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 Delaware University of Southern California Abstract While many moels of the TCP s ynamics have been evelope, few focus on the effects of timeout an high loss probability. Active queue management (AQM) is an important application of these ynamic moels. However, recent work has shown 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 winow. It is shown that, while the ynamics of the mean value of the congestion winow 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 willy oscillate. Finally, this paper inclues results from several million simulations proviing a etaile view of the ynamics of timeout. I. Introuction s of TCP are extensively use throughout networking research. For example, TCP moels are use in fiels such as AQM [], [], [3], [4], esign of TCP frienly transport protocols [5], planning an provisioning networks [6], preicting file transfer time [7], [8], etc. In this way, TCP moels are founation for research. The accuracy of these moels is critical to the accuracy an relevance of the work that rests upon them. s for TCP have evolve over time. Initially, a simple relationship between the sening rate an the loss probability was evelope. The well-known "/sqrt(p)" formula, or more specifically MSS p 3// T T p, was foun an verifie [9], []. While this formula is useful in many settings, it was foun to not be accurate in situations where 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 shown that timeout, as oppose to triple uplicate acknowlegement, is a significant way that TCP etects packet loss []. In orer to inclue the impact of timeout, the simple This work was supporte in part by NSF grant Simulations for the work escribe in this paper were supporte by the University of Southern California Center for High Performance Computing an Communications. "square root of p" formula was extene to inclue the effect of timeout []. This formula was shown to be more accurate for large loss probabilities where timeout is significant. Sikar et al. [] evelope a moel for TCP throughput that inclues time-out. While this moel is ifferentfrom[],theyreportthatitgivessimilarnumerical results. However, we have performe extensive simulations, over 5 million in total, an have foun that when applie to the ns- implementation of TCP-SACK an when loss is probabilistic, the moel given in [] results in errors as large as a factor of two. Consiering the importance of the relationship between 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 was to make them ynamic. That is, the initial moels assume that the loss probability was fixe. Inthecasethat the loss probability varie, ynamic moels were require. The first moel in this irection was given in [4], an several closely relate moels have followe [], [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 was first introuce, there have been a large number of researchers active in this area. ecently, several groups have investigate what, if any, benefit AQM has on the performance of the network (see [3] an references therein). A recent paper along these lines showe that the AQM, be it rop tail or any more elaborate version, has little impact on the performance of the network for utilization below 9%. However, for utilization above 9%, some AQM techniques were shown to improve the performance [4]. This work is significant as it inicates the environment where AQM might have a positive impact, specifically, in networks with 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 where AQM might have the largest impact on performance, the analysis an esign of AQM is least accurate.

2 Because of this inaccuracy, the possible impact of AQM on network performance at high utilizations has never been carefully explore. One reason that AQM esign has not consiere high utilization networks is that ynamics of TCP in such an environment are not known. By ynamics, we 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 willy varying bit-rates that can take several secons to ecay. For example, the lower right of Figure 4 shows such oscillations. This ynamics is far more complicate than the ynamics of the mean congestion winow use by most AQM esign. As a result, this work calls into question the proof of stability of AQM at high loss probabilities. As a by-prouct of this work, the variance an the istribution of the congestion winow is also foun. Specifically, it is shown that the winow size is accurately moele with a negative binomial istribution. An finally the huge number of simulations use to valiate this moel provie a etaile view of the behavior of the ynamics of TCP s timeout mechanism. The paper procees as follows: The next section iscusses the simulations use for verification. In Section III the istribution of the congestion winow is foun. Next, in Section IV, a moel of the probability of a flow 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 with 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 networks the sening rate of a TCP connection is complicate by interacting flows, 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 wil, it is necessary to thoroughly unerstan TCP in a controlle environment. Perhaps the most significant ifference between these simulations an the environment foun in the wil is that in these simulations packet losses are entirely ranom an never ue to queue overflow. In the case of AQM, such simulations are appropriate as the goal of AQM is to rop packets in a controlle fashion, not when the queue fills. The simulations presente in this paper are for a single flow over a single bottleneck topology where rops at the bottleneck are ranom. The ns- error moel was 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 acknowlegement an a maximum receiver winow were 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 were performe on the University of Southern California s Linux cluster [5] an the large simulation results were transferre via the Internet to the University of Delaware for post-processing an storage. There is little oubt that without such high-performance computing, this work woul not be possible. III. The istribution of the congestion winow This section presents a simple moel for the istribution of the congestion winow. While the istribution may be of interest in its own right, we are particularly intereste in this istribution since it is use to etermine the probability of a flow entering timeout. We justify the moel briefly with 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 winow size varies accoring to W t = T T W tn t () where N is a Cox process that counts the number of packet losses. This moel is for the evolution of the congestion winow when the flow is not in timeout. Thus, all probabilities shoul be conitione on the flow not in timeout. We enote this conition as TO. Thismoelwasfurther investigate in [7] where the partial ifferential equation of the winow size was foun. Specifically, if p w, t TO is the probability ensity of the congestion winow taking the value w at time t, thenp satisfies p w, t TO = Ã p w, t TO () t T T w +w δ (t T T ) 4p w, t TO p w, t TO, where δ is the loss probability. If δ is constant, then in p( w,t TO) steay state, i.e., t =, the istribution of the winow solves p w TO = w δ 4p w TO p w TO. (3) w From (3) it is straightforwar to show that E w m TO = c m δ m/,

3 for some constants c m. If m =, then this is the wellknown "square root of p" formula where c has been foun to be between. an.3. Consiering m =, we fin that the variance is of the form Var w TO := E w γ TO E w TO = δ, (4) where γ := c c. This relationship is borne out in simulations that inicates that γ.3. A close form solution has not been foun for (), whereas a complicate close form solution has been foun for (3). However, simple yet accurate approximate solution to (3) is the negative binomial istribution, i.e., p (w) = Γ (N + w ) Γ (N)(w )! ( q)n q w. Since the mean of this negative binomial ranom variable is Nq/( q)+an the variance is Nq/( q), q = E w TO Var w TO an N = q q E w TO (5) an from mean an variance of the w given above we get q = δ γ µ c + δ an N = ( q) q µ c. (6) δ In orer to etermine the istribution of the winow size, the parameters q an N must be etermine. This can be one by plugging the mean an variance into (5). However, if only the loss probability is known, then (6) must be use, in which case the parameters c an γ must beetermine. Wehavefounhatiftheobjectiveisto use the istribution to etermine the probability of timeout, the c = p 3/ an γ =.3 is sufficient. On the other han, if one is intereste in the istribution of the congestion winow (the objective of this section), then the selection of these parameters, especially c requires some more analysis. However, to keep this paper focuse, this issue is not iscusse. Once the parameters are etermine, the istribution of the congestion winow can be estimate. Figure shows the istribution of the congestion winow given the flow is not in the timeout state, i.e., p w TO. These figures shows the three types of curves, the observe istribution (histogram), the istribution with c = p 3/, anheistribution where the parameters are etermine by plugging in the observe mean an variance into (5). This last istribution coincies with the istribution given by (6) but with c =.7 for δ = 4 an c =.4 for δ =.5. For δ =., thevalueofc playslessofarole,c =.4, c = p 3/, anc =.7, or by using the observe mean an variance, all yiel nearly the same istribution. In all cases, γ =.3. probability. observe. observe.9 / /.9 / / (3/).8 /δ (3/).8 /δ.7.7/ δ / or.7 observe m.6 observe mean δ= -4.3 δ= w w probability Figure. Distribution of the congestion winow for loss probability of δ = 4 (left), δ =. (mile), an δ =.5 (right). IV. A moel of timeout In this section, we present a moel of the probability of a flow being in the timeout state. We say that a flow is in the timeout state if the next packet will 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 networks with ientical network characteristics (e.g., link spees, propagation elays, etc.). Our goal is to etermine the fraction of these flows that are in timeout. A flow can enter the timeout state in the following three ways: IIfaflow experiences so many losses that triple uplicate acknowlegements are not receive, i.e., if at least max (w, ) losses occur in one winow. II In the case of the ns- implementation of TCP-SACK, if more than w/ packets are roppe. III If a retransmitte packet is roppe. Consiering I an II, we see that a flow will timeout if a single rop is followe by at least max (min (bw/c,w 3), ) rops out of the next w packets. Given that rops occur at a rate w δ,ropsthat lea to timeout occur at a rate ρ (w, δ) = w w δ X k=max(min(bw/c,w 3),) µ w δ k ( δ) w k. k Let λ (δ) betheratethataflow moves congestion avoiance to timeout ue to I an II. This rate is foun taking the expecte value of ρ (w, δ), i.e., λ (δ) =E ρ (w, δ) TO = X w ρ (w, δ) p δ (w), where p δ (w) is given in Section III, however, we have shown the epenence on δ. This epenence is This way of entering time-out is iscusse in []. FC-357 eliminates this way to enter time-out. It is straightforwar to ajust the evelopment below to reflect FC-357.

4 mae explicit by (6). Note that the approximation E ρ (w, δ) TO ρ E w TO,δ will result in large errors. Consier the situation where δ =.9 an, using c = p 3/, the mean value of the congestion winow is 4. Inthiscase,E ρ (w, δ) TO =.3 while ρ E w TO,δ =.38, aifference aroun a factor of three. Next we etermine λ, the rate that a flow moves from congestion avoiance to timeout because a retransmitte packet is roppe. The rate that a packet is first roppe is δ w. The rop will lea to a retransmission only if triple uplicate acknowlegements are receive, i.e., if less than max (min (bw/c,w 3), ) packets are roppe out of the next w packets. In this case, the retransmission is roppe an timeout is entere with probability δ. Thus ρ (w, δ) :=δ w δ ( w X k=max(min(bw/c,w 3),) µ w δ k ( δ) w k. k The average rate that a flow enters timeout ue to roppe retransmissions is λ (δ) :=E (ρ (w, δ)) = δ µ (δ) Finally, the rate that a TCP flow moves into timeout is λ + λ. Up to this point we consiere the rate that a single flow moves into timeout. Now we will etermine the fraction of flows entering timeout. Let I (t), enotetherate that flows enter timeout at time t. That is, in small time interval t, the probability that some flow enters timeout is I (t) t. Let T O bethetimeinwhichflows remain inthetimeoutstateuponthefirst timeout. If a flow exits timeout, but immeiately experiences another rop, then this flow again enters timeout, but for T O secons. We enote the rate that flows enter timeout for this secon time with I (t). The fraction of flows in timeout are t t T O I (τ) τ + t t T O I (τ) τ. The rate that flows enter timeout is the prouct of the fraction of flows not in timeout an the rate that such flows enter timeout, i.e., I (t) =(λ (δ (t)) + λ (δ (t))) (7) µ µz t Z t I (τ) τ + I (t) τ. t T O t T O Whiletheratethatflows exit timeout only to reenter timeout is I (t) =δi (t T O). (8) We on t consier the case where a flow enters time-out a thir time for a time interval 4 T O. While it is straightforwar to exten the calculations, it oes not appear to improve the accuracy of the moel. In steay state, I (t) an I (t) are constant. So (7) an (8) reuce to I =( I T O I T O)(λ (δ)+λ (δ)) I = δi. This can be solve λ (δ)+λ (δ) I = +T O (λ (δ)+λ (δ)) ( + δ) δ (λ (δ)+λ (δ)) I =. +T O (λ (δ)+λ (δ)) ( + δ) Thus, the fraction of flows in timeout is Z t Z t P (TO)= I (τ) τ + I (τ) τ (9) t T O t T O = ( + δ) T O (λ (δ)+λ (δ)) +(+δ) T O (λ (δ)+λ (δ)). If the loss probability varies, then (7), (8), an the relationship P (in timeout at time t) = t t T O I (τ) τ + t t T O I (τ) τ must be use to etermine the probability that a flow is in timeout. To verify (9), a large number of ns- simulations were performe so that the probability of timeout coul be etermine even for small loss probabilities. Figure shows the observe relationship of P (in timeout) an the loss probability as well as the estimate of this probability given by (9). These calculations use c = p 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 flow spens in timeout, i.e., P (TO). Specifically, [] fins P (timeout) where Q (W (δ),δ) G (δ) T o δ 8 (W (δ)+)+q(w (δ),δ) G (δ) T o, δ () ³+( 3 ³ w 3 δ) ( δ) Q (w, δ) =min, ( ( δ) w ) / ³ 3 ( δ) r 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, while 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%, while () gives an

5 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 flow being in timeout. The ashe green line is the moel s estimate of this probability. Note that these two lines nearly coincie. The thir curve is the relationship erive from []. elative Error in Throughput (fraction).5 Pahye et al..4 FC δ(loss probability) Figure 3. elative Error of Preicte Throughput. error fraction of.5 implies a 5% error. Note that an estimate of 43%. However, this comparison must be qualifie. This paper focuses on TCP-SACK while the focus of [] is on TCP-ENO. Furthermore, this paper examines how TCP behaves uner ranom losses such as arises in AQM. In [], the queueing iscipline is rop tail. Furthermore, the loss probability in [] was not exactly the probability of a packet being roppe, but more along the lines of the probability of a rop event occurring where 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. However, this moel is useful as a benchmark as it was not known how accurate this moel woul be for TCP-SACK. From the probability of timeout, the average throughput can be foun via T = c ( P (TO)) MSS. δ Figure 3 shows the relative error of this moel when compare to the observe throughput. Figure 3 also shows the relative error of the moel in [] as well as the moel suggeste in FC-3448 for a TCP frienly sening rate. The figure shows only the relative error, an examination of the error shows that the moel of [] an FC-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 shows that for even moerately large loss probabilities, the probability of being in timeout is quite high. 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 show the observe an the ashe re curves shows the moel. In the lower right, the otte green inicates the bit-rate if only the mean size of the congestion winow is use (i.e., the ashe re curve in the upper left). Note the units of the bit-rate is in bps. Specifically, for δ>%, timeout plays an important role in the sening rate of TCP. For δ>5%, aflow will spen 6% of the time in timeout. Thus, timeout ominates the sening rate of TCP. There are many implications of this. ecently 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 with these moifications will be much more aggressive than implementations without these moifications. There has been extensive work 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 %. However, in the area of AQM, it is hope that these moels are applicable to higher loss probabilities. Next we evelop a moel for the ynamic behavior of TCP that inclues timeout. V. Dynamics of TCP Here we 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 we present a moel for the variance. Once the variance is known, the istribution is foun as in Section III. With the istribution, we can fin the probability of being in timeout as was one in the previous section. Of course, the paper to this point has only consiere a constant loss probability. Here we allow the loss probability to vary with time. Thus, this moel is appropriate for AQM esign.

6 8 6 5 variance(w(t)) E(W(t)) bit-rate Prob(timeout at time) 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 shown in Figure 4. However, here two sets of ynamics are shown, 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, however, here the loss probability jumps between % an 5%. Let w (t) = E w (t) TO an w (t) = E w (t) TO. An approximation of the ynamics of the mean value of the congestion winow is often given by w (t) = δ (t ) w (t ) w (t). c () This is often approximate as w (t) = δ (t ) w (t). c () Interestingly, this formula is often erive from (). However, if proper stochastic calculus is applie, the correct ynamics for the mean are w (t) = δ (t ) E (w (t ) w (t)), which can be approximate as w (t) = δ (t ) w (t). (3) Note that in (3) the secon moment is utilize, while () have c multiplie by the first moment square. For tunately, we have foun that () an () give goo approximation to the ynamics of the mean. It seems that the mean varies slowly enough that this ifference between w (t) an c w (t) is not significant. From () it is possible to etermine the ynamics of the secon moment of the congestion winow, 3 w (t) = w (t) δ (t ) E w3 (t). 4 Using the same iea as above, we approximate 3/ ³ an arrive at E w3 (t) 83 β 3/ w (t) (c.3) w (t) = w (t) (4) ³ 3/ 38 β. δ (t ) w (t) 3/ 4 3 (c.3) 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 winow can be foun as iscusse in Section III. With this approximation of the istribution, the probability of being in timeout can be foun as in Section IV, i.e., P (T O at time t) = t t I (τ ) τ + t T O I (τ ) τ, where I (τ ) is given t T O by (7) an I (τ ) is given by (8). Figures 4-6 provie some valiation of this ynamic moel of³ TCP. Through extensive simulation, E w (t) T O, E w (t) T O, an P (T O at time t) were foun uner ifferent time variations of the loss probability. Specifically, in each simulation, the loss probability took two values δ an δ. The simulation began an ran for secons with loss probability set to δ. In these simulations, secons was enough to ensure that the system was in steay state (i.e., E w (t) T O was constant). Then, secons after the beginning of the simulation, the loss probability was switche to δ. In the figures, this moment is labele as t =. This simulation was carrie out, times. With these simulations it is possi ble³ to etermine the value of the statistics E w (t) T O, E w (t) T O, an P (T O at time t) at each time point.

7 For example, to etermine P (TO at time 5 ms), we foun the fraction of the, flows that were in the timeout state at time t = 5 ms. This large number of simulations allows for high confience in the observations an allows the etails of the ynamics to be observe. Figures 4-6 inicate that the ynamic moel is quite accurate. They also show, as expecte, that using the ynamics of the mean congestion winow 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 winow perform rather poorly for very large loss probability. However, in this case, the behavior of TCP is completely ominate by the timeout, which is well moele. As a result, the bit-rate is accurately moele. Perhaps the most striking aspect of Figures 5-6 is the ynamics of the bit-rate at large loss probabilities. Most significantly, we see that the bit-rate experiences wil oscillations that can take several secons to ecay. In all cases, the moel for the mean congestion winow () oes not prouce any oscillations. To the best of the authors knowlege, 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 will impact the stability results of previous AQM research. Figure 6 shows some unexpecte behavior of the mean an the variance of the congestion winow. Specifically, we see that the mean has two istinct phases of growth with a transition at t = ms while the variance shows non-monotonic growth. The ODEs () an (4) o not moel this behavior. This behavior is ue to the feeback of the fraction of flows in timeout to the mean an the variance of the congestion winow. Notice that the nonmonotonic growth of w occurs just as flow exit timeout. Since the probability of a flow being in the timeout state is so high, a slight error in the mean congestion winow has a minor impact on the average bit-rate. VI. Conclusions We have evelope a new moel for the sening rate of TCP that inclues the effect of timeout. Previous moels have inclue the effect of timeout, but we 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 was shown to agree with simulations quite well. Furthermore, it was shown that timeout can cause the bit-rate to oscillate substantially for several secons after a change in the loss probability. It was also shown 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 work will examine the possibility of esigning an AQM that utilizes these moels for TCP. Perhaps, more substantial performance gains will be achieve for large loss probability. eferences [] S.Athuraliya,V.H.Li,S.H.Low,anQ.Yin, EM:Active queue management, IEEE Network, vol. 5, pp ,. [] C. V. Hollot, V. Misra, D. F. Towsley, an W. Gong, On esigning improve controllers for AQM routers supporting TCP flows, in INFOCOM,. [3] S. Kunniyur an. 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. Towsley, Stochastic ifferential equation moeling an analysis of tcp-winowsize behavior, in Proceeings of PEFOMANCE99, 999. [5] M.Hanley,S.Floy,J.Pahye,anJ.Wimer, FC TCP frienly rate control (TFC): Protocol specification, 3. [6] J.Beckers,I.Henrawan,.Kooij,an.V.erMei, Generalize processor sharing performance moels for internet access lines, in 9th IFIP CONFEENCE on PEFOMANCE MODELLING AND EVALUATION OF ATM an IP NET- WOKS, (Buapest),. [7] S. Bohacek, A stochastic moel of tcp an fair vieo transmission, in Infocom, 3. [8] N. Carwell, S. Savage, an T. Anerson, ing TCP latency, in INFOCOM (3), pp ,. [9] T.Ott,J.H.B.Kemperman,anM.Mathis, Winowsize behavior in TCP/IP with constant loss probability, in Proc. Of the DIMACS Workshop on Performance of ealtime 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. eview, vol. 7, no. 3, 997. [] J. Pahye, V. Firoiu, D. Towsley, 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 Networking, vol.,no.6,pp ,3. [3] S. Bohacek, K. Shah, G. Arce, an M. Davis, A review 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 web performance, in SIGCOMM, 3. [5]