Modelling TCP with a Discrete Time Markov Chain

Transcription

1 Modelling TCP with a Discrete Time Markov Chain José L Gil Motorola josegil@motorola.com ABSTRACT TCP is the most widely used transport protocol in the Internet. The end-to-end performance of most Internet applications depends largely on TCP. During the last decade researchers have devoted significant effort to model the performance of TCP as a way to understand better its operation. Modeling techniques range from the application of renewal theory, fixedpoint methods and fluid models to control theoretic approaches. In this paper we model TCP by means of a discrete time Markov chain, which allows us to model TCP in much greater detail than the other techniques. Specifically, rather than assuming an idealized stochastic evolution of the TCP congestion window size as other models do we model its exact evolution as dictated by the slow start, congestion avoidance and loss recovery algorithms. We model TCP features ignored in other models such as the details of the fast recovery phase, the slow start after a timeout and the dynamics of the slow start threshold. Our technique can be applied to model any variant of TCP such as TCP Tahoe, Reno and SACK. Keywords TCP, TCP Reno, performance,analytic modelling, Markov chain.. INTRODUCTION The TCP protocol continues to be by far the most used transport protocol by Internet applications. The performance of such applications depends to a large degree on the performance of TCP. Therefore the performance analysis of TCP is important to predict and understand the end-to-end performance of Internet applications. The performance analysis of TCP by means of analytical methods is important for several reasons. First, because it provides a mean to rapidly predict performance figures as opposed to simulations. Second, because the effort itself to abstract TCP into a mathematical model provides insights into the fundamental principles of the TCP algorithms difficult to gain through simulations. Third, because new protocols and congestion control techniques such as the equation-based [] are designed around TCP congestion control equations. And fourth, because analytical models can be integrated into simulations to speed them up [6]. Over the last decade several TCP analytical models have been developed motivated by one or more of the reasons above. In [8] TCP is simplified to its congestion avoidance algorithm operating under periodic losses. This resulted in the simplest of the TCP models. In [] a more realistic model of TCP that includes timeouts, fast retransmit and the receiver s window limitation is developed. By applying renewal theory and simple probability the authors obtained a closed form solution for the throughput of a TCP sender. More recently fixed-point methods and fluid models have been applied to analyse the performance of several TCP sources interacting with a network ([3],[9]). These models represent a major contribution since they depart from the single TCP source model to the modeling of a number of TCP sources within a network. However all these models do not include important TCP features such as fast recovery, slow start after a timeout and the impact of the slow start threshold. In this paper we are interested in modeling the impact of these features on the mean TCP throughput of a sender. In particular, we model the slow start phase after a timeout has occurred. We also model the exact dynamics of the slow start threshold variable as driven by the congestion window size value at the time a segment is detected as lost. We model in detail the fast retransmit and fast recovery phase including possible timeouts while fast recovery is in action. In the process of modelling the fast recovery phase we determine the mathematical conditions for fast recovery to be triggered and to succeed. We find out that the fast recovery mechanism under FIFO drop-tail buffers predominates over timeouts only for packet loss probabilities below %, which perhaps explains why more timeouts than fast recovery phases are seen in the Internet []. We model the congestion avoidance phase as soon as the congestion window size reaches the slow start threshold value or after a successful fast recovery phase. As other models we include the flow control mechanism imposed by the receiver, the average round trip time and the back-off period triggered by consecutive timeouts. Motivated by our interest of modeling the performance of TCP over mobile communication systems we consider the viewpoint of a mobile operator that knows the mean round trip time and packet loss of the network through network statistics and field measurements and desires to rapidly predict the mean throughput of a TCP sender. We model the performance of TCP over drop-tail buffers as these are the most typical in mobile communication systems such as GPRS and UMTS. In section we develop a discrete time Markov chain to model TCP Reno. In section 3 we present the results of our analytical model and compare them with the model in [] and with a model of TCP without fast recovery based on our Markov chain. We find that fast recovery increases modestly the TCP throughput when compared to TCP with only timeout. P0/

2 . MODEL OF TCP RENO As in [] we define the throughput B t of a TCP connection in the time interval [0,t] as: where N t is the number of segments transmitted by the TCP source in the interval [0,t]. In the long-term the TCP throughput B is: B Bt Nt = () t = lim B = lim t () t t N t t We are interested in deriving a mathematical relation for B as a function of the packet loss probability, the average round trip time, the timeout value and the receiver s advertised window. We consider the loss of packets to be caused by drop-tail buffers. We model drop-tail buffers by assuming that given a segment has been lost the next segments in the same round will be lost too. We assume the receiver s advertised window to remain constant during the TCP transfer and equal to its maximum value W max. To derive an expression for B we analyse the stochastic evolution of TCP in terms of cycles as in []. We define a statistical cycle as the period between two consecutive lost segments. The Figure illustrates our definition of cycle. We consider consecutive timeouts while the congestion window size is equal to one to be within the same cycle. cycle time=t cycle time=t cycle time=t 3 ssthresh slow start ssthresh slow start congestion avoidance ssthresh congestion avoidance slow start time Figure. TCP cycle definition. In order to derive an analytical expression for B we define the random variable R to be the number of segments transmitted in one cycle. And we define the random variable T to be the time duration of the cycle. If we think of R and T as a reward assigned at the end of each cycle we have what is called a renewal reward process. We denote by R i the reward assigned to the i th cycle. The R i form a sequence of random variables that we assume to be independent since the number of segments transmitted in one cycle does not depend on the number of segments transmitted in the previous cycle. We denote by T i the duration of the i th cycle. The T i also form a sequence of independent random variables. The sequences {R i } and {T i } might not be independent between them. As opposed to the general renewal process modeled in [] the sequence {R i }(and equivalently the {Ti}) are not identically distributed since they depend on the value of the congestion window size at the beginning of the cycle, the congestion control phase the cycle starts in (slow start or congestion avoidance) and the value of the slow start threshold ssthresh if the cycle starts in slow start. However the R i and T i follow one of the integer[w max /] possible different distributions since there are integer[w max /] different values for the slow start threshold (ssthresh). We identify each distribution as F j.therefore the average number of segments transmitted in a cycle is: J Fj j= ER [ ] = λ ER [ ] (3) where λ Fj is the probability that a cycle follows the F j distribution and E[R Fj ] is the average number of segments transmitted in a cycle when the F j distribution is followed. Similarly the average cycle time is: J Fj j= Fj ET [ ] = λ ET [ ] () By applying the renewal theory theorem we have that the long-term throughput B is equal to: Fj E[ R] B = (5) E[ T ] From equation (5) our approach to obtain B is to calculate the mean number of segments transmitted in a cycle E[R] and the mean duration of a cycle E[T]. E[R] and E[T] depend on the stochastic evolution of the congestion window size (). Hence we model the evolution of to derive E[R] and E[T]. This is where our model deviates from [], which does not model to obtain B. The evolution of in TCP Reno is governed by the slow start, congestion avoidance, fast retransmit, fast recovery and timeout algorithms. We describe now how we model all these algorithms. We assume the reader is familiar with these algorithms otherwise we recommend reading [] and []. We model the stochastic behaviour of as a discrete-time Markov chain. Our reason to model as a discretetime Markov chain is based on the observation that the future values of depend on the present value of but not P0/

3 the past values. We note that in [3] the authors develop a continuous-time Markov chain to model. However by considering a discrete-time Markov chain we eliminate their assumption that the transition time between states is exponentially distributed, which is difficult to justify in real TCP transactions. In our Markov chain the state is the congestion window size. There are slow start and congestion avoidance states, which we label as ss and ca respectively. The transitions among states are dictated by the slow start, congestion avoidance, timeout, fast recovery and the slow start threshold ssthresh. Figure illustrates the Markov chain when W max =6. The transitions are explained in the remainder of this section. Our Markov chain is composed of what we call horizontal branches. There is one branch for every possible ssthresh value. Therefore a Markov chain contains integer[w max /] - branches. We numerate the branches as x=ssthresh- and the states within a branch as ss(x,y) or ca(x,y) where the y corresponds to the number of the state starting from the left with the number one. A branch always starts in a slow start state with =. During slow start for every ack received the flow transitions to the next slow start state with =+. When =ssthresh if an ack is received the flow transitions to the first congestion avoidance state in the branch with =ssthresh+. As in [] we consider that congestion avoidance takes over slow start as soon as >ssthresh. For every new ack received the flow transitions to the next congestion avoidance state with =+n/, where n={,,}. When the flow reaches = W max it remains there as imposed by the receiver s flow control and as long as segments are not lost. If a segment is lost an ack will not be received and the flow will transition back to a slow start state with = if a timeout occurs or to a slow start state with =/ if fast recovery succeeds. We determine the probability of such transitions in the next subsection.. Timeout and fast recovery transition probabilities When a segment is lost depending on certain conditions a timeout or a fast recovery transition will happen. In this subsection we determine the necessary conditions and the probability for a fast recovery transition to occur. The states that do not satisfy the necessary conditions for fast recovery to succeed will experience a timeout transition with probability p, which is the probability that a segment is lost. The probability that a segment is not lost is q=-p. We assume segments are lost randomly before the first loss in a round of segments. When a timeout occurs there is a transition to the slow start state with = of the branch x=/- since when a timeout occurs ssthresh is halved and is set to one. The condition for a state to have a fast recovery transition depends on the number of segments that are lost and the value of. Our model of drop-tail buffers implies that when one segment is lost the remaining segments in the same burst are lost too. We determine the necessary conditions for fast recovery to be triggered with the help of the Figure 3. When a segment is lost its fast recovery is triggered when D duplicate acks are received and the lost segment is retransmitted. In general, if L=n segments are lost in a round the necessary condition for fast recovery to be triggered for the L=n lost segments is: ( + ) L- + (n- D+L+L L +L +...+L (6) We assume in our model that L + =L =L = =L (n- =0 and D=3 as recommended in [], therefore the necessary condition becomes: L- ( ) 3 + L (7) We note that the condition (7) is independent of the packet discard policy implemented by the queue discipline. However this condition together with our assumption of drop-tail buffers determine what states in our Markov chain might experience a fast recovery transition. Specifically we label a state as an L= state if it can only see one lost segment in the round and the condition (7) is satisfied. Equivalently a L=n state is that one that can only see n lost segments and satisfies the condition (7). The Figure 3 shows an L= state. The probability that a L= state will transition to the state with =/ of the branch x=integer[/]-, or equivalently that fast recovery will be successful when only one segment is lost is: P = p q (8) ca, For an L= state the fast recovery transition probability to a state with =/ is: P ca, + = p q (9) In general, the probability of a fast recovery transition from an L=n state to a state with =/ n is: P0/ 3

4 Figure. Markov chain that models the stochastic evolution of the TCP congestion window size. TCP segment legend: L+=0, F+=0 L'=0, F'= data packet lost packet ACK L=, F=8 first lost segment successfully recovered by fast recovery 8+/8 8+/8 8+6/8 8+5/8 8+/8 8+3/8 R ndupacks'=6 ndupacks''= not enough dupacks for second lost segment time Figure 3. Example of terminology used in the determination of the necessary conditions for a successful fast recovery transition. This specific example shows an unsuccessful fast recovery transition from the congestion avoidance state with =8+6/8 due to an insufficient number of duplicate acks. P ca, n +... ( ) + n n + + L = p q (0) Our assumption of considering L + =L =L = =L (n- =0 is well justified by the following argument. If during the n- first fast recovery phases one or more segments are lost the fast recovery transition will be to a state with =/ n+m where m is the additional number of lost segments. The fast recovery transition probability to this new state is: P0/

5 α β P = p q ca, n+m α ( ) β L L + m L If we compare (0) with () we conclude that for the range of packet loss probabilities of practical interest in the Internet, that is p 0., the fast recovery transition probability to the state with =/ n+m is at least 0 times smaller than the transition probability to the state with =/ n, which justifies our simplification. It is important to note that the timeout transition probabilities for the states that can have fast recovery transitions are not equal to p. This is because the fast recovery phase might not be successful and terminates in a timeout. For a generic state L=n we identify n+ different possible timeout transitions. The first one happens when fast retransmit and fast recovery fail to recover the first lost segment. In this case there will be a timeout transition to the first state of the branch with ssthresh=/ with probability: ( P = p q ) () TO, The second possible timeout transition occurs when the second lost segment has not been successfully recovered. The transition is to the first state of the branch with ssthresh=integer[/] with probability: - + P = p q -q TO, () (3) In general, the k th timeout transition is determined by the recovery of only k- lost segments. The transition is to the first state of the branch with ssthresh=integer[/ k ]and happens with probability: - 3 ( ) k- + + k + k- k- P = p q -q k () TO, k The last possible timeout transition from state L=n is determined by the successful recovery of the n lost segments and the loss of one or more segments during the previous fast recovery rounds. We identified this case previously when we were calculating the fast recovery transition probabilities and concluded that the probability of recovering successfully all the n+m lost segments was negligible with respect to the fast recovery transition =/ n. This essentially means that a timeout transition to the first state of the branch with =/ n+ will happen. The timeout transition probability for this last case can be derived from the fact that all departing transition probabilities from a state must equal one, therefore: n - TO, ca, i= TO, n+ n i P = p - P P (5) Figure shows the probabilities of the timeout and fast recovery transitions for an L= state with = as a function of the packet loss probability. In addition it shows the fast recovery transition probability for the case =/8, which is very small compared to the rest of the transition probabilities. It is interesting to note from Figure that successful fast recovery transitions only predominate over timeout transitions for packet loss probabilities below %.. TCP throughput In this subsection we explain how to apply our Markov chain to derive an expression for the mean TCP throughput B. To derive an expression for B we calculate the terms E[R] and E[T] as indicated in the equation (5). Because there are {integer[w max /]-} different cycle distributions E[R] and E[T] must be a weighted average over all the possible cycle distributions. Given that a cycle might start in a slow start state ss(x,) or in a congestion avoidance state ss(x,x+) the mean number of segments transmitted in a cycle is: πx, π (6) x, x+ x= ER [ ] = ER [ ssx (,)] + ER [ ssxx (, + )] where π x, and π x,x+ are the probabilities that the cycle starts in ss(x,) and ss(x,x+) respectively. These probabilities correspond to the steady-state probabilities of our Markov chain. Equivalently the mean time duration of a cycle is: P0/ 5

6 πx, π (7) xx, + x= ET [ ] = ET [ ss( x,)] + ET [ ss( x, x+ )] We next find out the steady-state probabilities of ss(x,) and ss(x,x+) and derive the analytical expressions for the conditional expectations E[R ss(x,y)] and E[T ss(x,y)]. Timeout and f ast recovery probabilit ies ( =, L=) probability Pto / Pto / Pca / Pto /8 Pca / packet loss rate (p) Figure. Timeout and fast recovery transition probabilities for an L= state as a function of the packet loss probability... Steady-state probabilities We find the probabilities π x, and π x,x+ by solving the steady-state equations of our Markov chain. For illustration purposes we develop the steady state equations for the L= case, which includes only L= states. The general case L=n can be developed as an extension of the L= case. We develop the steady state equations around the states ss(x,) and ss(x,x+). The first key element in the development of the steady-state equations is to establish the position y of the L= states: yl (, x' = ) = 0 + ( ) ( 5) yl (, x' = ) = y 0, L + [ ( x' + )] ( x ' + ) + [ ( x' + )] [ ( x ' + ) + ] k yl (, x' =, k = 3,,...) = y 0, L + [ ( x' + )] ( x' + ) + [ ( x' + )] [ ( x' + ) ] k yl (, x ', k = 3,,...) = y 0, L + [ ( x ' + )] ( x ' + ) + [ ( x ' + )] [ ( x ' + ) ] log ( x' + ) + y0, L = We can find π, from the requirement that all the steady-state probabilities must sum one: ( ) x ( x+ 3) x j j, = x, q x, x+ q x= j= x= j= ( ) ( x )( x+ ) x π xx, + q π π π x= We find π x, (x ) by considering all the timeout transitions to state ss(x,), that is from states with ={(x+),(x+)+} and ={(x+),,(x+)+3}: π x, = timeout ssthresh all states + L = timeout ssthresh + L = timeout ssthresh We find π x,x+ (x ) considering all the fast recovery transitions to the state ss(x,x+): x π xx, + = q π x, L = fast re covery ssthresh +.. Derivation of E[R ss(x,y)] E[R ss(x,y)] is the mean number of segments transmitted in a cycle that has started in the state ss(x,y). It can be calculated as the sum of the mean number of segments transmitted by each state visited during the cycle: (8) (9) (0) () P0/ 6

7 ymax ER [ ssxy (, )] = R = ER [ ] () ss( xy, ) xj, j= y The mean number of segments transmitted by each state in a cycle is equal to the product of the mean number of visits to the state by the mean number of segments transmitted by the state in each visit: ER [ x, y] = Ev [ x, y] Er [ x, y] (3) Based on our definition of cycle the mean number of visits to each state in a cycle that starts in slow start is: p + x = y = q x y = Ev [ xy, ] = y q x y y' max = ( ) ( x ) ( x + ) y q x y' max < y ymax = + x x+ W q max ( ) ( ) ( ) The mean number of visits to each state of a cycle that starts in congestion avoidance is: W max x y = x+ W Ev [ ] q x x y y' y x max xy, = + max y x q x y' W max < y ymax max q The mean number of segments transmitted per visit by each state when the cycle starts in slow start is: where for x : The specific terms in the expression (7) are: Er [, ] x= y= Er [ x,] x y= Er [ xy, ] = x y x+ x x + y withinteger ymax x x + y with non integer ymax E[ rx, ] = E[ rx, timeout L ] P[ timeout L TO] + E[ rx, timeout L = ] P[ timeout L = TO] + E[ rx, timeout L = ] P[ timeout L = TO] TO = timeout E[ rx, timeout L ] = () (5) (6) (7) 3 E[ rx, timeout L = ] = + q q + q q + q p+ q p (8) For x=: E[ rx, timeout L = ] = q 3 q q + q p + + P0/ 7

8 p E[ r, ] = exp ression (3) + E[ back off retransmissions] = exp ression (3) + (9) q The mean number of segments transmitted per visit by each state when the cycle starts in congestion avoidance is: x + x y = x + E[ rxy, ] = x x + y withinteger ymax x x + y with non integer ymax ymax = ( + ) ( x ) ( x+ ) E[ rxx, + ] is calculated belowsparetely..3 Derivation of E[T ss(x,y)] E[T ss(x,y)] is the mean time duration of the cycle when it starts in the state ss(x,y). It can be calculated as the sum of the mean time spent in each state during the cycle: ET [ x, y] Ev [ x, y] Et [ x, y] (30) = (3) The mean time spent in each state is equal to the product of the mean number of visits to the state during a cycle by the mean time spent in the state per visit: ymax ET [ ss( x, y)] = T = ET [ ] (3) ss( xy, ) xj, j= y The mean number of visits to each state has been already discussed in the subsection... We understand the mean state time per visit as the time elapsed between the transmission of the segment and the reception of its ack. The mean time spent in each state on a single visit varies on whether the cycle has started in slow start or in congestion avoidance. When the cycle starts in slow start the mean visit time for the states at the start of a round with exception of the ss(x,) state is one RTT. The mean visit time for the state ss(x,) of the branches with x is: where: E[ tx, ] = E[ tx, timeout L ] P[ timeout L = 0 TO] + E[ tx, timeout L = ] P[ timeout L = TO] + E[ tx, timeout L = ] P[ timeout L = TO] E[ tx, timeout L ] = To RTT 3 ( ) E[ tx, timeout L = ] = To + RTT q (3) E[ tx, timeout L = ] = To The mean state time for the ss(,) state considering consecutive timeouts is: 7 T o p E[ t, ] = exp ression (38) + E[ backoff time] = exp ression (38) + p 6 p q We assume for the rest of the states a zero mean time, which can be understood as all the acks belonging to the same round arriving at the same time. We note that asymmetry factors such as the ones described in [] could be captured by assigning non-zero times to these states since asymmetry could be modelled as acks not arriving at the same time. This modelling option offers an opportunity to explore the effects of asymmetry on the TCP throughput. We do not pursue this effort in this paper. When the cycle starts in congestion avoidance the mean visit time for ss(x,x+) state is RTT. The mean visit time for the states at the start of a round is RTT. For the rest of the states the mean time is zero. We consider that in the fast recovery transition to state ss(x,x+) x+ acks arrive together. Therefore the next round of segments starts at state ss(x, x+). 3. RESULTS The fundamental results of our model are the prediction of the TCP Reno steady-state throughput and the distributions of the congestion window size and the slow start threshold against the packet loss probability. The Figure 5 compares the average throughput of TCP Reno predicted by our model with the throughput predicted by [] (PFTK), which does not model the slow start phases, the ssthresh variable and the detail of fast recovery as we do. The comparison is done for the L= case with W max =6, RTT= and T o =3RTT. Both models predict very similar results though for packet loss probabilities below % the elegant model developed in [] is slightly more optimistic perhaps because it considers that the fast recovery phase is always successful when in fact it might terminate in a timeout (see Figure ). (33) (35) P0/ 8

9 6 TCP THROUGHPUT H=6, To=3RTTL t hroughput H p prt T L Our model PFTK Old Tahoe packet loss probability Figure 5. TCP Reno throughput against packet loss probability considering RTT=, To=3TT and W max =6. The model in [] has been validated with measurements and given the similarity in results with our model it is interesting to note that our assumption that the TCP congestion window size behaves as a Markov process is validated. In the Figure 5 we also show results derived from our model without fast recovery transitions (only timeout transitions), which is one of the early versions of TCP known as OldTahoe. The benefits of fast recovery under a drop-tail buffer environment are fairly modest and only manifested for packet loss probabilities below 5%. Figure 6 shows the distribution of the congestion window size against the packet loss probability for the case W max =0. This is an important new result that helps to validate assumptions made on the stochastic evolution of the congestion window size by other TCP models. For the packet loss range in which fast recovery dominates over timeouts the congestion window size is maintained within a 0% of its maximum. 0 TCP congestion window size distribution H=0,L=L 8 c wnd H# s egment sl packet loss probability Figure 6. TCP congestion window size distribution against packet loss probability (W max =0, RTT=, T o =3RTT). The distribution of the slow start threshold against packet loss is shown in the Figure 7. Observe the correlation between the congestion window size and the slow start threshold distribution graphs. 5 TCP ssthresh distribution H=0,L=L s sthresh H# s egment sl packet loss probability Figure 7. Slow start threshold against packet loss probability (W max =0, RTT=, T o =3RTT). P0/ 9

10 . CONCLUSIONS AND FUTURE WORK In this paper we have developed a new and detailed analytical model for TCP Reno in a drop-tail buffer environment that includes features not captured by previous models. Specifically, we have modelled the exact behaviour of the slow start phase after a timeout, the explicit evolution of the slow start threshold variable and its role as the switching point between slow start and congestion avoidance, and the details of the fast retransmit and fast recovery algorithms. The outcomes of this model are not only more accurate throughput results but valuable insights on the exact distributions of the congestion window size and the slow start threshold against the packet loss probability. The throughput results show that the fast recovery algorithm offers a modest increase in throughput over TCP implementations with only the timeout mechanism. In addition, the results show that fast recovery predominates over timeouts only in network environments with packet loss rates below %. As an indirect result our model proves that the stochastic evolution of the TCP congestion window size can be well modelled as a Markov process. Our model opens a new avenue to study analytically the impacts of network asymmetry on the TCP throughput by assigning an inter-ack time reward to the intermediate congestion window size states of our Markov chain. 5. REFERENCES [] Allman, M., Paxon, V., and Stevens, W. TCP congestion control. RFC 58, Internet Engineering Task Force (IETF), Apr 999. [] Balakrishnan H., Padmanabban V. N., and R. H. Katz. The Effects of Asymmetry on TCP performance. MOBICOM 97. [3] Casetti, C., and Meo, M. A new approach to model the stationary behavior of TCP connections. Proceedings INFOCOM, March 000. [] Floyd S., Handley M., Padhye J., and Widmer J. Equation-based congestion control for unicast applications: the extended version. Proc. ACM SIGCOMM 00, August 000. [5] Kumar A. Comparative Performance Analysis of Versions of TCP in a Local Network with a Lossy Link. IEEE/ACM Transactions on Networking, Vol.6, No., Aug 998. [6] Liu Y., Lo Presti F., Misra V., Towsley D., and Gu Y. Fuild Models and Solutions for Large-Scale IP Networks. In Proceedings of ACM SIGMETRICS 03. June 003. [7] Mahdavi, J., and Floyd, S. TCP-friendly unicast rate-based flow control. Note sent to endend-interest mailing list, January 997. [8] Mathis, M., Semke J., Mahdavi, J., and Ott, T. The macroscopic behaviour of thetcp congestion avoidance algorithm. Computer Communication Review,7(3), July 997. [9] Misra, A., Gong, W.B., and Towsley, D.F. Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. ACM SIGCOMM 00. [0] Ott, T., Kemperman, J., and Mathis, M. The stationary behaviour of ideal TCP congestion avoidance. [] Padhye, J., Firoiu, V., Towsley, D., and Kurose, J. Modeling TCP throughput: A Simple Model and its Empirical Validation. Proc. ACM SIGCOMM 98, September 998. [] Sikdar, B., Kalyanaraman, S., and Vastola, K.S. TCP Reno with Random losses: Latency, Throughput and Sensitivity Analysis. Proc. IEEE IPCCC, pp , April 00. [3] Sikdar, B., Kalyanaraman, S., and Vastola, K.S.. Analytic Models and Comparative Study of the Latency and Steady-State Throughput of TCP Tahoe, Reno and SACK. [] Stevens, W.R. TCP/IP Illustrated. Volume. Addison Wesley, 99. P0/ 0