TCP throughput and timeout steady state and time-varying dynamics

Transcription

1 TCP throughput an timeout steay state an time-varying ynamics Stephan Bohacek an Khushboo Shah Dept. of Electrical an Computer Engineering Dept. of Electrical Engineering University of Delaare University of Southern California Abstract While many moels of the TCP s ynamics have been evelope, fe focus on the effects of timeout an high loss probability. Active queue management (AQM is an important application of these ynamic moels. Hoever, recent ork has shon that AQM provies little performance benefit over rop-tail queueing for HTTP traffic, except possibly at high utilizations. It is at these utilizations that the ynamic moels of TCP are the least accurate. This paper presents a ynamic moel of TCP that accurately moels timeout. This moel is also applicable to the static case. This paper also presents a moel of the variance an the istribution of the congestion ino. It is shon that, hile the ynamics of the mean value of the congestion ino are rather mil, the ynamics of timeout isplay large oscillations that take several secons to ecay. These oscillations cause the average bit-rate to also illy oscillate. Finally, this paper inclues results from several million simulations proviing a etaile vie of the ynamics of timeout. I. INTRODUCTION s of TCP are extensively use throughout netorking research. For example, TCP moels are use in fiels such as AQM [], [], [3], [4], esign of TCP frienly transport protocols [5], planning an provisioning netorks [6], preicting file transfer time [7], [8], etc. In this ay, TCP moels are founation for research. The accuracy of these moels is critical to the accuracy an relevance of the ork that rests upon them. s for TCP have evolve over time. Initially, a simple relationship beteen the sening rate an the loss probability as evelope. The ell-knon /sqrt(p formula, or more specifically MSS 3// ( RT T p, as foun an verifie [9], []. While this formula is useful in many settings, it as foun to not be accurate in situations here the packet loss probability is high. The reason for this inaccuracy is that this simple moel oes not account for TCP s timeout mechanism. Some stuies have shon that timeout, as oppose to triple uplicate acknolegement, is a significant ay that TCP etects packet loss []. In orer to inclue the impact of timeout, the simple square root of p formula as extene to inclue the effect of timeout []. This formula as shon to be more accurate for large loss probabilities here timeout is significant. Sikar et al. [] evelope a moel for TCP throughput that inclues time-out. While this moel is ifferent from [], they report that it gives similar numerical results. Hoever, e have performe extensive simulations, over 5 million in total, an have foun that hen applie to the ns- implementation of TCP-SACK an hen loss is probabilistic, the moel given in [] results in errors as large as a factor of to. Consiering the importance of the relationship beteen sening rate an loss probability, these ifferences are significant an require closer examination. This paper provies an accurate metho to estimate TCP s sening rate even for large loss probabilities. Another important extension to the TCP moels as to make them ynamic. That is, the initial moels assume that the loss probability as fixe. In the case that the loss probability varie, ynamic moels ere require. The first moel in this irection as given in [4], an several closely relate moels have folloe [], [3], [7]. The principle application of these moels is for esign an analysis of AQM. These moels are extensively use to prove stability an to inicate the performance of AQM. Since AQM as first introuce, there have been a large number of researchers active in this area. Recently, several groups have investigate hat, if any, benefit AQM has on the performance of the netork (see [3] an references therein. A recent paper along these lines shoe that the AQM, be it rop tail or any more elaborate version, has little impact on the performance of the netork for utilization belo 9%. Hoever, for utilization above 9%, some AQM techniques ere shon to improve the performance [4]. This ork is significant as it inicates the environment here AQM might have a positive impact, specifically, in netorks ith high utilization. On the other han, this conclusion is troubling since the moels of TCP use to esign AQM neglect timeout. That is, in the environment here AQM might have the largest impact on performance, the analysis an esign of AQM is least accurate. Because of this inaccuracy, the possible impact of AQM on netork performance at high utilizations has never been carefully explore. One reason that AQM esign has not consiere high utilization netorks is that ynamics of TCP in such an environment are not knon. By ynamics, e mean the variation of the throughput in the face of a time-varying loss probability. This paper provies a simple moel of the ynamics of TCP s throughput that inclues timeout. Perhaps one of the most significant contributions of this paper is that the ynamics of timeout is quite complicate an can result in illy varying bit-rates that can take several secons to ecay. For example, the loer right of Figure 4 shos such oscillations. This ynamics is far more complicate than the Globecom /4/$. 4 IEEE

2 ynamics of the mean congestion ino use by most AQM esign. As a result, this ork calls into question the proof of stability of AQM at high loss probabilities. As a by-prouct of this ork, the variance an the istribution of the congestion ino is also foun. Specifically, it is shon that the ino size is accurately moele ith a negative binomial istribution. An finally the huge number of simulations use to valiate this moel provie a etaile vie of the behavior of the ynamics of TCP s timeout mechanism. The paper procees as follos: The next section iscusses the simulations use for verification. In Section III the istribution of the congestion ino is foun. Next, in Section IV, a moel of the probability of a flo being in timeout is evelope. The results from Sections III an IV yiel a static moel of TCP throughput. In Section V, this moel is then extene to the ynamic case. The paper closes ith some concluing remarks in Section VI. II. THE SIMULATIONS The goal of this effort is to closely examine the behavior of TCP. There is little oubt that in toay s netorks the sening rate of a TCP connection is complicate by interacting flos, router inuce packet reorering, faulty loa balancing inuce packet reorering, server stalls, implementation iiosyncrasies, etc. While these effects are critical an a better unerstaning of them is necessary for a complete unerstaning of TCP, this paper focuses strictly on the behavior of TCP. The rationale for this is that in orer to unerstan the behavior of TCP in the il, it is necessary to thoroughly unerstan TCP in a controlle environment. Perhaps the most significant ifference beteen these simulations an the environment foun in the il is that in these simulations packet losses are entirely ranom an never ue to queue overflo. In the case of AQM, such simulations are appropriate as the goal of AQM is to rop packets in a controlle fashion, not hen the queue fills. The simulations presente in this paper are for a single flo over a single bottleneck topology here rops at the bottleneck are ranom. The ns- error moel as use to prouce the rops. The results presente here are for a fixe roun-trip time of 3ms. Other roun-trip times have been explore an yiel the same results as the ones presente here. These simulations use ns- implementation of TCP-SACK in version.b8a. Delaye acknolegement an a maximum receiver ino ere not use. Our investigation into TCP inclue over 5 million simulations. This large number of simulations is require to estimate the ynamics of timeout. These simulations ere performe on the University of Southern California s Linux cluster [5] an the large simulation results ere transferre via the Internet to the University of Delaare for postprocessing an storage. There is little oubt that ithout such high-performance computing, this ork oul not be possible. III. THE DISTRIBUTION OF THE CONGESTION WINDOW This section presents a simple moel for the istribution of the congestion ino. While the istribution may be of interest in its on right, e are particularly intereste in this istribution since it is use to etermine the probability of a flo entering timeout. We justify the moel briefly ith some analysis an then present simulations results. Misra [4] introuce the iea of moeling TCP as a stochastic ifferential equation. Specifically, he suggeste that the ino size varies accoring to W t = RT T t W tn t ( here N is a Cox process that counts the number of packet losses. This moel is for the evolution of the congestion ino hen the flo is not in timeout. Thus, all probabilities shoul be conitione on the flo not in timeout. We enote this conition as TO. This moel as further investigate in [7] here the partial ifferential equation of the ino size as foun. Specifically, if p (, t TO is the probability ensity of the congestion ino taking the value at time t, then p satisfies p (, t TO t = ( p(, t TO RT T + δ (t RT T ( 4p (, t TO p (, t TO, here δ is the loss probability. If δ is constant, then in steay state, i.e., solves p ( TO p(,t TO t ( =, the istribution of the ino = δ ( 4p ( TO p ( TO. (3 From (3 it is straightforar to sho that E ( m TO = c m δ, m/ for some constants c m.ifm =, then this is the ell-knon square root of p formula here c has been foun to be beteen. an.3. Consiering m =, e fin that the variance is of the form Var ( TO := E ( TO E ( TO = γ δ, (4 here γ := c c. This relationship is borne out in simulations that inicates that γ.3. A close form solution has not been foun for (, hereas a complicate close form solution has been foun for (3. Hoever, simple yet accurate approximate solution to (3 is the negative binomial istribution, i.e., p ( = Γ(N + Γ(N(! ( qn q. Since the mean of this negative binomial ranom variable is Nq/( q+ an the variance is Nq/( q, q = E ( TO Var ( TO an N = q q ( E ( TO (5 an from mean an variance of the given above e get q = δ ( ( c ( q c + an N =. (6 γ δ q δ Globecom /4/$. 4 IEEE

3 In orer to etermine the istribution of the ino size, the parameters q an N must be etermine. This can be one by plugging the mean an variance into (5. Hoever, if only the loss probability is knon, then (6 must be use, in hich case the parameters c an γ must be etermine. We have foun that if the objective is to use the istribution to etermine the probability of timeout, the c = 3/ an γ =.3 is sufficient. On the other han, if one is intereste in the istribution of the congestion ino (the objective of this section, then the selection of these parameters, especially c, requires some more analysis. Hoever, to keep this paper focuse, this issue is not iscusse. Once the parameters are etermine, the istribution of the congestion ino can be estimate. Figure shos the istribution of the congestion ino given the flo is not in the timeout state, i.e., p ( TO. These figures shos the three types of curves, the observe istribution (histogram, the istribution ith c = 3/, an the istribution here the parameters are etermine by plugging in the observe mean an variance into (5. This last istribution coincies ith the istribution given by (6 but ith c =.7 for δ = 4 an c =.4 for δ =.5. Forδ =., thevalueofc plays less of a role, c =.4, c = 3/, an c =.7, or by using the observe mean an variance, all yiel nearly the same istribution. In all cases, γ =.3. IV. A MODEL OF TIMEOUT In this section, e present a moel of the probability of a flo being in the timeout state. We say that a flo is in the timeout state if the next packet ill be sent only after the retransmission timer expires. To unerstan the evelopment of this moel, one shoul consier a large collection of TCP experiments all running in parallel on ifferent netorks ith ientical netork characteristics (e.g., link spees, propagation elays, etc.. Our goal is to etermine the fraction of these flos that are in timeout. A flo can enter the timeout state in the folloing three ays: I If a flo experiences so many losses that triple uplicate acknolegements are not receive, i.e., if at least max (, losses occur in one ino. II In the case of the ns- implementation of TCP-SACK, if more than / packets are roppe. III If a retransmitte packet is roppe. Consiering I an II, e see that a flo ill timeout if a singleropisfolloebyatleastmax (min ( /, 3, rops out of the next packets. Given that rops occur at a rate Rδ, rops that lea to timeout occur at a rate ρ (, δ = R δ k=max(min( /, 3, ( k δ k ( δ k. This ay of entering time-out is iscusse in []. RFC-357 eliminates this ay to enter time-out. It is straightforar to ajust the evelopment belo to reflect RFC-357. Let λ (δ be the rate that a flo moves from congestion avoiance to timeout ue to I an II. This rate is foun taking the expecte value of ρ (, δ, i.e., λ (δ =E ( ρ (, δ TO = ρ (, δ p δ (, here p δ ( is given in Section III, hoever, here e have shon the epenence on δ. This epenence is mae explicit by (6. Note that the approximation E ( ρ (, δ TO ρ ( E ( TO,δ ill result in large errors. Consier the situation here δ =.9 an, using c = 3/, the mean value of the congestion ino is 4. In this case, E ( ρ (, δ TO =.3 R hile ρ ( E ( TO,δ =.38 R, a ifference aroun a factor of three. Next e etermine λ (i.e., III, the rate that a flo moves from congestion avoiance to timeout because a retransmitte packet is roppe. The rate that a packet is first roppe is δ R. The rop ill lea to a retransmission only if triple uplicate acknolegements are receive, i.e., if less than max (min ( /, 3, packets are roppe out of the next packets. In this case, the retransmission is roppe an timeout is entere ith probability δ. Thus ρ (, δ :=δ R δ ( ( δ k ( δ k. k k=max(min( /, 3, The average rate that a flo enters timeout ue to roppe retransmissions is λ (δ :=E (ρ (, δ Finally, the rate that a TCP flo moves into timeout is λ +λ. Up to this point e consiere the rate that a single flo moves into timeout. No e ill etermine the fraction of flos entering timeout. Let N I (t, enote the rate that flos enter timeout at time t, N is the total number of flos. That is, in small time interval t, the probability that some flo enters timeout is N I (t t. LetRT O be the time in hich flos remain in the timeout state upon the first timeout. If a flo exits timeout, but immeiately experiences another rop, then this flo again enters timeout, but for RT O secons. We enote the rate that flos enter timeout for this secon time ith N I (t. The fraction of flos in timeout are t RT O I (τ τ + t RT O I (τ τ. The rate that flos enter timeout is the prouct of the number of flos not in timeout an the rate that such flos enter timeout, i.e., I (t =(λ (δ (t + λ (δ (t (7 ( ( I (τ τ + I (t τ. t RT O t RT O We on t consier the case here a flo enters time-out a thir time for a time interval 4 RT O. While it is straightforar to exten the calculations, it oes not appear to improve the accuracy of the moel. Globecom /4/$. 4 IEEE

4 probability. observe. observe.9.9 (3/.8 /δ (3/.8 /δ.7.7/ δ or.7 observe mean.6 observe mean δ= -4.3 δ= probability probability. observe.8 (3/.6 /δ.4.4/ δ or. observe mean..8.6 δ= Figure. Distribution of the congestion ino for loss probability of δ = 4 (left, δ =. (mile, an δ =.5 (right. While the rate that flos exit timeout only to reenter timeout is I (t =δi (t RT O. (8 In steay state, I (t an I (t are constant. So (7 an (8 reuce to I =( I RT O I RT O(λ (δ+λ (δ I = δi. This can be solve λ (δ+λ (δ I = +RT O (λ (δ+λ (δ ( + δ δ (λ (δ+λ (δ I = +RT O (λ (δ+λ (δ ( + δ Thus, the fraction of flos in timeout is P (TO= t RT O I (τ τ + t RT O I (τ τ (9 = ( + δ RT O (λ (δ+λ (δ +(+δ RT O (λ (δ+λ (δ. If the loss probability varies, then (7, (8, an the relationship P (in timeout at time t = t RT O I (τ τ + t RT O I (τ τ must be use to etermine the probability that a flo is in timeout. To verify (9, a large number of ns- simulations ere performe so that the probability of timeout coul be etermine even for small loss probabilities. Figure shos the observe relationship of P (in timeout an the loss probability as ell as the estimate of this probability given by (9. These calculations use c = 3/. Note that the fit is quite goo, the observe probability an the moel are essentially the same. Pahye et al. [] provies an estimate of the throughput that accounts for timeout. This formula also provies an estimate of the fraction of time a flo spens in timeout, i.e., P (TO. Specifically, [] fins P (timeout Q (W (δ,δ G (δ T o δ R 8 (W (δ++q(w (δ,δ G (δ T o δ, ( probability of timeout observe.4 (3/ /δ.3 Pahye et al δ(loss probability probability of timeout δ(loss probability Figure. Probability of Timeout. The soli blue line represents the observe probability of a flo being in timeout. The ashe green line is the moel s estimate of this probability. Note that these to lines nearly coincie. The thir curve is the relationship erive from []. here (+( δ 3( ( δ 3 Q (, δ = min, ( ( δ / ( ( δ 3 8( δ W (δ =+ + 3δ G (δ =+δ +δ +4δ 3 +8δ 4 +6δ 5 +3δ 6. Figure inclues this estimate. We see that this moel, hile proviing a qualitative fit in that the shape of the relationship is correct, it over estimates the probability of being in timeout. For example, for δ =%, the observe probability of being in timeout is 5%, hile ( gives an estimate of 43%. Hoever, this comparison must be qualifie. This paper focuses on TCP- SACK hile the focus of [] is on TCP-RENO. Furthermore, this paper examines ho TCP behaves uner ranom losses such as arises in AQM. In [], the queueing iscipline is rop tail. Furthermore, the loss probability in [] as not exactly the probability of a packet being roppe, but more along the lines of the probability of a rop event occurring here a rop event leas to the rest of the packets in the roun being roppe. Thus, the δ use in ( is less than the packet loss probability. Hoever, this moel is useful as a benchmark as it as not knon ho accurate this moel oul be for TCP-SACK. From the probability of timeout, the average through-put Globecom /4/$. 4 IEEE

5 Relative Error in Throughput (fraction Pahye et al. RFC δ(loss probability Figure 3. Relative Error of Preicte Throughput. Note that an error fraction of.5 implies a 5% error. can be foun via T = c R ( P (TO MSS. δ Figure 3 shos the relative error of this moel hen compare to the observe throughput. Figure 3 also shos the relative error of the moel in [] as ell as the moel suggeste in RFC-3448 for a TCP frienly sening rate. The figure shos only the relative error, an examination of the error shos that the moel of [] an RFC-3448 unerestimate the sening rate (this is ue to the over estimation of the timeout probability. As mentione, these other moels are for benchmarking purposes only. Figure shos that for even moerately large loss probabilities, the probability of being in timeout is quite high. Specifically, for δ>%, timeout plays an important role in the sening rate of TCP. For δ>5%, a flo ill spen 6% of the time in timeout. Thus, timeout ominates the sening rate of TCP. There are many implications of this. Recently there have been some propose moifications to TCP such as limite transmit an ECN. These moifications significantly affect the behavior of timeout, specifically, they enter timeout much less frequently. Thus, for δ>%, one can expect that TCP implementations ith these moifications ill be more aggressive than implementations ithout these moifications. There has been extensive ork in ynamic moeling of TCP, but most has neglecte timeout. For example, many approaches to AQM utilize a moel of TCP that neglects timeout. We see that such an approach is only reasonable if the loss probability is less than %. Hoever, in the area of AQM, it is hope that these moels are applicable to higher loss probabilities. Next e evelop a moel for the ynamic behavior of TCP that inclues timeout. V. DYNAMICS OF TCP Here e exten the ynamics that is commonly use to moel the ynamics of TCP. The commonly use moel is for only the mean sening rate of TCP. Here e present a moel for the variance. Once the variance is knon, the istribution is foun as in Section III. With the istribution, e can fin the probability of being in timeout as as one in the previous section. Of course, the paper to this point has only consiere a constant loss probability. Here e allo the loss probability to vary ith time. Thus, this moel is appropriate for AQM esign. E(W(t Prob(timeout at time variance(w(t bit-rate Mean Only - 34 Figure 4. Simulation an moeling of a jump in loss probability from % to %. The jump occurre at t =. The soli blue curves sho the observe an the ashe re curves shos the moel. In the loer right, the otte green inicates the bit-rate if only the mean size of the congestion ino is use (i.e., the ashe re curve in the upper left. Note the units of the bit-rate is in bps. Let (t =E ( (t TO an (t =E ( (t TO. An approximation of the ynamics of the mean value of the congestion ino is often given by t (t = R c δ (t R (t R (t. ( R This is often approximate as t (t = R c R δ (t R (t. ( Interestingly, this formula is often erive from (. Hoever, if proper stochastic calculus is applie, the correct ynamics for the mean are t (t = R δ (t R E ( (t R (t, R hich can be approximate as t (t = R R δ (t R (t. (3 Note that in (3 the secon moment is utilize, hile ( have multiplie by the first moment square. Fortunately, c e have foun that ( an ( give goo approximation to the ynamics of the mean. It seems that the mean varies sloly enough that this ifference beteen (t an (t is not c significant. From ( it is possible to etermine the ynamics of the secon moment of the congestion ino, t (t = R (t 3 4 R δ (t R E ( 3 (t. Globecom /4/$. 4 IEEE

6 bit-rate Prob(timeout at time variance(w(t E(W(t 8 7 Mean Only variance(w(t Mean Only bit-rate Prob(timeout at time E(W(t Figure 5. These figures are similar to the ones shon in Figure 4. Hoever, here to sets of ynamics are shon, the loss probability jumping from.% to % an falling from % to.%. In both cases, the changes occur at t = Figure 6. Valiation of the. These figures are near ientical to Figure 5, hoever, here the loss probability jumps beteen % an 5%. Using the same iea as above, e approximate E 3 (t 3/ β 8 an arrive at 3 (c.33/ (t (t = (t (4 t R 3/ 38 β δ (t R (t. 3/ 4 3 (c.3 R As in the case of (, this approximation is exact in steay state. Equations ( an (4 form a system of orinary ifferential equations (ODEs. With the mean an secon moment, the istribution of the congestion ino can be foun as iscusse in Section III. With this approximation of the Globecom 4 istribution, the probability of being in timeout can be foun t as in Section IV, i.e., P (T O at time t = t RT O I (τ τ + t I (τ τ, here I (τ is given by (7 an I (τ is t RT O given by (8. Figures 4-6 provie some valiation of this ynamic moel of TCP. Through extensive simulation, E (t T O, E (t T O, an P (T O at time t ere foun uner ifferent time variations of the loss probability. Specifically, in each simulation, the loss probability took to values δ an δ. The simulation began an ran for secons ith loss probability set to δ. In these simulations, secons as enough to ensure that the system as in steay state (i.e., E (t T O as constant. Then, secons after the beginning of the simulation, the loss probability as sitche to δ. In the figures, this moment is labele as t =. This simulation as carrie out, times. With these simulations it is possible to etermine the value of the statistics E (t T O, E (t T O, an P (T O at time t at each time point. For example, to etermine P (T O at time 5 ms, e foun the fraction of the, flos that ere in the timeout state at time t = 5 ms. This large number of simulations allos for high confience in the observations an allos the etails of the ynamics to be observe. Figures 4-6 inicate that the ynamic moel is quite accurate. They also sho, as expecte, that using the ynamics of the mean congestion ino given by ( is not aequate for large loss probability, but is aequate for small loss probability. We also see that the ynamics for the mean an the variance of the congestion ino perform rather poorly for very large loss probability. Hoever, in this case, the behavior of TCP is completely ominate by the timeout, hich is ell moele. As a result, the bit-rate is accurately moele. Perhaps the most striking aspect of Figures 4-6 is the ynamics of the bit-rate at large loss probabilities. Most significantly, e see that the bit-rate experiences il oscillations that can take several secons to ecay. In all cases, the moel for the mean congestion ino ( oes not prouce any oscillations. To the best of the authors knolege, this behavior has not been previously observe, much less moele. Oscillating step-responses are often a sign of instability, thus it is likely that these ynamics ill impact the stability results of previous AQM research. Figure 6 shos some unexpecte behavior of the mean an the variance of the congestion ino. Specifically, e see that the mean has to istinct phases of groth ith a transition at t = ms hile the variance shos nonmonotonic groth. The ODEs ( an (4 o not moel this behavior. This behavior is ue to the feeback of the fraction of flos in timeout to the mean an the variance of the congestion ino. Notice that the non-monotonic groth of occurs at RTO, just as flo exit timeout. Hoever, since the probability of a flo being in the timeout state is so high, a slight error in the mean congestion ino has a minor impact on the average bit-rate /4/$. 4 IEEE

7 VI. CONCLUSIONS We have evelope a ne moel for the sening rate of TCP that inclues the effect of timeout. Previous moels have inclue the effect of timeout, but e have foun these moels to greatly over estimate the probability of being in timeout. The moel presente is easily extene to a ynamic moel of timeout. This ynamic moel as shon to agree ith simulations quite ell. Furthermore, it as shon that timeout can cause the bit-rate to oscillate substantially for several secons after a change in the loss probability. It as also shon that for large loss probabilities, the behavior of TCP is ominate by the behavior of timeout. This moel is significantly ifferent from those use in previous esign of AQM. Future ork ill examine the possibility of esigning an AQM that utilizes these moels for TCP. Perhaps, more substantial performance gains ill be achieve for large loss probability. REFERENCES [] S. Athuraliya, V. H. Li, S. H. Lo, an Q. Yin, REM: Active queue management, IEEE Netork, vol. 5, pp ,. [] C. V. Hollot, V. Misra, D. F. Tosley, an W. Gong, On esigning improve controllers for AQM routers supporting TCP flos, in IN- FOCOM,. [3] S. Kunniyur an R. Srikant, Analysis an esign of an aaptive virtual queue (AVQ algorithm for active queue management, in Proceeings of ACM SIGCOMM,. [4] V. Misra, W. Gong, an D. Tosley, Stochastic ifferential equation moeling an analysis of tcp-inosize behavior, in Proceeings of PERFORMANCE99, 999. [5] M. Hanley, S. Floy, J. Pahye, an J. Wimer, RFC TCP frienly rate control (TFRC: Protocol specification, 3. [6] J. Beckers, I. Henraan, R. Kooij, an R. V. er Mei, Generalize processor sharing performance moels for internet access lines, in 9th IFIP CONFERENCE on PERFORMANCE MODELLING AND EVALUATION OF ATM an IP NETWORKS, (Buapest,. [7] S. Bohacek, A stochastic moel of tcp an fair vieo transmission, in Infocom, 3. [8] N. Carell, S. Savage, an T. Anerson, ing TCP latency, in INFOCOM (3, pp ,. [9] T. Ott, J. H. B. Kemperman, an M. Mathis, Wino size behavior in TCP/IP ith constant loss probability, in Proc. Of the DIMACS Workshop on Performance of Realtime Applications on the Internet, 996. [] M. Mathis, J. Semke, J. Mahavi, an T. Ott, The macroscopic behavior of the TCP congestion avoiance algorithm, ACM Comput. Comm. Revie, vol. 7, no. 3, 997. [] J. Pahye, V. Firoiu, D. Tosley, an J. Krusoe, ing TCP throughput: A simple moel an its empirical valiation, Proceeings of the ACM SIGCOMM 98 conference on Applications, technologies, architectures, an protocols for computer communication, pp , 998. [] K. S. V. Biplab Sikar, Shivkumar Kalyanaraman, Analytic moels for the latency an steay-state throughput of TCP tahoe, reno, an SACK, IEEE/ACM Transactions on Netorking, vol., no. 6, pp , 3. [3] S. Bohacek, K. Shah, G. Arce, an M. Davis, A revie of central issues - signal processing challenges in active queue management, IEEE Signal Processing Magazine, vol., pp , 4. [4] L. Le, J. Aikat, K. Jeffay, an F. D. Smith, The effects of active queue management on eb performance, in SIGCOMM, 3. [5] Globecom /4/$. 4 IEEE