TCP-Illinois: A Loss and Delay-Based Congestion Control Algorithm for High-Speed Networks

Transcription

1 TCP-Illnos: A Loss and Delay-Based Congeston Control Algorthm for Hgh-Speed etworks Shao Lu, Tamer Başar and R. Srkant Abstract We ntroduce a new congeston control algorthm for hgh speed networks, called TCP-Illnos. TCP-Illnos uses packet loss nformaton to determne whether the wndow sze should be ncreased or decreased, and uses queueng delay nformaton to determne the amount of ncrement or decrement. TCP-Illnos acheves hgh throughput, allocates the network resource farly, and s ncentve compatble wth standard TCP. We also buld a new stochastc matrx model, capturng standard TCP and TCP-Illnos as specal cases, and use ths model to analyze ther farness propertes for both synchronzed and unsynchronzed backoff behavors. We fnally perform smulatons to demonstrate the performance of TCP- Illnos. Keywords: Congeston Control, TCP, farness, stablty, synchronzaton I. ITRODUCTIO TCP-Reno [2], TCP-ewReno [9], and SACK TCP [2] are the standard versons of TCP congeston control protocols currently deployed n the Internet, and they have acheved great success n performng congeston avodance and control. The key feature of standard TCP s ts congeston avodance phase, whch uses the addtve ncrement multplcatve decrement (AIMD) algorthm []. Beng a wndow-based algorthm, TCP controls the send rate by mantanng a wndow sze varable W, whch lmts the number of unacknowledged packets n the network from a sngle user. Ths wndow sze s adjusted by the AIMD algorthm n the followng manner: W s ncreased by α/w (α = for standard settng) for each ACK, and thus s ncreased by a constant α/b per round trp tme (RTT) f all the packets are acknowledged wthn an RTT, where b s the number of packets acknowledged by each ACK (b = for orgnal TCP, and b = 2 for delayed ACK [26]). On the other hand, W s decreased by a fxed proporton βw (β = /2 for standard settng) once some packets are detected to be lost n the last RTT. Under ths algorthm, senders gently probe the network for spare bandwdth by cautously ncreasng ther send rates, and sharply reduce ther send rates when congeston s detected. Along wth other features lke slow start, fast recovery, and fast retransmsson, TCP All the three authors are wth the Department of Electrcal and Computer Engneerng and Coordnated Scence Laboratory, Unversty of Illnos at Urbana-Champagn, 38 West Man Street, Urbana, IL , USA. Emals: (shaolu,basar,rsrkant)@uuc.edu Research supported by the SF ITR Grant CCR Ths s an extended verson of [9], a paper wth the same ttle, whch was presented at Frst Internatonal Conference on Performance Evaluaton Methodologes and Tools (VALUETOOLS), October, 26. Wthn one RTT, W may decrease multple tmes n Reno and can decrease only once n ewreno and SACK. acheves congeston control successfully n the current low speed networks. However, the current TCP can perform poorly n networks wth hgh bandwdth-delay product (BDP) paths, snce the AIMD algorthm, beng very conservatve, s not desgned for large wndow sze flows. Frst, t takes too long tme for a large wndow sze user to recover after a backoff and the bandwdth s not effectvely utlzed [8]. Second, TCP s tme average wndow sze W s related wth the loss event probablty 2 p n the followng manner [23] W 3/2bp or p 3 2b( W) 2. () Snce TCP nterprets all packet losses as congeston sgnals, W s upper bounded by 3/2bp t, where p t s the transmsson error rate [8]. p t s around 7 n optcal fber networks, and much hgher n other lossy networks, lke wreless networks. So TCP, and ts AIMD algorthm n partcular, should be modfed n hgh bandwdth delay product networks. Several alternatves to current versons of TCP have been proposed for mplementaton n hgh-speed networks. Some requre the modfcaton to router algorthms also, lke XCP [4], and some modfy the sender sde only, lke HS- TCP [8], Scalable TCP [5], TCP-Westwood [3], H-TCP [7], BIC-TCP [3], TCP Vegas [7], FAST TCP [3] and Compound-TCP [27]. Although each of these has shown advantages over standard TCP n some aspects, none of them have yet provded convncng evdence that they are overwhelmngly better than standard TCP and are sutable for general deployment. In ths paper, we frst lst some desrable desgn specfcatons that a hgh speed TCP varant should meet, and then ntroduce the TCP-Illnos algorthm, whch uses packet loss nformaton as the prmary congeston sgnal to determne the drecton of wndow sze change (whether wndow sze should be ncreased or decreased), and uses queueng delay nformaton as the secondary congeston sgnal to adjust the pace of wndow sze change (the amount of wndow sze ncrement or decrement). We then show that TCP- Illnos satsfes all the desgn requrements we lsted, and outperforms standard TCP and many other TCP varants. To study the farness, stablty, and responsveness propertes of TCP-Illnos, we extend the stochastc matrx model [2] [4], [6], [25], [29] by allowng wndow sze backoff 2 All the packet losses wthn one RTT are regarded as one loss event. Loss event probablty s the number of loss events dvded by the number of packets sent.

2 probabltes to be functons of flow arrval rates at congeston events. Our contrbuton to ths modelng technque ncludes the followng: () we show that a large class of general AIMD algorthms, ncludng standard TCP and TCP- Illnos, have smlar farness propertes, and the farness propertes only depend on the backoff behavors for these algorthms; () the backoff behavor can be characterzed by a functon f( ), where f( ) s a user s backoff probablty as a functon of ts send rate: f f( ), the backoffs of all the users are completely synchronzed; f f( ) s lnear, the backoffs are completely unsynchronzed; and n general, the partally-unsynchronzed backoff les n the mddle; () f( ) s determned by the number of packets dropped n each congeston event: heavy congeston causes synchronzaton and lght congeston leads to unsynchronzed backoff; (v) the heavness of congeston (the number of packets dropped n one congeston event as compared to the number of flows) depends on the wndow sze ncrement of these flows just before the congeston event, and thus a smaller (respectvely, larger) ncrement before a congeston event causes the backoff to be more unsynchronzed (respectvely, synchronzed). The paper s structured as follows. In secton II, we lst the requrements for a new verson of TCP, compare the exstng protocols and pont out ther shortcomngs, and descrbe our desgn objectve. We next ntroduce the TCP-Illnos protocol n secton III, and study ts farness and stablty propertes usng a new stochastc matrx model for a class of general AIMD algorthms n secton IV. We further explore some other propertes of TCP-Illnos n secton V and provde ns- 2 smulaton results n secton VI for a comparatve study of TCP, HS-TCP and TCP-Illnos. II. BACKGROUD AD MOTIVATIO As we have mentoned above, several new protocols have been ntroduced to replace standard TCP n hgh speed networks. To compare these protocols and to provde nsght nto the development of an deal protocol, we lst below some requrements that a new protocol should satsfy. Ths lst broadens the lst of requrements n [7]. Intra-protocol requrements: The requrements that the protocol should satsfy n a network consstng of a sngle protocol are the followng: Effcency. The average throughput for the new protocol should be larger than that of standard TCP n hgh speed networks. Intra-Protocol Farness. etwork resources should be farly allocated to all flows. Farness here does not necessarly mean that all flows sharng the same lnk acheve the same throughput. Instead, ths means that the new protocol should not be sgnfcantly more unfar than the current TCP. For example, under the current TCP, flows wth dfferent RTTs acheve smlar average wndow szes, and ther average throughput are nversely proportonal to ther RTTs. The new protocol should not be sgnfcantly more based aganst long RTT flows. Responsveness. The congeston control algorthm should reach the far operatng pont quckly, startng from any ntal condton. Heavy Congeston 3 Avodance. A smple dea to acheve a larger average wndow sze for a gven loss event probablty s to choose a large value for α (wndow ncrement parameter) and a small value for β (wndow reducton parameter). However, rapd ncrease and small decrease n wndow sze may cause large number of packets to be dropped durng a congeston event, whch we call heavy congeston, and thus may lead to some undesrable consequences. Frst, heavy congeston causes more tmeouts and makes TCP enter the slow start phase more often, and causes underutlzaton. For example, HS-TCP faces tmeouts regularly f SACK s not used. Second, heavy congeston causes synchronzaton more often, whch makes the resource allocaton very unfar for large RTT users, as wll be dscussed later. Router Independence. The new protocol should work well regardless of router characterstcs, lke the buffer sze at the router, and the queue management algorthm of the router (Droptal or some Actve Queue Management (AQM) schemes). Wth a more advanced router, lke wth a larger buffer or an AQM support, the new protocol mght acheve better performance, but the performance wth Droptal and small buffer should also be good. Robustness. The new protocol should be robust aganst the nose n congeston sgnal measurements, especally f ths new protocol uses queueng delay as the congeston sgnal, snce queueng delay measurements are typcally nosy. Inter-protocol requrements: The requrements on the protocol when t co-exsts n a network wth standard TCP are the followng: Compatblty. In low speed networks, the new protocol should acheve a smlar rate as that of standard TCP; and n hgh-speed networks, standard TCP should not suffer sgnfcant throughput loss when t coexsts wth the new protocol. Incentve to swtch. By swtchng to the new protocol from standard TCP, the users should acheve a hgher average throughput n a network that accommodates both protocols. We now brefly dscuss exstng TCP varants to see whether they satsfy all these requrements. Frst, t s mpractcal to modfy routers f the beneft s margnal or can be acheved by sender sde modfcatons, and thus algorthms whch need router sde modfcatons, lke XCP, are not deal. Wthout modfyng the router, a sender has only two congeston sgnals: packet loss and queueng delay. We can 3 In our context, heavy congeston means that many packets are dropped when congeston happens. It only concerns the tme when congeston happens and t does not necessarly mean that the packet loss probablty or the loss event probablty s hgh. In some other papers, t s called heavy synchronzaton.

3 thus classfy the pror sender-sde protocols nto one of two classes. Loss-based congeston control algorthms, lke HS- TCP and Scalable TCP, use packet loss as prmary congeston sgnal, ncrease wndow sze for each ACK and decrease wndow sze for packet loss. Loss-based algorthms can be regarded as generalzatons of TCP s AIMD algorthm, and we call them general AIMD algorthms, snce the only dfference from AIMD s that they set dfferent α and β values and allow them to be varables. On the other hand, delay-based congeston control algorthms, lke TCP-Vegas and FAST TCP, are fundamentally dfferent from AIMD, as they use queueng delay as the prmary congeston sgnal, ncrease wndow sze f delay s small and decrease wndow sze f delay s large. The advantage of delay-based algorthms s that they acheve better average throughput, snce they can keep the system around full utlzaton. As a comparson, the lossbased algorthms purposely generate packet losses and oscllate between full utlzaton and under utlzaton. However, exstng delay-based algorthms suffer from some nherent weaknesses. Frst, they are not compatble wth standard TCP. TCP-Vegas gets a very small share of the lnk capacty f competng wth TCP-Reno [22], []; and FAST TCP yelds non-unque equlbrum pont f competng wth TCP-Reno: the allocaton of the bandwdth between FAST and Reno users depend on whch users enter the network frst [28]. Second, they requre the buffer sze at the router to be larger than a specfed value and ths value ncreases wth the number of users. Both Vegas and FAST control the number of packets queued n the router for each flow, and ths number cannot be too small. The requrement for the router buffer s thus tmes ths number. For a fxed buffer sze, there s an upper bound on for Vegas or FAST to work effcently. Fnally, the performance of these delaybased algorthms deterorates f the delay measurements are nosy [6], [2], [24]. On the other hand, none of the exstng loss-based algorthms satsfy all the requrements ether. Scalable TCP sets α proportonal to W, but t has been demonstrated to be unfar (see [7], Fg. 2). HS-TCP sets α to be a stepwse ncreasng functon of W, and β a step-wse decreasng functon of W, but ts convergence speed s very slow (see [7], Fg. ). H-TCP ams at a faster convergence and better utlzaton by settng α to be an ncreasng functon of the tme elapsed snce last backoff and settng β to be such that the lnk s always around full utlzaton, even after the backoff. For all the above algorthms, the ncrease s ntally slow, when the wndow sze s small and the network s far from congeston, but becomes fast later, when the wndow sze s large and the network s close to congeston. As a result, the wndow sze curve between two consecutve loss events s convex. Ths convex nature s not desrable. Frst, the slow ncrement n wndow sze when the network s far from congeston s neffcent. For a gven β, the convex wndow curve gets an even smaller average throughput than tradtonal lnear ncrease, and thus these algorthms have to choose a smaller β < /2, whch s not frendly to standard TCP. Second, the fast ncrement n wndow sze when the network s close to congeston causes heavy congeston more easly. As we have mentoned before and wll further dscuss later, heavy congeston causes more frequent tmeouts, more synchronzed wndow backoffs, and s more unfar to large RTT users. In summary, the man problem wth exstng general AIMD algorthms s the convexty of the W curve. An deal wndow curve should be concave, whch s more effcent and avods heavy congeston. An objectve of our work s to desgn a general AIMD algorthm whch results n a concave wndow curve. III. THE TCP-ILLIOIS PROTOCOL To acheve the concave wndow curve, we should set α large when far from congeston and set t small when close to congeston. To acheve a better throughput n networks wth packet losses not due to congeston and to be far wth standard TCP, we should also set β small when far from congeston and set t large when close to congeston. The dffculty s n judgng whether the congeston s mmnent or not, snce t requres an estmaton of the current congeston level. Before congeston (packet loss) really happens, the only congeston ndcatng nformaton s queueng delay. So our key dea s the followng: when the average queueng delay d a s small, the sender assumes that the congeston s not mmnent and sets a large α and small β ; when d a s large, the sender assumes that the congeston s mmnent and sets a small α and large β. As a result, α = f (d a ) and β = f 2 (d a ), where f ( ) s decreasng and f 2 ( ) s ncreasng. Any combnaton of ncreasng f ( ) and decreasng f 2 ( ) functons results n a concave wndow curve and therefore, we call such algorthms Concave-AIMD or C-AIMD algorthms. ote that C-AIMD algorthms use loss to determne the drecton and use delay to adjust the pace of wndow sze change. So loss s the prmary congeston sgnal and delay s the secondary congeston sgnal. Ths makes C-AIMD fundamentally dfferent from another recently proposed algorthm, called Compound-TCP [27], whch uses both loss and delay nformaton as prmary sgnals (determnng drecton of wndow sze change). As we have mentoned, one problem n usng delay to control congeston s that delay cannot be measured accurately snce usually the RTT measurements are nosy. If delay determnes the drecton of wndow sze change, nosy RTT measurements could degrade the performance sgnfcantly. Our C-AIMD algorthms whch use delay only as a secondary sgnal, are much more robust to nose n RTT measurements, as dscussed n subsecton VI-D. There are numerous choces for f ( ) and f 2 ( ). TCP- Illnos s a specal case of C-AIMD algorthms whch uses the followng choces for f ( ) and f 2 ( ): { αmax f d a d α = f (d a ) = otherwse. κ κ 2 +d a (2)

4 alpha_max alpha_mn alpha d dm da beta_max beta_mn beta Fg.. α and β curves Vs d a. d2 d3 dm β mn f d a d 2 β = f 2 (d a ) = κ 3 + κ 4 d a f d 2 < d a < d 3 β max otherwse. We let f ( ) and f 2 ( ) be contnuous functons and thus κ κ 2 +d = α max, β mn = κ 3 + κ 4 d 2 and β max = κ 3 + κ 4 d 3. Suppose d m s the maxmum average queueng delay and let κ α mn = f (d m ); then we also have κ 2 +d m = α mn. From these condtons, we have κ = (d m d )α mn α max α max α mn and κ 2 = (d m d )α mn α max α mn d, κ 3 = β mnd 3 β max d 2 d 3 d 2 and κ 4 = β max β mn d 3 d 2. Ths specfc choce s shown n Fg.. We now descrbe the TCP-Illnos protocol n more detals: All the features of TCP-ewReno except the AIMD algorthm are retaned. In the congeston avodance phase, the sender measures RTT T for each acknowledgement, and average the RTT measurements over the last W acknowledgements (one RTT nterval) to derve the average RTT T a. The sender records the maxmum and mnmum (average) RTT 4 ever seen as T max and T mn, respectvely, and computes the maxmum (average) queueng delay d m = T max T mn and the current average queueng delay d a = T a T mn. The sender pcks the followng parameters: < α mn α max, < β mn β max /2, W thresh >, η <, η 2 η 3. The sender sets d = η d m ( =,2,3), computes κ ( =,2,3,4) from (4), and computes α and β values from (2) and (3), respectvely. The standard settngs of these parameters are gven n Secton VI. α and β /2 f W < W thresh. The κ ( =,2,3,4) values are updated f T max or T mn s updated. The α and β values are updated once per RTT. W W + α/w for each ACK. W W βw, f n the last RTT there s packet loss detected through trple duplcate ACK. Once there s a tmeout, the sender sets the slow start threshold to be W/2, enters slow start phase, and resets α = and β = /2, and α and β values are unchanged untl one RTT after the slow start phase ends. 4 The are two optons here. In the default opton, the maxmum and mnmum average RTTs are recorded. In an alternatve opton, the maxmum and mnmum nstantaneous RTTs are recorded. These two optons yeld almost dentcal results, unless the delay sgnal s bured wth nose and the nose s n a hgh level. Under ths case, the default opton s a better choce. da (3) (4) TCP-Illnos retans the fast recovery and fast retransmsson features of ewreno n standard opton. If the recevers support selectve acknowledgement, TCP-Illnos can also back off ts wndow sze when packet loss s detected through selectve ACK and adopt features from SACK TCP. However, the SACK support s not needed, snce TCP-Illnos avods heavy congeston effectvely. In addton to the above major features of the protocol, TCP-Illnos also contans another feature to mprove the robustness aganst sudden fluctuatons n delay measurements that can result from measurement nose, bursty packet arrval process, etc. To understand ths feature, note that deally once d a becomes greater than d, t should stay above d untl some users reduce ther wndow szes. However, due to bursty packet arrval process or measurement nose, t s possble for d a to drop rather suddenly below d before some users reduce ther wndow szes. In ths case, we should not set α = α max unless we are really sure that the network s not n a congested state. Therefore, once d a > d, we do not allow α to ncrease to α max unless d a stays below d for a certan amount of tme. TCP-Illnos chooses another parameter θ, and lets θ tmes RTT be ths amount of tme. The standard settng for θ s agan gven n Secton VI. We note that the adaptaton of α s the key feature of TCP- Illnos, whereas the adaptaton of β as a functon of average queueng delay s only relevant n networks where there are non-congeston-related losses, such as wreless networks or extremely hgh speed networks. In wreless networks, some packet losses arse from channel fluctuatons. In extremely hgh speed networks, congeston loss probablty s so small that t s at the same level as or even smaller than the probablty of packet transmsson error at the lnk, and as a consequence, a non-trval proporton of packet losses are from transmsson error. For these non-congestonrelated packet losses, we wsh to avod a sharp wndow sze reducton. Then, the β adaptaton of TCP-Illnos shows ts advantage: although t stll reduces wndow sze, the reducton percentage s very small, snce the queueng delay s very small. IV. FAIRESS AD STABILITY In ths secton, we study the farness and stablty of TCP-Illnos. Ths nvolves both the ntra-protocol farness between dfferent TCP-Illnos users and also nter-protocol farness wth standard TCP,.e., the resource allocaton between TCP-Illnos users and standard TCP users. We frst develop a new stochastc matrx model for a class of general AIMD algorthms, whch nclude standard TCP and TCP- Illnos as specal cases, and then study the farness and stablty propertes of these algorthms usng ths new model. A. Stochastc Matrx Model for general AIMD Algorthms There have been several recent papers on the stochastc matrx model of AIMD algorthms; see [2] [4], [6], [25], [29]. We frst provde an overvew of ths model, and then extend ths model by modfyng one of the assumptons n the earler work. Throughout, we consder networks wth a sngle

5 bottleneck lnk whch uses Droptal, analyze the congeston avodance phase only, and assume that all packet losses are caused by congeston. Suppose a lnk wth capacty C and queue lmt B s shared by users, ndexed by ( =,2,,). User has a transmsson rate (or throughput) x, a wndow sze W, a wndow ncrement parameter α, a wndow backoff factor β, and RTT T. We defne W := [W,...,W ] T, and x := [x,...,x ] T. When the lnk s congested and one or more packets are dropped, we call ths a congeston event, and denote by the tme at the k-th congeston event. At a congeston event, one or more flows see packet losses and backoff ther wndow szes, and we say that a loss event 5 happens for these flows. For any varable v, we use v(t) to denote ts value at tme t, use v[k] (respectvely, v[k + ]) to denote ts value just before (respectvely, after) the k-th congeston event, use E[v] to denote the expected value of v[k], and use v to denote the average of all v[k] s. Here, v could stand for W, x, α, β, T, W, x, as well as some other varables to be ntroduced later. We now consder the congeston even for Droptal queue. When congeston happens, the buffer s full, so every user experences a maxmum queueng delay d m = B/C, and thus T [k] ˆT := T p + d m, k, where T p s the propagaton delay of user. At the congeston event, the nstantaneous throughput for user s W [k]/ ˆT. Ignorng the burstness of packet arrval process, we can assume that the outgong packets from one partcular user are evenly dstrbuted along the path. For user, altogether there are W [k] packets, and thus the number of packets from user queued n the lnk buffer should be W [k]d m / ˆT = x [k]d m. The sum of the queued packets from all users should be the lnk buffer lmt B, and thus we have B = = x [k]d m = whch leads to followng equaton: = = x [k] B C, (5) x [k] = C, k {,,2, }. (6) As mentoned earler, n ths analyss we have gnored the burstness of packet arrval process; f we had consdered ths burstness, then = x [k] would not be a constant, and would be ether greater than or less than C. We now defne Σ = {z = [z,,z ] T R : z, = z = C}; then Σ s the set of all possble x[k] s, and we call Σ the feasble set of x[k]. Between two consecutve congeston events, W (t) s ncreased at rate α (t)/t (t), and thus If we defne W [k+] = W [k + ]+ T [k] := tk+ tk+ tk+ α (t) T (t) d t. (7) α(t)d t α(t) T (t) d t, (8) 5 In our termnology, a congeston event s for a lnk, whle a loss event s for an ndvdual user. then, we have W [k+] = W [k + ]+ T [k] tk+ α (t)d t. (9) For any user and congeston even, T [k] [T p, ˆT ]. In general, queueng delay s much smaller than propagaton delay, and T [k] vares n a very small range, so we can assume that T [k] T, k. At each congeston even and for each flow, we defne the loss event random varable D [k]: D [k] := { f flow sees at least one packet loss, otherwse, () and defne D[k] := [D [k],,d [k]] T. ote that D [k] and D j [k] are correlated, snce = D [k]. Wth the loss event random varables defned, we have W [k + ] = W [k]( β [k]d [k]). () Combnng (9) and (), and usng the fact that x [k] = W [k]/ ˆT, we have x [k+] = x [k]( β [k]d [k])+ ˆT T tk+ α (t)d t. (2) Equatons (6) and (2) descrbe the dscrete-tme stochastc model of all general AIMD algorthms. B. Markov Chan for Identcal α (t) and constant β [k] We consder the class of AIMD algorthms whch have the followng propertes: () α (t) = α(t),, where α(t) s the common wndow ncrement for all users at tme t; () β [k] ˆβ,,k, where ˆβ s a constant ndependent of and k. Ths class ncludes standard TCP obvously, snce t satsfes α (t),β [k] ˆβ = /2,,k. Ths class also ncludes TCP- Illnos. Frst, α (t) = α(t), snce the queueng delay s the same for all users. Here, we gnore the dfferences n α among dfferent flows due to feedback delays, snce the queueng delays are averaged to compute α. Then, β [k] ˆβ = β max,,k, snce the average queueng delay d a s larger than the threshold parameter d 3 when congeston happens, f the parameters are carefully chosen. Recall our modelng of congeston events n the prevous subsecton: we know that the maxmum queueng delay d m s reached at each congeston event. Even consderng the averagng process, d a [k] s close to d m and stll larger than d 3. For ths class of AIMD algorthms, from (6), we have tk+ and thus tk+ Defne α(t)d t = ( T ˆT ) = α(t)d t = = ( ˆT T ) γ := ( T ˆT ) / j= = ˆβ D [k]x [k], (3) ˆβ D [k]x [k]. (4) = ( T ˆT ), (5)

6 and γ = [γ,,γ ] T. Then = γ =. We now have x [k+] = x [k]( ˆβD [k])+γ j=x [k] ˆβ D [k]. (6) In vector form, we have where x[k+] = A[k]x[k], (7) A[k] = A(D[k]) = dag( ˆβD [k],, ˆβD [k]) +γ ˆβ(D [k],,d [k]). (8) We see that x[k] forms a dscrete-tme Markov Chan on the contnuous state space Σ. For any k, A[k] s a non-negatve, random, column stochastc matrx [5], []. The property of ths Markov Chan s determned by the A matrx, and thus determned by D[k]. ote that, although α (t) determnes the wndow curve, the recovery tme after a congeston event, and the utlzaton of the bandwdth, once all users see the same α(t) at any tme, α(t) does not nfluence the dscrete-tme Markov Chan at congeston events, and thus the exact form of α(t) s not mportant to understand the macroscopc farness propertes of ths class of algorthms. So ths stochastc matrx model apples to the entre class of such algorthms, and the specal choce of α(t) n TCP-Illnos s not mportant when analyzng the farness of TCP-Illnos. Ths specal choce of α(t) ndeed nfluences many other propertes, such as effcency and synchronzaton, as we wll dscuss later. C. Stablty and Farness: General Case In ths subsecton, we study the stablty and farness propertes of the Markov Chan defned n (7) and (8). Let S be the set of all non-empty subsets of {,2,,}, and suppose s[k] S s the set of users that experence a loss event at congeston even. Defne ρ s [k] := Prob(s[k] = s), where s S. Then, the dstrbuton of D[k] s determned by the values of ρ s [k], s S. Let q [k] := Prob(D [k] = ), then q [k] = s: s ρ s [k], and q [k] denotes the wndow backoff probablty of user at congeston even. Most pror work assumed that D [k] s ndependent of x[k],.e., ρ s [k] s a constant ndependent of k, for each s S, and s[k] s dentcally ndependently dstrbuted (..d.). From ths assumpton, q [k] s also a constant ndependent of k and x[k] for all user. However, n realty, at dfferent congeston events, a flow s more lkely to see a loss event when t has a larger throughput than when t has a smaller throughput. Therefore, we modfy the stochastc matrx model by allowng D[k] to be dependent on x[k], and allowng q [k] and ρ s [k] to be functons of x[k] as well: ρ s [k] = ρ s (x[k]), l,k, and q [k] = q (x[k]),,k. (9) We make the followng assumpton on ρ s,q, and D: Assumpton. () ρ s ( ) and q ( ) are contnuous functons n x[k]. () For any realzaton of the nfnte length Markov Chan defned n (7) and (8) and for any user, D [k] = for nfntely many k s almost surely,.e., for any J >, Prob(D [k] =, k J) =, {,2,,}. We now state the followng theorem. Theorem IV.. Under Assumpton, the Markov Chan defned n (7) and (8) has a unque nvarant dstrbuton, and startng from any ntal state, the dstrbuton of x[k] converges to ths nvarant dstrbuton. Moreover, the Markov Chan s ergodc,.e., for any contnuous functon h( ) : Σ R, h(x[k]), the tme average of h(x[k]), equals E[h(x[k])], the expected value of h(x[k]) under the nvarant dstrbuton. Proof. See [8], a longer verson of ths paper. Wth the exstence and unqueness of the nvarance dstrbuton establshed, we now study the farness among dfferent users,.e., the resource allocaton under the nvarant dstrbuton. We have the followng theorems on farness. Theorem IV.2. If users sharng one lnk have homogeneous RTTs, under the unque nvarant dstrbuton of the Markov Chan defned n (7) and (8), all flows share the same expected throughput E[x [k]]. Proof. When all users have the same RTT, ˆT, T and γ are the same for all users. Then, from (8), the A matrx does not depend on or T. If we swap user and user j, the Markov Chan s the same as f we do not swap user and user j, but swap x [] and x j []. From Theorem IV., we know that the nvarant dstrbuton s unque, ndependent of the ntal condton. Therefore, the nvarant dstrbuton s unchanged f we swap user and user j, and thus E[x [k]] = E[x j [k]],, j {,2,,}. Theorem IV.3. If users sharng one lnk have heterogeneous RTTs, under the unque nvarant dstrbuton of the Markov Chan defned n (7) and (8), the followng equaton holds: ˆT T E[x [k]q (x[k])] = E[W [k] T q (x[k])] = C,, (2) where C s a constant ndependent of. Proof. Takng expectaton of x [k+] gven x[k] n (6), we have E[x [k+] x[k]]= x [k] ˆβ q (x[k])x [k]+γ j= Under the nvarant dstrbuton, E[x [k]] = E[x [k+]] = E[E[x [k+] x[k]]] = E[x [k]] ˆβE[q (x[k])x [k]] +γ ˆβ j= E[q j (x[k])x j [k]]. ˆβ q (x[k])x j [k]. (2) (22) So we have E[x [k]q (x[k])]/γ = ˆT T E[x [k]q (x[k])] = E[W [k] T q (x[k])] s ndependent of and we have proved (2).

7 D. Models for ρ s ( ) and q ( ) From Theorem IV.3, we see that the resource allocaton depends on the form of q ( ). If q ( ) s constant, t s exactly the same as the pror results n [29]. In our model, q [k] s allowed to be a functon of x[k]. We need to specfy the q ( ) functon to further analyze the farness property. Recall that the dependence of q [k] on x[k] arses from the fact that a flow wth a larger throughput s more lkely to see a loss event than a flow wth a smaller throughput. Accordngly, we make the followng assumpton: Assumpton 2. At each congeston event, the total number of packets dropped s a random varable that takes values n {,2,,M max }, and ts dstrbuton s ndependent of k. Furthermore, for any packet dropped at congeston even, the probablty that t belongs to flow s x [k]/c. Ths assumpton s justfed by the followng reasonng: snce the total arrval rate s ndependent of k, so s the dstrbuton for the total number of packets dropped; snce at least one packet s dropped and only a fnte number of packets are dropped, there are lower and upper bounds for the total number of packets dropped; snce the probablty of an arbtrary packet belongng to flow s x [k]/c, so s the probablty of a dropped packet belongng to flow. Lemma IV.. Assumpton holds gven Assumpton 2 Proof. We frst prove that Assumpton () holds. Let M be the random varable ndcatng the total number of packets dropped n one congeston event, let P M (m) =Prob(M = m) for all m {,2,, M}, and let ˆM = E[M]. Then, we have q = Prob(no dropped packets from flow ) = M max m= P M(m)[ ( x [k] C )m ] = f(x [k]), (23) where f(x) := M max m= P M(m) f m (x), and f m (x) := ( x C )m. Both f m (x), m and f(x) are strctly ncreasng contnuous functons n x [,C]. ote that Assumpton allows q [k] to be functons of all users rates, whle Assumpton 2 further tells us that q [k] s only a functon of ts own rate x [k], and ths relatonshp f( ) s common for all users. We then study ρ s ( ). For a specfc s S, suppose s = {, 2,, H }, where < 2 < < H. Then, Prob(s[k] = s M = m) = f m < H. If m H, we have ρ s,m (x[k]) := Prob(s[k] = s M = m) = m,,m H ( x [k] C )m ( x H [k] ( m ) C )m H, m,m 2,,m H where the summaton s over all m h, h {,2,,H}, and H h= m h = m. And we have ρ s (x[k]) = Prob(s[k] = s) = m=h P M (m)ρ s,m (x[k]). (24) So both q (x[k]) and ρ s (x[k]) are contnuous functons of x[k]. ext we prove Assumpton (). At each congeston event, at least one user wll decrease ts wndow sze by at least, and thus = x [k] ˆβ D [k] /T m, where T m = max ˆT. Hence, f user does not back off at congeston even, x [k + ] γ /T m. Snce at least one packet s lost at congeston even + and the probablty of a lost packet belongng to flow s x [k + ]/C, we know that q [k+] x [k+]/c. Thus q [k+] ε := γ /(CT m ) >. So the probablty that user backs off at least once n any two consecutve congeston events s lower bounded by ε >. As a consequence, for any user {,2,,}, D [k] = for nfntely many k s almost surely. Snce Assumpton 2 mples Assumpton, we know that Theorem IV., Theorem IV.2 and Theorem IV.3 hold under Assumpton 2 also. In the next subsecton, we analyze the farness property for the specfc f(x ) functon gven by (23). E. Synchronzaton and Farness From Theorem IV.3 and equaton (23), the f( ) functon unquely determnes the backoff behavor and the farness property. Dfferent f( ) functons lead to dfferent backoff behavors: for example, f f( ), the backoffs are completely synchronzed; otherwse, they are not. The exact form of f( ) depends on the dstrbuton of M, and s thus unknown f P M ( ) s unknown. However, we can bound f(x) n general and approxmate f(x) for some specal cases. Snce x C ( x C )m mx C, x C, we have Mmax x C f(x) P M (m)[ ( mx [k] ˆMx )] = C C. (25) m= ote that the loss event for a flow s the unon of the events that each dropped packet belongs to ths flow, therefore the bound on f(x) n (25) s just the unon bound. Snce x and f(x) always, we have x 2 ˆMx 2 E[x f(x)] C C. From Theorem IV.3, we have T j W 2 ˆM ˆT j ˆM ˆT, j. (26) T W 2 j The bounds are tght and W 2 W j 2 f ˆM s close to (very lght congeston). If the varance of W [k] s much smaller than (E[W [k]]) 2, then W 2 ( W ) 2, and thus the average wndow szes of all flows are almost the same under very lght congeston case. From smulatons whch wll be presented later, we observe that Var(W [k]) << (E[W [k]]) 2 ndeed. If ˆM s very large (heavy congeston), the bounds are meanngless and W 2 and W j 2 can be sgnfcantly dfferent. We then consder the approxmaton of f(x) under some specal cases. When x s small, such that mx/c <<, the probablty that more than one dropped packets belong to one flow (wth rate x) s very small, and thus the unon upper bound s nearly reached: f m (x) mx/c. If M max x/c <<, then f(x) ˆMx/C, and f(x) x. When x s large, such that mx/c >>, then f m (x). If M >> c/x wth very hgh probablty, then f(x). In general, f(x) s a concave

8 curve, and f(x) x λ, where λ, and λ as M max, and λ as ˆM. As we have mentoned, smulatons show that the standard devaton of x [k] s very small compared wth E[x [k]], so wth hgh probablty, x [k] les not far away from E[x [k]]. If the RTTs of dfferent users do not dffer sgnfcantly, E[x [k]] s not sgnfcantly dfferent from C/, and the probablty of x [k] << C/ or x [k] >> C/ s very small. So f M max <<, whch we call lght congeston, almost always Mx [k]/c <<, and thus f(x [k]) x [k] (λ ), and the wndow backoffs of dfferent users are completely unsynchronzed. If ˆM >>, whch we call heavy congeston, almost always Mx/c >>, and thus f(x) (λ ), and the wndow backoffs are completely synchronzed. In the mddle of these two extreme cases, < λ <, and the wndow backoffs are partally-unsynchronzed. Pluggng f(x) x λ nto (2), we get ˆT T E[(x [k]) λ ] s the same for all users. If the varance of x [k] s small, and f the dfference between T and ˆT s small also, we get the followng farness property: x ˆT +µ, and W ˆT µ, (27) where µ = ( λ)/( + λ). We know that µ n general; µ for lght congeston and completely unsynchronzed wndow backoff; and µ for heavy congeston and completely synchronzed wndow backoff. So when the varance of x [k] s small and when the RTTs are not sgnfcantly dfferent, the farness depends on the synchronzaton, whch further depends on the heavness of congeston. Lght congeston leads to unsynchronzed wndow backoff and equalty of wndow szes (µ = ); heavy congeston leads to synchronzed wndow backoff and an nverse proportonal relatonshp between wndow sze and RTT (µ = ); and the general case ( < µ < ) les n the mddle of these two extreme cases. We then explore the factors that nfluence the dstrbuton of M. Consder the homogeneous RTT case and suppose the system s slotted wth each slot beng one RTT. Snce the ppe can hold at most CT + B packets and W ncreases by α wthn each slot, = W [CT + B+ = α,ct + B] n the slot just before congeston, and = W [CT + B+,CT + B + = α ] n the slot of congeston. As a result, anywhere from to = α packets could be dropped at one congeston event, and we know that the congeston s heaver f the ncrement before congeston s larger. Approxmately, we can assume that M takes values from to = α wth equal probablty, and thus M max = max(, = α ), and ˆM = max(,(+ = α )/2). Snce TCP-Illnos chooses very small α << just before congeston, M max << and lght congeston condton s satsfed. From ths analyss, one advantage of TCP-Illnos s that t avods heavy congeston and synchronzed backoff, and t reaches a far resource allocaton between dfferent users. On the contrary, convex curve algorthms, lke HS-TCP, yeld heavy congeston regularly, and ths further causes synchronzaton and unfarness, as shown n Secton VI. The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 2. = 4, lght congeston, M [,/2], unformly dstrbuted. We fnally perform Matlab smulatons to support our assumpton of small x [k] varance and valdate our analyss on the relatonshp between heavness of congeston and farness. We have performed a large number of smulatons on the evoluton of the Markov Chan defned n (7) and (8). We vary (the number of users) from 4 to. For each, we select three probablty dstrbutons of M: () lght congeston, M s unformly dstrbuted n [, /2]; () medum level congeston, M s unformly dstrbuted n [, ]; () heavy congeston, M s unformly dstrbuted n [, 2]. For each scenaro, we perform 5 smulatons. For each smulaton, x[] and T, are randomly generated ntally, γ, are computed by (5), and M[k] and s[k] are randomly generated accordng to Assumpton 2 at each congeston even, and thus each A[k] and the sample path of the Markov Chan are derved. For each sample path, we average congeston events after the dstrbuton converges, to compute average throughput x, standard devaton of throughput Std(x ) := Var(x ), for each user. We also compute x for all by assumng that (27) holds for µ =,.e., all users share the same wndow sze. We plot x /x and Std(x )/ x for all users and all smulatons performed, and plot log(x ) Vs log(t ) for all users n each smulaton. The results are shown from Fg From the fgures, we have the followng observatons: () the x /x rato s very close to for lght congeston, whch ndcates that all users share almost the same wndow sze under lght congeston, and as the congeston becomes heavy, the range of ths rato becomes wder and thus the dfference between W s becomes larger; () the Std(x )/ x rato s always much smaller than for any and any heavness of congeston, whch supports our assumpton of small varance; () log(x ) s lnear wth log(t ), whch valdates the farness property n (27): x /T +µ, where µ [,], and µ ncreases as the congeston becomes heavy. Remark. In equaton (27), the average of W and x s over ther values at the congeston events, and not over all tme. Snce a general AIMD algorthm can yeld any wndow sze curve, t s a challengng problem to compute 6 Due to space lmtaton, we only provde the case for = 4 and. For other values of, the results turn out to be smlar.

9 The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 3. = 4, medum level congeston, M [,], unformly dstrbuted. The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 5. =, lght congeston, M [,/2], unformly dstrbuted. The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 4. = 4, heavy congeston, M [,2], unformly dstrbuted. The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 6. =, medum level congeston, M [,], unformly dstrbuted. the average W and x over all tme, and t s an open problem whether the above concluson on farness holds for tme averages of W and x. However, for general AIMD algorthms, snce the all tme average of W les between E[W [k]] and E[W [k + ]] = E[( ˆβ q [k])w [k]] ( β max )E[W [k]], we know that tme average of W for one user s smlar to the average at congeston events. For TCP-Illnos, n partcular, snce x ncreases to near full utlzaton very quckly and stays around full utlzaton for a long tme, the tme average of x s very close to E[x[k]], and thus the farness property should hold approxmately for tme average also. From our ns-2 smulatons n Secton VI, we observe that the tme averages of W for dfferent users are also approxmately the same. F. Compatblty wth the Standard TCP We now consder the equlbrum allocaton when TCP- Illnos coexsts wth TCP-Reno (ewreno and SACK are the same) and all flows share the same RTT 7. If we use ˆT to denote the common maxmum RTT for all flows, then W [k] = x [k] ˆT. The users are dvded nto two classes, Illnos user set I and Reno user set R, and α (t) = α IL (t),β [k] = β IL [k], t, k, I, α j (t),β j [k] /2, t, k, j R. 7 Due to lack of space, we consdered here only the homogeneous RTT case. For the heterogenous RTT case, smlar method can be used to derve the equlbrum allocaton. Defne ᾱ [k] = tk+ α (t) T(t) d t tk+ T(t) d t. (28) Then, we know that ᾱ [k], k, R and α mn ᾱ [k] = α j [k] α max, k,, j I, and we have tk+ W [k+] = ( β [k]d [k])w [k]+ᾱ [k] T(t) d t From smlar steps n (3) to (6), We have W [k+] = ( β [k]d [k])w [k]+ ᾱ[k] = ᾱ[k] β [k]d [k]w [k]. = (29) Suppose ᾱ [k] s ndependent of W [k], and defne ˆα = E[ᾱ [k]], and ˆβ = E[β [k]]. Then, ˆα =, ˆβ = /2, R, and ˆα j = ˆα k, ˆβ j = ˆβ k, j,k I. We use αil and β IL to denote the common ˆα j and ˆβ j values for TCP-Illnos users. If all packets dropped are due to congeston, and Droptal s used, βil β max. Takng condtonal expectaton of W [k+] gven W [k], we get ( j= ˆα j)e[w [k+] W[k]] = ( j= ˆα j)(w [k] ˆβ f( W [k] ˆT )W [k])+ ˆα ˆβ j= j f( W j[k] ˆT )W j[k]. Equatng E[W [k+]] and E[W [k]], we have E[W [k] f( W [k] ˆT )] ˆβ = C 2, (3) ˆα

10 The rato between measured average throughput and computed average throughput The rato between the standard devaton of throughput and average throughput Log(average throughput) Vs Log(RTT). The slope v= Fg. 7. =, heavy congeston, M [,2], unformly dstrbuted. where C 2 s a constant ndependent of user. If the congeston s lght and the backoff s unsynchronzed ( ˆM or ˆM << C/x), f(x) s approxmately proportonal to x, and thus we have ˆβ ˆα E[(W [k]) 2 ] s the same for all user. And as Var(W [k]] << (E[W [k]]) 2, approxmately we have ˆβ ˆα (E[W [k]]) 2 s the same for all user. Snce ˆα and ˆβ are the same wthn each protocol, we know that at equlbrum, all Reno users share the same average wndow sze W R and all Illnos users share the same average wndow sze W IL, and W IL W R αil 2βIL α IL 2β max, In smulatons presented later, for typcal scenaros, α IL s slghtly larger than and β max s usually pcked to be /2 for frendlness wth TCP-Reno. Thus, W IL s usually slghtly larger than W R. Ths means that n a network wth both TCP- Illnos and TCP-Reno users, the TCP-Reno users wll not suffer a sgnfcant degradaton n performance. Furthermore, unlke TCP-Vegas whch performs poorly when used wth TCP-Reno, TCP-Illnos actually performs better than TCP- Reno, thus provdng the rght ncentve for users to swtch to TCP-Illnos. V. TCP-ILLIOIS PROPERTIES In Secton II, we lsted some requrements for the new TCP varant to satsfy, and n Secton IV, we showed that TCP-Illnos mantans the ntra protocol farness the same way as standard TCP, satsfes the stablty and scalablty requrement, avods heavy congeston, and s compatble wth the current TCP. In ths secton, we consder the remanng requrements. Snce q a ncreases wth ncreasng W, α decreases wth ncreasng W, and thus the W curve s concave. We can show that the curve s actually frst lnear, and then close to a parabola, and fnally lnear agan. The proof s omtted due to space lmtatons and s avalable n [8]. In [8], we also show that TCP-Illnos acheves a better average throughput than standard TCP for any route buffer sze B, and ts average throughput ncreases as B ncreases, snce compared wth standard TCP, TCP-Illnos ncreases ts rate to full utlzaton faster and stays around full utlzaton longer (the length of tme t stays around full utlzaton ncreases wth ncreasng B). Thus the requrements of effcency, router buffer ndependence, and ncentve to swtch are all met. From Secton IV, we see that convergence speed n k for TCP-Illnos s the same as that for standard TCP. So the response tme s only determned by the tme nterval between two consecutve congeston events. We can show that ths tme nterval of TCP-Illnos s smlar to or smaller than that of standard TCP for a wde range of α mn values (see [8] for a proof), and thus the responsveness requrement s also satsfed. In lossy networks such as wreless networks, many packets are dropped not due to congeston. These packet drops greatly reduce the throughput for standard TCP, but for TCP- Illnos, the degradaton s not as severe, snce when a packet s dropped before congeston, the average queueng delay s always almost zero, and thus β β mn and α α max always, and TCP-Illnos s essentally an AIMD algorthm wth a larger α = α max and smaller β mn. Snce W α/β p, the rato of the average wndow sze of TCP-Illnos over that of standard TCP can be up to αmax /(2β mn ). (3) Ths mprovement s sgnfcant. For example, f α max = 9,β mn = /8, then W Illnos can be up to 6W Reno. VI. SIMULATIO RESULTS In ths secton, we provde ns-2 smulaton results to valdate the propertes of TCP-Illnos and compare ts performance wth TCP-Reno and HS-TCP. Throughout, one bottleneck lnk s shared by one or multple users, whch may choose TCP-Reno, HS-TCP, TCP-Illnos, or TCP-Vegas. For HS-TCP, all the default parameter settngs are used. For TCP-Vegas, W ncreases f d f f < γ and decreases f d f f > γ. For TCP-Illnos, wthout explct explanaton, we set α max =,α mn =.,β max = /2,β mn = /8,W thresh =,η =.,η 2 =.,η 3 =.8, and θ = 5. A. Sngle User: Effcency Property We frst perform smulatons for a sngle user scenaro, wth C = Mbps, B = packets 8, and T p = ms. The wndow szes are plotted n Fg. 8. The smulatons clearly demonstrate the concave nature of the curve of TCP-Illnos and show that TCP-Illnos acheves a larger average wndow sze than TCP-Reno. For HS-TCP, we have chosen the Reno base, ewreno base and SACK base, and we have found that HS-TCP generates tmeouts frequently for Reno and ewreno bases, and only works well f SACK s used. Ths supports our clam that HS-TCP causes heavy congeston. umercally, the average send rates for TCP-Reno, SACK based HS-TCP 9, and TCP-Illnos are 78.32, and 9.34 Mbps, respectvely. As a comparson to HS-TCP, f 8 The packet sze s bytes throughout. 9 Henceforth, we mean SACK based HS-TCP when we menton HS-TCP wthout specfyng ts base.

11 Illnos, beta=.25 Illnos, beta=.5 Reno SACK ewreno Reno Average throughput (Mbps) Average throughput of 4 homogeneous RTT users 4 Reno 4 Illnos 4 HS TCP,2 Reno & 3,4 Illnos,2 Reno & 3,4 HS TCP Average wndow sze (pkt) Reno 4 Illnos 4 HS TCP Wndow Sze Flow ID Flow RTT Tme Fg.. Left: Average throughput of 4 homogenous RTT users. Rght: Average wndow of 4 heterogeneous RTT users. Fg. 8. Sngle user, TCP-Reno, HS-TCP, and TCP-Illnos. Top plot: Reno and Illnos (beta=.25 and beta=.5). Bottom plot: HS-TCP, wth Reno, ewreno, and SACK bases B= B=2 B=3 B=4 B= Fg. 9. The wndow sze curve for dfferent buffer szes. TCP-Illnos sets β max = β mn =.25 (β.25 for HS- TCP n ths capacty range), then the average throughput s Mbps. We then study the effect of the buffer lmt on the wndow sze curve. We fx C = Mbps, T p = 6 ms, and vary B from to 5 packets, and the wndow curve s plotted n Fg. 9. It s clear that as B ncreases, there s more tme around full utlzaton and the average wndow sze ncreases. B. Multple Users: Farness Property We now perform smulatons for multple ( = 4) users, whch may choose Reno, HS-TCP or TCP-Illnos. We demonstrate the nter-protocol farness (compatblty to Reno) of TCP-Illnos and HS-TCP n the homogeneous RTT scenaro, where C = Mbps, B = packets, and T p = ms; and demonstrate the ntra-protocol farness of Reno, TCP-Illnos and HS-TCP n the heterogeneous RTT scenaro, where C and B are unchanged, but the RT Ts for the four flows are 6,8,, and 2 ms, respectvely. The average throughput n the homogeneous scenaro and average wndow sze n the heterogeneous scenaro are plotted n Fg.. From ths fgure, we see clearly that TCP-Illnos s more far to competng Reno user and large RTT users than HS-TCP. To further demonstrate the performance of these protocols, we also plot the wndow curves of these smulatons n Fg. and Fg. 2. C. Performance n Lossy/Wreless etworks We then perform smulatons for lossy/wreless lnks. It s a sngle lnk sngle user scenaro, wth the user choosng ether TCP-Reno or TCP-Illnos, and the lnk randomly droppng packets wth droppng probablty p d much larger than the congeston loss probablty (snce p d s large, the lnk s under utlzed and there s no congeston loss at all n many cases). The capacty and buffer length of the lnk are 4 Mbps and 2 packets, respectvely, and the propagaton delay for the sngle user s ms. Instead of choosng the default settng, the TCP-Illnos user sets η =.2. We vary p d values from.5 to.5, and plot the average wndow sze for TCP-Illnos and Reno and the rato of these two multpled by 2, as n the left plot of Fg. 3. From the plot, we see that W Illnos 4W Reno n most cases. From (3), the rato should be α max /(2β mn ) = The dfference of the rato between smulaton and analyss can be explaned f we observe the wndow curve plot of TCP-Illnos and Reno, as n the rght plot of Fg. 3. From the wndow curve plot, we see that tmeout happens frequently for TCP-Illnos user, snce the ncrement amount α s very large before a packet loss happens. Equaton (3) only consders the congeston avodance phase, and tmeout s the reason that W Illnos /W Reno s around 4 nstead of 6. Though, TCP-Illnos acheves a much better throughput than Reno n wreless networks. D. Performance wth osy RTT Measurement We fnally perform smulatons to compare the performance of TCP-Illnos and TCP-Vegas when the delay measurement s naccurate. We consder a sngle lnk and two user scenaro. For the lnk, C = Mbps and B = 5 packets (correspondngly, the maxmum queueng delay d m = 4 ms). For the users, ether both choose TCP-Illnos or both choose TCP-Vegas, and the propagaton delay s T p = 6 ms for each user. We now suppose that there s an extra whte nose term n the RTT measurement, denoted by n, and let n be unformly dstrbuted between [,2σ] (the nose term s an extra delay due to reasons other than propagaton and queueng, so t s nonnegatve). Then, RTT = T p + d + n, where d s the queueng delay. We vary the value of σ to vary the nose level and study the performance of TCP-Vegas and TCP-Illnos under nosy RTT measurement. For each protocol, we have two groups of smulatons. In group one, both users face the nose term n the RTT measurement; and n group two, only one user faces the nose term and the other user measures RTT accurately. The average throughput of the users under dfferent nose levels are plotted n Fg. 4. From Fg. 4, we see that as the nose level ncreases, TCP-