared to a threshold, whh n turn an be related to a threshold on the per-paket delay seen by the flow T z s less than the buffer sze here Now, whle a flow annot hange We wll refer to pre-nterpretaton funtons and protools nterhangeably

3 3 ts dsutlty funton parameterzed by τ t an hoose to use a ombnaton of protools as t fnds approprate A partular flow s hoe ould take the form q p z T ϵ z l= L R l m z p l 9 where z T ϵz =, and ϵz 0 The onvex ombnaton models the dea that a flow sometmes measures pre n one way eg, delay-based and sometmes n another way eg, loss-based ϵ z an be thought of as the probablty wth whh flow uses protool z For example, ths stuaton mght orrespond to a flow usng delay and loss measurements smultaneously, and respondng to ongeston sgnals loss or delay probablstally We refer to the hoe [ϵ, ϵ2, ϵt ], made by flow as ϵ E {ϵ : z T ϵz =, ϵz 0} Further, we denote aggregate hoes of all flows by ϵ E Π N E, and wll refer to ϵ E as a protool-profle We frst show n Seton III that for a gven protool-profle, the bandwdth alloatons and hene the payoffs are unque Further, a prmal-dual type ontrol wll onverge to ths unque bandwdth alloaton The result s essentally a onssteny hek that allows us to analytally determne the payoffs as a funton of the protool-profle hosen We show n Seton IV that all bandwdth alloatons that are attanable by a protool-profle over T protools wth m p m 2 p m T p are attanable by a protool-profle over just the two protools m p and m T p The result has the appealng nterpretaton that when m z p = p/t z, t s suffent to only onsder the strtest nterpretaton smallest T z, whh an be thought of as delay-based feedbak and the most lenent largest T z, assoated wth loss-based feedbak We next show that wth two protools wth <, the bandwdth alloaton reeved by a flow s dereasng n the weght t plaes on the strt protool Although the proof s nvolved, the result s ntutve sne a strt protool would always nterpret p as a larger ongeston than the lenent protool However, sne payoffs are the sum of utlty and dsutlty, t does not follow that all flows would hoose the protool wth the hgher threshold We show n Setons V and VI that n many ases, the total system value s maxmzed when all flows hoose to use only m p = p/ On the one hand f flows have pre-nsenstve payoffs, the protool-profle used does not matter as long as all of them use the same profle On the other hand, f there s a mx of flows, some of whh have a large dsutlty funton pre-senstve and others whh do not pre-nsenstve, usng the strt pre-nterpretaton m p = p/, ensures that the pre does not beome too large for all flows, whh maxmzes system value In Setons V and VI, we also onsder the ase flows use selfsh optmzatons to hoose ther protool-profles and study the Nash equlbrum If all flows have pre-nsenstve payoffs, then they all hoose the lenent pre-nterpretaton m 2 p = p/ Ths ase an be mapped to throughput maxmzng flows all hoosng TCP Reno If we have a mx of flow types sharng a lnk, t turns out that the pre-senstve flows wth dsutlty funton parametrzed by τ, hoose the strt pre-nterpretaton m p = p/, regardless of the hoe of others Smlarly, the pre-senstve flows wth dsutlty threshold τ, hoose the lenent prenterpretaton m 2 p = p/ Whle the other flows may employ mxed strateges When the number of flows n the system s large, a flow wth dsutlty threshold τ pks a mxed strategy that yelds an effetve pre nterpretaton p/τ The result s nterestng sne t suggests that a delay senstve applaton annot do any better n terms of overall payoff even f t hooses a more lenent protool We also haraterze the rato of system value n the game versus the soal optmum for the sngle-lnk ase to determne an effeny rato, whh an be qute hgh Fnally, n Seton VIII we ntrodue vrtual networks, eah of whh s assgned a ertan fraton of the apaty, and hooses a toll Flows an hoose a network and protools The dea s smlar to Pars Metro Prng PMP [7] [9], and we show that the system value at Nash equlbrum an be hgher overall n spte of tollng The result suggests that the Internet mght beneft by havng separate ters of serve for delay-senstve and loss-senstve flows III PROBLEM FORMULATION We assume that for eah lnk, there exsts at least one flow that uses only that lnk The assumpton mples that all lnks have a non-zero pre We hypothesze from 3 and 5 that the payoffs should be determned by the protool-profle ϵ as T L x p, ϵ = U R l m z p l, 0 wth ϵ E and for all l L ϵ z z= l= N R l x p, ϵ = l p l > 0, = Note that although we have denoted x as dependng on both ϵ and p, the pres themselves depend on ϵ through x, and the soluton x ϵ, p ϵ f t exsts s solely a funton of ϵ We show that the equlbrum exsts, and an be reahed usng Prmal-Dual dynams We have the followng proposton Proposton Gven any protool-profle ϵ, Prmal-Dual dynams onverge to the unque soluton x, p of the ondtons 0 and Proof: For pre-nterpretaton funtons of the form p/t z, the soure dynams n 3 an be re-wrtten as T L + ẋ t = κ U x T R l m p l z= ϵ z where m p l = p l T Let U x = ζ U x where ζ = T z= ϵz T T z, and let κ = ζ Then the above equaton an be modfed as ẋ t = ζ U x t T z l= L R l m p l t l= + x x 2

4 4 Now, n 5 hoose ρp l = m p l, where m s dervatve of m Then the pre-update equaton an be re-wrtten as, N ṁ p l t = 3 R l x t l+ = p l Equatons 2 and 3 orrespond to the prmal-dual dynams of the followng onvex maxmzaton problem N U x max x>0 subjet to = N R l x l, l L = The above s a onvex optmzaton problem wth a unque soluton satsfyng 0 and Thus, by the usual Lyapunov argument [2] [5] Prmal-Dual dynams onverge to ths soluton Note that our hoe of pre nterpretaton makes t a speal ase of the result n Appendx A Case- of [] We are now n a poston to ask questons about what the flows payoffs would look lke at suh an equlbrum, and how ths would mpat the hoe of the protool-profle Reall that the payoff obtaned by a flow when the system state s at x ϵ, p ϵ s gven by F ϵ = U x ϵ Ũp ϵ 4 We defne a system-value funton V, whh s equal to the sum of payoff funtons of all flows n the network, N V ϵ = F ϵ 5 = Our frst objetve s to fnd an optmal protool-profle that maxmzes the system-value funton Opt: max V ϵ 6 ϵ E Let ϵ S be an optmal profle vetor for the above problem Then we refer to V S = V ϵ S as the value of the soal optmum An alternatve would be for flows to ndvdually maxmze ther own payoffs However, suh a proeedng mght not not lead to an optmal system state that maxmzes the value funton 5 We haraterze the equlbrum state of suh a selfsh behavor by modelng t as a strateg game Let G =< N, E, F > be a strateg game, where N s the set of flows players, E s the set of all protool profles aton sets and F = {F, F 2,, F N }, where F : E R s the payoff funton of user defned n 4 Defne ϵ = [ϵ, ϵ 2,, ϵ, ϵ +, ϵ N ], e, ths represents the hoes of all flows exept Then ϵ = [ϵ, ϵ ] For any fxed ϵ, flow maxmzes ts payoff as shown below Game: max F ϵ, ϵ N 7 ϵ E The game s sad to be at a Nash equlbrum when flows do not have any nentve to unlaterally devate from ther urrent state We defne ϵ G as a Nash equlbrum of the game G f ϵ G = arg max ϵ E F ϵ, ϵ G, N We refer to V G = V ϵ G as the value of the game Fnally, we defne the Effeny Rato η as η = V G V S 8 IV BASIC RESULTS We frst show that a T -protool network an be replaed wth an equvalent 2-protool network Consder a T -protool network wth pre nterpretaton funtons [m, m 2,, m T ] Let ϵ E T be a profle state n the T -network Then the equlbrum rate vetor x ϵ and pre vetor p ϵ satsfy the equlbrum ondtons 0 and Now, onsder a 2- protool network wth pre nterpretaton funtons m and m T Note that m m z m T, z = 2,, T Let µ E 2 be a profle state n the 2-protool network Proposton 2 For any equlbrum x ϵ, p ϵ n a T - protool network, a protool-profle µ st x ϵ, p ϵ s also an equlbrum for the 2-protool network Proof: For any gven ϵ E T, let x ϵ, p ϵ be an equlbrum par that satsfes the equlbrum ondtons 0 and, whh are reprodued below for larty x ϵ = U T z= ϵ q z Rx ϵ =, p l, N, > 0, l L where q z = L l= R lm z p l ϵ The fat that mt m z m, mples, q T q z q, N, Z T Sne both m and m T are strtly nreasng funtons, there exsts a unque µ [0, ], suh that, T z= Now, we have ϵ z q z = µ q x ϵ = U T z= = U µ q + µ q T ϵ z q z + µ q T, N, Rx ϵ =, p l > 0, l L The above equatons orrespond to the equlbrum ondtons of a 2-protool network wth pre nterpretaton funtons m and m T Therefore, there exsts a protool-profle µ = [µ,, µ N ] suh that x ϵ, p ϵ s an equlbrum par of 2-protool network The above proposton shows that any equlbrum state of a T -protool network an be obtaned wth an equvalent 2- protool network Therefore we restrt our study to 2-protool networks wth a strt pre nterpretaton m s = p and a lenent pre nterpretaton m l = p, e, < Also, we redefne the protool profle of flow, ϵ, as s ϵ = ϵ, where ϵ s the weght appled on the strt pre nterpretaton Fnally, the equlbrum rate of flow an be wrtten n terms

5 5 of m s and m l as follows: L x ϵ = U R l ϵ m s p l + ϵ m l p l = U l= ϵ + ϵ L R l m s p l l= 9 where ϵ = [ϵ, ϵ 2,, ϵ N ] s the system protool-profle The above result follows from 0 We next show that the bandwdth alloaton reeved by a flow s dereasng n the weght t plaes on the strt protool m s p = p/ Proposton 3 Let x ϵ be the equlbrum rate of flow for any ϵ E 2 Then, x 0, N, ϵ Proof: From 9, we have L U x = R l m s p l ϵ + ϵ l= Then, dfferentatng above equaton wth respet to ϵ j, we get, where A j = x = A j + L l= p l B l, 20 T L s l= R lm s p l U x δ j, and B l = R lm s p l ϵ + ϵ U x Also, δ j = f = j, and zero otherwse At equlbrum, N = R lx ϵ = l, l L Now, dfferentatng ths equaton wth respet to ϵ j, we get N = Replang x wth 20, we obtan N ϵ + ϵ T R l l = R l x = 0, l L 2 x + R lj k= L k= L k= R kjm s p k R k m s p k p k j x j = 0 Now, rearrangng terms n the above expresson, we get, L N m s p k p ϵ k + ϵ T R l R l k U x = R lj = L k= R kjm s p k j x j We an represent the above n a matrx form as RW R T ζ = r, where W = dag ϵ + ϵ ζ = r = x [ m s p p m s p ϵ 2 p 2 m s p L p L j ϵ j L k= R kjm s p k ] T j x j [R j R Lj ] T Note that U s a strtly onave funton and hene x < 0 Therefore, RW R a postve defnte matrx Now, we have p L l k= = R T kjh lk s m s p l ζ = RW R T r 22 Let H = RW R T, where H s an L L matrx Let us represent ts elements usng h lm Thus, from 22, we have L k= R kjm s p k j x j 23 Let V = W R T RW R T R Then, from 20 and 23, we get x j x = T l L l= R kjm s p k = T l L j x j v jj, 24 k= R kjm s p k j x j v j, 25 where v j represent elements of V Now, we show that x j s negatve gven the assumpton n the lemma Note that V s a projeton matrx The dagonal elements of a projeton matrx are postve and less than or equal to unty e, v jj Then, from 24, we onlude that x j 0 and hene have proved the proposton The above proposton s ntutve n that a strt protool would fore the flow to ut down ts rate for the same pre as a lenent protool Corollary 4 In the sngle lnk ase, the lnk-pre p and the rate vetor x satsfes, p < 0 and x > 0 f j,, j N where Proof: From 23, 24 and 25, we have x = p T = l m s p m s p U j x j N r= ν, 26 r ν j δ j N r= ν, 27 r m s p j x j ν = ϵ + ϵ x U x = m s p The above result follows from 9 and the fat that U x = x U x Note that U x < 0 sne U s strtly onave Now, the orollary s straghtforward from the above results Now, we now study dfferent mxes of flow types n order to understand the system value n eah ase

6 6 V FLOWS WITH PRICE-INSENSITIVE PAYOFF We assoate eah flow N to a lass, based on ts dsutlty funton of the form l L R lp l /τ x We begn by onsderng a system of flows that have a pre-nsenstve payoff, e, τ = N Ths means that payoff s solely a funton of bandwdth, and we have F ϵ = U x ϵ However, even n ths stuaton, flows must employ ongeston ontrol, e, they must hoose a protool-profle From Seton IV, reall that sne we only have two protools, the flow s hoe of protool profle s defned by a salar value ϵ Also note that T z for eah protool z =, 2 The system-value s equal to the sum of user payoffs, V ϵ = N = U x ϵ We then have the followng result Proposton 5 The system-value s maxmzed when the protool hoes made by all users are the same Thus, f ϵ S = arg max ϵ E V ϵ, and ϵ S s used to denote the protool hoe made by-profle of user, then ϵ S = ϵ S j,, j N Proof: We frst derve an upper bound for system-value V ϵ and then show that the upper bound s aheved when all soures hoose the same protool Suppose that X = {x Rx = N } Let ˆx = arg max Rx= = U x Note that equlbrum rate x ϵ X, sne Rx = Then the value of N = U x evaluated at x ϵ satsfes N N V ϵ = U x ϵ U ˆx = We showed n Proposton 2 that the equlbrum rate x ϵ, s the unque maxmzer of the onvex problem N max x>0,rx= = ζ U x, where ζ = ϵ + ϵ Then, x ϵ an be made equal to ˆx, the optmal pont n set X, by hoosng ζ = ζ j, j N Suh a hoe means that = ζ = ζ j ϵ + ϵ = ϵ j + ϵ j, ϵ = ϵ j Thus, f ϵ S = arg max ϵ E V ϵ ϵ S = ϵ S j,, j N Therefore, the system value s maxmzed when the protool hoes made by all the users are dental Also, the maxmum value does not depend on the parameters of the seleted protool We next onsder the game n whh flows are allowed to hoose ther protools selfshly Proposton 6 Let G =< N, E, F > be a strateg game wth payoff funton of user s gven as F ϵ = U x ϵ Then there exsts a Nash equlbrum for game G, and the equlbrum profle for any user N s ϵ G = 0 Proof: Dfferentatng F wrt ϵ, and usng Proposton 3 F = U x ϵ x ϵ 0 ϵ ϵ Hene, F ϵ s maxmzed when ϵ = 0 Therefore, ϵ G = 0, N Effeny Rato: We showed n Proposton 5 that the value funton s maxmzed when all flows pk the same protoolprofle In Proposton 6 we saw that when eah flow selfshly maxmzes ts own payoff, there exsts a Nash equlbrum under whh every soure hooses the lowest pred protool, e, the protool wth the hgher value of T Suh a profle s a speal ase of all flows hoosng the same protool-profle Thus, value of the soal optmum and the value of the game are dental and Effeny Rato η s unty Example-: Consder the ase n whh a sngle lnk wth apaty = 0 s shared by 2 pre-nsenstve flows Users have -far utlty funtons wth = 2, w = 00 and w 2 = 00 We use pre-nterpretaton funtons p 2 2 and p 5 2 Note that the smulaton parameters, and threshold values are hosen arbtrarly These parameters may not orrespond to any partular protool used n prate Nevertheless, the observatons made here hold true for any values of, > and,, τ > 0 In Fgure we show the system value for dfferent hoes of protool profles The plot llustrates that system value s maxmzed when both flows hoose the same profle Fgure 2 shows how the payoff funton of a flow vares wth ts protool profle We fnd that regardless of the value of the protool profle hosen by the other flow, the payoff funton s maxmzed when t pks the lower pre protool System Value System Value Vs Protool Profle ǫ = 0 ǫ = ǫ = 025 ǫ = ǫ 2 Fg System Value wth pre-nsenstve flows as a funton of the protool-profle We observe that the system value s maxmzed when both flows hoose the same protool-profle Payoff User Payoff User Vs Protool Profle for User ǫ 2 = ǫ 2 = 0 ǫ 2 = ǫ Fg 2 Payoff of a pre-nsenstve flow as a funton of ts protool-profle We observe that payoff s maxmzed when the flow hooses the more lenent pre nterpretaton, regardless of the other flow VI MIXED ENVIRONMENT We now onsder the ase where a network s shared by flows wth dfferent dsutltes We dentfy the optmal

7 7 protool profle that maxmzes the system value, and ompare t wth and the Nash equlbrum We frst study the ase of a network onsstng of a sngle lnk A Sngle Lnk Case Consder a sngle lnk system wth apaty shared by N flows The payoff of user N s F ϵ = U x ϵ p ϵ x ϵ Then, the system value s V ϵ = N = F ϵ τ Proposton 7 The system- value s maxmzed when all users pk the protool wth lowest threshold, e, f ϵ S = arg max ϵ E V ϵ, then ϵ S =, N Proof: Sketh Reall that by our assumpton Gven ths assumpton, t an be shown through straghtforward dfferentaton that Ũϵ s a monotonally dereasng funton of ϵ Now, the value funton V s maxmum when Uϵ s maxmzed and Ũϵ s mnmzed We already know from Proposton 5 that Uϵ s maxmzed when all flows hoose the same protool-profle Couplng ths result wth the fat that Ũϵ s dereasng n ϵ, we see that system value s maxmzed when ϵ =, N We now study the strateg game n whh users ndvdually maxmze ther payoff as n 7 We show that there exsts a Nash equlbrum and haraterze the protool-profle Proposton 8 Let G =< N, E, F > be a strateg game wth payoff of user s F ϵ = U x ϵ p ϵ τ x ϵ Then there exsts a Nash equlbrum NE for Game G At NE, flows wth greatest senstvty to pre hoose the strt protool, e, f τ =, then ϵ = Proof: We wll show that F ϵ s quas-onave, and use the Theorem of Nash to show exstene of a NE Dfferentatng F wrt ϵ, F ϵ = U x d p x ϵ τ d p x p ϵ, 28 where d p = p and d p s ts dervatve Now, substtutng the results from 26 and 27, n the above equaton, we get F = BU ϵ x d p ν j N 29 r= ν r B m s p d p x m s p N r= νr, 30 x where B = x and ν = m s p Note that B < 0 sne U s a negatve funton From 9 along wth the defntons of ν and d p, the above expresson an be smplfed as follows: F ϵ = Bm s p N N r= r=,r x r x r Bms p N r= ϵ + ϵ x r τ τ x 3 We show that f the above expresson has a root, then t s unque The roots are haraterzed by ϵ + ϵ = τ + x N r=,r x r 32 Frst observe that the left sde of the above expresson s strtly nreasng n ϵ sne < Sne x ϵ < 0 and x r ϵ > 0 f r from Proposton 3 and Corollary 4, the rght sde of the above expresson s strtly dereasng F Therefore, the set of roots of the equaton, ϵ x = 0 s a sngleton or null set Thus, F s unmodal or monoton n ϵ for any fxed ϵ and hene quas onave Sne ϵ [0, ] s a non-empty ompat onvex set, by the theorem of Nash, the quas onavty of F ϵ, ϵ guarantees that there exsts a ϵ G, suh that for all =,, N, ϵ G = arg max ϵ [0,] F ϵ, ϵ G Hene, the frst part of the proof s omplete Now, onsder a flow wth dsutlty per unt rate p τ, where τ = Replang τ wth n 3, we observe that F ϵ > 0 Note that B < 0 Therefore, payoff s maxmzed when ϵ = In the next seton, we study the haratersts of the NE and show that t s unque B Nash equlbrum haratersts We have establshed the exstene of NE of the strateg game 7 n the prevous seton We ondut further studes on the propertes of NE n ths seton Frst, we derve ondtons for the NE system protool profle Then, n Proposton 9, we show that the game has a unque NE Fnally, n Proposton 0, we derve the NE strateges of flows when there are large number of flows n the system Let ˆϵ be a Nash equlbrum system protool profle aton profle Then, by defnton, t must satsfy the ondton that ˆϵ = arg max ϵ [0,] F ϵ, ˆϵ, N Then, from the frst order optmalty ondton, we have F ˆϵ ϵ ϵ ˆϵ 0 Consequently, from 3, we get that, N, γˆϵ = x T + ˆϵ N r=,r x r ˆϵ T l 33 where a b = mn{a, b}, a b = max{a, b} and γϵ = ϵ + ϵ In addton, the Nash equlbrum profle must also satsfy, x w ˆϵ =, 34 γˆϵ p N x ˆϵ = 35 = Here, 34 follows from 9 and the defnton of U x Also, 35 follows from the assumpton that every lnk has one flow

8 8 usng that lnk alone Now, we show that the set of Nash equlbra, haraterzed by 33-35, s sngleton Proposton 9 The strateg game, G =< N, E, F >, has a unque Nash equlbrum Proof: To prove by ontradton, assume multple Nash equlbra exst Let two dstnt NE system protool profles be ˆϵ and ˆϵ 2 Also, let x = x ˆϵ, x 2 = x ˆϵ2, p = p ˆϵ, p 2 = p ˆϵ 2, γ = γˆϵ and γ 2 = γˆϵ2 Then, by reorderng the flow ndes, we get that, for some k {0,,, N}, γ > γ 2 for =, 2,, k, 36 γ γ 2 for = k +,, N 37 Also, f k = 0, there exst a flow N suh that γ < γ2 We show that the above ondton are nfeasble for all values of k, under the NE ondtons gven by Intally, onsder the ase when k = N Then, from 33, for =, 2,, N, we have x N r=,r > x r x 2 N x x 2 r N r=,r r= N x r N x 2 r r= r= N > r x N r r= r= > x r x 2 N r= x 2 r x 2 r whh s a ontradton Hene, ths ase s not feasble Smlarly, we an show that the ase when k = 0 s also not feasble Now, onsder the ase when k < N Also, suppose that p p 2 Then, from 34, we have Let x < x 2, for =, 2,, k = arg max Note that > k and hene, γ that x > x2 Observe that, x x = x x 2 x 2 x x x 2 γ2 Also, from 35, note x 2 x 2 x2 x 2, and strt nequalty holds f k It follows from the above result that, x x 2 N > x r N x 2 r r= r= x N r=,r > x r x 2 N r=,r x 2 r Fnally, from 33 and the above result, we get γ γ2 But, from the defnton of, we know that γ γ2 In ase γ = γ2, then, from 34 and the assumpton that p p 2, we get x x2, whh also rases a ontradton Hene, ths ase s also not feasble In smlar fashon, we an show that the ase n whh p < p 2 s also not feasble Hene, our assumpton that multple NE exst s not true Therefore, NE s unque Next, we haraterze the NE n the asymptot regme Proposton 0 When the number of flows n the system, N, s large, the protool profle of flow at NE, ˆϵ, satsfes ˆϵ + ˆϵ = Proof: Reall from 33 that, the NE protool profle of flow, satsfes, x γˆϵ = + ˆϵ τ T s N r=,r τ x r ˆϵ In order to prove the proposton, we lam that, lm N x ˆϵ = 0, 40 x r ˆϵ N r=,r holds true Before provng the above result, we ntrodue a few notatons: Let max = max, mn = mn, w max = max w and w mn = mn w Now, the proof of the lam 40 s as follows: From 35, we an show that, x ˆϵ N x r ˆϵ r=,r Also, from 34, we have, x w ˆϵ = γˆϵ p ˆϵ max x ˆϵ w max T mn l p ˆϵ 4 The above result follows from the fat that γˆϵ From Corollary 4, we observe that the lnk-pre s a dereasng funton of protool profle of eah flow and hene, the system protool profle ϵ Therefore, the lnk pre aheves the lowest value, when every flow adopts the strt protool Then, from 34 and 35, t s easy to show that N p ˆϵ mn w mn 42 Fnally, from 4 and 42, we have x N r=,r x r max max x max NK where K s a onstant The upper bound n the above expresson goes to zero for large values of N Therefore, the lam n 40 holds true and hene, the proof s ompleted Example-2: We onsder a lnk wth apaty = 0 shared by two flows wth dsutltes p 2 2 and p 5 2, respetvely, and w = w 2 = The other parameters are unhanged from Example- We show the system value for dfferent hoes of protool-profles n Fgure 3 The value s maxmzed when both flows hoose the strt protool Fgure 4 shows how the payoff of eah flow vares wth ts hoe of protool profle, gven other s s fxed We fnd that for the frst senstve flow, the payoff funton s maxmzed when t hooses the strt protool, regardless of the other flow But the payoff of the seond less-senstve flow s maxmzed for some ombnaton of protools The results valdate our fndngs Example-3: We onsder a lnk wth apaty = 000 There are 40 flows sharng the lnk The strt and lenent

9 9 06 System Value Vs Protool Profle System Value ǫ = ǫ = 0 ǫ = 025 ǫ = 05 Payoff of a Class 4 flow Fg 3 Fg ǫ 2 System value aganst protool hoes ϵ : Two flows sharng a lnk Payoff Class 2 User Payoff Class User Payoff User Class Vs Protool Profle ǫ Payoff User Class 2 Vs Protool Profle ǫ 2 = ǫ 2 = 0 ǫ 2 = 05 ǫ = ǫ = 0 ǫ = ǫ 2 Payoff aganst protool hoe ϵ : Two flows sharng a lnk thresholds are = 2 and = 7 respetvely In our smulatons, we have set = 2, = 2 for half of users and = 3 for the other half There are 0 lasses of flows, wth eah lass ontanng 4 flows The dsutlty threshold of a Class flow, gven by τ, s hosen aordng to the followng relaton: τ = + /0 We hoose a anddate flow that belongs to Class 4 We assume that every other flow has hosen ther NE protool profle That means, the effetve pre nterpretaton of a flow belongng to Class s p τ Fgure 5 plots the payoff of the anddate flow as a funton of ts effetve protool hoe γϵ 4 = ϵ 4 / + ϵ 4 /, where ϵ 4 s ts protool profle As lamed by Proposton 0, the payoff s maxmzed when γϵ 4 = τ 4 = 07 C Network Case We onsder a system of flows wth log utlty funtons, whh s a speal lass of an -far utlty funton wth The payoff of flow N s F ϵ = w logx ϵ L l= R l p l ϵ τ x ϵ Then the system-value s V ϵ = N = F ϵ Proposton The System-Value funton s maxmzed when all flows pk the hgher pred protool, namely m = p Let ϵ S = arg max ϵ V ϵ, then ϵ S =, =,, N, Proof: We an show through straghtforward dfferentaton that, the dsutlty funton, Ũ ϵ, s a monotonally γ 4 ε 4 = ε 4 / + ε 4 / Fg 5 Payoff of a Class 4 flow s maxmzed when γϵ 4 = T 4 = 07 dereasng funton of ϵ The rest of the proof s smlar to that of Proposton 7 We now onsder a game wth two types of flows: prensenstve flows wth zero dsutltes, and pre-senstve flows wth dsutlty per unt rate p l In ths speal ase, there exsts a unque Nash equlbrum In Proposton 6 we saw that pre-nsenstve flows pk the lenent protool at Nash equlbrum rrespetve of the hoes of the other players We wll now show that pre-senstve flows pk the strt protool at Nash equlbrum Proposton 2 At the unque Nash equlbrum, any flow wth dsutlty per rate p/ e τ = pks ϵ = Proof: It an be shown through straghtforward dfferentaton that F ϵ > 0 for any flow N wth dsutlty per rate p/, whh ompletes the proof VII EFFICIENCY RATIO We now haraterze the loss of system value at Nash equlbrum, as ompared to the value of the soal optmum We fous on the ase of a sngle lnk wth apaty Proposton 3 Assume >, N When the number of flows n the system s large, where ˆ = max η = V G V S < ˆ Proof: Let ϵ = [ϵ, ϵ 2,, ϵ N ] be the system protool profle at soal optmum From Proposton 7, every user hooses the strt protool at soal optmum, e ϵ =, Hene, from 9, and the defnton of U, we have x ϵ = w p ϵ, x ϵ = 43 Interpretng p ϵ as the dual varable, the above equatons an be dentfed as the KKT ondtons of the optmzaton problem gven below: max x w x, subjet to x =

10 0 And, x ϵ s the unque maxmzer of the above problem The payoff of a flow at soal optmum, from 8 and the above results, s gven by F ϵ = U x ˆϵ + τ 44 where = f flow s a pre senstve flow and zero otherwse The system value at soal optmum s V S = F ϵ Now, let ˆϵ = [ˆϵ, ˆϵ 2,, ˆϵ N ] be the system protool profle at Nash equlbrum From Proposton 0, equaton 9 and the defnton of U, we have x ˆϵ = w Tl τ p ˆϵ, x ˆϵ = 45 Reall that a b = mn{a, b}, a b = max{a, b} Interpretng p ˆϵ as the dual varable, the above equatons an be dentfed as the KKT ondtons of the optmzaton problem gven below: max x w τ x, subjet to x = Also, x ˆϵ s the unque maxmzer of the above problem Fnally, the payoff of a flow s F ˆϵ = U x ˆϵ + Tl τ τ 46 The system value at NE s V G = F ˆϵ Now, from the above results and the fat that U s are negatve, sne > by the assumpton of ths proposton, we an show that T V G ˆ T s Tl > ˆ = ˆ V S, U x ˆϵ ˆ T U x ϵ + T U x ϵ 47 where ˆ = max and T = τ Sne V G and V S are negatve, the effeny rato η, an be bounded as η = V G V S < ˆ Tl, whh ompletes the proof Example-4: The exat expresson for effeny rato s derved for the followng speal ase: We assume that every flow has the same utlty funton, e, n 6, w = w and =, N We assoate the flows, havng dsutlty funtons of the form p τ j x wth Class-j Assume that there are J suh lasses wth τ < τ 2 < < τ J and τ j [, ], j The flows havng zero dsutlty funton s lassfed as Class J For algebra onvenene, we defne τ j = Let N be the number of flows belongng to Class Effeny Rato Effeny Rato Mx of Class and Class 2 Users / =4 / = 6 / = Fraton of Class Users Fg 6 Effeny Rato η n the sngle lnk ase, plotted aganst the fraton of Class- flows for dfferent ratos of / Sne V S and V G were negatve n ths example, a hgher rato s worse and n = N /N Then, the Value of soal optmum V S and value of game equlbrum V G are gven by V S = N J n j + j, 48 N and j= j= V G = N N S S 2, 49 respetvely, where J τj S = Tl n + n J and J S 2 = j= τj Tl n j + nj τ j Also, j = 0 when j = J and one otherwse The effeny rato, η, s gven by η = S S 2 J j= n j + j τ j 50 Now, we plot the effeny rato for the followng ase Let two lasses of flows, namely Class and Class 2, are sharng a lnk Also, let ther dsutlty thresholds be τ = and τ 2 = respetvely Lettng = 2 and = 3, we plot the effeny rato η, gven by 50, n Fgure 6 The Fgure 6 shows that η nreases wth Note that a hgher rato s worse Hene, the performane deterorates wth VIII PARIS METRO PRICING We have shown n the prevous seton that when the flows selfshly hoose protools to maxmze ther own payoff, the system performane at the resultng equlbrum, ompared to the soally optmal ase, an be muh worse Ths s due to the fat that, as shown by Proposton 0, the flows wth relatvely lower dsutlty funtons hoose relatvely lenent protools, and hene apture a larger fraton of hannel bandwdth leavng not enough for the ones wth larger dsutlty funtons

11 who hoose strter protools As a soluton to the aforementoned problem, we propose a sheme n whh the network s parttoned nto vrtual subnetworks eah havng ts own queung buffer, ndependent pre queue-length dynams and fxed entrane toll A flow s free to hoose a protool along wth a subnetwork so as to maxmze hs own payoff Ths sheme s smlar to Pars Metro Prng PMP [7] We show that the effeny of ths sheme s superor to the onventonal, untolled, sngle network sheme We haraterze the performane of the proposed sheme n a sngle lnk ase The sngle lnk, wth apaty bts/se, s parttoned nto J vrtual subnetworks Let S j represent the j th sub-network The bandwdth and toll assoated wth S j are denoted by j and λ j respetvely Also, let = [,, J ] and λ = [λ,, λ J ] We refer to and λ as bandwdth vetor and toll vetor respetvely We assume that every flow has the same utlty funton, e, n 6, w = w and =, N We assoate the flows havng dsutlty funtons of the form p τ j x to Classj We assume that there are J suh lasses and τ < τ 2 < < τ J wth τ j [, ] The pre nsenstve flows are lassfed as Class-J For algebra onvenene, we defne τ J = We also assume that there are a large number of flows n eah lass Let N j represent the number of flows n Class-j A flow that seeks to maxmze ts payoff pks a subnetwork that yelds the maxmum payoff Thus, f ˆk s the subnetwork hosen by flow, ˆk = arg max k {,,J} F jk j =, J where F jk s the payoff of a Class-j flow n S k A Nash equlbrum NE here s a state from whh none of the flows has an nentve to devate from ts urrent hoe of subnetwork Note that we already know the flow s hoes of protools n eah network so no devatons n protool are possble The desred NE s one n whh all Class-j flows selet S j, e F jj F jk, j, k 5 Note that the payoffs reeved are unquely determned by the PMP system parameters and λ Now, we derve suffent ondtons on the par, and τ, so that 5 holds true Assume that the system s at the desred equlbrum, e, every Class-j flow s sendng ts traff over S j Let p k be the equlbrum pre per unt rate n S k The throughput reeved by a Class j flow or antpated by a Class j flow f t shfted to S k s gven by, x jk = τj p k and x Jk = Tl p k, k 52 The above results are due to the fat that the entry of a Class- flow nto S k may not sgnfantly hange ts pre, p k, sne there are large number of flows n S k In 52, the frst result follows from Proposton 0, 34 and the assumpton that T j [, ] when j < J, whle the seond one follows from Proposton 6 The lnk pre p k n S k, follows from the above results and the fat that rates of flows sharng a sub-network add up to ts bandwdth alloaton, s gven by p Nk k = τk, f k < J, and p NJ J = Tl 53 k The payoff of Class j flow n S k, from 8, s gven by F jk, λ = x jk p k τ j x jk λ k, J = A k N λ, k, 54 where A k = τ τ k for, k < J, A J = τ,, A Jk = τ k, k < J and A JJ = Also, 54 follows from 52 and 53 The followng lemma derves ondtons on the par, λ for 5 to hold true Before statng the lemma, we ntrodue some notaton Let l k u k = A k N A kk k N k 55 = A N A k k N k, 56 Lemma 4 Suppose the par, λ satsfy the followng ondtons: f k < J, k+ N k+ τk+ N k τ k, 57 k J τ J N J τl N J, J j= j =, 58 J l kk+ λ k λ k+ u kk+, 59 Then, 5 hold true and the state where all the Class-j flows hoosng S j, j, s a Nash equlbrum Proof: See Appendx The system-value s sum of payoffs of all the flows, whh s gven by, J J V T, λ = N F = N A λ 60 N =0 We must hoose and λ that maxmze 60 satsfyng the NE ondtons, Let ĉ, ˆλ be one suh optmal par Note that 60 s a dereasng funton of toll vetor, λ Hene, from 58 and 64, we get J ˆλ J = 0, and ˆλk = l + 6 =k Substtutng the optmal toll values n 60, we get N J J V T = + N J N J J Tl + J 2 k= N k N J k N k where N k = k = N Then, defne, τk+ τ k τ J, 62 V T = max V T subjet to

12 2 We refer to V T as System value wth tollng Now, we have the followng proposton, whh asserts that the system value aheved by the tolled mult-ter regme s superor to that of the untolled sngle ter regme Proposton 5 The system value wth tollng s no less than the value of sngle ter network game e, V T V G Also, the strt nequalty holds f there exsts a k < J suh that N J N k N J Nk τ k τk+ τ k, 64 Proof: Suppose attans equalty n 57-58, e a orner pont of the onstrant set Note that the elements of, the bandwdths alloated to eah subnetwork, that means to eah flow lass, s equal to the total bandwdth reeved by the orrespondng flow lass at the NE of the un-tolled sngle network game Also, from 6 and 55, the optmal entrane toll n eah subnetwork drops to zero Then, V T = V G Hene, we onlude that V T V G Note that V T s strtly onave and hene, 63 has a unque maxmzer When 64 holds true, the unque maxmzer les n the nteror of the onstrant set of 63 Then, V T > V G whh ompletes the proof Next, we derve a bound on the effeny of the mult-ter tollng sheme Let J η = + k= = where n = N N Then, we lam that k n 65 η T = V T V S mn{η G, η} 66 where η G s the effeny of sngle ter sheme wthout tollng The lam an be proved as follows: Let j = N j N for all j J Then, = [,, J ] les n the feasble set of the optmzaton problem, 63 Then, V T V T It an be shown that V T V S < η where V S s gven by 48 Therefore, η T < η Also, from Proposton 5, we get that η T η G Together, we get the lam Note that, η, does not depend on the rato, ; but t sales up wth the number of lasses n the system Nevertheless, η no more than the effeny of the sngle ter networks wthout tollng Therefore, we onlude that when the number of lasses n the system s not arbtrarly large, the effeny of mult-ter tollng shemes are superor to the sngle ter T networks and, t does not sale up wth the rato, s Note that there mght be Nash equlbra other than the one stated by Lemma 4 Therefore, 66 may be better than the effeny of the worst Nash equlbrum Now, we present a numeral example to valdate our analytal observatons Example-6: Let two flow lasses, namely Class and Class 2, wth dsutlty thresholds τ = and τ 2 = are sharng a lnk wth apaty unts The lnk s parttoned nto two subnetworks, namely S and S 2 Let N be the number of flows n Class and defne n = N /N + N 2, for =, 2 The optmal bandwdth alloaton to subnetwork S, that Effeny Rato Effeny Rato Mx of Class and Class 2 Flows Sngle Ter Network, / =4 Two Ter Network, / = 4 Sngle Ter Network, / =6 Two Ter Network, / = Fraton of Class Flows Fg 7 Comparson of Effeny Rato η between PMP sheme and Game n a network wth pre-nsenstve flows and delay senstve flows Sne V S and V G were negatve n ths example, a hgher rato s worse maxmzes the system value wth tollng, s gven by ĉ = + n 2 Tl n n 2 η T = V T = V S + n2 n Also, [ the optmal toll n S s gven by ˆλ = ] N 2 ĉ N ĉ Note that S 2 has no entrane toll and the optmal alloaton to S 2 s ĉ 2 = ĉ We defne Effeny Rato η T here as the rato of System- Value wth tollng V T to Soal optmum V S From 60 and V S,from 48, we an show that n + n 2 ĉ n + ĉ n n K + n + n 2, where K = Tl In Fgure 7, we have ompared η attaned usng the PMP sheme versus that of a sngle-ter We have used = 2, = 3 and = 4 n our smulaton We observe that nspte of tollng, the PMP sheme always performs better than the sngle-ter sheme Also, note that, unlke the sngle ter sheme, the effeny of the PMP sheme does not sale wth IX CONCLUSION In ths paper we examned the onsequenes of the dea that a protool s smply a way of nterpretng Lagrange multplers We showed that flows ould hoose the nterpretatons, based on rtera suh as delay or loss senstvty We determned the soally optmal protool, as well as the hoe that would result by flows takng ther own selfsh desons We showed that the soal good s maxmzed by usng the strtest possble pre nterpretaton However, based on dfferent mxes of flow types a mx of nterpretatons ould be the Nash equlbrum state We haraterzed the loss of effeny for some spef ases, and showed that a mult-ter network wth tollng s apable of ahevng superor system value The result suggests the onsderaton of multple tolled vrtual networks, eah geared towards a partular knd of flow In the future we propose to explore the dea of vrtual, tolled subnetworks further

13 3 X APPENDIX- Proof of Lemma 3: The Nash equlbrum ondtons, 5, are equvalent to l k λ λ k u k, k >,, 67 whh follows from the defnton of F k gven by 54 Reall the defntons of, l k and u k from 55 and 56 respetvely Therefore, we prove the lemma by showng that 67 hold true when are satsfed Suppose are true Then, t s easy to observe that l k u k, k > Also, we have m t=k l tt+ λ k λ m, m > k, k 68 From the defntons of l k s and the fat that τ < τ k f < k, t s easy to show that [5] M Podlesny and S Gornsky, RD network serves: dfferentaton through performane nentves, n ACM SIGCOMM Computer Communaton Revew, vol 38, no 4, 2008, pp [6] J Mo and J Walrand, Far end-to-end wndow-based ongeston ontrol, IEEE/ACM Transatons on Networkng, vol 8, no 5, pp , Otober 2000 [7] A M Odlyzko, Pars Metro Prng for the Internet, n Proeedngs of the ACM Conferene on Eletron Commere EC 99, 999 [8] S Shakkotta, R Srkant, A Ozdaglar, and D Aemoglu, The pre of smplty, IEEE Journal on Seleted Areas n Communatons, vol 26, no 7, pp , September 2008 [9] C-K Chau, Q Wang, and D-M Chu, On the vablty of Pars Metro Prng for ommunaton and serve networks, n Proeedngs of IEEE Infoom, San Dego, CA, Marh 200 l kk+j l kk+j l k+j k+j, 69 for k < J and < j J k Then, we have, l km = l kk+ + l kk+2 l kk+ + + l km l km m t=k l tt+ λ k λ m 70 In smlar fashon, we an show that u km λ k λ m Then, 67 s proved and hene the lemma Vnod Ramaswamy reeved hs Bahelor of Engneerng degree n eletrons and ommunaton from Kerala Unversty, Inda, n 2004 and hs MTeh degree n eletral engneerng, wth spealzaton n ommunaton systems, from Indan Insttute of Tehnology, Madras, Inda, n 2006 He s urrently workng towards the PhD degree n Eletral Engneerng at Texas A&M Unversty, College Staton Hs researh nterests nlude game theory, optmzaton methods, and ther applaton to ommunaton networks REFERENCES [] F P Kelly, A Maulloo, and D Tan, Rate ontrol n ommunaton networks: shadow pres, proportonal farness and stablty, Journal of the Operatonal Researh Soety, vol 49, pp , 998 [2] S H Low and D E Lapsley, Optmzaton flow ontrol, I: Bas algorthm and onvergene, IEEE/ACM Transatons on Networkng, pp , Deember 999 [3] S Kunnyur and R Srkant, A tme-sale deomposton approah to adaptve ECN markng, n Proeedngs of IEEE Infoom, Anhorage, AK, Aprl 200 [4] R Srkant, The Mathemats of Internet Congeston Control Brkhauser, 2004 [5] S Shakkotta and R Srkant, Network optmzaton and ontrol, Found Trends Netw, vol 2, no 3, pp [6] R Stevens, TCP/IP Illustrated, Volume Addson-Wesley, 994 [7] T Kelly, Salable TCP: Improvng performane n hghspeed wde area networks, Deember 2002 [8] D X We, C Jn, S H Low, and S Hegde, FAST TCP: motvaton, arhteture, algorthms, performane, IEEE/ACM Transatons on Networkng, De 2006 [9] S Lu, T Başar, and R Srkant, TCP-Illnos: a loss and delay-based ongeston ontrol algorthm for hgh-speed networks, n valuetools 06: Proeedngs of the st nternatonal onferene on Performane evaluaton methodolges and tools, Psa, Italy, Otober 2006 [0] A Tang, J Wang, S Hegde, and S Low, Equlbrum and farness of networks shared by TCP Reno and Vegas/FAST, Teleommunaton Systems, vol 30, no 4, pp , 2005 [] A Tang, J Wang, S Low, and M Chang, Equlbrum of heterogeneous ongeston ontrol: Exstene and unqueness, IEEE/ACM Transatons on Networkng, vol 5, no 4, pp , August 2007 [2] A Tang, X We, S Low, and M Chang, Equlbrum of heterogeneous ongeston ontrol: Optmalty and stablty, IEEE/ACM Transatons on Networkng, vol 8, no 3, pp , June 200 [3] J Mo, R La, V Anantharam, and J Walrand, Analyss and omparson of TCP Reno and Vegas, n IEEE Infoomm, 999, pp [4] E Altman, R El-Azouz, Y Hayel, and H Tembne, The evoluton of transport protools: An evolutonary game perspetve, Computer Networks, vol 53, no 0, pp , 2009 Dganto Choudary reeved hs Bahelors of Sene degree n Eletral Engneerng wth a mnor n Mathemats from Texas A&M Unversty n 202 He has reeved the dstnton of Undergraduate Researh Fellow and Engneerng Sholar He s urrently workng n Maxm Integrated n Dallas TX Ths work was done as part of hs Undergraduate Summer Researh Program Hs researh nterests nlude peer-to-peer systems, queung theory, onneton between nformaton theory and eletromagnet theory, and ongeston ontrol Srnvas Shakkotta S 00-M 08 reeved hs Bahelor of Engneerng degree n eletrons and ommunaton engneerng from the Bangalore Unversty, Inda, n 200 and hs MS and PhD degrees from the Unversty of Illnos at Urbana- Champagn n 2003 and 2007, respetvely, both n eletral engneerng He was Postdotoral Sholar at Stanford Unversty untl Deember 2007, and s urrently an Assstant Professor at the Dept of ECE, Texas A&M Unversty He has reeved the Defense Threat reduton Ageny Young Investgator Award 2009 and the NSF Career Award 202 as well as researh awards from Cso 2008 and Google 200 Hs researh nterests nlude ontent dstrbuton systems, prng approahes to network resoure alloaton, game theory, ongeston ontrol, and the measurement and analyss of Internet data