This article reviews the current transmission. Internet Congestion Control. By Steven H. Low, Fernando Paganini, and John C. Doyle.

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1 DIGITAL VISION LTD. Internet Congeston Contro By Steven H. Low Fernando Pagann and John C. Doye Ths artce revews the current transmsson contro protoco (TCP congeston contro protocos and overvews recent advances that have brought anaytca toos to ths probem. We descrbe an optmzaton-based framework that provdes an nterpretaton of varous fow contro mechansms n partcuar the utty beng optmzed by the protoco s equbrum structure. We aso ook at the dynamcs of TCP and empoy near modes to exhbt stabty mtatons n the predomnant TCP versons despte certan but-n compensatons for deay. Fnay we present a new protoco that overcomes these mtatons and provdes stabty n a way that s scaabe to arbtrary networks nk capactes and deays. Introducton Congeston contro mechansms n today s Internet aready represent one of the argest depoyed artfca feedback systems; as the Internet contnues to expand n sze dversty and reach payng an ever-ncreasng roe n the ntegraton of other networks (transportaton fnance etc. havng a sod understandng of how ths fundamenta resource s controed becomes ever more cruca. Gven the scae and compexty of the network however and the heurstc ntrcate nature of many depoyed contro mechansms (whch we summarze n the next secton unt recenty ths probem appeared to be we beyond the reach of anaytca modeng and feedback contro theory. Theoretca research on ths topc (e.g. [1]-[5] has deat mosty wth smpe scenaros (e.g. snge-botteneck per-fow queueng; see aso recent surveys n [6] and [7]. Meanwhe the Internet communty has turned to sma-scae smuatons to vadate desgns. A of these eave huge gaps n our understandng of rea network behavor. In the ast few years arge strdes have been taken n brngng anaytca modes nto Internet congeston contro Low (sow@catech.edu s wth the Departments of Computer Scence and Eectrca Engneerng Caforna Insttute of Technoogy Pasadena CA 9115 U.S.A. Pagann s wth the Department of Eectrca Engneerng Unversty of Caforna Los Angees CA 995 U.S.A. Doye s wth the Departments of Eectrca Engneerng and Contro and Dynamca Systems Caforna Insttute of Technoogy Pasadena CA 9115 U.S.A //$17. IEEE 8 IEEE Contro Systems Magazne February

2 (see e.g. [8]-[] and the references theren. Key to these advances has been the expct modeng of the congeston measure that communcates back to data sources the nformaton on congeston n network resources beng used; more precsey t s assumed that each network nk measures ts congeston by a scaar varabe (termed prce and that sources have access to the aggregate prce of nks n ther path. These assumptons are mpcty present n many varants of today s TCP protocos; ths framework exhbts the prce sgna beng used n these protocos (e.g. oss probabty queueng deay. Aso t s the natura settng for exporng aternatve protocos based on more expct prce sgnang (e.g. bt markng. Two types of studes are of fundamenta nterest. On the one hand t s mportant to characterze the equbrum condtons that can be obtaned from a gven congeston contro protoco from the pont of vew of farness effcency n use of resources dependence on network parameters etc. In ths regard the above-mentoned prces can be nterpreted n economc terms (e.g. [9] and the congeston contro system as seekng the goba optmum of a certan aggregate utty functon subject to network capacty constrants. By descrbng the utty mpct n exstng TCP protocos [11] [1] equbrum propertes are nferred some of whch corroborate emprcay observed features. Ths methodoogy w be revewed n the secton Equbrum Structure and Utty Optmzaton. A second ne of nqury concerns the dynamcs of congeston contro protocos drecty n the doman of contro theory. In partcuar we are nterested n the stabty of the postuated equbra especay n the presence of feedback deay and n performance metrcs such as speed of convergence capacty trackng etc. In fact by ncorporatng expct measures of congeston recent anayss [1] [] has shown that the predomnant TCP mpementaton (caed Reno and ts varants are prone to nstabtes when combned wth network deays and more surprsngy wth ncreases n network capacty. We w show smar studes n the secton tted Dynamcs and Stabty. Ths rases the queston of whether more effcent and stabe protocos coud be deveoped wth the hep of ths anaytc framework. We note that the constrant of decentrazaton makes t mpossbe for a controer to be syntheszed by for exampe optma contro; st one can try to postuate a pausbe contro aw and support t wth proofs of stabty and performance. In ths regard goba stabty resuts that appy to arbtrary networks have been gven for new prce-based fow controers [8] [1] [14] [15] n the absence of deay; further deay can be studed n terms of condtons on contro gans to retan stabty (e.g. [8] [1] [] [3]. Now as noted n for exampe [4] and [5] wndow-based protocos contan an automatc compensaton for deay ( sef-cockng ; ths has ed to recent work seekng protocos that woud reman stabe for suffcenty sma deays [6]-[8]. In ths ven we descrbe n the fna secton A Scaabe Contro some of our recent work n fndng a protoco that can be mpemented n a decentrazed way by sources and routers and that provdes near stabty for arbtrary deays capactes and routes. Athough TCP Reno has performed remarkaby above expectatons and s wdey depoyed we emphasze a key mtaton of ths congeston contro mechansm: by usng packet oss as a congeston measure hgh utzaton can be acheved ony wth fu queues.e. when the network operates at the boundary of congeston. Ths seems partcuary -suted to hande the types of traffc that have been observed n recent studes. Indeed Internet traffc 9% of whch s TCP based (see measurements at exhbts burstness at many tme scaes whch s due to the heavy-taed nature of fe szes [9]-[33]. In smpe terms ths means that most TCP connectons are mce (short but requrng ow atency but a few ong TCP connectons ( eephants whch can toerate atency generate most of the traffc. By controng the network around a state wth fu queues the eephants subject the mce to unnecessary oss and queueng deays. Ths probem can be avoded by decoupng oss from prce sgnang. Another mtaton of usng oss to measure congeston s the degradaton of performance n the cases where osses are often due to other effects (e.g. wreess nks. These consderatons are motvatng a new ook at congeston contro protocos; our am n ths artce s to argue that a more sound anaytca perspectve now avaabe shoud be brought to bear on ths nvestgaton. Current TCP Protocos TCP uses wndow fow contro where a destnaton sends acknowedgments for packets that are correcty receved. A source keeps a varabe caed wndow sze that determnes the maxmum number of outstandng packets that have been transmtted but not yet acknowedged. When the wndow sze s exhausted the source must wat for an acknowedgment before sendng a new packet. Two features are mportant. The frst s the sef-cockng feature that automatcay sows down the source when a network becomes congested and acknowedgments are deayed. The second s that the wndow sze contros the source rate: roughy one wndow of packets s sent every round-trp tme. The frst feature was the ony congeston contro mechansm n the Internet before Van Jacobson s proposa n 1988 [4]. Jacobson s dea s to dynamcay adapt wndow sze to network congeston. In ths secton we w revew how TCP nfers congeston and adjusts wndow sze. TCP aso provdes other end-to-end servces such as error recovery and round-trp tme estmaton but we w mt our attenton to the congeston contro aspect. TCP Tahoe and Reno The predomnant TCP mpementatons are caed Tahoe and Reno. The basc dea of these protocos s for a source to February IEEE Contro Systems Magazne 9

3 Congeston contro mechansms n the Internet represent one of the argest depoyed artfca feedback systems. genty probe the network for spare capacty by neary ncreasng ts rate and exponentay reducng ts rate when congeston s detected. Congeston s detected when the source detects a packet oss. A connecton starts cautousy wth a sma wndow sze of one packet (up to four packets have recenty been proposed and the source ncrements ts wndow by one every tme t receves an acknowedgment. Ths doubes the wndow every round-trp tme and s caed sow-start. When the wndow reaches a threshod the source enters the congeston avodance phase where t ncreases ts wndow by the recproca of the current wndow sze every tme t receves an acknowedgment. Ths ncreases the wndow by one n each round-trp tme and s referred to as addtve ncrease. The threshod that determnes the transton from sow-start to congeston avodance s meant to ndcate the avaabe capacty n the network and s adjusted each tme a oss s detected. On detectng a oss the source sets the sow-start threshod to haf of the current wndow sze retransmts the ost packet and re-enters sow-start by resettng ts wndow to one. Ths agorthm was proposed n [4] and mpemented n the Tahoe verson of TCP. Two refnements caed fast recovery were subsequenty mpemented n TCP Reno to recover from oss more effcenty. Ca the tme from detectng a oss (through dupcate acknowedgments to recevng the acknowedgment for the retransmtted packet the fast retransmt/fast recover (fr/fr phase. In TCP Tahoe the wndow sze s frozen n the fr/fr phase. Ths means that a new packet can be transmtted ony a round-trp tme ater. Moreover the ppe from the source to the destnaton s ceared when the retransmtted packet reaches the recever and some of the routers n the path become de durng ths perod resutng n oss of effcency. The frst refnement aows a Reno source to temporary ncrement ts wndow by one on recevng each dupcate acknowedgment whe t s n the fr/fr phase. The ratonae s that each dupcate acknowedgment sgnas that a packet has eft the network. When the wndow becomes arger than the number of outstandng packets a new packet can be transmtted n the fr/fr phase whe t s watng for a (nondupcate acknowedgment for the retransmtted packet. The second refnement essentay sets the wndow sze at the end of the fr/fr phase to haf of the wndow sze when fr/fr starts and enters congeston avodance drecty. Hence sow-start s entered ony rarey n TCP Reno when the connecton frst starts and when a oss s detected by tmeout rather than dupcate acknowedgments. TCP Vegas TCP Vegas [34] mproves upon TCP Reno through three man technques. The frst s a new retransmsson mechansm where tmeout s checked on recevng the frst dupcate acknowedgment rather than watng for the thrd dupcate acknowedgment (as Reno woud and resuts n a more tmey detecton of oss. The second technque s a more prudent way to grow the wndow sze durng the nta use of sow-start when a connecton starts up and t resuts n fewer osses. The thrd technque s a new congeston avodance mechansm that corrects the oscatory behavor of Reno. The dea s to have a source estmate the number of ts own packets buffered n the path and try to keep ths number between α (typcay 1 and β (typcay 3 by adjustng ts wndow sze. The wndow sze s ncreased or decreased neary n the next round-trp tme accordng to whether the current estmate s ess than α or greater than β. Otherwse the wndow sze s unchanged. The ratonae behnd ths s to mantan a sma number of packets n the ppe to take advantage of extra capacty when t becomes avaabe. Another nterpretaton of the congeston avodance agorthm of Vegas s gven n [1] n whch a Vegas source perodcay measures the round-trp queueng deay and sets ts rate to be proportona to the rato of ts round-trp propagaton deay to queueng deay the proportonaty constant beng between α and β. Hence the more congested ts path the hgher the queueng deay and the ower the rate. The Vegas source obtans queueng deay by montorng ts round-trp tme (the tme between sendng a packet and recevng ts acknowedgment and subtractng from t the round-trp propagaton deay. FIFO DropTa and RED A Vegas source adjusts ts rate based on observed queueng deay; n other words t uses queueng deay as a measure of congeston. Ths nformaton s updated by the FIFO (frst-n-frst-out buffer process and fed back mpcty to sources through round-trp tme measurement. A Reno source uses oss as a measure of congeston. Ths nformaton s typcay generated and fed back to sources through DropTa a queueng dscpne that drops an arrva to a fu buffer. RED (random eary detecton [35] s an aternatve way to generate the congeston measure (oss to Reno sources. Instead of droppng ony at a fu buffer RED mantans an exponentay weghted queue ength and drops packets wth a probabty that ncreases wth the average queue ength. When the average queue ength s ess than a mnmum threshod no packets are dropped. When t exceeds a maxmum threshod a packets are dropped. When 3 IEEE Contro Systems Magazne February

4 t s n between a packet s dropped wth a probabty that s a pecewse near and ncreasng functon of the average queue ength. Ths type of strategy s caed actve queue management (AQM. Anaytca Modes In ths secton we descrbe anaytca modes that were deveoped n the ast few years. A arge body of terature exsts on congeston contro but we w focus narrowy on these recent modes. A network s modeed as a set of L nks wth fnte capactes c = ( c L. They are shared by a set of N sources ndexed by n set I. Each source uses a set L Lof nks. The sets L defne an L N routng matrx R 1 = f L otherwse. A frst consderaton s that we w use determnstc fow modes to descrbe transmsson rates n contrast to much of cassca queueng theory whch rees on stochastc (e.g. Posson modes for traffc. Whe randomness n network arrvas s a natura assumpton when modeng entre connectons [36] t s ess sutabe at the packet eve where transmsson tmes for congeston-controed sources are determned predomnanty by feedback as descrbed n the prevous secton. Furthermore the dstrbutons to be used n such random modes have recenty come nto queston [9]-[33] and n any event there are few tractabe resuts on queueng theory n the presence of feedback (but see [37]. For these reasons we w study feedback at a hgher eve of aggregaton than packets modeng rates as fow quanttes. Each source has an assocated transmsson rate x ( t; the set of transmsson rates determnes the aggregate fow y ( t at each nk by ( f y( t = Rx t τ (1 f n whch the forward transmsson deays τ from sources to nks are accounted for. We assume each nk has a capacty c n packets per second. The next step s to mode the feedback mechansm that communcates to sources the congeston nformaton about the network. The key dea n the ne of work we are dscussng s to assocate wth each nk a congeston measure p ( t whch s a postve rea-vaued quantty. Due to the economc nterpretatons to be dscussed n the next secton we w ca ths varabe a Sources prce assocated wth usng nk. The fundamenta assumpton we make s that sources have access to the aggregate prce of a nks n ther route Here agan we aow for backward deays τ b n the feedback path. As we w dscuss ater ths feedback mode ncudes to a good approxmaton the mechansm present n exstng protocos wth a dfferent nterpretaton for prce n dfferent protocos (e.g. oss probabty n TCP Reno queueng deay n TCP Vegas. The precedng equatons can be represented n the Lapace doman n terms of the deayed forward and backward routng matrces: [ R ( s] f [ R ( s] b Then we have n vector form ( T f τ s e f L = otherwse τ e b s f L = otherwse. (3 denotes transpose: y( s = R ( s x( s (4 f T q( s = R ( s p( s. (5 b To specfy the congeston contro system t remans to defne how the sources adjust ther rates based on ther aggregate prces (the TCP agorthm and how the nks adjust ther prces based on ther aggregate rates (the AQM agorthm. At the source sde we can n genera postuate a dynamc mode of the form z = F( z q x = G( z q (6 where z woud be a oca state varabe. We w however mosty encounter two speca cases: the statc case where there s no z and we have x( t = G( q( t and the frst-order case wth z = x. Smary at the nk eve one can wrte a dynamc aw x q v = H( y v p = K( y v. (7 R (s f T R b(s y p Lnks ( b q( t = R p t τ. ( Fgure 1. Genera congeston contro structure. February IEEE Contro Systems Magazne 31

5 The key restrcton n the above contro aws s that they must be decentrazed.e. sources and nks ony have access to ther oca nformaton. The overa structure of the congeston contro system s now depcted n Fg. 1 where the dagona structure of the source and nk matrces represents the decentrazaton requrement. We w dscuss TCP modes wthn ths genera framework. In the next subsecton we w focus on equbrum propertes; dynamc ssues are tacked n the foowng subsecton Dynamcs and Stabty. Any prcng scheme that stabzes queues or queueng deays soves the dua probem. Equbrum Structure and Utty Optmzaton In ths secton we study the above feedback at equbrum.e. assumng the rates and prces are at some fxed vaues x y p q. We w see how an optmzaton formuaton heps understand the propertes of such equbra. T The equbrum reatonshps y = Rx q = R p foow mmedatey from (4-(5. Here R s the statc routng matrx; snce we are dscussng equbrum we can set s = n the mode (3 (equvaenty settng a deays to zero. The frst basc assumpton we make s that the equbrum rates satsfy The above equaton can be nterpreted n economc terms: f U( xs the utty the source attans for transmttng at rate x and q s the prce per unt fow t s hypothetcay charged the above represents a maxmzaton of an ndvdua source s proft. We emphasze that ths nterpretaton requred mnma assumptons about the protoco; gven a mode of the source (TCP contro one can derve from t the utty functon assocated wth the protoco as we w see beow. The roe of prces s to coordnate the actons of ndvdua sources so as to agn ndvdua optmaty wth soca optmaty.e. to ensure that the soutons of (8 aso sove the probem max U ( x (9 x subject to Rx c; (1 n other words maxmze aggregate utty across a sources subject to nk capacty constrants. Ths probem formuated n [38] s a convex program for whch a unque optma rate vector exsts. The chaenge s to sove t n a dstrbuted manner over a arge network. A natura way to ntroduce prces n regard to the above optmzaton s the duaty approach ntroduced n [1] (see aso [39] for reated deas and [] for mutcast extensons. Here prces appear as Lagrange mutpers for the probem (9-(1. Specfcay consder the Lagrangan Lxp ( = U( x p( y c = U ( x q x + pc. x = f ( q where f ( s a postve strcty monotone decreasng functon. Ths functon can be found by fndng the equbrum pont n (6; n the statc case t s just gven by the source statc aw. Monotoncty s a natura assumpton for a protocos: f q represents congeston n the source s path the equbrum source rate shoud be a monotoncay decreasng functon of t. We now see that ths assumpton aone aows us to ntroduce an optmzaton nterpretaton for the equbrum by ntroducng a source utty functon. Namey consder the nverse f 1 ( xof the above functon and et U( xbe ts ntegra;.e. U x = f 1 ( ( x. By assumptonu( xhas a postve decreasng dervatve and s therefore tsef monotone ncreasng and strcty concave. Now by constructon the equbrum rate w sove max U ( x x q. (8 x The dua probem s where mn B ( q + pc (11 p B( q = max U ( x x q. (1 x Convex duaty mpes that at the optmum p s (whch need not be unque the correspondng x maxmzng ndvdua optmaty (1 s exacty the unque souton to the prma probem (9-(1. Note that (1 s dentca to (8; therefore provded the equbrum prces p can be made to agn wth the Lagrange mutpers the ndvdua optma computed n a decentrazed fashon by sources w agn wth the goba optma of (9-(1. The smpest nk agorthm that guarantees these equbrum prces are ndeed Lagrange mutpers as shown n [1] s based on appyng the gradent projecton agorthm to the dua probem (11: p γ( y( t c f p( t > = + γ[ y( t c] f p( t = (13 3 IEEE Contro Systems Magazne February

6 where [ z] + = max{ z }. The fact that the gradent of the Lagrangan ony depends on aggregate rates y s key to the above decentrazed mpementaton at the nks. If the above equatons are at equbrum we have y c wth nonzero prces p correspondng to the actve constrants. It foows that equbrum prces are the Lagrange mutpers. Ths property however s not excusve to ths agorthm; rather any prcng scheme that stabzes queues or queueng deays (e.g. RED or Vegas see beow so that fow rates y are matched to capactes c w aow for the same nterpretaton. Ths s because matchng rate drves the gradent of the dua probem (11-(1 wth respect to p to zero sovng the dua probem and mpyng that the resutng prces are Lagrange mutpers. Another approach to prce varabes proposed n [8] (and used aso n [15] s to treat them as a penaty or barrer functon for the constrant (1. Here prces are assumed to be a statc ncreasng functon of y p h y = ( whch becomes arge as y approaches c. It foows that the resutng equbrum maxmzes the goba utty y U( x h( y dy (14 whch can be seen as an approxmaton to the above probem (9-(1. To summarze the dscusson so far under very genera assumptons the equbrum ponts of source protocos can be nterpreted n terms of sources maxmzng ndvdua proft based on ther own utty functons. Lnk agorthms generate prces to agn exacty or approxmatey these sefsh strateges wth soca wefare. Dfferent protocos correspond to dfferent utty functonsu and to dfferent dynamc aws (6-(7 that attempt n a decentrazed way to reach the approprate equbrum. We now take a coser ook at modeng Reno and Vegas n ths context. TCP Reno/RED We focus ony on the congeston avodance phase of TCP Reno n whch an eephant typcay spends most of ts tme. We take source rates as the prma varabe x and nk oss probabtes as prces p. In ths secton we assume the round-trp tme τ of source s constant and that rate x s reated to wndow w by We aso make the key assumpton that oss probabtes p ( tare sma so that the end-to-end probabtesq ( tsatsfy q ( t = 1 ( 1 p( t ~ p( t L L for a and for a t. We now mode the addtve-ncrease-mutpcatve-decrease (AIMD agorthm of TCP Reno n an average sense at tme scaes above the round-trp tme. In partcuar our modes do not attempt to predct a wndow jump of the sort The objectve s to have oca dynamc stabty for arbtrary network deays nk capactes and routng topooges. observed under MD; rather they shoud track the mean evouton of the wndow as a functon of ACKs and osses. We ntay gnore feedback deays snce we are nterested n equbrum ponts. At tme t x ( ts the rate at whch packets are sent and acknowedgments receved. A fracton( 1 q( t of these acknowedgments are postve each ncrementng the wndow w ( t by1/ w ( t ; hence the wndoww ( tncreases on average at the rate of x( t( 1 q( t/ w( t. Smary negatve acknowedgments are returnng at an average rate of x( t q( t each havng the wndow and hence the wndow w ( tdecreases at a rate of x( t q( t w( t/. Hence snce x ( t = w ( t/ τ we have for Reno the average mode x 1 q( t = 1 q( t x( t. τ We now consder the equbrum of (16: q (16 = + τ ( x. (17 From t we can obtan the utty functon of TCP Reno by dentfyng the above wth the Karush-Kuhn-Tucker condton U ( x = q. Ths gves the utty functon w( t x ( t =. τ (15 U( x = tan τ 1 τx (18 The wndow of outstandng packets at tme t actuay refects the average rate n the nterva [ t τ t] ; the above approxmaton s vad snce our modes are not ntended to provde accurate descrpton at fner tme scaes than the round-trp tme. whch seems to appear frst n [9] and [11]. Our descrpton here s sghty dfferent from that n [11] n that here oss probabty s taken as the dua varabe regardess of the nk agorthms whch from ths pont of vew affect the dynamcs but not the utty functon. February IEEE Contro Systems Magazne 33

7 The reaton (17 between equbrum source rate and oss probabty reduces to the we-known reaton (see e.g. [4] and [41] x a = τ q when the probabty q s sma or equvaenty when the wndow τ x s arge compared wth. (Ths corresponds to repacng( 1 q( t n (16 by 1 as done n []. The vaue of the constant a around 1 has been found emprcay to depend on mpementaton detas such as TCP varant (e.g. Reno versus NewReno versus SACK and whether deayed acknowedgment s mpemented. Equatng U ( x wth q the utty functon of TCP Reno becomes a U( x =. τ x Ths verson s used n [15] and [4]. We now turn to the nk agorthm.e. how oss probabtes p ( t are generated as a functon of nk rate y. Both DropTa and RED produce osses as a functon of the state of the queue so the queston nvoves modeng of the queue dynamcs as a functon of the nput rate y. In the current context of fow modes t s natura to mode queues as ntegratng the excess capacty; ths opton s dscussed beow and n the case of RED one can then easy reate queues to oss (or markng probabty. Unfortunatey a determnstc mode of DropTa that ncudes the queue dynamcs does not appear to be tractabe. At the other extreme many references mode queues as beng n steady state and postuate a statc aw p = h( yfor oss probabty as a functon of traffc rate e.g. [8] [15]. The queston as to whether the steady-state queue assumpton can be justfed s st a subject of current research [43]; however dynamc smuaton studes of Reno/RED as n [] or those descrbed beow (see Fg. (b ndcate that queue dynamcs are ndeed sgnfcant at the tme scae of nterest. For ths reason we w consder the frst opton and mode queues as ntegrators focusng on the RED queue management to obtan smpe modes of oss probabtes. Letb ( tdenote the nstantaneous queue ength at tme t; ts dynamcs s then modeed by ( y( t c f b( t > b = + [ y( t c] f b( t =. (19 RED averages the nstantaneous queue by an exponentay weghted average; denotng r ( tto be the averaged queue ength we can mode t as a ow-pass fter r = α c( r( t b( t ( for some constant < α < 1. Gven the average queue ength r ( t the markng (or droppng probabty s gven by a statc functon r( t b ρr( t ρb b < r( t < b p( t = m( r( t: = η r( t ( 1 p b r( t < b 1 r( t b (1 where b b and p are RED parameters p ρ : = and η : = b b 1 p. b Now (19 ( and (1 fa nto the genera form (7 where the nterna state v s composed of b and r. Assume now that these equatons are n equbrum. It s not dffcut to see that n ths case y c and that the nequaty s ony strct when p =. These facts mpy that the equbrum p consttute a set of Lagrange mutpers for the dua probem (11-(1. Thus we concude that f the Reno/RED reaches equbrum the resutng source rates x w sove the prma probem (9-(1 wth the source utty functons gven n (18. Moreover the oss probabtes p are Lagrange mutpers that sove the dua probem (11-(1. We remark agan that the precedng anayss refers to the averaged mode of TCP as descrbed n (16; we are not attemptng to mpose equbrum on the detaed evouton of a wndow under AIMD but rather on ts mean evouton as woud happen wth the average of many dentca sources (see smuatons beow. St even n ths mean sense we have not gven any ndcaton yet that TCP reaches equbrum. Indeed n the next secton we w fnd that t often does not snce the equbrum s unstabe and a mt cyce s observed; n partcuar average wndows and queues can oscate dramatcay. Nevertheess the above equbrum anayss s usefu n understandng the state that s amed for by the current protocos.e. the resource aocaton pocy mpcty present n the current Internet. Furthermore there s evdence that some of the nsghts derved from the equbrum modes do refect emprca propertes of the Internet. Ths suggests that the modes mght have vaue n descrbng the protoco s ong-term behavor even n an oscatory regme. We now dscuss some of these nsghts. Deay and Loss The current protoco (Reno wth DropTa fs rather than emptes botteneck queues when the number of eephants becomes arge eadng to a hgh oss rate and queueng deay. What s more ntrgung s that ncreasng the buffer sze does not reduce oss rate sgnfcanty but ony ncreases queueng deay. Ths deay and oss behavor s exacty opposte the mce-eephant contro strategy we am for: to maxmay utze the network n a way that eaves network 34 IEEE Contro Systems Magazne February

8 queues sma so that deay-senstve mce can fy through the network wth tte queueng deay. Accordng to the duaty mode oss probabty under Reno s the Lagrange mutper and hence ts equbrum vaue s determned soey by the network topoogy and the number of sources ndependent of nk agorthms and buffer sze. Increasng the buffer sze but eavng everythng ese unchanged does not change the equbrum oss probabty and hence a arger backog must be mantaned to generate the same oss probabty. Ths means that wth DropTa the buffer at a botteneck nk s aways cose to fu regardess of buffer sze. Wth RED snce oss probabty s ncreasng n average queue ength the queue ength must ncrease steady as the number of sources grows. Farness It s we known that TCP Reno dscrmnates aganst connectons wth arge propagaton deays. Ths s cear from (17 whch mpes that Reno equazes wndows for sources that experence the same oss probabty and hence ther rates are nversey proportona to ther round-trp tmes. The equbrum characterzaton (17 aso exposes the beat down effect where sources that go through more congested nks seeng arger q receve ess bandwdth. Ths effect s hdden n snge-nk modes and n mutnk modes s often confused wth deay-nduced dscrmnaton of TCP as expressed n (17. It has been observed n smuatons [44] and has ong been deemed unfar but the duaty mode shows that t s an unavodabe and even desrabe feature of end-to-end congeston contro. For each unt of ncrement n aggregate utty a source wth a onger path consumes more resources and hence shoud be beaten down. If ths s undesrabe t can be remeded by weghtng the utty functon wth deay. TCP Vegas The dynamc mode and utty functon U of TCP Vegas have been derved and vadated n [1]. We brefy summarze the resuts here. The utty functon of TCP Vegas s U ( x =α d ogx where α s a protoco parameter and d s the round-trp propagaton deay of source. In equbrum source buffersα d packets n the routers n ts path. The utty functon mpes that Vegas acheves proportona farness n equbrum. The prce varabe n TCP Vegas s queueng deay whch evoves accordng to 1 p ( ( c y t c = ( Tabe 1. Modes of TCP/AQM. (x ( t = U 1 ( q ( t TCP Reno Source contro x wth an addtona nonnegatvty constrant exacty as n (13 wth γ repaced by 1/c. Therefore agan we can nterpret equbrum prces as Lagrangan mutpers. To descrbe the rate adjustment (6 et 1 x ( t = U ( q ( t = α d q ( t be the target rate chosen based on the end-to-end queueng deay q ( tand the margna utty U. Then Vegas s source agorthm s x Mode 1 q( t = τ 1 f x( t < x ( t τ = 1 f x( t > x( t τ movng the source rate x ( ttoward the target rate x ( tat a pace of1 / τ. The modes of TCP Reno and Vegas are summarzed n Tabe 1. Dynamcs and Stabty As dscussed n the prevous secton statc or dynamc contro aws at sources and nks attempt to drve the system to a desrabe equbrum pont. So far we have ony used dynamc modes to derve the equbrum ponts and we note that there can be dfferent dynamc aws wth the same equbrum ony dstngushed by ther dynamc propertes whch we now dscuss. Many we woud ke to determne whether the equbrum s (ocay or gobay stabe. We 1 q ( t x ( t RED Lnk contro ( y( t c f b( t > b = + [ y( t c] f b( t = r = α c ( r ( t b ( t p = m( r Vegas FIFO Utty Source contro Lnk contro 1 τx U( x = tan τ 1 f x( t < x ( t x τ = 1 f x( t > x( t τ 1 ( ( ( c y t c p t f > p = 1 [ ( ] ( c y t c + p t f = Utty U ( x =α d ogx February IEEE Contro Systems Magazne 35

9 begn wth a bref overvew of some dynamc aws that have been proposed n the optmzaton framework and that aow for anaytca stabty resuts. We w then move to study n deta the dynamcs of TCP Reno/RED. In genera one coud have dynamcs at both sources and nks; however most anaytca resuts refer to systems where ony one of the aws s dynamc and the other statc. In ths regard [8] denotes by prma agorthms those where the dynamcs are at the sources and by dua agorthms when the dynamcs are at the nks. An exampe of dynamcs at the source s the frst-order aw (used n [8] for a partcuar utty functon x = κ ( U ( x q x. Combned wth a statc nk aw p = h( y t s shown n [8] that the system n the absence of deays has a snge goba attractor whch optmzes the cost functon n (14; n fact ths modfed utty serves as a Lyapunov functon for the system. An exampe of dynamcs at the nks was aready gven n (13; combned wth the statc source contro 1 [ ] x ( t = U + ( q ( t (3 t s shown n [1] that goba stabty s obtaned. Another nk agorthm proposed n [45] and [46] s p γ( y c + αb f p( t > = + γ[ y c + αb] f p( t = where b s the queue ength as n (19. Goba stabty n the absence of deay for ths scheme together wth (3 has been proved n [14] by a Lyapunov argument. Ths protoco can be mpemented n a smar fashon to RED but smuatons n [45] and [46] have shown a marked mprovement over RED n terms of achevng fast responses and ow queues. Recenty [16] has proposed a scheme wth dynamcs at both nks and sources but workng at dfferent tme scaes; thus stabty anayss reduces to two probems one wth statc nks and one wth statc sources. We emphasze that the Lyapunov-based stabty proofs n [8] and [14] whe goba do not consder network deays. Some oca nearzed studes n [8] and [3] examne toerance to deay and n genera yed bounds on the varous gan parameters of the agorthms (e.g. κ γ must be nversey proportona to deay to mantan oca stabty. In turn the goba stabty anayss n [1] s done n dscrete tme possby asynchronousy and does aow for deays but agan stabty s guaranteed ony f gan parameters are suffcenty sma. Note that deays are the ony dynamcs of the open-oop system descrbed n Fg. 1; were t not for deay the rate and prce adaptaton coud be performed arbtrary fast. Thus t s natura that gan parameters shoud be chosen nversey proportona to deay. Snce sources measure ther round-trp tme and a scang by1/τ s aready mpct n a wndow-based protoco due to (15 (the sef-cockng feature ths rases the ntrgung possbty that compensaton for deay coud be done automatcay n a protoco such as Reno. We now turn to a detaed study of Reno/RED that contans dynamcs at both sources and nks. Dynamcs of Reno/RED The dynamc mode of Reno/RED has so far been used ony to understand the equbrum propertes; we now study the dynamc propertes around an equbrum by nearzng the mode deveoped earer. Before we do that we must refne the nonnear mode to ncude the effect of deays whch are essenta to stabty anayss. For ths purpose we must account for forward and backward deays n the propagaton of rates and prces as was done n (1-(. For the wndow dynamcs a frst approxmaton woud be 1 w ( t w = x( t τ( 1 q( t x( t τ q( t w( t (4 wthq ( tas n (. Here we ncorporate the fact that the rate of ncomng ACKs s determned by the source rate τ unts of tme ago. However a subte ssue that arses n these modes s that the round-trp tme s tsef tme varyng snce t depends on queueng deays. In partcuar round-trp tme can be expressed as τ ( = + t d R b c where d s the round-trp propagaton deay and b s the backog at nk at the tme the packet arrved at the nk. These arrva tmes are dffcut to capture exacty snce they depend themseves on queueng deays earer n the path; ths woud mean that the tme argument n b above woud depend recursvey on other queues themseves at earer tmes and so on. We avod ths ssue by assumng a common tme t for a the queues ( = +. τ ( t d R b t c (5 Ths smpfcaton s acceptabe provded we do not attempt to mode tme resouton smaer than round-trp tmes. St dffcutes reman. In partcuar f one generazes (15 to w( t x( t = τ ( t (6 the wndow equaton (4 woud contan x( t τ( t = [ w ( t τ ( t] / [ τ ( t τ ( t] wth a nested tme argument n the 36 IEEE Contro Systems Magazne February

10 round-trp tme whch s not easy to nterpret. For ths reason we adopt the foowng conventon: whenever round-trp tme or forward and backward deay appears n the argument of a varabe we w repace t by ts equbrum vaue τ f τ b τ. Ths accounts for equbrum queueng deays but at that eve does not ncude ther varaton n tme. However when round-trp tme appears n the dependent varabe as n (6 we w consder t tme varyng and use for t the mode n (5. Ths avods recursve tme arguments but s admttedy an approxmaton done excusvey for mode tractabty. (Smar though sghty more restrctve approxmatons were made n []. As ths paper frst noted and we corroborate beow retanng deay varatons n the round-trp tme s essenta for the predctve power of ths mode. Confdence n these approxmatons can ony be obtaned through comparsons wth packet-eve smuatons. Wth these approxmatons we obtan the foowng nonnear tme-deayed mode for Reno expandng on (4: w = 1 Rp t 1 Rp t τ ( τ ( w ( t τ 1 τ ( t τ w ( t b w ( t τ w ( t. τ ( t τ b Assumng the routng matrx R has fu rank there s a unque equbrum ( w p. Lnearzng around t we have (varabes now denote perturbatons 1 b qw w = Rp ( t ( τ * w t. τ q τ Now consder the nk dynamcs of RED as descrbed n Tabe 1. For the purposes of nearzaton we note that nonbotteneck nks (wth empty equbrum queues can be gnored. For botteneck nks we make the assumpton that rate ncreases of a source affect a bottenecks n ts path (we do not mode the throttng effect that upstream nks may have on downstream ones and wrte ḃ R w t = τ t = R d ( f τ c ( f τ w( t f τ f + Rk ( b t τ / c k k k c. Let τ = d + R k k bk / ck be the equbrum round-trp tme (ncudng queueng deay. Lnearzng we have (varabes now denote perturbatons: ( f τ b R w t = RR τ k k w f bk t τ (. ( τ c k summaton sums over a nks k that share a source wth nk. It says that the nk dynamcs n the network are couped through shared sources. The term ( w / τck bk( t τ f s roughy the backog at nk k due to packets of sourceunder FIFO queueng. Hence the backog b ( t at nk decreases at a rate that s proportona to the backog of ths shared source at another nk k. Ths s because the backog n the path of source reduces the rate at whch source packets arrve at nk and hence decreases b ( t. Puttng everythng together Reno/RED s descrbed n the Lapace doman by 1 T w( s = ( si + D1 D Rb ( s p( s 1 ps ( = ( si+ D3 Dbs 4 ( T 1 b( s = ( si + Rf ( s D5R D6 Rf ( s Dw 7 ( s (7 where the dagona matrces are qw D1 = dag τ D3 = dag( α c w D5 = dag τ ( 1 D = dag τq D4 = dag( αcρ D6 = dag 1 D dag 1 7 = c τ and R ( f s and R s b ( are defned n (3. To gan some nsght nto the system s behavor et us specaze to the case of a snge nk wth N dentca sources (see generazaton to heterogeneous sources n [47]. The transfer functons are (droppng a subscrpts and emnatng bs ( e b * τ s w( s = ps ( p s p w ( τ + f * τ s αρ c Ne ps ( = w( s f * * s s + τ αc τ s+ e (8 (9 where w = cτ / N s the equbrum wndow sze and p s the equbrum oss probabty. When forward deay τ f = the mode (8-(9 reduces to that n []. It s easy to show f that τ < τ mpes that (9 s open-oop stabe. Hence the cosed-oop system s stabe f and ony f the oop functon αρ c 1 N Ls ( = e f s + τ s αc τ s+ e p ( τ s+ p w τ s does not encrce ( 1 as s traverses the cosed D contour n the compex pane. Substtuton wth the equbrum vaues of p and w yeds (usng p ~ / w The second term above woud be gnored f we dd not ncude queueng deay n the round-trp tme. The doube 3 3 c c Ls ( = αρ 1 τ s c s e N c s N e + + f τ s α τ ( τ + τ s. (3 February IEEE Contro Systems Magazne 37

11 Deay (Mode Frequency (Mode Data Ponts Deay (NS (a Statc-Lnk Mode Dynamc-Lnk Mode Data Ponts Frequency (NS (b Fgure. Vadaton: comparson of crtca (round-trp propagaton deay and crtca frequency computed from near mode and measured from smuaton (NS. (a Crtca deay ms. (b Crtca frequency Hz. The frst factor above s due to queue averagng n RED. The second factor descrbes the reaton between wndows f τ s and buffer sze; the term e arses due to the effect of queueng deays n (7. If τ f = t reduces to a ow-pass fter of tme constant1/τ. The thrd term s due to Reno; t has a dc gan of ( c 3 τ 3 / 4N = ( w 3 N / 4 and a poe at ( / N cτ = ( / w τ. Under typca condtons the equbrum wndow satsfesw >> so the system has hgh gan at ow frequences and a poe that s sower than that of the second term. Fnay we have the round-trp feedback deay. The above oop gan conssts of a stabe functon tmes a pure deay. The atter w provde sgnfcant phase startng at frequences of the order of1/τ. Therefore a cassca Bode pot anayss says that cosed-oop stabty w requre that the oop gan at those frequences be beow unty. (Equvaenty one mght say that t s mpossbe for a stabe oop to track varatons that are faster than the pure deay of the oop. Ths suggests that t s dffcut for RED to stabze Reno as extensve smuaton experence has shown. In partcuar assumng τ or c become arge at a fxed frequency ω the term w be approxmatey c τ Njω whch grows n magntude and has 9 of phase and s thus destabzng. Smar concusons happen when N s sma. Note that nstabty for hgh τ s not surprsng; however here t shows that Reno s not successfu n scang down gans by τ as was suggested as a stabzng method n the begnnng of the secton based on the sef-cockng property. Perhaps more strkng s the destabzng effect of hgh capacty; as routers become faster Reno s bound to go nto an unstabe regme. Intutvey ths s because hgher capacty eads to hgher equbrum rate. Ths produces hgher gan n the mutpcatve term of AIMD snce sources update more frequenty on each acknowedgment wth a arger amptude. Smuaton Studes The equbrum mode has been vadated n [48] for Reno and n [1] for Vegas. In ths secton we present smuaton resuts to vadate our near dynamc mode when the system s stabe or barey unstabe. They aso ustrate numercay the stabty regon of Reno/RED. We consder a snge nk of capacty c pkts/ms shared by N sources wth dentca round-trp propagaton deay d ms. For N = 3 6 sources capacty c = 89 15pkts/ms and propagaton deay d = ms we examne the Nyqust pot of the oop gan of the feedback system (L( jω n (3. For each ( N c par we determne the deay d ( m N c at 5-ms ncrements at whch the smaest ntercept of the Nyqust pot wth the rea axs s cosest to 1. Ths s the deay at whch the system( N c transts from stabty to nstabty accordng to the near mode. For ths deay we compute the crtca frequency f ( m N cat whch the phase of L( jω s π. Note that the computaton of L( jω requres equbrum round-trp tme τ the sum of propagaton deay d ( m N cand equbrum queueng deay. The queueng deay s cacuated from the equbrum mode. Hence for each ( N c par that becomes barey unstabe at a deay between 5 ms and 1 ms we obtan the crtca (propagaton deay dm ( N cand the crtca frequency f ( m N c. We repeat these experments n ns- usng persstent FTP sources and RED wth ECN markng. The RED parameters are (.1 4 pkts 54 pkts1 4. For each ( N c par we examne the queue and wndow trajectores to determne the crtca deay d ( ns N cwhen the system transts from stabty to nstabty. We measure the crtca frequency f ( ns N c the fundamenta frequency of queue oscaton from the FFT of the queue trajectory. Thus correspondng to the near mode we obtan the crtca deay d ( ns N c and frequency f ( ns N cfrom smuatons. We compare mode predcton wth smuaton. Fg. (a pots the crtca deay d N c computed from the near m ( 38 IEEE Contro Systems Magazne February

12 mode versus the crtca deay d ( ns N c from packet-eve smuatons. Each data pont (there are 3 of them corresponds to a partcuar( N c par. The dotted ne s where a ponts shoud e f the near mode agrees perfecty wth the smuaton. Fg. (b gves the correspondng pot for crtca frequences f ( m N c versus f ( N c ns. The agreement between mode and smuaton seems qute reasonabe (reca that deay vaues have a resouton of 5 ms. Aso shown n Fg. (b are crtca frequences predcted from the near mode f the nk dynamcs are gnored and congeston probabty s proportona to fow rate pt ( =ρ yt ( usng the same Nyqust pot method descrbed above. It shows that queue dynamcs s sgnfcant at the tme scae of nterest. Fg. 3 ustrates the stabty regon mped by the near mode. For each N t pots the crtca deay d ( m N c versus capacty c. The curve separates stabe (beow from unstabe regons (above. The negatve sope shows that TCP/RED becomes unstabe when deay or capacty s arge. As N ncreases the stabty regon expands.e. a sma oad nduces nstabty. A arger deay or capacty or a smaer oad eads to a arger equbrum wndow; ths confrms the fokore that TCP behaves poory at arge wndow sze. A Scaabe Contro The above dscusson suggests that the current protoco may be -suted for future networks where both deay and capacty can be arge. Ths has motvated the search for protocos that scae propery so as to mantan stabty n the presence of these varatons. Wth regard to deay stabty scang of source gans by 1/τ was frst suggested n [5] and subsequenty proved n [6]-[8] to provde stabty for prma aws nvovng frst-order source contro and statc nk markng provded deay or contro gan s suffcenty sma. The concept of scang down gans by deay was aready mentoned n the context of optmzaton-based stabty proofs; when the scang s done by a common goba constant ths can be very conservatve. In contrast ndvduazed scang as suggested here has the appeang feature that sources wth ow round-trp tmes can respond qucky and take advantage of avaabe bandwdth and t s ony those sources (wth ong deays whose fast response compromses stabty that must sow down. In ths secton we descrbe a protoco deveoped n [48] that can be mpemented n a decentrazed way by sources and nks and that satsfes some basc objectves: hgh network utzaton n equbrum and oca stabty for arbtrary deays capactes and routng. These requrements mpose certan constrants on the nearzed dynamcs: ntegraton at nks and condtons on the gan at sources and nks. We w present a goba mpementaton by nonnear agorthms at sources and nks that are consstent wth the nearzaton requrements. We w aso dscuss sgnang mpcatons of ths protoco and concude wth a packet-eve smuaton that vadates the theoretca resuts. Objectves and Lnear Desgn We now ay out a seres of objectves for the feedback contro aws n purey oca (nearzed terms. These w ead us to a oca contro aw whch s proved n [48] to acheve these objectves. A frst objectve s that the target capacty c s matched at equbrum; as n (13 ths can be acheved when prces ntegrate the excess capacty (varabes denote perturbatons n ths subsecton: ṗ =γ y. Is ths the ony choce? If exact trackng of capacty s desred there must be an ntegrator n the oop; and as expaned n [48] to have stabty we must perform the ntegraton at the ower-dmensona end of the probem.e. at the nks. So the above s the smpest aw consstent wth ths objectve; the constant γ w be chosen ater. The next man objectve s to have oca dynamc stabty for arbtrary network deays nk capactes and routng topooges. We now argue more carefuy for the desred scang as a functon of deay. As remarked earer deays are the ony dynamcs of the open oop; therefore they are the ony quantty that sets a tme scae to our cosed-oop behavor. Thus we am here for a system where a scang of a deays by a common factor woud resut n dentca tme responses except for tme scae. Consder frst a snge-nk snge-source probem. The network deay and the above nk ntegrator w yed a term τs e s n the oop-transfer functon. Thnkng for exampe n terms of ts Nyqust pot nstabty w aways occur at hgh vaues of τ uness the gan s made a functon of τ. Indeed n- Deay [ms] N = 5 N = 6 N = N = N = Capacty [pkts/ms] Fgure 3. Stabty regon: for each N the regon above the curve s unstabe and the one beow s stabe. February IEEE Contro Systems Magazne 39

13 troducng a gan K/τ n the oop (specfcay at the source eads to a oop gan τs K e τs whch s scae nvarant: namey ts frequency response s a functon of τω so Nyqust pots for a vaues of τ woud fa on a snge curve and by choosng K appropratey one can ensure stabty for a τ. Further the tme responses of such a oop w gve the desred nvarance up to scae. Inspred by ths for the mutnk mutsource case we are ed to ncude a factor1/τ n the contro gan at each source. Now we turn to nvarance to capacty and routng. It s cruca that the overa oop gan ntroduced by the routng matrces R f R b be kept under contro; ntutvey as more sources or nks partcpate n the feedback the gan must be appropratey scaed down. The dffcuty s mpementng ths n a decentrazed fashon wthout access to the goba routng nformaton. To scae down the gan due to R f at nks we expot the fact that at equbrum the aggregate source rates add up to capacty so Rf ( x = c. Snce sources know ther rates and nks ther capactes one s ed to ntroduce a gan 1/c at each nk and a gan x at each source. To scae down the gan due to R b at sources we ntroduce a gan1/m at each source M beng the number of botteneck nks n the source s path. Summarzng the above requrements n ther smpest possbe form we propose the foowng near contro aws: For the source a statc gan (from q to x of κ α x : = M τ (31 where α s a gan parameter to be chosen x s the equbrum rate τ s round-trp tme and M s a bound on the number of botteneck nks n source s path. The sgn s used to provde negatve feedback from prces to rates. For the nk an ntegrator wth gan normazed by (vrtua capacty p = 1 cs y. (3 Note that the normazaton gves ths prce unts of tme. In practce c s chosen to be strcty ess than the actua nk capacty n order to mantan zero buffer n equbrum. If c were the actua nk capacty p woud represent the queueng deay at the nk and s the prce sgna used n TCP Vegas (see (; so here we can thnk of p as a vrtua queueng deay (see [16] for reated deas. It s proved n [48] that provded the routng matrx Rhas fu row rank and the gans α < 1 the feedback system wth source agorthm (31 and nk agorthm (3 s neary stabe for arbtrary deays nk capactes and routng. Goba Nonnear Impementaton We now descrbe a goba mpementaton by nks and sources that have sutabe equbrum ponts around whch the nearzaton satsfes the requrements ad out above. The prce dynamcs can be mpemented by the nk agorthm p 1 c = 1 c ( y ( t c f p( t > ; + [ y ( t c ] f p( t =. That s prces ntegrate excess capacty n a normazed way and are saturated to be aways nonnegatve. At equbrum bottenecks wth nonzero prce w have y = c as requred. Nonbottenecks wth y < c w have zero prce. For the sources the nearzaton requrement (31 eads to a dfferenta equaton f α f( q = q M τ whch can be soved anaytcay and gves the contro aw x = f( q: = xmax e α q M τ (33 as the statc source aw. Note that the smaer the deay the more responsve a source s n varyng ts rate as a functon of prce. The arger the deay the more conservatve the source contro s n order to avod nstabty. Here x max s a maxmum rate parameter whch can vary for each source and n fact can aso be schedued to depend on M τ. A that s requred s that t does not depend onq. For nstance x max can be chosen to compensate for the effect of deay τ on equbrum rate. The utty functon correspondng to the source contro s M τ x U( x = x 1 og α x max for x x. max The above s the smpest nonnear source aw that gves the requred scang; as dscussed n [48] more degrees of freedom are avaabe by ettng α be a functon of q ; however t s mportant to emphasze that the stabty requrements do pose constrants on the famy of utty functons. 4 IEEE Contro Systems Magazne February