Can Congestion Control and Traffic Engineering Be at Odds?

Transcription

1 Can Congeston Control and Traffc Engneerng Be at Odds? Jayue He Dept. of EE, Prnceton Unversty Emal: Mung Chang Dept. of EE, Prnceton Unversty Emal: Jennfer Rexford Dept. of CS, Prnceton Unversty Emal: Abstract In the Internet today, traffc engneerng s performed assumng that the offered traffc s nelastc. In realty, end hosts adapt ther sendng rates to network congeston, and network operators adapt the routng to the measured traffc. Ths rases the queston of whether the jont system of congeston control and routng s stable and optmal. Usng establshed optmzaton models for TCP and traffc engneerng as a bass, we fnd the jont system s stable and typcally maxmzes aggregate user utlty through smulaton. The jont system may devate from ths soluton when the topology s not unform. A modfcaton to the jont system wll guarantee stablty and optmalty for applcatons that are suffcently elastc, but at the cost of robustness. Keywords: Network utlty maxmzaton, Optmzaton, Congeston control, Routng, Traffc Engneerng, Robustness. I. INTRODUCTION In the Internet today, end hosts runnng the Transmsson Control Protocol (TCP) adapt ther sendng rates n response to network congeston. Separately, network operators montor ther networks for sgns of overloaded lnks and adapt the routng of traffc to allevate congeston, n a process known as traffc engneerng. TCP congeston control assumes that the network paths do not change, and traffc engneerng assumes that the offered traffc does not change. Due to the layered network archtecture, congeston control and routng operate ndependently, though ther ndvdual decsons are nevtably coupled. In ths paper, we nvestgate whether the jont system s stable and optmal and there s a good alternatve system. Traffc engneerng and congeston control both solve, explctly or mplctly, optmzaton problems defned for the entre network. Traffc engneerng conssts of collectng measurements of the traffc matrx the observed load between each par of entry and ext ponts and performng a centralzed mnmzaton of a cost functon that consders the resultng utlzatons on all lnks (e.g., [1], [2]). In contrast, TCP congeston control can be vewed as mplctly solvng an optmzaton problem n a dstrbuted fashon (e.g., [3], [4], [5], [6]), where the many varants of TCP dffer n the shape of user utlty as a functon of the source rate. Prevous analyss of congeston control and routng used congeston prce as lnk weghts (e.g., [7], [8]) rather than modelng the current traffc engneerng practces. In ths paper, we use the establshed optmzaton models n (e.g., [1], [2], [3], [4], [5], [6]) to study the nteracton between traffc engneerng and congeston control, and examne the followng key questons through both analyss and smulaton: Stablty: Do the jont dynamcs of congeston control and routng converge to an equlbrum? Optmalty: If the jont system does converge, does the equlbrum maxmze the aggregate user utlty, over both the routng parameters and source rates? Better desgn: Can we modfy the current system to guarantee stablty and optmalty? In our jont congeston control and traffc engneerng (CC- TE) model TCP converges under a fxed routng confguraton, before any routng changes are made. From our analyss and smulaton experments, we obtan the followng nsghts: Confrmng the ntuton of network operators: Our smulaton results show the CC-TE model s stable for a varety of topologes. Tenson between performance and robustness: A modfcaton to the CC-TE model can guarantee stablty and optmalty (Theorem 1), but at the cost of robustness. The rest of the paper s organzed as follows. Secton II ntroduces the network topology, congeston control and our jont congeston control and traffc engneerng model. We smulate the CC-TE model n Secton III and extend t wth analyss n Secton IV. Fnally, Secton V concludes the paper and ponts to future work. II. NETWORK MODEL For analytc tractablty, our study makes some smplfyng assumptons. Frst, we focus on routng and congeston control n a sngle Autonomous System, where the operator has full vew of the offered traffc load and complete control over routng. Second, we consder a routng model where traffc between source-destnaton pars can be splt arbtrarly across multple paths. Ths s not the OSPF [9] or IS-IS [1] protocols used today, but can be mplemented usng the emergng MPLS [11] technology. Thrd, we assume that the sources have nfnte backlog,.e., long sessons modelng elephant traffc n the Internet. In other words, we gnore sesson level stochastc dynamcs, and focus on the average behavor of TCP traffc profle, even though the actual sessons themselves are dfferent. Our notaton follows the work n [7], [8]: n general, small letters are used to denote vectors, e.g., x wth x as ts th component; captal letters to denote matrces, e.g., R, or 1

2 constants, e.g., L, N. Also t s used to denote the teraton number, e.g., x(t). A. Network Topology and Routng A network s modeled as a set of L bdrectonal lnks wth fnte capactes c = (c l, l = 1,...,L), shared by a set of N source-destnaton pars, ndexed by ; we often refer to a source-destnaton par smply as source. The routng matrx R l specfes the fracton of s flow that traverses each lnk l. We consder a model of routng that closely reflects today s operatonal practce [1], [2]. The operators measure the offered load between each ngressegress par x. Based on the known network topology and the traffc matrx, the operators try to fnd the best routng matrx R to mnmze network congeston. For a gven routng confguraton, the utlzaton of lnk l s u l = R lx /c l. To penalze routng confguratons that congest the lnks, canddate routng solutons are evaluated based on a cost functon f(u l ) that ncreases steeply as u l approaches 1 (whle stayng fnte): f(u l ) s strctly convex and ncreasng. In addton, f should map zero to zero, snce there should be no penalty for a lnk wth % utlzaton. The routng update R(t + 1) s the soluton to the followng optmzaton problem over R for fxed x and c: mnmze l f( R lx /c l ). (1) By consderng the total lnk cost rather than tryng to mnmze a sngle bottleneck, the optmzaton framework would prefer a soluton that utlzes a sngle lnk at 91% over one that loads many lnks at 9%. In practce, the network operators often use a pecewse-lnear f for faster computaton tme [1], [2]. In ths paper, we allow f to be any strctly convex, ncreasng, and contnuous functon that maps zero to zero. B. TCP Model Whle the varous TCP congeston-control algorthms were orgnally desgned based on engneerng heurstcs, recent work such as [5], [6] has shown through reverse engneerng that they mplctly solve a convex optmzaton problem n a dstrbuted fashon. Consder a network where each source has a utlty functon U (x ) as a functon of ts total transmsson rate x. The basc (concave) network utlty maxmzaton problem over source rate vector x, for a gven fxed routng matrx R, s maxmze U (x ) (2) subject to Rx c. The goal s to maxmze aggregate user utlty by varyng x (but not R), subject to the lnear flow constrant that lnk loads cannot exceed capacty. TCP congeston-control algorthms mplctly solve (2), wth dfferent TCP varants maxmzng dfferent ncreasng and concave utlty functons. The utlty functon can be used to descrbe the user s degree of satsfacton wth a partcular throughput, and can also be vewed as a measure of the elastcty of the traffc. The aggregate utlty capture both the effcency and farness of the system n allocatng bandwdth to the traffc. A partcular famly of wdely-used utlty functons s parameterzed by α [12]: { log x, α = 1 U α (x) = (1 α) x 1 α (3), α 1. Maxmzng these α-far utltes over lnear flow constrants leads to rate-allocaton vectors that satsfy the defntons of α-farness n the economcs lterature. A utlty functon wth α = 2 was examned by [13], and then lnked to TCP Reno [14]. TCP Vegas can be nterpreted as α = 1, see [4]. Other TCP varants that can be nterpreted as α = 1 are STCP [15] and FAST [16]. XCP has α n the sngle-lnk case [17]. One wdely-deployed TCP varant that s not modeled by an α-far utlty functon s TCP Tahoe, whch has been reverse engneered and shown to be maxmzng the utlty functon U(x) = arctanx [5]. C. Jont Congeston Control and Traffc Engneerng Model Fg. 1. A detaled vew of the CC-TE model. The CC-TE model, as shown n Fgure 1, has two steps n each teraton of the feedback loop. At tme t + 1, the congeston-control step frst computes new source rates based on the routng confguraton from tme t: x(t + 1) = argmax x U (x ), subject to R(t)x c. (4) Then the routng step computes new paths based on the source rates: ( ) R(t + 1) = argmn R f R l x (t + 1)/c l. (5) l The teratons of (4,5) repeat over tme, wth congeston control adaptng the source rates to the new routes, and traffc engneerng adaptng the routes to the measured traffc. III. SIMULATION RESULTS We frst llustrate some nterestng numercal observatons before presentng theorems on stablty and optmalty. Our numercal experments use a combnaton of the Matlab and MOSEK [18] envronments. 2

3 FGHIJ yz{ }~u ztu z D E 123 # $ 127 %&' ( stuvw q r ^_` ^_d RST U )*+, L M ^_a ^_d e` VWX Y ^_b ^_d e a / NO ^_c ^_d e b Z[\]!" 12C FGHIK :;< PQ ^_p stuvx ^_d e c f gh jk l mno (a) N nodes, N sources (b) N nodes, 1 destnaton Fg. 2. Two N-node rng topologes wth dfferent traffc patterns. ŠŠ ˆ Š Œ Ž ƒ (a) Access-Core topology (b) Ablene topology Fg. 3. Two realstc topologes. A. Smulaton Set-up We evaluate two varants of TCP congeston control: α = 2 (e.g., TCP Reno) and α = 1 (e.g., TCP Vegas). For the costfuncton f(u l ), we use an exponental functon, whch s the contnuous verson of the functon used n varous studes of traffc engneerng [1], [2]. Our ntal experments evaluate a smple N-node rng topology, where we can easly scale the sze of the network. To evaluate the nfluence of the traffc patterns, we consder two scenaros. In the frst scenaro, each node s a source sendng to ts clockwse neghbor; each source has two possble paths: a drect one-hop path and an ndrect (N )-hop path. In the second scenaro, node 1 s the destnaton and the remanng N 1 nodes are sources; each source x has an -hop path and an (N )-hop path. Our experments vary the number of nodes N and the capacty of lnk 1 (between nodes 1 and N). To study realstc topologes wth greater path dversty, we also experment wth the two networks n Fgure 3. On the left s a tree-mesh topology, whch s representatve of a common network structure. In the mddle s a full mesh representng the core of the network wth rch connectvty. On the edge are three access tree subnetworks. Of the twelve possble sourcedestnaton pars, 1 3, 1 5, 2 4, 2 6, 3 5, and 4 6 are chosen, and for each source-destnaton par, the three mnmum-hop paths are chosen as possble paths. On the rght s the Ablene backbone network [19]. Of the many possble source-destnaton pars, we choose 1 6, 3 9, 71, and For each source-destnaton par, we choose the four mnmum-hop paths as possble paths. For the accesscore and Ablene topologes, the smulatons assume the lnk capactes follow a truncated (so as to avod negatve values) Gaussan dstrbuton, wth an average of 1 and a standard devaton that vares from to 5. We smulate twenty random confguratons for each value of the standard devaton. In all experments, we start wth an ntal routng confguraton that splts traffc evenly among the K paths for each sourcedestnaton par. B. Suboptmalty Gap Smulatons Gven the structure of (2), t s natural to wonder f the nteracton of congeston control and traffc engneerng maxmzes aggregate user utlty. Prevous work [2], [7] has proposed the followng jont optmzaton problem: maxmze U (x ) (6) subject to Rx c, x where both R and x are varables. Our experments quantfy the gap n aggregate utlty between the jont system and the optmal aggregate utlty of (6). Table I summarzes the key results. In Fgure 4, we vary the capacty of lnk 1 and plot the gap n aggregate utlty for rng topologes wth three, fve, and ten nodes, where each node communcates wth ts clockwse neghbor. The two graphs plot results for α = 1 (e.g., TCP Vegas) and α = 2 (e.g., TCP Reno), respectvely. 3

4 N=1.2.4 N= capacty of lnk one (a) α = 1 (e.g., TCP Vegas) capacty of lnk one (b) α = 2 (e.g., TCP Reno) Fg. 4. Aggregate utlty gap for the N-node, N-source rng capacty of lnk one (a) α = 1 (e.g., TCP Vegas) capacty of lnk one (b) α = 2 (e.g., TCP Reno) Fg. 5. Aggregate utlty gap for the N-node, 1-destnaton rng standard devaton (a) Access-core topology standard devaton (b) Ablene topology Fg. 6. Aggregate utlty gap for two realstc topologes (wth α = 1). A -x- marker denote an ndvdual test pont and a -o- marker denote the average. Fgure(s) 4a versus 5a 4a versus 4b 5a versus 5b All fgures All fgures Key Message Traffc pattern has a sgnfcant effect. TCP varants gve the same trend. Relatvely small suboptmalty gap. Homogenety mnmzes suboptmalty gap. TABLE I SUMMARY OF RESULTS ON SUBOPTIMALITY GAP. The graphs show trends that are very smlar across a range of topology szes, suggestng that the number of sources alone does not have a sgnfcant nfluence on the suboptmalty gap. Smlarly, the two TCP varants lead to very smlar results. The vertcal lne n the mddle of the two graphs hghlghts the confguraton where all lnks have unt capacty. The suboptmalty gap s zero for a wde range of capacty confguratons. When one lnk has much lower capacty than the other lnks, a suboptmalty gap emerges. Ths occurs because the traffc-engneerng step n the jont system stops makng use of ths low-capacty lnk, snce the penalty for placng even a small amount of load on ths lnk exceeds the cost of forcng the traffc on a longer path that places load on multple lnks. When lnk 1 has an extremely low capacty, even the optmal soluton cannot place much traffc on ths lnk, leadng to a small suboptmalty gap. The graphs n Fgure 5 confrm that varatons n lnk capactes affect the suboptmalty gap. These graphs evaluate 4

5 the N-node rng wth one destnaton node, for two values of N and two TCP varants. In contrast to Fgure 4, havng ether a smaller or a larger capacty on lnk 1 leads to a suboptmalty gap. Ths s not surprsng because lnk 1 s a bottleneck lnk for ths traffc pattern. If the lnk has a small capacty, the traffc-engneerng step does not make use of the lnk, makng the left part of these curves closely resemble the plots n Fgure 4. If the lnk has a hgh capacty, the traffc-engneerng step tres to drect more sources through the lnk; however, ths s not the best soluton when the capacty of lnk 1 s just slghtly larger than the other lnks because traffc traverses longer paths, placng load on a larger number of lnks. Comparng Fgures 4 and 5 llustrates the mportant role the traffc pattern plays n determnng whether the jont system successfully maxmzes aggregate utlty. The graphs n Fgure 6 llustrate the effects of a varaton n lnk capactes on realstc topologes. We show how the suboptmalty gap depends on the standard devaton of the lnk capactes, whch are all vared accordng to a truncated Gaussan dstrbuton; we plot separate ponts for each of the 5 experments for each value of standard devaton, as well as a curve that hghlghts the mean values. The trend that a more homogeneous capacty dstrbuton (smaller standard devaton) would lead to a smaller suboptmalty gap exsts, but t s much more subdued than n the rng topology and t s domnated by the varance. Ths suggests wth realstc topologes, the relatonshp between lnk capacty and utlty gap s more complex. One possble explanaton s that the bottleneck lnk on each path s what matters and whle that s easly correlated wth varyng a sngle lnk n the rng topology, the effect s coupled n a more complex topology. In addton, for the Ablene topology, a suboptmalty gap exsts even for a homogenous capacty dstrbuton. Whle the results of the rng topology suggested network operators can favor certan confguratons to mprove network effcency, t s more challengng when dealng wth realstc topologes. IV. CONVERGENCE AND OPTIMALITY ANALYSIS Our smulatons showed that the CC-TE model s stable and close to optmal for a range of topologes. We speculate, but cannot yet show t s provably stable for general topologes. In ths secton, we show how a change n the cost functon can lead to a provably stable and optmal jont system. Ths comes at the cost of robustness, however, and s not recommended for mplementaton. Theorem 1: If the cost functon f s zero untl u l = 1, and postve afterwards, then the CC-TE model converges for suffcently concave utltes (.e., suffcently elastc traffc): U (x ) U (x) x. In partcular, t converges for α-far utltes when α 1 and for arctan utlty of TCP Tahoe. Proof: The proof conssts of three man steps. Frst we show that there exsts an unconstraned optmzaton over both x and R such that the jont congeston control and routng system s equvalent to a successve, alternatng optmzaton over x and then R. Then we provde a suffcent condton to guarantee convergence. Fnally the condton s examned for α-far utltes and arctan utlty. Consder the unconstraned mnmzaton of g(x, R) = U (x ) + γ ( ) f R l x /c l (7) l for some γ. The two steps n the alternatng optmzaton method of Gauss-Sedel algorthm [21] are as follows: x(t + 1) = argmn x U (x ) + γ f( R l (t)x /c l ) l R(t + 1) = argmn R g(x(t + 1), R(t)) = argmn R f( R l (t)x (t + 1)/c l ). l The mnmzaton of g(x, R) over R s clearly equvalent to (1). We need to further show that mnmzng g(x, R) over x (an unconstraned problem) s equvalent to the utltymaxmzaton problem mplctly solved by TCP congeston control (2) over x (a constraned problem), for suffcently large γ. By the penalty functon method (see [22] for detals), there exsts a penalty functon P and a constant γ so that (2) s equvalent to (8): maxmze x U (x ) γ ( ) P R l x /c l 1, (8) l provded that γ s suffcently large and P s convex, ncreasng, and zero for Rx c (postve otherwse). Essentally, n (8) γ l P ( R lx /c l 1) n the objectve functon replaces the constrant Rx c when γ s suffcently large. We now just need to establsh a mappng between the lnk cost functon f and penalty functon P whle preservng the desred propertes. Indeed, convexty of P mples convexty of f (convexty s preserved through a lnear operaton). If P s ncreasng, so s f. If operators chooses a cost functon f whch s zero untl u l = 1, then t can match P exactly. So far we have constructed an optmzaton problem (mnmzaton of g(x, R)) whose Gauss-Sedel soluton algorthm s equvalent to the system model of jont congeston control and routng as descrbed n the prevous subsecton. Now we wll examne the condtons for convergence of ths Gauss- Sedel Algorthm. From [21], the Gauss-Sedel Algorthm wll converge to the mnmzer of g f g s bounded from below, dfferentable, margnally strctly convex n x and R, and jontly convex n x and R. The frst three condtons are already satsfed through the constrants placed n the system model defnton. Condton 1 s satsfed snce x, R by defnton. Condton 2 s satsfed snce U and f are dfferentable, so s g. The thrd condton s satsfed snce U s strctly concave n x, and f s margnally strctly convex n x and R. The last condton s not satsfed n general snce the functon f( l R lx /c l ) s not jontly convex n R and x. In order to satsfy the condton on jont convexty n x and R, consder a log change of varable. Let x = log x, Rl = log R l, then R l x = exp( R l + x ). Wth the change of 5

6 varable, t can be readly verfed that f s stll jontly convex n x and R l, but the utlty functon may no longer be concave n x. If the utlty functon s concave n x, then g would be strctly convex n x snce f s strctly convex n x. Denote the new utlty functon (after the log change of varable) as W ( x ). A suffcent condton for convergence of the Gauss-Sedel algorthm s for W to be concave n x. A smple dervaton shows that such a condton reduces to the followng smple bound on the curvature of the utlty functon: U (x ) U (x )/x. Now we specalze to the α-farness model for U whch covers TCP Reno (currently deployed) and several proposed varants. In ths case, W α ( x) can be wrtten as follows: { x, α = 1 W α ( x) = (1 α) exp(x) 1 α (9), α 1. Examnng W ( x) shows that W( x) s concave for α 1. Fnally, TCP Tahoe s examned. Recall that U(x) = arctan(x) for TCP Tahoe, and W ( x) = arctan(x). It follows that W ( x) = (exp( x) exp(3 x))/(1 + exp(2 x)) 2 and W s concave. Therefore, convergence of the system model s guaranteed for TCP algorthms wth α 1 and TCP Tahoe. Followng a smlar argument as n ths proof, Theorem 1 can be extended to utlty functons W whch are not concave n x, as long as lnk-cost functons f are suffcently convex. Whle ths change to the objectve functon of traffc engneerng can guarantee stablty and optmalty, we do not recommend t n practce for robustness reasons. By lettng the cost functon be zero up untl lnk utlzaton hts capacty, there s a rsk of havng multple lnks operate at capacty. Ths s a fragle pont of operaton for the network snce a small burst n traffc would cause the traffc on certan lnks to exceed capacty. Once the traffc exceeds capacty, congeston s nevtable and so s the subsequent packet loss and delay ncrease. So here we have a trade-off between performance metrcs on one hand (stablty and optmalty) and robustness on the other. V. CONCLUSION In ths paper, we have studed the nteracton of congeston control and routng from a network operator s perspectve. TCP and traffc engneerng both try to make effcent use of lnk bandwdth to mprove network performance for end users. In today s IP networks, however, these two mechansms operate ndependently, though they are coupled because they both adapt to network congeston. In ths paper, we fnd through smulaton that TCP and traffc engneerng work effectvely together to reach a stable equlbrum that maxmzes aggregate user utlty under most network confguratons. A modfcaton to the operator s cost functon leads to a provably stable and optmal system, but at the cost of robustness. Ths hghlghts the potental tenson between performance and non performance metrcs. In our ongong work, we have defned an optmzaton problem where the objectve s a weghted dfference of enduser utltes and network operator penalty functon. A dstrbuted soluton to ths problem and ts mplementaton over exstng TCP and traffc-engneerng systems have recently been presented [23]. Ths helps to balance the tenson between robustness and optmalty n two ways. Frst, by ncorporatng the operator s penalty functon nto the objectve, t protects the network from short traffc bursts. Second, by fndng a dstrbuted soluton, the algorthm can react to traffc shfts on a smaller tmescale. ACKNOWLEDGMENT We would lke to thank Ma ayan Bresler for her help wth the smulaton re-runs and her suggestons to the proof. Ths work has been supported n part by NSF grants CNS and CCF-44812, and a Csco grant GH7265. REFERENCES [1] B. Fortz and M. Thorup, Optmzng OSPF weghts n a changng world, IEEE JSAC, vol. 2, pp , May 22. [2] J. Rexford, Route optmzaton n IP networks, n Handbook of Optmzaton n Telecommuncatons, Sprnger Scence + Busness Meda, February 26. [3] F. P. Kelly, A. Maulloo, and D. Tan, Rate control for communcaton networks: Shadow prces, proportonal farness and stablty, J. of Operatonal Research Socety, vol. 49, pp , March [4] S. H. Low, L. Peterson, and L. Wang, Understandng Vegas: A dualty model, J. of the ACM, vol. 49, pp , March 22. [5] S. H. Low, A dualty model of TCP and queue management algorthms, IEEE/ACM Trans. Networkng, vol. 11, pp , August 23. [6] R. Srkant, The Mathematcs of Internet Congeston Control. Brkhauser, 24. [7] J. Wang, L. L, S. H. Low, and J. C. Doyle, Cross-layer optmzaton n TCP/IP networks, IEEE/ACM Trans. Networkng, vol. 13, pp , June 25. [8] J. He, M. Chang, and J. Rexford, TCP/IP Interacton Based on Congeston Prce: Stablty and Optmalty, n Proc. Internatonal Conference on Communcatons, June 26. [9] J. Moy, OSPF Verson 2. RFC 2328, Aprl [1] R. Callon, Use of OSI IS IS for Routng n TCP/IP and Dual Envronments. RFC1195, December 199. [11] E. Rosen, A. Vswanathan, and R. Callon, Multprotocol Label Swtchng Archtecture. RFC 331, January 21. [12] J. Mo and J. C. Walrand, Far End-to-end Wndow-based Congeston Control, IEEE/ACM Trans. Networkng, vol. 8, pp , October 2. [13] L. Massoulé and J. Roberts, Bandwdth sharng: Objectves and Algorthms, IEEE/ACM Trans. Networkng, vol. 1, pp , June 22. [14] S. Kunnyur and R. Srkant, End-to-End Congeston Control: Utlty Functons, Random Losses and ECN Marks, IEEE/ACM Trans. Networkng, pp , October 23. [15] T. Kelly, Scalable TCP: Improvng Performance n Hghspeed Wde Area Networks, ACM SIGCOMM Computer Communcaton Revew, vol. 33, pp , February 23. [16] C. Jn, D. X. We, and S. H. Low, FAST TCP: Motvaton, Archtecture, Algorthms, Performance, n Proc. IEEE INFOCOM, March 24. [17] S. H. Low, H. Andrew, and B. P. Wydrowsk, Understandng XCP: Equlbrum and Farness, n Proc. IEEE INFOCOM, March 25. [18] MOSEK Optmzaton Software. [19] Ablene Backbone. [2] X. Ln and N. B. Shroff, Utlty Maxmzaton for Communcaton Networks wth Mult-path Routng, IEEE Trans. Automatc Control, 26. To appear. [21] D. P. Bersekas and J. N. Tstskls, Parallel and Dstrbuted Computaton: Numercal Methods. Athena Scentfc, second ed., [22] D. P. Bersekas, Nonlnear Programmng. Athena Scentfc, second ed., [23] J. He, M. Chang, and J. Rexford, DATE: Dstrbuted Adaptve Traffc Engneerng. Poster sesson at INFOCOM 26. 6