TCP/IP Interaction Based on Congestion Price: Stability and Optimality

Transcription

1 TCP/IP Interacton Based on Congeston Prce: Stabty and Optmaty Jayue He Eectrca Engneerng Prnceton Unversty Ema: Mung Chang Eectrca Engneerng Prnceton Unversty Ema: Jennfer Rexford Computer Scence Prnceton Unversty Ema: Abstract Despte the arge body of work studyng congeston contro and adaptve routng n soaton, much ess attenton has been pad to whether these two resource-aocaton mechansms work we together to optmze user performance Most anayss of congeston contro assumes statc routng, and most studes of adaptve routng assume that the offered traffc s fxed In ths paper, we anayze the nteracton between congeston contro and adaptve routng, and study the stabty and optmaty of the jont system Prevous work has shown that the system can be modeed as a jont optmzaton probem that naturay eads to a prma-dua agorthm wth shortest-path routng usng congeston prces as the nk weghts In practce, the agorthm s commony unstabe We consder three aternatve tmescae separatons and examne the stabty and optmaty of each system Our anaytc characterzatons and smuaton experments demonstrate how the step sze of the congestoncontro agorthm affects the stabty of the system, and how the tmescae of each contro oop and homogenety of nk capactes affect system stabty and optmaty The strngent condtons mposed for stabty suggests that congeston prce woud be a poor feedback mechansm n practce Keywords: Network utty maxmzaton, Congeston contro, Dynamc routng, TCP/IP I INTRODUCTION There are two man ways n the Internet to adapt the aocaton of network resources to maxmze user utty: congeston contro n TCP) and routng n IP) Congeston contro aocates the mted capacty on each nk to competng fows, whe routng determnes whch fows pass through whch nks Optmzaton frameworks have provded rgorous characterzatons of TCP and IP performance n soaton For exampe, recent work has shown that TCP congeston contro mpcty soves network-utty maxmzaton probems [], [], [3], [4], [5], but these studes assume a statc mappng of traffc to network paths Smary, research on traffc engneerng [6], [7] and oad-senstve routng [8], [9], [0] nvestgate how to optmze the assgnment of traffc to paths, but assume that the sources do not adapt ther sendng rates to the prevang network condtons In practce, however, these two resource-aocaton mechansms do nteract wth each other n potentay compcated ways Optmzaton-theoretc anayss of the TCP/IP nteracton s scarce n the terature For exampe, [] examnes the nteracton between congeston contro and adaptve routng based on centray mnmzng the maxmum nk utzaton However, congeston contro s not modeed anaytcay and the resuts are mted to networks wth a snge botteneck The ony paper wth a detaed anaytc mode of TCP/IP nteracton s the recent work [] Ths study vews the jont optmzaton probem as maxmzng user utty wth both the source rates and network paths as optmzaton varabes In partcuar, the nteracton between TCP and IP s modeed as dynamc routng based on congeston prces on the nks, where congeston prce can be nterpreted as nk metrcs ke packet oss or queung deay The work n [], however, assumes a partcuar separaton of tmescaes: congeston contro converges nstantaneousy, foowed by one step of dynamc route optmzaton, and the process repeats In reaty, the jont system conssts of two dstrbuted contro oops runnng concurrenty, wth tmescaes determned by many compex factors eg, round-trp tme, TCP sesson duraton, routngprotoco tmer, and traffc-engneerng practce) In ths paper, we present a comprehensve framework to study TCP/IP nteracton based on congeston prce, and examne the foowng key questons through both anayss and smuaton Stabty: Does the TCP/IP system converge? Optmaty: If the system converges, does t converge to a jont optmum? Further, what knd of genera concusons can we draw to gude the desgn and operaton of IP networks? Gven that nether shortest-path routng nor network-utty maxmzaton has cosed-form soutons n genera, the anaytc resuts n ths paper focuses on stabty condtons for a rng topoogy Smuaton resuts are used to further quantfy the ntutons on optmaty gap, genera topooges, and gudenes for system mprovement We study three dfferent tmescae separatons between congeston contro and routng, a motvated by Internet reaty: ) System Mode One mtates traffc engneerng today where the operator tunes nk weghts, [6], [7] In ths mode, congeston contro woud terate unt convergence to produce source rate and congeston prce, then routng woud aso terate unt convergence to produce a new routng, and so on ) System Mode Two s motvated by oad-senstve dynamc routng assumng that congeston contro adapts at a much smaer tmescae than routng, coverng the mode n [] as a speca case In ths mode, congeston contro terates unt convergence to produce source rates

2 and congeston prces, but routng terates ony once 3) System Mode Three s aso motvated by oad-senstve routng but assumes that congeston contro and routng adapt on the same tmescae In ths mode, congeston contro and routng nteract competey dynamcay, each teratng once wth overap n ther contro oops For the rng topoogy, we fnd the condtons for a three systems to converge to mnmum-hop routng System Mode One requres the nta routng confguraton to be mnmumhop routng Theorem ) for convergence By choosng a sma enough step sze representng the steepness of the congestoncontro adaptaton), convergence can be guaranteed Theorem ) for System Mode Two In addton to a sma step sze, a certan capacty dstrbuton s aso requred for System Mode Three to converge Theorem 3) System Mode One and Two can ony converge to shortest-hop routng, whch may be suboptma Unke the other two systems, System Mode Three may aso converge to other routng confguratons Theorem 3) Smuaton shows that, when System Mode Three converges, t s cose to the optmum From our anayss and smuaton experments, we observe that congeston prce s not an approprate ayerng prce for TCP/IP nteracton, gven the strngent stabty condton The foowng new nsghts are aso obtaned: ) Sma step sze mproves system convergence : The cassca tradeoff between convergence sma step sze heps) and speed of convergence arge step sze heps) for congeston contro carres over to TCP/IP nteracton ) Shorter tmescae enhances optmaty : The more dynamc the nteracton between congeston contro and routng, the smaer the suboptmaty gap between the convergent pont and the jonty optma TCP/IP souton, because nformaton passed between TCP and IP s ess stae 3) Homogenety enhances optmaty : Optmaty of TCP/IP nteractons are enhanced by homogenety of nk capactes The rest of the paper s organzed as foows Secton II provdes the network topoogy, routng, and congeston contro modes, foowed by Secton III that descrbes the detas of three System Modes Anayss and smuaton are presented n Sectons IV and V, respectvey Future research drectons are then outned n the concuson secton VI II MODELS AND NOTATION We start wth some genera assumptons Frst, we w ony consder a snge Autonomous System, so that shortest-path mnmum-cost) routng based on nk weghts nk costs) s a reasonabe mode Second, we w consder a routng mode where traffc between source-destnaton pars can be spt arbtrary between mutpe paths Ths s not the OSPF [3] or IS-IS protocos used today, but can be easy mpemented usng the emergng MPLS [4] technoogy Thrdy, we assume the sources have nfnte backog The notaton foows []: n genera, sma etters are used to denote vectors, eg, x wth x as ts th component; capta etters to denote matrces, eg, H, W, R, or constants, eg, L, N, K ; and scrpt etters to denote sets of vectors or matrces, eg, W m, R m Superscrpt s used to denote vectors, matrces, or constants pertanng to source, eg, w, H, K A Network and Routng A network s modeed as a set of L b-drectona nks wth fnte capactes c = c, =,, L), shared by a set of N source-destnaton pars, ndexed by we w aso refer to a source-destnaton par smpy as source ) There are a tota of K acycc paths for each source, represented by a L K 0- matrx H, where { Hj, f path j of source uses nk = 0, otherwse Let H be the set of a coumns of H that represents a the avaabe paths for Defne the L K matrx H as H = [H H N ], where K := K H defnes the topoogy of the network Let w be a K vector where the jth entry represents the fracton of s fow on ts jth path such that w j 0 j, and T w =, where s a vector of an approprate dmenson wth the vaue n every entry We aow wj [0, ] for mutpath routng Coect the vectors w, =,, N, nto a K N bock-dagona matrx W Defne the correspondng set W m for mutpath routng as {W W = dagw,, w N ) [0, ] K N, T w = } In summary, H defnes the set of acycc paths avaabe to each source, and represents the network topoogy W defnes how the sources oad baance across these paths Ther product defnes a L N routng matrx R = HW that specfes the fracton of source s fow that traverses each nk B Revew TCP Mode As n [4], we nterpret the equbra of varous TCP congeston-contro agorthms as soutons of a network utty maxmzaton probem defned n [], [?] Suppose each source has a utty functon U x ) as a functon of ts tota transmsson rate x We assume U s ncreasng and strcty concave as s the case for TCP agorthms [4]) The constraned utty maxmzaton probem over x for a fxed R s maxmze U x ) ) subject to Rx c The duaty gap for the above optmzaton probem s zero Zero duaty gap means that the mnmzed objectve vaue of the Lagrange dua probem s equa to the maxmzed tota utty n the prma probem ) We brefy revew the souton to ) Frst form the Lagrangan of ): Lx, p) = U x ) + p c y )

3 where p 0 s the Lagrange mutper e, congeston prce) assocated wth the near fow constrant on nk, and y = R x s the oad on nk It s mportant that the Lagrangan can be decomposed for each source: Lx, p) = [ ) ] U x ) R p x + c p = L x, q ) + c p where q = R p s the end-to-end prce for source The Lagrange dua functon gp) s defned as the maxmzed Lx, p) over x for a gven p Ths net utty maxmzaton can be conducted dstrbutvey by each source, as ong as the aggregate nk prce q s feedback to source : x q ) = argmax x [U x ) q x ], ) The Lagrange dua probem of ) s to mnmze gp) over p 0 An teratve gradent method can be used to update the dua varabes p n parae on each nk to sove the dua probem: [ p t + ) = p t) α c + R x q t)))], 3) where t s the teraton number and α > 0 s step sze It can be shown [4] that, for suffcenty sma step sze, the above updates of x, p) through,3) converge to the jonty optma rate aocaton and congeston prces for ) and ts Lagrange dua probem At equbrum, the foowng Karush- Kuhn-Tucker KKT) optmaty condtons [4] are satsfed: q = U x ) y { c f p = 0 = c f p > 0 x 0, p 0 III PROBLEM FORMULATIONS We start the nvestgaton by consderng the jont TCP/IP optmzaton probem and motvate the usage of congeston prce Then we defne three modes, comparng and contrastng ther tmescae assumptons We concude ths secton by motvatng the usage of a rng topoogy for our anayss and some of the smuaton experments A Jont Optmzaton Mode What knd of TCP/IP nteractons woud work together to maxmze end-user uttes over both rate aocaton x and routng matrx R, sovng the foowng probem: U x ) maxmze subject to Rx c, x 0 R R, where both R and x are both varabes? Consder the dua probem of 5) n the form of optmzng the Lagrangan Lp, x, R): mn p 0 max x 0 U x ) x mn R R ) R p + 4) 5) c p 6) It hnts that dynamc shortest-path routng mn R R p, where nk cost s based on congeston prces p, may be desgned to jonty maxmze network utty wth TCP Ths possbty was frst nvestgated n [], whch shows that, under a partcuar tmescae separaton, TCP/IP woud jonty sove 5) f an equbrum exsts Such an equbrum exsts f mutpath routng s aowed, but t can be unstabe It can be stabzed by addng a statc component to nk weght, but at the expense of a reduced utty at equbrum Before gvng the detaed descrpton of the modes, we hghght the foowng basc ntuton: TCP adjusts x, IP adjusts R, each affected by the other through the congestonprce vector px, R), whch s ceary a functon of both x and R, and jonty determnng the objectve of U x ) Snce the tmescae of TCP s affected by the round-trp tme and that of IP determned by routng protocos and operatona practce, there can be four dfferent modes of the above nteracton Gven that IP rarey operates faster than TCP convergence, we have three System Modes, ncudng the one n [] as a speca case, descrbed beow B System Mode Defntons The progresson offered by Fgures, and 3 shows a trend toward a tghter coupng of the two contro oops: ) System Mode One: The TCP oop shows the steps taken for congeston contro as descrbed n Secton IIB, and, gven x, p ) from TCP, the IP oop s as foows: ) update the congeston prce p for nk, gven the nk oad y, ) update the routng per source gven the nk weghts set to the congeston prces p, and ) update the nk oads y based on the new routng matrx R Then the TCP oop s repeated, foowed by another round of the IP oop, and so on ) System Mode Two: TCP s exacty the same as n Mode One, but the IP oop terates ony once Ths s smar tmescae separaton proposed by [] However, each round of IP mode n [] gnores the change n nk oad y due to change n routng and can aso be vewed as settng the step sze to zero) Each IP round n our Mode Two takes a fu teraton of an IP round n Mode One by takng nto account the effect on nk oad due to the antcpated routng change 3) System Mode Three: The TCP and IP oop are nteractng at the same tmescae Each TCP/IP round conssts of maxmzaton over x and mnmzaton over R of the Lagrangan 6) for the same gven p, whch then s updated based on both the change n x and that n R C Rng Topoogy and Traffc Mode One of the goas of ths paper s to derve cosed-form soutons for the stabty condtons of TCP/IP nteractons However, when nk cost s a combnaton of both congeston prce and a statc component, anaytc souton or even proof of the exstence of an equbrum s an open probem[] We thus focus on purey dynamc routng where the nk cost s the congeston prce Accordng to the KKT optmaty condton 3

4 -qx y= y R qconverges TCP IRx x= U ) y= IRx q= Rp pconverges mn IPpR Fg Iustraton of System Mode One -qx y= y R qconverges TCP IRx x= U ) max Ux), st Rx c q= Rp p pt+) mn [p*-αc-rx*)]+r = [pt)-αc-y)]+ Fg Iustraton of System Mode Two -qx y= IRx x= U ) y qq= Rp p pt+)= mn [pt)-αc-y)]+ prr Fg 3 Iustraton of System Mode Three xpath 4), congeston prce has to be zero when nk oad s strcty ess than nk capacty Therefore, to avod the case of random routng due to zero nk costs, we need a topoogy and traffc mode that can x = avod = xn=n xn- zero congeston prces xn- x3=3 =N- xn-3 x4 =4 =N-3 =N- x5 path xn-5 xn-4 Fg 4 =5 =N-4 N-node rng topoogy wth N sources Consder a rng topoogy wth N nodes, each of whch beng a source wth a destnaton beng the cockwse neghbor node as shown n Fgure 4 Note that we can nterchange and ndces n ths case Each source has two possbe paths: a onehop path and an N )-hop path For the probem defned by ) at optmaty, the KKT condtons 4) aows for the constrant Rx c beng satsfed to be a potenta souton If R s nvertbe, then the constrant woud be satsfed wth equaty and the source rates woud be x = R c In addton, congeston prces woud be non-zero and p = qr, where q = U x ) There are degenerate cases where R s not nvertbe, eg, when two sources have the same spt between paths Those routng confguratons woud be changed n the next TCP/IP round snce there woud be at east one nk wth zero congeston prce and the routng adaptaton w change the routng matrx to take advantage of the zero-congestonprce nk IV STABILITY ANALYSIS In ths secton, stabty anayss s performed on each System Mode for the rng topoogy and traffc mode descrbed n Fgure 4 We fnd that for System Mode One, strngent nta condtons are requred for convergence For System Mode Two, for sma enough step sze, convergence to mnmumhop routng) s guaranteed Reca that even for TCP to converge, α needs to be suffcenty sma For System Mode Three, convergence to mnmum-hop routng) s guaranteed f there s a nk whose capacty domnates those of other nks, whe other capacty confguratons may aso ead to converge to non-mnmum-hop routng) A Anayss of System Mode One Fgure ) Each TCP/IP round conssts of the foowng two oops: TCP: Compete teratons ) and 3) to generate x t) and p t), where t ndexes the teraton of the jont TCP/IP system IP: Update the prces p k+) = [p k) αc y k))] +, where k ndexes the teraton wthn the IP oop Then, for each source, sove mn R p k)r The new R w update y k), whch n turn updates p k + ) We frst present smpe exampes ustratng three possbe system behavors x = TCP/IP stabe: Consder a three-node = rng topoogy wth unt capacty on a nks, startng wth shortest-path routng Then the TCP/IP system converges to x = [ ]; p = [ ] T ; R = R0) and t s stabe Fg 5 Two-node topoogy wth a snge source IP unstabe: From Fgure 5, there are two parae nks wth unt capacty and ony one source-destnaton par Let a = 05 + ɛ, b = a and c = b/a, where 0 < ɛ Gven R0) = [a b] T, TCP converges to x 0) = /a, p 0) = [ 0] T Insde the IP oop, each successve teraton produces: pk) = [ kα, kαc] T ; Rk) = [0 ] T, unt kα < kαc, at whch pont we have Rk + ) = [ 0] T Ths, however, trggers the congeston prce of the top nk to decrease wth each teraton whe the congeston prce of the bottom nk to ncrease unt routng R = [0 ] T So n ths case, the IP oop tsef never converges 3 IP stabe, TCP/IP unstabe: Ths exampe uses the same topoogy as exampe two Intay, the top path s chosen, e, R0) = [ 0] T From TCP, x 0) = []; p 0) = [0 0] T The IP teraton converges to R = [0 ]; p = [ 0] T snce a the traffc w be routed to the path wth the ower 4

5 congeston prce In the next TCP teraton, however, x ) = []; p ) = [0 ] T It s easy to see the system ends up oscatng between routng on the top path and routng on the bottom path and never converges Theorem : For the rng topoogy and traffc mode n Fgure 4, System Mode One converges and necessary to mnmum-hop routng) f and ony f the nta routng s mnmum-hop routng on at east N nodes Proof: Insde the IP oop, pk+) = pk) f a nks are fuy utzed It s aso easy to see that shortest-path routng n the IP oop means that each source does a comparson between ts two paths, wth three possbtes: ) If p < j p j, then choose the one-hop path ) If p = j p j, then spt arbtrary between the two paths snce the probem has many optmzers 3) If p > j p j, then choose the onger-hop path Snce p ) > 0,, for any source, f p j p j for some nk, then a other sources must be dong mnmum-hop routng So there are ony three possbe routng confguratons: ) A sources choosng one-hop paths ) N sources choosng one-hop paths, one spttng 3) N sources choosng one-hop paths, one source gong on the onger-hop path Ths s an unstabe confguraton Let R IP be the set of a routng confguratons IP can produce For the f drecton of the theorem: If R0) R IP, then TCP w generate a source rate whch fuy utzes a nks under such a routng confguraton Insde the IP oop, shortest-path routng woud produce R = R0), and the TCP/IP system s stabe For the ony f drecton : If R0) / R IP, then TCP w generate a source rate whch cannot fuy utze a nks for R R IP Then nsde the IP oop, there w be aways be nks wth zero congeston prce, and the IP oop w never converge Note that the stabe souton s not necessary optma As a smpe exampe, consder N = 3, where nk has capacty 0 whe a other nks have unt capacty Uttes are og functons Mnmum-hop routng acheves an aggregate utty of og 0 If x s spt to have / on the one-hop path and 9/ on the onger-hop path, however, then a hgher aggregate utty of 3 og 055 can be acheved B Anayss of System Mode Two Fgure ) Due to speca propertes of the rng topoogy and traffc mode, as expaned n Secton IIIC, when routng ony terate once, after a few system teratons, t s safe to assume the congeston prce s nonzero on every nk We can choose α < max p /c to ensure p > 0,, for a subsequent teratons The optmzaton probem thus becomes: mnmze [p t) α c k x k j H j w j, j H j w j )] subject to w j 0,, j; j w j =, 7) Theorem : For the rng topoogy and traffc mode n Fgure 4, TCP/IP System Mode Two converges and necessary to mnmum-hop routng) f the step sze s suffcenty sma Proof: We can rewrte 7) as foows: mnmze subject to αw T XH T Hw + s T w A T w = b w 0 where the symbos are defned beow H s smpy the topoogy matrx Construct a stacked-up verson of w as w = [w w w w w 3 w 3 w N w N ] T X s a N N matrx where row and row + are fed wth x + for = 0 to N, e, X = x x x x x x x x x x x x x x x x x x x x x N x N x N x N x N x N x N x N x N x N, s s the near term of the optmzaton objectve and t depends on p and c : αc p N )αc p ) αc p s = N )αc p ) αc N p N N )αc N p N ) A = b = [ ] T Ths s an equaty-constraned convex quadratc mnmzaton probem where the KKT optmaty condtons can be wrtten as a system of near equatons: [ αxh T H A T A 0 ] [ w v ] = T [ s b Sovng for w through matrx nverson, we obtan w = N N a c N p ) cj b N p ) j x αx x j αx j 4 N 4 N 4N ) where a N = 4 + 4N ), b N = Projectng the above souton to the nonnegatve quadrant, we map a w > to and a w < 0 to 0 Lemma : w =,, s a stabe souton Proof: Snce TCP w produce x = c, x = c, x 3 = c 3,, x N = c N, and a the nks w be fuy utzed Then the routng adaptaton w resut n w = + xp /αx ) + y j p j/αx j ), whch mpes w =, j, ] 8) 9) 5

6 For convergence to mnmum-hop routng, the foowng must hod: N a c N p ) cj b N p ) j 0) x αx x j αx j j There are two cases dependng on the sze of the rng: ) N 4: a N 0, b N 0 and for suffcenty sma α, 0) w hod ) N > 4: a N 0, b N < 0, n ths case the b N term heps wth achevng nequaty 0) A suffcenty sma α but bgger than the argest α aowed n the N 4 case) w enabe 0) to hod As n the prevous secton, we note that mnmum-hop routng s not necessary the optma souton C Anayss of System Mode Three Fgure 3) Theorem 3: For the rng topoogy and traffc mode n Fgure 4, TCP/IP System Mode Three converges to mnmumhop routng f the capacty of one nk n the rng s suffcenty arge and the step sze s suffcenty sma Proof: System Mode Three can ony converge to x, R ) f the congeston prces after a certan tme ndex evove to mantan R If the R s constant, System Mode Three reduces to a TCP oop, and w converge to the optma x for the gven R Wthout a oss of generaty, we may assume after a number of teratons, at east one nk becomes congested, then, foowng drecty from the anayss of System Mode One, there s at most one source spttng or gong on the onger-hop path Let the potentay spttng source be node and et a = w, 0 a, parameterze a a ap t)+ a) N possbe R It foows y t) = ; y t) = p t) a + N ap t)+ a) p t) p t), There are three cases: ) One source takng onger-hop path a = 0): p t + ) = [p t) αc ] +, ths s a monotoncay decreasng functon, so after a number of teratons, p > N p w no onger hod and R w change ) One source spttng 0 < a < ): Convergence requres p t + k) = N p t + k), k > 0, gven p t) = N p t) Ths hods when c N c ) = y N y ) Snce y s changng wth change n R and x, whe c stay constant Ths confguraton s unstabe 3) A sources takng one-hop path a = ): Convergence requres p t + ) < N p t + ) gven p t) < N p t) Snce N p t + ) p t + ) = N p t) p t)) α N p t) p t)))+αc N c ) If α s chosen suffcenty sma so that > α N p t) p t)) and the capacty dstrbuton s such that c > N c, then p t + ) < N p t + ) s guaranteed In summary, for suffcenty sma step sze and a capacty dstrbuton domnated by c, convergence to mnmum-hop routng s guaranteed apha Fg a) System Mode Two apha b) System Mode Three Convergence whte) and dvergence shaded) for fve-node rng V SIMULATION RESULTS Frst, we smuate over the rng topoogy for a three systems to confrm our stabty anayss resuts Secondy, the acheved aggregate utty s compared to the TCP/IP jont optmum The resuts demonstrate that ncreased homogenety and faster tmescae nteractons shrnk the gap to jont optmum Fnay, an access-core network topoogy s smuated for System Modes Two and Three We use og-utty n a our smuatons In a pots, the x-axs s capacty of nk of the rng, shown on a og scae We use a combnaton of Matab and MOSEK wwwmosekcom) envronments to numercay study the nteractons of TCP congeston contro and IP routng Most of the mpementaton s straght-forward, except for the jont optmzaton probem 5) that s a non-convex optmzaton n x, R) Wth a smpe change of varabe = x w, however, the probem can be transformed to a convex optmzaton probem n y: maxmze U T y ) subject to Hy c y 0 y A Stabty of Rng Topoogy Ony System Mode Two and Three are shown, because System Mode One s convergence depends heavy on the nta routng confguraton In Fgure 6, the shaded regon represents the dvergent regon Fgure 6a) confrms our fndngs that a smaer step sze heps wth convergence for System Mode Two Fgure 6b) aso confrms the anaytc resuts n showng that, for a certan capacty range, convergence s very dffcut, for the other capacty range, smaer step sze heps wth convergence For the rng topoogy and traffc mode n Fgure 4, stabty s certany dependent on tmescae as the attracton regons for the System Modes are qute dfferent System Mode Two appears to be the best tmescae nteracton for a stabe souton Havng a more dynamc nteracton System Mode Three) or a more statc nteracton System Mode One) reduces stabty B Optmaty of Rng Topoogy In ths secton, we examne the optmaty gap of each System Mode at a stabe pont) For System Mode One, we 6

7 5 4 3 Jont System System Mode One System Mode Two System Mode Three Jont System System Mode One System Mode Two System Mode Three aggregate utty 0 aggregate utty a) Three-node rng N = 3) b) Ten-node rng N = 0) Fg 7 Aggregate utty for optma souton and the three System Modes aggregate utty gap 5 aggregate utty gap 05 5 System Mode One System Mode Two System Mode Three System Mode One System Mode Two System Mode Three a) Three-node rng N = 3) b) Ten-node rng N = 0) Fg 8 Aggregate utty gap for the three System Modes assume the nta routng s such that source s spt 995% on the one-hop path and 05% on the N )-hop path In the pots n Fgure 7, the dotted ne sgnfes the jont optmum souton to the jont optmzaton probem 5)), and the other nes represent the three System Modes It can be seen that whe s cose to that of the other nks, e, the system s homogeneous, there s no utty gap between the dstrbuted and jont system Ths hods for both the threenode rng and the ten-node rng cases As to be expected, the effect of heterogenety s hgher for the three-node rng snce the standard devaton for the dstrbuton woud be hgher for the same vaue of capacty on nk one The effect of nk capacty homogenety s best seen n Fgure 8, where the dfference between each system and the jont optmum s potted A four fgures demonstrate that the more dynamc the tmescae nteracton, the coser a System Mode acheves the jont optmum when t converges to a stabe pont C Stabty and Optmaty of Access-Core Topoogy We next smuate over a tree-mesh topoogy, eg, n Fgure 9, to gan further nsghts on behavors of jont system modes for access-core type of topoogy In the mdde s a fu mesh representng the core of the network wth rch connectvty On the edge are three access tree subnetworks There are sx possbe source nodes and tweve possbe source-destnaton pars Of the tweve pars, 3, 5, 4, 6, 3 5, 4 6 are chosen, and for each source-destnaton par, the three mnmum-hop paths are chosen as possbe paths The smuatons were performed by assumng the capacty of the nks foows a truncated so as to avod negatve vaues) Gaussan dstrbuton, wth an average of 00 and a standard devaton that we vary from 0 to 50 Ten reazatons at each standard devaton are tested System Mode One s not smuated snce t does not converge except under strngent nta condtons System Mode Two converges for the range of step sze from 00 to 00 It has a sgnfcant gap from optmaty, however, as can be seen n Fgure 0 where each ndvdua experment s shown wth an x and the sod ne ndcates the averages From the sod ne, t s easy to observe that, once agan, homogenety heps attanng optmaty For System Mode Three, the smuatons graphs not shown) show that t s prone to beng stuck n an nfeasbe regon for a arge range of step szes In such cases, at each routng update, routng swngs from one confguraton to another, whch n turn causes the nk utzaton to swng from one nfeasbe pont to another, 7

8 Fg engneerng practce, routng woud be tryng to centray mnmze a penaty functon of nk utzaton based on a network-wde vew of the current offered traffc [7] Turnng from anayss to desgn, we can aso defne an optmzaton where a weghted dfference of end-user uttes and network operator penaty functon s maxmzed over both routes and source rates that are constraned by nk capactes A dstrbuted souton to ths probem and ts mpementaton over exstng TCP and traffc engneerng systems have recenty been presented [8] An access-core network topoogy ACKNOWLEDGMENT We woud ke to thank Steven Low, Jantao Wang, Lun L and Ao Tang of Catech for umnatng dscussons on ths topc Ths work has been supported n part by NSF grants CNS and CCF-04480, and a Csco Unversty Research Program grant Fg 0 aggregate utty gap standard devaton of nk capacty Aggregate utty gap for access-core network, System Mode Two causng constant congeston, route oscatons and packet oss VI CONCLUSIONS AND FUTURE WORK Whe congeston prce s used by TCP for dstrbuted congeston contro and may seem to be a natura choce of nk weghts for dynamc routng, t s prone to oscatons f depoyed n practce In partcuar, for stabty n a rng topoogy, strngent nta condtons are requred for System Mode One and specfc capacty confguratons are requred for System Mode Three Even when the jont system does converge, there exsts arge optmaty gap for reastc topooges Usng termnoogy n the unfyng framework of Layerng As Optmzaton Decomposton [5], congeston prce s a poor ayerng prce for TCP/IP nteracton Compared to a other cross-ayer desgns based on Layerng As Optmzaton Decomposton, ths s so far the ony excepton where congeston prce or queung deay) s not an approprate coordnaton across ayers Whe we have not addressed stochastc traffc or feedback deay ssues [6], [7] n our mode, t s unkey that such features n the mode woud enhance stabty of the TCP/IP system There are severa drectons for future work To avod nstabty of TCP/IP jont system, we can ether adopt the heurstcs of addng a statc component to the nk weght as n the eary ARPANET work [9]), or change the feedback metrc and route optmzaton probem For exampe, n current traffc REFERENCES [] F P Key, A Mauoo, and D Tan, Rate contro for communcaton networks: Shadow prces, proportona farness and stabty, J of Operatona Research Socety, vo 49, pp 37 5, March 998 [] R J La and V Anantharam, Utty-based rate contro n the nternet for eastc traffc, IEEE/ACM Trans Networkng, vo 0, pp 7 86, Apr 00 [3] S H Low, L Peterson, and L Wang, Understandng Vegas: A duaty mode, J of the ACM, vo 49, pp 07 35, March 00 [4] S H Low, A duaty mode of TCP and queue management agorthms, IEEE/ACM Trans Networkng, vo, pp , August 003 [5] R Srkant, The Mathematcs of Internet Congeston Contro Brkhauser, 004 [6] B Fortz and M Thorup, Optmzng OSPF weghts n a changng word, IEEE JSAC, vo 0, pp , May 00 [7] J Rexford, Route optmzaton n IP networks, n Handbook of Optmzaton n Teecommuncatons, Sprnger Scence + Busness Meda, February 006 [8] D Bertsekas, Dynamc behavor of shortest-path routng agorthms for communcaton networks, IEEE Trans Automatc Contro, pp 60 74, February 98 [9] J M McQuan and D C Waden, The ARPA network desgn decson, Computer Networks, vo, pp 43 89, August 977 [0] Z Wang and J Crowcroft, Anayss of shortest-path routng agorthm n dynamc network envronment, ACM SIGCOMM Computer Communcaton Revew, vo, pp 63 7, Apr 99 [] E J Anderson and T E Anderson, On the stabty of adaptve routng n the presence of congeston contro, n Proc IEEE INFOCOM, Apr 003 [] J Wang, L L, S H Low, and J C Doye, Cross-ayer optmzaton n TCP/IP networks, IEEE/ACM Trans Networkng, vo 3, pp , June 005 [3] J Moy, OSPF Verson RFC 38, Apr 998 [4] E Rosen, A Vswanathan, and R Caon, Mutprotoco Labe Swtchng Archtecture RFC 303, January 00 [5] M Chang, S H Low, R A Caderbank, and J C Doye, Layerng as optmzaton decomposton To appear n Proceedngs of IEEE, 006 Shorter verson appeared n Proc Conf Inform Scence and Sys, March 006 [6] X Ln and N B Shroff, Utty Maxmzaton for Communcaton Networks wth Mut-path Routng, IEEE Trans Automatc Contro, 006 To appear [7] F Key and T Voce, Stabty of end-to-end agorthms for jont routng and rate contro, ACM SIGCOMM Computer Communcaton Revew, vo 35, pp 5, Apr 005 [8] J He, M Chang, and J Rexford, DATE: Dstrbuted Adaptve Traffc Engneerng Poster sesson at INFOCOM 005 8