Inter-protocol fairness between

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1 Inter-protocol farness between TCP New Reno and TCP Westwood+ Nels Möller, Chad Barakat, Konstantn Avrachenkov, and Etan Altman KTH, School of Electrcal Engneerng SE- 44, Sweden Emal: INRIA 24 route des Lucoles, 692 Sopha Antpols, France Emal: {Chad.Barakat k.avrachenkov Abstract In ths paper we nvestgate the effect of ntroducng TCP Westwood+ on regular TCP New Reno. By means of analytcal modelng and ns-2 smulatons, we demonstrate that the two protocols get dfferent shares of the avalable bandwdth n the network. Our man result s that the bandwdth sharng between the two protocols depends on one crucal parameter: the rato between the bottleneck router buffer sze and the bandwdth delay product. If the rato s smaller than one, TCP Westwood+ takes more bandwdth. On the contrary, f the rato s greater than one, t s TCP New Reno whch gets the larger part. Inspred by our results, we propose a smple modfcaton to the wndow decrease algorthm n TCP Westwood+ that solves the unfarness problem for large buffer szes. For small buffers, the unfarness problem s stll open. I. INTRODUCTION In recent years, several new proposals and mplementatons of TCP congeston control algorthms have been developed, motvated by a growng heterogenety of networks such as wreless networks, hgh speed networks, ad-hoc and sensor networks. In all these types of networks, the transmsson s subject to losses due to unrelable lnks transmsson losses n addton to losses due to congeston congeston losses. The Westwood+ TCP verson [] has appeared to be partcularly useful when transmsson losses cannot be neglected. TCP Westwood+ s novel wth respect to TCP Westwood because of a new, smpler and unbased estmator of the avalable bandwdth. It behaves exactly as TCP New Reno verson n ncreasng ts wndow when there are no packet losses. However when a loss occurs, the behavor s dfferent: nstead of employng the classc TCP by half wndow decrease, Westwood+ decreases the wndow sze to a new value that exactly matches the bandwdth avalable at the tme of congeston. In partcular, the wndow sze s set equal to the avalable bandwdth tmes the smallest RTT t has been observed so far. The ratonale for ths choce s to keep full the avalable ppe, where the avalable ppe s the avalable bandwdth tmes the mnmum round trp tme. When evaluatng proposed TCP mprovements, one mportant ssue s that of farness, and n partcular farness n a mxed envronment where the old and new TCP versons The work was supported by the France Telecom R&D Grant Modélsaton et Geston du Trafc Réseaux Internet no coexst and share the same resources. The major goal of the present work s to study the nter-protocol farness between TCP Westwood+ and TCP New Reno. As TCP Westwood+ can reduce the congeston wndow less than TCP New Reno, one mght suspect that TCP Westwood+ takes a larger share of the avalable resources. On the other hand, t s known that TCP New Reno under-utlzes the bandwdth n the case of small buffer szes. So does TCP Westwood+ just take ths un-used part of the bandwdth? In [2], the mpact of the buffer sze on the performance of TCP Westwood+ was studed, and t was shown that unlke TCP New Reno, Westwood+ can acheve full lnk utlzaton wth arbtrarly small lnk buffers. Thus, one can expect that the buffer sze at the bottleneck wll be one of the most mportant parameters n the analyss. Moreover, the problem of the choce of the buffer sze for regular TCP traffc has drawn a sgnfcant attenton [3] [7]. Here we show that the router buffer sze has a sgnfcant nfluence not only on the effcency of the network but also on the farness between dfferent TCP versons. In partcular, we show that n order TCP Westwood+ to take a far share of the avalable bandwdth one needs to choose the buffer sze of the bottleneck router not smaller than half of the bandwdth-delay product. We provde a model that descrbes the nteracton of TCP Westwood+ wth the bottleneck router buffer as well as wth the conventonal TCP New Reno. Snce queueng delays and packet losses due to congeston are coupled to the queue and wndow szes, t becomes necessary to take queue and buffer sze nto account. The evoluton of the wndow szes of both flows s modeled as a hybrd system [8], wth a contnuous ncrease between congeston events, and dscontnuous reductons at the congeston event. Ths model lets us calculate analytcally the throughput for the flows for any buffer sze. The results are also valdated usng ns2 smulatons. Both the analytcal model and smulatons gve the same results: If the buffer sze equals the bandwdth-delay product, TCP Westwood+ and TCP New Reno wll share avalable capacty almost equally. For smaller buffers, Westwood+ wns the battle for capacty, and for larger buffers, TCP New Reno wns. Furthermore, we show that the results are only senstve to a sngle parameter: the rato between the buffer sze and

2 C Capacty of bottleneck lnk [bytes/s] B Buffer sze [bytes] T Round trp tme of flow, excludng [s] queueng delay at the bottleneck m Segment sze of flow [bytes] qt Queue sze [bytes] w t Wndow sze of flow [bytes] r t Sendng rate of flow [bytes/s] TABLE I NOTATION the bandwdth delay product. Fnally, we propose a smple modfcaton to TCP Westwood+ that makes TCP Westwood+ effcent even n the case of large buffers. The paper s organzed as follows. In the next Secton II we descrbe the system model and n the ensung Secton III we provde ts mathematcal analyss. Then, n Secton IV we provde the numercal as well as smulaton results and dscusson. We conclude the paper wth Secton V. II. SYSTEM MODEL We use a hybrd flud flow model for a network wth two competng TCP sources: TCP New Reno and TCP Westwood+. They share a sngle bottleneck router wth a drop-tal buffer of sze B and wth transmsson capacty C. The state varables of nterest are the queue sze qt at the bottleneck router and the congeston wndow szes w t, =,2, of each TCP source. We assgn ndex to the TCP New Reno flow and ndex 2 to the TCP Westwood+ flow. The varable qt s regarded as contnuous and w t as pece-wse contnuous, and the sendng rate of each sources s one full wndow of data per round trp tme, where the round trp tme conssts of a constant propagaton delay T and a queueng delay qt/c that s determned by the current queue sze. The wndow sze corresponds to the amount of data that a source has transmtted, but not yet receved any acknowledgment ACK for. The notatons are summarzed n Table I. For the wndow dynamcs, we focus on the congeston avodance mode of TCP. In TCP New Reno, congeston avodance s based on Addtve ncrease / Multplcatve decrease: The wndow s ncreased by one packet per round trp tme as long as ACKs are receved. When a packet loss s detected, the wndow sze s set to one half of the value before the loss. In TCP Westwood+ [], the addtve ncrease s the same. The dfference s that the multplcatve decrease s replaced by a dfferent mechansm based on estmaton of the avalable bandwdth and of the propagaton delay. The orgnal motvaton was to mprove TCP performance over wreless lnks, wth a sgnfcant rate of losses that are due to transmsson errors rather than congeston. When Westwood+ detects a packet loss, t sets ts wndow sze to the product of the estmated bandwdth before the loss, and the end-to-end propagaton delay. The dea s that ths wndow sze s suffcently small to allow queues on the path to dran, but not smaller. The propagaton delay estmate s smple: t s just the mnmum observed round trp tme. As long as at least one packet has been sent when the queue s close to empty, ths estmate s accurate. The bandwdth estmaton s more complex. We have to refer to the Westwood lterature for the detals [], but the man dea s to obtan bandwdth samples from the stream of ACKs, and form the estmate by lowpass flterng of these samples. The frst proposed verson of Westwood formed a bandwdth sample for each ACK, whle Westwood+ collects a round trp tme worth of ACKs before formng a bandwdth sample. Thus, accordng to the flud flow approxmaton of wndow based congeston control schemes such as TCP New Reno and TCP Westwood+, the sendng rate of a flow s one full wndow per RTT: r t = w t T + qt/c. The queue sze s related to the sendng rates by dqt/dt = r t C, or, substtutng, dqt dt = w t C. 2 T + qt/c The above two equatons gnore sgnalng delays, but t ncludes the dependence on the queue sze, va the RTT T + qt/c. Snce the queue sze must be non-negatve, 2 s vald only when qt > or the rght hand sde s postve; otherwse dq/dt =. The evoluton of the wndow szes s governed by the addtve ncrease of TCP s congeston avodance mode, whch s the same for both New Reno and Westwood+. The wndow s ncreased by one packet, m bytes, each RTT. dw t dt = m T + qt/c. 3 At the congeston event, the flow or flows that lose packets make a dscontnuous change n ts wndow sze. For TCP New Reno, w t + = w t /2. For TCP Westwood+, w 2 t + = RTT mn ĉ 2, where ĉ 2 s the estmate of the flow s bandwdth, and RTT mn s the smallest observed RTT. For TCP Westwood+, there are several ssues wth how ths should be modeled. RTT mn : Ths s ntended to be the round trp delay, excludng queueng delay. So t makes sense to put RTT mn = T. But on the other hand, f the Westwood+ flow s started when the bottleneck s already loaded, Westwood cannot observe T, and t may be more accurate to set RTT mn = T + mn qt/c. ĉ 2 : The bandwdth s sampled once per round trp tme, and then these samples are low-pass fltered to form a smoother estmate. The smplest model s to put ĉ 2 = r 2 t = w 2 t /T+B/C. Ths s an assumpton of optmstc estmaton, and t neglects the delay and the bas whch are present n the real flter. Wth RTT mn = T and ĉ 2 = w 2 t /T + B/C, t follows that w 2 t + = βw 2 t, wth the constant β = CT/CT + B.

3 III. ANALYSIS To analyze the system evoluton, we use separate models for the evoluton between congeston events, and for the congeston events. A congeston event s a short perod of tme when the router queue s full, and one or more packets are dropped. Fnally, we put these two models together to fnd the statonary behavor, and the correspondng throughput. A. System evoluton between congeston events Assume that a congeston event ends at tme t =. At ths tme q = B,.e., the buffer s full, and the wndow szes are gven by the ntal condtons w = w. After a congeston event, the evoluton of the rates and the queue can be dvded nto three phases. Phase : Durng the frst phase, the total rate, r + r 2, s smaller than C, and ncreasng. In ths phase, the queue s shrnkng, and t may even become empty. Phase 2: Durng the second phase whch s present only for small buffer szes, the queue s empty, and sendng rates are ncreasng. Durng the second phase, the lnk s under-utlzed, and the phase ends when the total sendng rate reaches the lnk capacty, r + r 2 = C. Phase 3: Durng the thrd phase, the ACK-clock mechansm forces the total sendng rate to stay essentally constant, barely larger than C, and the growng wndows result n a growng queue, not ncreasng sendng rates. The thrd phase s termnated by the next congeston event, whch happens when the queue sze reaches the buffer sze, qt = B. The objectve of the analyss n ths secton s to fnd the smallest t > such that qt = B, and to express t and the correspondng wndow szes w t as functons of the ntal wndow szes w. These functons are needed n Sec. III-B, when dervng the evoluton over a large number of congeston events. When nvestgatng farness between two flows, usng T T 2 would ntroduce a prejudce, favorng one of the flows aganst the other. To gve the two flow equal opportuntes, we assume T = T 2 = T. Ths assumpton lets us to ntroduce a vrtual tme, whch can be thought of as measurng tme n number of round trps. Ths tool makes t possble to fnd explct solutons to the dfferental equatons. Vrtual tme s s defned by dt = T + qt/cds. Ths change of varables transforms Equatons 2 and 3 nto dw s = m 4 ds dqs = qs CT + w s 5 ds The second equaton s stll vald only when qs > or the rght hand sde s postve. Let s denote the vrtual tme correspondng to t,.e., the next congeston event. Gven s, the amount of data that s transmtted up to tme t can be computed as follows. Frst solve 4, whch gves, w s = w + sm 6 Then the volume of data transmtted up to tme s s t s D = rtdt = wsds = s w + m s 2 Ths says smply that the amount of data s the number of round trps, s, tmes the average wndow sze. Here, the average s not a proper tme average, but an average wth respect to the vrtual tme s. We wll compute the throughput as D /t. The calculaton of s and t depends on the buffer sze. In general, t can computed gven qs and s, by ntegratng t s t dt = dt = ds ds = Ts + s qsds 8 C The calculatons n absolute tme, t, are smplfed by the remarkable fact that 3 and 2 can be solved for q n terms of w. Substtuton of 3 nto 2 gves dqt dt = C + { = d dt Ct + w t dw t m dt 2m w t 2 7 } 9 Integratng, we get qt 2 qt = Ct 2 t + w t 2 2 w t 2 2 m For any tme nterval, durng whch the queue stays non-empty, ths allows us to compute the length of the nterval gven only the ntal and fnal state, { } t 2 t = qt qt 2 + w t 2 2 w t 2 C 2 m Full utlzaton: The lnk wll be fully utlzed f and only f qt > for all tme except possble for an solated nstant. In other words, the second phase s empty. Ths secton derves condtons on the ntal condtons for ths to happen. Introduce the notaton W = w + w2 and M = m + m 2. Substtutng w s = w + sm nto 5 gves dqs = CT qs + W + sm 2 ds whch together wth the ntal condton q = B can be solved explctly, qs = Be s +W CT e s +e s +sm 3 Ths equaton s vald only as long as q >, snce the dfferental equaton doesn t model queue underflow. Lookng more closely at ths equaton, t can be dvded nto a transent, related to the ntal buffer B and a new equlbrum sze W CT, and a lnear growth wth rate M. Asymptotcally,

4 Fg.. The two real branches of Lambert s W functon. W sold s defned for all x /e, whle W dotted s defned on the nterval /e x <. for large s, we have qs W + s M CT,.e., the queue s the dfference between the total wndow sze one RTT earler, and the bandwdth-delay product CT. Proposton : The lnk s fully utlzed f and only f ether of these two nequaltes s satsfed: W CT 4 B CT W M exp CT W 5 M M Proof: Frst observe that f W CT, then qs > for all s note that e s + s >. So assume that W < CT. Then the functon qs, defned by 3, s ntally decreasng, and has a sngle mnmum at s = log+ct +B W/M. We have full utlzaton f and only ths mnmum value qs s non-negatve. By substtutng s nto 3, we see that qs s equvalent 5, whch concludes the proof. 2 Large buffer: If the buffer s large,.e., the condton n Prop. s satsfed, then the queue never underflows, and we can fnd s by puttng qs = B n 3. The soluton can be expressed n terms of Lambert s functon [9], defned as the nverse of z ze z. We use the two real branches, denoted W and W, and llustrated n Fg.. Proposton 2: When the full-utlzaton condton of Prop. s satsfed, the values of s and t are gven by where s s defned by s = s + + W + se + s 6 t = W s + Ms 7 C 2 s = CT + B W/M 8 Proof: Put qs = B n 3. After some smplfcatons, ths equalty mples + s = + se s + s Ths equaton has two solutons, the trval one s =, and a second soluton whch can be expressed usng the W functon, resultng n 6. Wth ths value for s, 7 follows from. The throughput can be computed from 7, D t = C w + m s /2 W + Ms /2 9 As expected, wth full utlzaton, the total throughput, D + D 2 /t, equals C. 3 Small buffers: When computng the throughput for a small buffer, we must handle the three phases separately, snce nether the dfferental equaton for q, nor the tme nterval equaton, s vald durng the second phase, when the queue s empty and the lnk s under-utlzed. Proposton 3: When the full-utlzaton condton of Prop. s not satsfed, the values of s and t are gven by s = s + + W { e B/M} 2 t = Ws + CTs 3 s + Ms 2 + s 2 3/2 2 C where s and s 2 are defned by s = + CT W M + W CT + se W/M s 3 = + B M + W e B/M Proof: Denote the duraton of the three phases, n absolute tme and vrtual tme, by t, t 2, t 3, s, s 2, and s 3. We handle one phase at a tme. The frst phase: We fnd s by puttng qs = n 3 and solvng for s. Then t s found by substtutng ntal and fnal state n. Ths procedure gves: s = + CT W M + W CT + se W/M t = B + Ws + Ms 2 /2 C The second phase: Throughout ths phase, q =. The phase starts wth w t < CT, and ends when w t = CT. Intally, the total wndow sze s W +s M, so t follows that s 2 = CT W M s t 2 = Ts 2 The thrd phase: Ths s smlar to the frst phase, but wth dfferent start and stop condtons. Intally, w s + s 2 = CT and qs + s 2 =. Wth these ntal condton, the dfferental equatons 4 and 5 have the soluton qs = e s s s2 + s s s 2 M 22 The phase ends when qs = B. The soluton of ths equaton, whch s found usng the same method as for the frst phase, gves the value of s 3. t 3 s found by substtutng ntal and fnal state n. The result s s 3 = + B M + W e B/M t 3 = B + CTs 3 + Ms 2 3/2 C

5 Fnally, addton of the values for each phase, s = s +s 2 +s 3 and t = t + t 2 + t 3, results n 2 and 2. 4 Throughput: Propostons 3 let as compute t and s as functons of the ntal wndow szes. Wth these values, the data volume D follows from 7, and the throughput D /t for each flow can be computed. For reference n the next secton, we defne the functons sw and s W as follows: sw = CT + B W M{ s W + se + s W = s + + W e B/M B. System evoluton at congeston events 23 full utlzaton otherwse 24 The prevous secton analyzed the queue evoluton between congeston events. To fnd the average throughput over a longer tme, we need to know the statonary behavor that results from a large number of congeston events. We frst consder what happens at the sngle congeston event at tme t = t. As shown at the end of Secton II, at the congeston event f TCP New Reno loses a packet: w t + =.5w t, and f TCP Westwood+ source loses a packet: w 2 t + = βw 2 t, wth the constant β = CT/CT + B. Then, an mportant ssue s whch flow loses packets at a congeston event. It could be one of the flows, or both. In the TCP farness lterature, t s common to use stochastc modelng for ths see e.g., [], []. Under the farly general assumpton that the probabltes of the dfferent possble outcomes at a congeston event depends only on the wndow szes of the nvolved flows just pror to the congeston, the evoluton can be descrbed as a Markov chan. Let X k denote the vector of the two wndow szes just after a congeston event. If the frst flow uses TCP New Reno and the second uses TCP Westwood+, the evoluton can be descrbed as X k+ = D lk GX k 25 In ths equaton G w w 2 = w w 2 + s w + w 2 m m 2 26 represents the wndow growth between congeston events, and the functon s W s defned by 24. The actual packet loss s represented by the random varable l k, wth three possble values,, and 2. The wndow reducton s represented by the constant dagonal matrces D, D = /2 β D = /2 D 2 = β 27 Of the three outcomes, D represents a loss event where each flow loses a packet, D represents an event where only the frst flow loses a packet, and D 2 represents an event where only the second flow loses a packet. If we make the further assumpton that the probablty that l k = j depends only on the state X k,.e., Pl k = j X k,l k,l k 2,... = p j X k 28 then 25 clearly descrbes a Markov chan. To gan nsght nto the problem, and to be able to solve t analytcally, we have to make a few smplfyng assumptons and approxmatons. Our frst assumpton s that flows are fully synchronzed,.e., that at every congeston event, both flows lose packets. Ths means that l k = for all k, and t actually makes the process fully determnstc. The second smplfcaton s droppng the non-lnear terms of the functon s w,w 2. In both expressons for s, 6 and 2 are of the form s = s + + W, where the fnal term s non-lnear, and bounded between - and. So by replacng s by s + /2, the error n GX k s at most half a packet. To gve both flows equal opportuntes, we also put m = m 2 = m. The result s the followng approxmaton of the evoluton of the state X k. X k+ = D X k + sx k,x2 k + /2m /4 /4 = X k /4 + CT + B + m β/2 β/2 β/2 29 Snce β <, the matrx has all egenvalues wthn the unt crcle. Then, for any ntal values, the X k sequence converges to /4 /4 X = I = CT + B + m 3 2β β/2 β/2 β β CT + B + /4 β/2 3 To fnd the throughput for both flows n the statonary regme, we plug n the wndow szes X as the ntal wndow szes n the procedure of Sec. III-A.4. C. Lmts Let us study the farness n the lmtng cases when buffers are very small or very large. Of partcular nterest s the throughput rato D 2 = 2w 2 + ms D 2w + 3 ms Proposton 4: In the lmt as B, Westwood+ gets one quarter of the capacty, whle New Reno gets three quarters. In the lmt as B, Westwood+ gets all the capacty, and New Reno gets. Proof: For the large buffer case, note that CT + B = CT/β. We parameterze the expressons n terms of β, and let β. We have from 3 that w = X = CT/β + O 32 3 CT + Oβ w 2 and W = CT/3β + O. It s clear that we have full utlzaton, and we get s = 2CT + O 33 3Mβ Fnally, D 2 /D = + Oβ/3 + Oβ /3.

6 D Mbt/s 5 ms R Mbt/s ms Mbt/s ms Fg. 2. Expermental setup. Parameter values correspond to scenaro #. For the small buffer case, frst note that 3 mples that W CT m > as B, so by Prop., we have full utlzaton for all suffcently small B. Takng lmts n Prop 2, we fnd that s as B. Furthermore, Eq. 3 also mples that w and w2 CT + m as B and β. It follows that D 2 /D as B. Ths result s dfferent from what one could obtan by usng the throughput formulas for TCP New Reno and Westwood+ n [2]: T Reno = RTT 2 p, T Westwood = p S S 2 p pt q RTT. 34 In these equatons, T q s the average queueng tme and RTT = T+T q s the average round-trp tme. Let s form the throughput rato: T Westwood T Reno = + T 35 2 T q We see that ths rato predcts farness when T q = T,.e., when the average queung delay equals the propagaton delay. Ths s slghtly dfferent from our model, whch predcts farness when the maxmum queung delay, B/C, equals the propagaton delay. In the small buffer lmt, T q, the rato tends to nfnty. So here we have perfect agreement. In the large buffer lmt, T q, the rato tends to / 2, sgnfcantly hgher than the predcton /3 from our analyss. The smulaton results presented n the next secton wll support our analyss n predctng the trend of the throughput rato for large buffers. Our explanaton for ths dscrepancy s n the random loss assumpton made by the square root formula, whch does not hold n our settng. IV. RESULTS AND DISCUSSION We use the network topology llustrated n Fg. 2. There are two source nodes, S New Reno and S 2 Westwood+, sendng data to a sngle destnaton node, D. There s one ntermedate router, R, and the lnk between the router and the destnaton s the network s bottleneck. For the bottleneck lnk, we use two values for the capacty C, Mbt/s and Mbt/s, and two dfferent values of the propagaton delay, correspondng to round trp propagaton delay T of 5 and 2 ms. Table II shows the parameter values for the four scenaros, and the correspondng bandwdth-delay product. Both sources use a packet sze of 5 bytes. For all scenaros, we vary the buffer sze of the router between 2 packets and roughly twce the bandwdth-delay product, 2CT. Scenaro # C [Mbt/s] T [ms] CT [packets] TABLE II NETWORK PARAMETERS FOR FOUR DIFFERENT SCENARIOS Throughput Buffer sze Fg. 3. Normalzed throughput vs. buffer sze n packets for scenaro #. The resultng throughputs accordng to the analyss of Sec. III, are shown n Fgs. 3 and 4. In Fg. 3, we plot the normalzed throughput a fracton of the capacty of the bottleneck lnk for scenaro #. The sold lne represents TCP New Reno, and the dashed lne represents TCP Westwood+. The topmost dotted curve s the sum of the normalzed throughputs,.e, the lnk utlzaton. In [2] t was shown that Westwood+ acheves almost full utlzaton for arbtrary small buffer szes, so t s not surprsng that TCP Westwood+ domnates over New Reno for small buffer szes. When the buffer sze equals the bandwdth-delay product, 42 packets n scenaro #, both flows get the same throughput. Ths s expected from the model, snce for ths buffer sze, β = /2, both flows use the same decrease factor after a loss, and both elements of X are equal. For larger buffer szes, Westwood+ suffers from ts estmaton of RTT mn. It uses the smallest observed RTT, 5 ms, even though n statonarty, the buffer never gets empty, and the actual RTT stays sgnfcantly hgher than Westwood s RTT mn at all tmes. Ths forces Westwood+ to reduce ts wndow sze, after a packet loss, even more than New Reno does. In Fg. 4, the correspondng curves for all four scenaros are shown n the same fgure. To ad the comparson, the horzontal axs has been scaled so that for each scenaro, corresponds to the bandwdth-delay product. The theoretcal throughput curves for all four scenaros are plotted on top of each other, the ncreasng curve represents New Reno, the decreasng curve represents Westwood+, and the uppermost curve close to one s the utlzaton. The curves for the four scenaros are barely dstngushable. Scenaro #3, wth the largest bandwdth-delay product, s shown wth sold curves, and for ths we see that the lnk s slghtly under-utlzed for buffer szes sgnfcantly smaller than the bandwdth-delay product. We can conclude that the sngle varable defned by B/CT determnes the bandwdth sharng between TCP

7 Throughput Throughput Normalzed buffer sze Buffer sze Fg. 4. Normalzed throughput vs. normalzed buffer sze B/CT. The theoretcal throughput curves for all four scenaros are plotted together. Throughput Buffer sze Fg. 5. Valdaton for scenaro # - Normalzed throughput vs. buffer sze n packets. Westwood+ and TCP New Reno. To valdate the present model and the assumptons such as flud flow approach and full synchronzaton, we smulate the same scenaros usng ns2, and compare these results to the theoretcally computed values. We assgn random start tmes to the two flows, measure the actual throughput for a long transmsson, excludng transents at flow startup, and average over several realzatons. For lack of space we only show the throughputs for scenaro # n Fg. 5 wth 95% confdence ntervals. The results for the other three scenaros lead to the same conclusons. In Fgure 5, the vertcal axs corresponds to the normalzed throughput and the horzontal axs corresponds to the buffer sze n packets. The ns2 smulatons are dsplayed for New Reno wth marks, Westwood+ wth marks +, and ther sum wth marks. The curves show the theoretcal results. The ns2 smulaton and the analyss gve throughput values that match remarkably well, except for very small buffer szes B < 5. The most lkely cause of the msmatch for small values of B s the flud model approach. Consder the rato between Westwood+ and New Reno throughput. As can be seen n the fgure, the rato decreases as the buffer sze s ncreased and t tends to /3 rather than the / 2 that one obtans by applcaton of the square root formula n [2]. For TCP Westwood+ to functon properly, t needs to es- Fg. 6. Ths fgure shows the result for scenaro #, c.f., Fg. 5, but wth the start of the Westwood+ flow delayed around 5 s. tmate RTT mn. It shares ths requrement wth delay based congeston control methods such as TCP Vegas and Fast TCP. If a Westwood+ flow s started at a tme when the bottleneck lnk s hghly loaded, the buffer may stay close to full for the whole duraton of the flow. Then, n effect, part of the queueng delay wll be erroneously accounted for as propagaton delay, whch results n an overestmaton of the real bandwdth delay product and hence n a less dramatc reducton of the congeston wndow. In ths case, the dsadvantage that Westwood+ has when competng wth New Reno over large buffer, s actually reduced. Ths s llustrated n Fg. 6, whch shows the result of a smulaton of scenaro # c.f., Fg. 5, wth the only dfference beng that the start of the Westwood+ flow s delayed around 5 seconds so that the TCP New Reno flow gets enough tme to fll the buffer. For the larger buffer szes, Westwood s RTT mn estmate has values up to 8ms, to be compared to the true round trp propagaton delay of 5ms. Furthermore, when the number of flows over the bottleneck lnks s ncreased, t becomes less probable that the router buffer emptes. Thus, also n ths case, one can expect the RTT mn estmate to nclude some of the queueng delay. And consequently, the dsadvantage of Westwood+ can be expected to be reduced when more flows are multplexed. To mprove farness n the large buffer case, the wndow decrease of Westwood+ n response to a packet loss could be modfed, so that the new wndow s never set to a value smaller than half of the prevous value. In other words, modfy Westwood+ to never decrease ts wndow more than New Reno would have done. To valdate ths clam, we modfed the Westwood+ code n ns-2 accordngly and rerun the smulaton for scenaro #. The results are presented n Fg. 7. Clearly, for buffers larger than the bandwdth delay product when t s very probable that TCP Westwood+ dvdes ts wndow by more than 2, our modfcaton solves the unfarness problem. At the same tme, the performance of both protocols for buffers smaller than the bandwdth delay product stays the same. For successful deployment of TCP Westwood+ as a general purpose congeston control mechansm over the Internet wthout sgnfcant negatve effects on New Reno flows, t seems prudent not to set the sze of buffers n routers to small values. It s known see e.g., [3], [7] that for small

8 NewReno Westwood+ Lnk utlzaton Fg. 7. The performance of TCP Westwood+ n scenaro # after our modfcaton - Normalzed throughput vs. normalzed buffer sze B/CT Regular TCP vs TCP WestWood Regular TCP vs Regular TCP Regular TCP vs TCP WestWood Regular TCP vs Regular TCP Fg. 8. The normalzed throughput of TCP New Reno when competng wth ether TCP New Reno or TCP Westwood+ as a functon of the normalzed buffer sze. buffers, TCP New Reno under-utlzes the avalable bandwdth at the bottleneck lnk. In contrast, TCP Westwood+ s able to utlze all avalable bandwdth even when the buffers are small. One could ask a queston f TCP Westwood+ takes only un-used bandwdth when t competes wth TCP New Reno, or t steals ths bandwdth from TCP New Reno. To answer ths queston, we consder two TCP New Reno flows nstead of the mx of TCP Westwood+ and TCP New Reno and we vary the rato between the buffer sze and the bandwdth delay product. The results are depcted n Fg. 8. In ths fgure, we plot normalzed throughput of TCP New Reno when the competng TCP connecton s ether TCP New Reno or TCP Westwood+. The two topmost curves correspond to lnk utlzaton. We can read n ths fgure that when two TCP New Reno flows compete for the avalable bandwdth and the bottleneck buffer s small, the lnk s not fully utlzed, as expected. The fgure demonstrates that TCP New Reno does suffer from the presence of TCP Westwood+ flow when the buffer sze s smaller than the bandwdth-delay product. For buffers larger than the bandwdth delay product, regular TCP realzes better performances after the ntroducton of TCP Westwood+. Note that our modfcaton to TCP Westwood+ ntroduced earler can prohbt regular TCP from stealng bandwdth from TCP Westwood+ when buffers are large and hence, solve the unfarness problem n ths regon. V. CONCLUSIONS In ths paper, we studed analytcally and by the means of ns-2 smulatons, the nter-protocol farness between TCP Westwood+ and TCP New Reno. Untl the present, the effect of the ntroducton of TCP Westwood+ on TCP New Reno was not thoroughly nvestgated. We explaned why these protocols when they compete for avalable bandwdth get dfferent shares. Our man concluson s that the bandwdth sharng only depends on the rato between the buffer sze at the bottleneck router and the bandwdth delay product. In partcular, f the rato s smaller than one, TCP Westwood+ takes more bandwdth. On the contrary, f the rato s greater than one, t s TCP New Reno whch gets the larger part. The ntroducton of TCP Westwood+ allows to solve the well known problem of network under-utlzaton by regular TCP when buffer szes n routers are set to small values. Unfortunately, ths gan n the utlzaton comes at the expense of regular TCP whch loses some of ts throughput. Inspred by our results, we proposed a smple modfcaton to TCP Westwood+ that solves the unfarness problem for large buffer szes. For small buffers, the unfarness problem s stll open. REFERENCES [] S. Mascolo, C. Casett, M. Gerla, M. Y. Sanadd, and R. Wang, TCP Westwood: bandwdth estmaton for enhanced transport over wreless lnks, n MobCom, Rome, Italy, 2. [2] S. Mascolo and F. Vacrca, Congeston control and szng router buffers n the nternet, n IEEE Conference on Decson and Control and European Control Conference. Sevlle: IEEE CSS, 25, pp [3] K. Avrachenkov, U. Ayesta, E. Altman, P. Nan, and C. Barakat, The effect of router buffer sze on the TCP performance, n Proceedngs of LONIIS workshop, 22. [4] G. Appenzeller, I. Keslassy, and N. McKeown, Szng router buffers, n SIGCOMM. Portland: ACM, September 24. [5] A. Dhamdhere, H. Jang, and C. Dovrols, Buffer szng for congested nternet lnks, n Proceedngs of IEEE INFOCOM, 25. [6] S. Gornsky, A. Kantawala, and J. Turner, Lnk buffer szng: A new look at the old problem, n Proceedngs of IEEE Symposum on Computers and Communcatons, June 25, pp [7] K. Avrachenkov, U. Ayesta, and A. Punovsky, Optmal choce of the buffer sze n the nternet routers, n IEEE Conference on Decson and Control and European Control Conference. Sevlle: IEEE CSS, 25. [8] J. P. Hespanha, S. Bohacek, K. Obraczka, and J. Lee, Hybrd modelng of TCP congeston control, n Hybrd Systems: Computaton and Control, ser. Lecture Notes n Computer Scence, M. D Benedetto and A. Sangovann-Vncentell, Eds. Berln, Germany: Sprnger-Verlag, 2, vol. 234, pp [9] R. M. Corless, G. H. Gonnet, D. E. H. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W functon, Advances n Computatonal Mathematcs, vol. 5, pp , 996. [] F. Baccell and D. Hong, AIMD, farness and fractal scalng of TCP traffc, n Proceedngs of IEEE INFOCOM, New York, June 22, pp [] A. Lezarowtz, R. Stanojevc, and R. Shorten, Tools for the analyss and desgn of communcaton networks wth markovan dynamcs, to appear n Proceedngs of IEE, Control Theory and Applcatons, 25. [2] L. A. Greco and S. Mascolo, Performance evaluaton and comparson of Westwood+, New Reno and Vegas TCP congeston control, ACM Computer Communcaton Revew, vol. 34, no. 2, Aprl 24.