A Fluid-Based Model of Time-Limited TCP Flows 1

Transcription

1 A Fud-Based Mode of Tme-Lmted TCP Fows Maro Barbera DIIT - Unversty of Catana V.e A. Dora Catana -Itay hone: fax: mbarbera@.unct.t Afo Lombardo DIIT - Unversty of Catana V.e A. Dora Catana -Itay hone: fax: ombardo@.unct.t Govann Schembra DIIT - Unversty of Catana V.e A. Dora Catana -Itay hone: fax: schembra@.unct.t ABSTRACT Desgn methodooges for TCP/IP based networs s one of the most chaengng toc n research on teecommuncatons networs. The man dffcuty conssts n modeng congeston contro mechansms because t nvoves feedbac from the networ. In ths ersectve, n ths aer we deveo an accurate anaytca framewor for networed TCP acatons suortng both Sow Start and Congeston Avodance agorthms. To ths end we enhance the TCP fud mode ntroduced n the revous terature consderng aso TCP sources amed at transmttng a refxed quantty of data such as fe transfer or networ browsng. The roosed mode addresses a networ of routers suortng any Actve Queue Management (AQM technques, rovded that the equatons descrbng the AQM rues memented n the routers are ntroduced n the fud framewor. The roosed framewor aows desgners to study not ony the steady-state behavor of the networ, but aso the transent behavor when a set of TCP sources start to transmt or fnsh transmttng. Moreover the mode resuts rovde the average vaues of the queue ength n each router even when some of them are not bottenec ones. Keywords Internet, TCP, Sow-Start, Performance Anayss, Fud-fow Anayss.. INTRODUCTION In the ast few years, the need for a desgn methodoogy for TCP/IP based networs has stmuated chaengng research nto TCP traffc modeng. The man dffcuty n ths fed s the modeng of congeston contro mechansms because t nvoves feedbac from the networ. Athough a varety of TCP congeston contro agorthms have been roosed (see for exame[9], [4], [6], [], [2], n most of the current TCP mementatons the nut traffc oad s adjusted to ft the networ oad conons by means of two mean agorthms: the Sow Start agorthm at the connecton start u, and the Congeston Avodance agorthm when the rght throughut s acheved. A remarabe attenton has been gven n the terature to TCP modeng, but a ot of the roosed modes ([3], [3], [4], [], [2] rovde steady-state averages of throughut, queue sze and dro robabty. They do not rovde any ndcaton of ther varabty n tme. Recenty, a very nterestng fud mode has been roosed for studyng the dynamc behavor of TCP congeston wndow n Msra et a. []. That mode consders a set of N mutexed TCP fows, and descrbes the behavor over tme of both the average TCP wndow szes and queue ength by a set of dfferenta equatons. That mode was aso extended to networs wth arbtrary toooges. Accordng to the authors the two man mts of the framewor resented n [] are the hyothess that the offered oad s generated by greedy sources ony, and the fact that the Sow Start hase s not consdered. The use of greedy sources s obvousy an aroxmaton that s very often far from reaty; a sgnfcant art of Internet connectons, n fact, concern wth the web envronment where sma fes are transferred foowed by a sent erod that causes the TCP to reset ts wndow to the nta sze. For ths reason, the Sow Start mechansm has to be modeed as we as Congeston Avodance n order to study a more comex scenaro. In [8] there s an enhancement of the TCP fud mode resented n [], where the effects of short-ved TCP fows on the erformance of ony one bottenec AQM router are consdered. However n ths framewor the aggregate short-ved traffc s modeed as a shot nose rocess, and so the average vaue of ts throughut s consdered ony. Further n [8] the short-ved TCP fows are defned as never eavng the Sow Start hase of TCP, so sources gong from Sow Start hase to Congeston Avodance one, and comng bac n Sow Start after a tmeout exraton, are not consdered. In ths aer we deveo an anaytca framewor for networed TCP acatons suortng both Sow Start and Congeston Avodance agorthms. To ths end we enhance the TCP fud mode ntroduced n [] but, une [], we aso consder TCP sources amed at transmttng a refxed quantty of data such as fe transfer or networ browsng. As n [] the roosed mode addresses a networ of routers suortng Actve Queue Management (AQM technques. Athough we consder a networ of RED [7] routers ony, the roosed framewor can be used wth any AQM technques, even n a non-homogeneous stuaton,.e. when dfferent AQM technques are memented n the networ smutaneousy, rovded that the equatons descrbng the AQM rues memented n the routers are ntroduced n the Ths wor was artay suorted by the Itaan Mnstry for Unversty and Scentfc Research (MIUR through the TANGO FIRB roject.

2 fud framewor. The roosed framewor aows desgners to study not ony the steady-state behavor of the networ, but aso the transent behavor when a set of TCP sources start to transmt or fnsh transmttng. Moreover the mode resuts rovde the average vaues of the queue ength n each router even when some of them are not bottenec ones. The rest of the aer s organzed as foows. In Secton 2, we deveo our anaytca mode. In order to mae the TCP modeng and the mode of AQM technque used n the routers ndeendent, n Secton 2. we ntroduce the TCP behavor mode and n Secton 2.2 we ntroduce the RED mode we w use through the aer. In Secton 3, we consder an acaton of our mode to a networ scenaro and comare our resuts wth those obtaned usng the ns-2 smuator [5]. Fnay, we resent our concusons n Secton MODEL AND ANALYSIS In ths secton we w resent a fud-fow mode to evauate the average erformance of TCP sources n a networ of AQM routers. In Secton 2. we w derve the set of dfferenta equatons that characterze the behavor of TCP sources wthout secfyng the artcuar AQM technque used by the networ routers. In Secton 2.2 we w secfy our mode n the case of RED routers, and derve the equatons that characterze ther behavor. Let us consder a set of AQM routers mang u a networ and et L be the set of a the outut buffers of the routers, whose am s to store acets before transmsson on the assocated undrectona outut ns. The generc n has a transmsson caacty of C acets er second, and a constant roagaton deay of d seconds. Let (B (t be the dscard robabty functon of the generc buffer L at tme t, and q (t the ength of the queue n the same buffer, exressed n acets. 2. TCP behavor modeng Consder a woroad of N TCP fows abeed as =, K, N, and et W (t denote the congeston wndow rocess of the fow. Because we are nterested n networ anayss and desgn, n our framewor we w not consder the end-to-end fow contro agorthm, assumng TCP throughut s bounded by the congeston contro agorthm ony. For ths reason W (t aso reresents the transmsson wndow of the TCP fow. Furthermore, et T (t be the vaue of the threshod searatng the Sow Start range and the Congeston Avodance range for the congeston wndow of the TCP fow at the nstant t. Snce we are nterested n anayzng not ony the behavor of greedy sources, but aso that of sources whch have to transmt fnte amounts of data, we aso need to consder the rocess D (t reresentng the number of acets successfuy sent by source from ts startng to the tme t. In order to characterze our mode cometey, we need to derve the mean vaues of rocesses W (t, T (t and D (t. To ths end, et us frst cacuate the exresson of R (t reresentng the round-tr tme (RTT for the generc TCP fow. The RTT for a generc fow s the sum of the queung tmes n a the buffers aong ts ath and the roagaton tmes assocated wth the outut n of these buffers. Therefore, f we ndcate the set of buffers assed through by the beongng to the fow as L, we obtan: q R = d + =,2,..., N C L Now et us cacuate the oss robabty rocess, (, of acets beongng to fow aong ts ath. Assumng statstca ndeendence between the oss rocesses n the router queues, can be cacuated as one mnus the robabty that no oss occurs n a the queues beongng to the same ath, that s: ( ( t ( t = L Let us now derve the equaton descrbng the adve-ncrease mutcatve-decrease (AIMD behavor of the TCP wndow sze rocess. We can wrte the varaton of the wndow sze W (t as the sum of two contrbutons: the frst term, A (t, corresonds to the adve-ncrease art, the second, B (t, corresonds to the mutcatve-decrease art: dw = A + B =,2,..., N (3 Frst we w cacuate the adve-ncrease term A (t. To ths end et us note that the wndow sze s doubed durng the Sow Start hase, whe t neary ncreases by one acet er RTT when the source s n the Congeston Avodance hase. In other words, for each ACK acet that reaches a generc TCP source, ts congeston wndow sze W (t ncreases by one acet durng the Sow Start hase, whe t ncreases by W (t acets durng the Congeston Avodance hase. So, f we ndcate the arrva rate of ACKs for the fow as λ, we can wrte: λ f W < T A = =,2,..., N λ f W T W The rocess λ s equa to the emsson rate of acets one RTT before, muted by the no dscardng robabty, that s: ( ( =,2 N W ( τ λ = τ,..., R ( τ where τ s the nstant at whch the generc acet n fow that arrves at tme t, was emtted. The term τ s reated to the current nstant t by the foowng mct equaton: τ It s evdent that τ s a functon of t, but to smfy the notaton n the rest of aer we w not ndcate ths deendence excty. To derve the mutcatve-decrease term B (t n (3, t s ( L necessary to evauate the acet oss rate, λ, for the generc fow. Snce the RED ocy tends to dstrbute osses n (2 (4 (5 + R (τ t (6 =

3 roorton to bandwh share, we can estmate the oss rate for fow as ts throughut muted by the dscardng robabty of the acets beongng to the same fow : ( L W λ = =,2,..., N (7 R It s mortant to observe that the tme nstant when a oss s detected s dfferent from the nstant at whch ths oss haened. Here we consder that the dstance n tme between the two events s aroxmatey equa to a round tr tme. Ths s a common aroxmaton n terature [],[8]. Consequenty, the rate of oss ndcatons receved by the source of fow s gven by (7 at the ( L nstant τ, λ ( τ. The comete dervaton of the term B (t n (3 deends on the TCP verson of the sources, because of the dfferent agorthms adoted to cacuate the new congeston wndow vaue when a oss s detected. In the rest of the aer we w refer to one of the most common TCP versons, that s, the New-Reno TCP, but our dervaton coud be extended to the other TCP versons. Because a New-Reno TCP source behaves dfferenty f t detects a acet oss by recevng a tre ducate ACK (TD_oss, or a tmeout (TO_oss exres, we need to dstngush between the two dfferent oss causes. Let Q(w be the robabty that, when the transmsson wndow sze s w, a oss s due to tmeout exraton, rovded that a oss has occurred. For Q(w we w adot the exresson derved n [9], and aso adoted n [], accordng to whch a TO oss occurs f the transmsson wndow has a vaue of ess than three, or f the ost acet n the sequence of w acets sent n an RTT s one of the ast three acets. Therefore we have: 3 Q( w = mn, w A generc New-Reno TCP source haves ts congeston wndow when a TD_oss occurs, whe t sets ts congeston wndow to one when a tmeout exres. Consequenty the varaton of the congeston wndow W (t s equa to ( W (t 2 when a TD_oss s detected, whe t s equa to ( W (t when a TO_oss s detected. From these consderatons we obtan the fna exresson of B (t : W B = 2 + ( Q( W ( τ ( L [( W Q( W ( τ ] λ ( τ =,2,..., N λ ( L ( τ + Now, tang exectaton of the two sdes n (3, after agebrac manuatons, we obtan: dw ( + ( ( τ ( ( τ W ( τ W ( τ QW R ( τ 2 W ( τ R ( τ W ( τ W R ( τ P f W < T f W T ( ( τ for W + 2 =,2,..., N (8 (9 ( In the above equaton we made two tyes of aroxmaton. Frst we assumed that E{ f ( x } f ( E{ x}, and secondy we assumed E{ x y} = E{ x} E{ y}, whch mes statstca ndeendence between the two rocesses x and y. As n [], we cannot anaytcay demonstrate the accuracy of these aroxmatons, but the resuts shown n Secton 3 w mcty suort the correctness of our dervaton. Now we w derve the equaton that reguates the behavor of the threshod T (t, searatng the Sow Start wndow range and the Congeston Avodance wndow range. Ths threshod s set to haf the congeston wndow every tme a oss s detected; so ts varaton s equa to zero when there s no oss ndcaton, whe s equa to W T ( otherwse. Consderng that the rate of 2 t ( L λ oss ndcatons s equaton: ( τ, from (7 we derve the foowng dt W ( L = T λ ( τ =,2 N 2,..., ( If we consder the exected vaues for both the sdes n ( and tang nto account (7, wth the same aroxmatons adoted n (, we have: dt W ( τ W ( τ T =,2 N R,..., ( τ 2 (2 Fnay we derve the reatonsh to cacuate the average number of acets D (t successfuy sent by source from ts startng unt the tme t. Let us note that when D (t s equa to the sze of the fe to be sent by source, ths source has ended ts transmsson. The varaton n the average number of acets successfuy sent by the generc source s gven by the average emsson rate of the source, W R, muted by the robabty of none of ts acets beng dscarded aong ther ath, that s: ( P W ( =,2 N dd =,..., R (3 U to now we have derved a set of 3 N dfferenta equatons that descrbe the average behavor ( W, T, D of N TCP sources (n artcuar usng the New-Reno verson n an AQM networ. To comete our mode we have to derve the equatons that aow us to cacuate the average vaue of the two rocesses q (t and for each buffer n the networ, and we w dea wth ths n Secton 2.2. Now t s mortant to ont out that we can reduce the number of equatons descrbng the sources f we grou the fows havng the same average behavor. More secfcay, a grou s consttuted by the set of fows foowng the same ath n the networ and startng n a tme nterva that s shorter (3 4 tmes than ther average RTT. In ths way we can dvde the N fows nto M grous ( M N. The generc grou contans n fows, where =,2,K M, wth the conon n + n + K + nm = N. 2 For each grou of fows we w consder a functon f (s that reresents the number of sources n the grou that have to

4 transmt a fe of a sze ess than or equa to s, exressed n acets. Consequenty the number of sources beongng to grou whch are actve at tme t, s f ( D (t n. For the next dervatons t s necessary to cacuate the average tota throughut at the generc tme nstant t for each grou, that s: W [ n f ( D ] for,2 M Th = =,..., R 2.2 AQM modeng: an acaton to RED To comete the mode resented n Secton 2. we st have to derve the reatonshs to evauate the rocesses and q (t, whch reresent the dscard robabty functon for the generc buffer L and the queue ength n the same buffer at tme t, resectvey. Because t has not, u to now, been necessary to consder exct form of these rocesses, the equatons that descrbe the average behavor of the TCP sources are vad for a the networs of AQM routers. More generay, t s not necessary for a the routers n the networ to adot the same AQM technque, because our mode s aso vad when the reatonshs to cacuate and q (t are dfferent for dfferent routers. As an acaton, we w consder a networ n whch a the routers adot the RED agorthm [7]. For ths reason we w frst derve the equaton for the average vaue of q (t and then the one for the average vaue of L. n the generc RED buffer If we ndcate the average acet arrva rate at a generc buffer at tme t as λ, we have: dq = [ ] + C + ( λ [ C + ( λ ] + f q > f q = L (4 (5 where f (t s equa to f (t when t s ostve, and t s nu otherwse. In (5, we note that f the average dscardng robabty n the RED buffer s Q (t, λ reresents the actua ( ( arrva rate n the queue, tang the RED dscardng rocess nto account. Snce we are aso nterested n studyng the transent behavor of the networ we w derve an exresson for λ that s vad not ony for the queue n a bottenec n (as n [], but for a the queues n the networ. For ths reason, for each buffer queue, we w consder λ as sum of two contrbutons: λ F The frst term n (6, grous of fows λ ( NA = λ + λ L (6, s the contrbuton of the set of concernng TCP sources drecty attached to the queue ; the second, λ of grous of fows F assng through the queue, but generated by sources not drecty attached to t. The frst term n (6 can be cacuated as the sum of the average throughut of grous of fows beongng to F, that s: λ = j ( NA (NA F Th j, s the contrbuton of the set L (7 When the sources of acets beongng to the generc grou are not drecty attached to the queue, the rate of ther acets cannot be exressed, n genera, as Th (t, because t coud be mted or shaed by the n caacty of one of the ns aong ther ath. For ths reason, f we assume that the avaabe outut bandwh for each buffer s roortonay dvded among ts nut fows, we can aroxmate the average outut rate µ, of grou, as foows: µ mn t = ( C ( λ,, ( t λ Th ( (8 Now, et us ndcate the generc nut n h for the queue as v ( h, and the set of nut ns for the queue as ( ( ( V = v v,, v (the notaton s shown n Fg.. { }, 2 K v ( ( v h ( v n n ( NA F F Fgure : Inut connectons for the generc buffer We obtan the foowng reatonsh for : where F ( vh λ ( NA ( vh = F F n ( NA λ ( through the buffer v h. The ndex h ndcates the generc nut n of the queue, and the generc grou of fows gong through ( the queue, and arrvng from the queue- nut n v. = µ ( NA ( vh, h= ( NA F ( F v h (9 s the comete set of fows that go h

5 The ast reatonsh that we need to derve concerns the rocess at tme t. reresentng the dscard robabty n the generc buffer If we assume that queue sze s suffcent to avod osses due to overfow, the robabty dscard functon drecty from the RED agorthm as: m t = tmax t mn mn max f m < t f t mn f m > t mn m t max can be derved max (2 where m (t s the estmated average queue ength rocess of the RED buffer assocated to the n, at tme t, and tmn, t max, are the RED arameters 2. Because the rocess max deends on the rocess descrbng the behavor of the average queue ength estmated by RED, m (t, we need an equaton for t. To ths end we adot the resut obtaned n []: dm n( α = ( m q L (2 δ where α reresents the weght n the EWMA fter to cacuate the average queue ength estmated by RED, whe δ reresents the tme nterva between two consecutve acet arrvas at the buffer. In [], where the goa was to study the networ n a stabe conon, t was assumed that the steady-state arrva rate s equa to the servce rate, and so t was set δ = C. Here, on the contrary, we want to study the transent conon of the networ; therefore we w set δ equa to the nverse of the buffer arrva rate λ. In adon we w aso consder that, when the queue ength s zero, the generc RED router adots a dfferent formua to cacuate ts estmated average queue ength (see [7] for reference. Consequenty the fna equaton for m (t s: ( m q n( α m t d ( = n( α C m λ f q > f q = L The comete dervaton of (22 s shown n the Aendx. (22 The equatons (5, (2 and (22 comete the set of dfferenta equatons that, n ths way, s determnated, and t can be soved numercay to yed the transent behavor of the networ 2 Note that the defnton of the rocess (B (t can easy be extended to the case of osses due to buffer overfow as we. Further ths equaton s not vad for the gente verson of RED [5], but the extenson s mmedate. 3. NUMERICAL RESULTS In order to demonstrate the ower of the mode roosed n ths aer, and the robems due to aroxmaton resent n the revous terature of TCP modeng wth fud aroach, n ths secton we comare the resuts obtaned wth our mode wth those acheved wth the mode ntroduced n [], here referred to as roosed mode and revous mode, resectvey. We w ay our mode to the networ tooogy resented n Fg. 2. L 2 A B C Fgure 2. Networ tooogy We consder four RED routers named A, B, C and D, and for each router there are one or more LANs drecty attached to t. The RED arameters are set as n [], that s, t mn = 5 acets, tmax = 2 acets, max =. and α =., for a the routers. In the networ we dstngush 5 aths, whch are sted n Tabe. Each ath s foowed by 2 fows from greedy sources and fows from non-greedy sources. Tabe. Descrton of the consdered aths through the networ Path name P3 L 3 L D Source-Routers-Destnaton L B C D L4 2 L2 A B L 3 L2 A C D L4 4 L3 C B L 5 L3 C D L4 For the sae of smcty, we assume that the non-greedy sources foowng the same ath, start to transmt aong each ath at aroxmatey the same nstant; consequenty n our system we can characterze grous of fows, 5 referrng to greedy sources and 5 to non-greedy sources. In adon, we w assume that non-greedy sources beongng to the same grou have a fe of the same sze to transmt, even though ths s not a restrctve hyothess. To hghght the effects of the Sow Start agorthm we w assume that the grous of non-greedy sources start to transmt at dfferent onts n tme, as secfed n Tabe 2. L 4

6 Tabe 2. Informaton about grous of fows Index of Number of Path Fe sze Startng tme grou ( fows foowed (acets (sec greedy greedy greedy greedy greedy The frst anayss of the networ shown n Fg. 2 was carred out assumng that a the ns had a caacty of 5 Mb/s whch corresonds, f the acets are consdered to have a fxed sze of 5 bytes, to a caacty of 25 acets/sec. The resuts gven by our mode are shown n Fgs. 3, 5, 6 and 8, where they are comared wth those obtaned usng the ns-2 smuator. Fg. 3 ony shows the average queue engths n the two buffers reresentng bottenecs for the networ, as anayss usng both ns-2 and our mode shows that the queues n the other buffers are ractcay nu throughout the observaton erod. As can be seen n Fg. 3, our mode catures the average queue ength dynamcs n both buffers qute we. What s more sgnfcant s that t catures the nta eas bascay due to the Sow Start hase for a the sources. In adon, t aso foows transent networ henomena, as can be seen for exame n Fg. 3a where, around nstants t = 6 sec and t = 8 sec, greater varatons n the average queue ength are detected, due to actvaton of the non-greedy sources. 4 2 ns smuaton roosed mode 4 2 ns smuaton roosed mode average queue ength (acets average queue ength (acets tme (sec. (a Average queue ength of the router B outut buffer towards L tme (sec. (b Average queue ength of the router C outut buffer towards router D Fgure 3. Average queue ength : comarson between smuaton and roosed mode 4 2 ns smuaton revous mode 4 2 ns smuaton revous mode average queue ength (acets average queue ength (acets tme (sec. (a Average queue ength of the router B outut buffer towards L tme (sec. (b Average queue ength of the router C outut buffer towards router D Fgure 4. Average queue ength : comarson between smuaton and revous mode

7 RED estmated average queue ength (acets ns smuaton roosed mode RED estmated average queue ength (acets ns smuaton roosed mode tme (sec. (a Average estmated queue ength of the router B outut buffer towards L tme (sec. (b Average estmated queue ength of the router C outut buffer towards router D Fgure 5. Average estmated queue ength : comarson between smuaton and roosed mode average congeston wndow (acets ns smuaton roosed mode average congeston wndow (acets ns smuaton roosed mode tme (sec. (a Average congeston wndow for non-greedy sources on ath tme (sec. (b Average congeston wndow for greedy sources on ath 2 Fgure 6. Average congeston wndow on ath 2 : comarson between smuaton and roosed mode 45 4 ns smuaton revous mode 7 6 ns smuaton revous mode average congeston wndow (acets average congeston wndow (acets tme (sec. (a Average congeston wndow for non-greedy sources on ath tme (sec. (b Average congeston wndow for greedy sources on ath 2 Fgure 7. Average congeston wndow on ath 2 : comarson between smuaton and revous mode

8 To demonstrate the benefts of ntroducng modeng of the Sow- Start agorthm, Fg. 4 comares the resuts gven by ns-2 and those rovded by the system of dfferenta equatons n [], to whch equaton (3 has been added to tae the resence of nongreedy sources nto account. It s evdent that ths mode s unabe to cature the henomena that occur n the networ when the varous grous of sources are actvated, that s when the effects of the Sow-Start agorthm are more fet. Fg. 5 resents a comarson between the resuts gven by ns-2 and our mode, referrng to the average queue ength estmated by RED on both the bottenec routers. Here agan, the good match shows that our mode gves an exceent estmate of the robabty of acet oss nstant by nstant. Fnay, Fgs. 6 and 8 show the average congeston wndow for some TCP fows, these resuts are comared n Fgs. 7 and 9 wth the corresondng resuts obtaned by the mode n []. As can be seen, aso n ths case our mode better aroxmates the resuts obtaned va smuaton, both n the nta hase whch, once agan, s due to the Sow Start agorthm, and n the oscaton dynamcs. Ths s due to the better aroxmaton of the average queue engths n the varous buffers, and consequenty a better estmate of the oss robabty n each buffer. It shoud aso be onted out that n Fg. 6a our mode shows that t s caabe of redctng wth suffcent accuracy the nstant at whch the non-greedy sources fnsh transmttng the reestabshed amount of data. The same cannot be sad of the mode resented n [], whch tends to shft the average end of transmsson tme for non-greedy sources further on than t reay s. The same consderatons ay to the congeston wndows for the fows beongng to the other grous, whch are not resented here average congeston wndow (acets ns smuaton roosed mode average congeston wndow (acets ns smuaton roosed mode tme (sec. (a Average congeston wndow for non-greedy sources on ath tme (sec. (b Average congeston wndow for greedy sources on ath 3 Fgure 8. Average congeston wndow on ath 3 : comarson between smuaton and roosed mode 45 4 ns smuaton revous mode 35 3 ns smuaton revous mode average congeston wndow (acets average congeston wndow (acets tme (sec. (a Average congeston wndow for non-greedy sources on ath tme (sec. (b Average congeston wndow for greedy sources on ath 3 Fgure 9. Average congeston wndow on ath 3 : comarson between smuaton and revous mode

9 for reasons of sace. In order to better evauate the comarson between the mode we roose, the exstng mode and the resuts obtaned wth the smuator, Fg. comares the throughut for some grous of fows, as cacuated by ns-2, our mode and the one resented n []. To obtan these ots we consdered the same networ tooogy shown n Fg. 2 and the same scenaro resented n Tabe 2. We then cacuated the throughut of a the grous of fows (n Fg. we resent ony four cases for reason of sace for dfferent n caacty, assumng that a the ns n the networ shown n Fg. 2 had the same n caacty. These resuts aso demonstrate the accuracy of our fud mode n redctng the average throughut of TCP sources n the networ. 4. CONCLUSIONS In ths aer we have used a methodoogy deveoed n [] to construct an accurate fud mode for TCP sources that are not necessary greedy, aso tang the Sow-Start hase nto consderaton n order to hghght the effects on networ erformance of short connectons. We have consdered TCP fows n a networ whose nodes wor accordng to the RED ocy and comared the resuts gven by the mode wth the resuts obtaned va smuaton, obtanng a good match. The too deveoed to sove the system of dfferenta equatons mang u the mode gves resuts n a much shorter tme than smuaton (for the tooogy n Fg. 2 ess than sec usng a norma -GHz PC Pentum III, so t coud be used to fnd the otma networ arameter confguraton for dfferent traffc conons. 5. APPENDIX: Comete dervaton of dfferenta equaton for average queue ength estmated by RED. In Secton 2 we resented (22 as the equaton to cacuate the varatons n the average queue ength m(t n a RED buffer. The frst art, whch s vad when q >, has been demonstrated n [] and we w not reeat the roof. Here we are ony nterested 4 35 ns smuaton revous mode roosed mode 2 ns smuaton revous mode roosed mode average throughut (acets/sec average throughut (acets/sec n caactes (acets (a Average throughut for non-greedy sources on ath n caactes (acets (b Average throughut for greedy sources on ath ns smuaton revous mode roosed mode 2 ns smuaton revous mode roosed mode average throughut (acets/sec average throughut (acets/sec n caactes (acets n caactes (acets (a Average throughut for non-greedy sources on ath 4 (b Average throughut for greedy sources on ath 4 Fgure. Average throughut: comarson between smuaton, roosed mode and revous mode

10 n demonstratng the foowng reatonsh: dm( t = n( α C m( t f q( t = (23 where α reresents the weght n the EWMA fter and C s the n caacty. We reca that the RED router s cacuatons of the average queue sze tae nto account the erod when the queue s emty (the de erod by estmatng the number n of acets that coud have been transmtted by the router durng the de erod. After the de erod the router comutes the average queue sze as f n acets had arrved n an emty queue durng that erod [7]. When a new acet arrves on the RED buffer and the queue s emty, the cacuaton s as foows: t n = s de m ( α n m where tde s the queue de nterva, and s s the transmsson tme for a acet, that s, s = / C. So f we observe that tde s the nterva between the arrva tmes of two acet at the buffer (when the queue s emty, we have: m + m C ( t + t (24 = ( α m (25 It s ossbe to wrte (25 n the foowng form: where + = e A ( t + t m (26 A = C n( α (27 Because (26 s the samed souton of the foowng contnuous dfferenta equaton: dm( t = A m( t (28 Substtutng (27 n (28, we fnay obtan reatonsh ( REFERENCES [] Bramo, L.S., Peterson, L. TCP Vegas: New technques for congeston detecton and avodance. ACM SIGCOMM, (August 994. [2] Bu, T., Towsey, D. Fxed Pont Aroxmatons for TCP Behavor n an AQM Networ. ACM SIGMETRICS (June 2. [3] Cardwe, N., Savage, S., Anderson, T. Modeng TCP atency. IEEE INFOCOM (2. [4] Frou, V., Borden, M. A Study of actve queue management for congeston contro. IEEE INFOCOM (2. [5] Foyd. S., Recommendaton on usng the gente_ varant of RED. htt:// (March 2. [6] Foyd, S., Henderson, T. The NewReno Modfcaton to TCP s Fast Recovery Agorthm. RFC 2582 (Ar 999. [7] Foyd, S., Jacobson, V. Random Eary Detecton Gateways for Congeston Avodance. IEEE/ACM Transactons on Networng (993. [8] Hoot, C.V., Lu, Y., Msra, V., Towsey, D. Unresonsve Fows and AQM Performance. IEEE INFOCOM (23. [9] Jacobson, V., Kares, M.J. Congeston Avodance and Contro. ACM SIGCOMM (August 988. [] Msra, V., Baras, J., Ott, T. The wndow dstrbuton of mute TCPs wth random oss queues. GLOBECOM 99 (December 999. [] Msra, V., Gong, W., Towsey, D. Fud-based anayss of a networ of AQM routers suortng TCP fows wth an acaton to RED. SIGCOMM (August 2. [2] Maths, M., Mahdav, J., Foyd, S., Romanow, A. TCP Seectve Acnowedgment Otons. RFC 28 (October 996. [3] Padhye, J., Frou, V., Towsey, D., Kurose, J. Modeng TCP throughut: A sme mode and ts emrca vadaton. ACM SIGCOMM (998. [4] Stevens, W. TCP Sow Start, Congeston Avodance, Fast Retransmt, and Fast Recovery Agorthms. RFC 2 (January 997. [5] The networ smuator ns-2. LBL, URL: htt://