Analysis of Binomial Congestion Control Λ

Transcription

1 Analysis of Binomial Congestion Control Λ Y. Richard Yang, Simon S. Lam Deartment of Comuter Sciences The University of Texas at Austin Austin, TX fyangyang, TR June 8, Introduction Binomial congestion control was roosed by Bansal and Balakrishnan in [2]. However, the sending rate derivation in [2] is greatly simlied and does not consider the effect of timeouts. Further, even though the authors use ff = 1 and =0:6 for TCP-friendliness in their exeriments; this selection is not justied by their analysis. On the contrary, according to the authors, for ff = 1, they should select such that ff = 1, therefore, =0:5. 0:5 The motivation of this aer is to analyze the sending rate of binomial congestion window adjustment olicy, considering both trili-dulicate loss indications and timeout loss indications. We also consider the selection of ff and for IIAD and SQRT congestion control strategies [2] to be TCP-friendly. This aer suggests that the authors of Binomial should test their rotocol under higher loss scenarios. The balance of this aer is as follows. In Section 2, we describe the Binomial congestion control and state the analysis assumtions. The detail of the derivations is ut in the Aendix. In Section 3, we use the sending rate formula to derive conditions under which a Binomial flow is TCP-friendly. Λ Research sonsored in art by National Science Foundation grant No. ANI and grant no. ANI Exeriments were erformed on euiment rocured with NSF grant no. CDA

2 2 Model and Analysis Assumtions Formally, the Binomial window adjustment olicy is ρ wt+r ψ w t + ff=w k t if no loss w t+ft ψ w t w l t when loss (1) We can see that TCP is a secial case when k = 0, l = 1. In this aer, we analysis the two cases considered by the authors: when k = 1, l = 0, whichis called IIAD (inverse-increase/additive decrease) and k = l = 0:5, which is called SQRT (suare-root). Window adjustment olicy, however, is only one comonent of a comlete congestion control rotocol. Other mechanisms such as congestion detection and round-tri time estimation are needed to make a comlete rotocol. Since TCP congestion control has been studied extensively for many years, Binomial adots these other mechanisms from TCP Reno [5, 6, 8, 1]. In the next subsection, we give a brief descrition of the Binomial congestion window adjustment algorithm. All other algorithms are the same as those of TCP Reno. 2.1 Congestion window adjustment A Binomial session begins in the slowstart state. In this state, the congestion window size is doubled for every window of ackets acknowledged. Uon the rst congestion indication, the congestion window size is cut in half and the session enters the congestion avoidance state. In this state, the congestion window size is increased by ff=w k in each round-tri time, where W is the current congestion window size. Notice that in this analysis we assume that the receiver returns one new ACK for each received data acket. It is straightforward to extend the analysis to consider delayed ACK. Binomial reduces the window size when congestion is detected. Same as TCP Reno, Binomial detects congestion by two events: triledulicate ACK and timeout. If congestion is detected by a trile-dulicate ACK, Binomial changes the window size to W W l. If the congestion indication is a timeout, the window size is set to Modeling assumtions The assumtions and simlications made in this analysis are summarized below. ffl We assume that the sender always has data to send (i.e., a saturated sender). The receiver always advertises a large enough receiver window size such that the send window size is determined by the Binomial congestion window size. 2

3 ffl The sending rate is a random rocess. We have limited our efforts to modeling the mean value of the sending rate. An interesting future toic will be to study the variance of the sending rate which is beyond the scoe of this aer. ffl We focus on Binomial s congestion avoidance mechanisms. The imact of slowstart has been ignored. ffl We model Binomial s congestion avoidance behavior in terms of rounds. A round starts with the back-to-back transmission of W ackets, where W is the current window size. Once all ackets falling within the congestion window have been sent in this back-to-back manner, no more acket is sent until the rst ACK is received for one of the W ackets. This ACK recetion marks the end of the current round and the beginning of the next round. In this model, the duration of a round is eual to the round-tri time and is assumed to be indeendent of the window size. Also, it is assumed that the time needed to send all of the ackets in a window is smaller than the round-tri time. ffl We assume that losses in different rounds are indeendent. When a acket in a round is lost, however, we assume all ackets following it in the same round are also lost. Therefore, is dened to be the robability that a acket is lost, given that it is either the rst acket in its round or the receding acket in its round is not lost [7]. ffl To void having too many arameters, we assume that the receiver returns one new ACK for each received data acket, i.e., no delayed ACK. To model the effect of delayed ACK, we can simly relace all ff with ff=b,whereff is the increasing arameter, and b is the number of data ackets before an ACK is sent. ffl To derive an analytic result, sometimes in the analysis we assume E[W t ] ß E[W ] t,wherew is the window size and t 2 (0; 1). 3 TCP-friendly Binomial Congestion Control As derived in Aendix, the sending rate of both IIAD and SQRT can be exressed as T Binomial (ff; ; ; R; T 0 ) ß R ff + T 0 min 1 1; 3 ff ( ) (2) 3

4 where is the loss rate, R the mean round-tri time, and T 0 the timeout. We should emhasize that to derive (2), in some cases we have assumed is small. For detail, refer to the Aendix. To be TCP-friendly, we need to match T Binomial (ff; ; ; R; T 0 ) to that of TCP sending rate formula, which is T TCP (; R; T 0 ) ß R T 0 min 1 1; ( ) (3) Under low loss scenario, the rst terms in the denominators of (2) and (3) dominates, and we have the exression: ff = 2 (4) 3 For examle, when the Binomial congestion control uses ff = 1, we select =0:66 so that the control is TCP-friendly. To consider the sensitivity of the TCP-friendliness on the arameters, we dene F (ff; ) = = R R r (5) ff Under small loss rate, F is the relative throughut of a IIAD/SQRT flow and a TCP flow. Figure 1 lots F as a function of when ff =1. Comare Figure 1 with the exerimental results in Figure 16 of [2], we nd that the two gures are very similar. This can be considered a validation of (2). However, it is imortant to oint out that F is valid only when loss rate is small. When loss rate is high, we should use the comlete sending rate formula to derive the TCP-friendly ff and, using the methods as in [?]. It also suggests that the authors of Binomial should evaluate Binomial under high loss scenarios. 3 2ff (6) 4

5 Fairness sensitivity to beta F(beta) 1 F = TCP / Binomial beta Figure 1: F as a function of when ff =1 A Sending Rate Derivation We carry the derivation in two stes. In the rst ste, we only consider the case when congestion indications are exclusively of tye trile dulicate ACK (TD). In the next ste, we consider both TD and timeout loss indications. A.1 Congestion indications are exclusively trile-dulicate ACKs We rst consider the case when congestion indications are exclusively of tye trile dulicate ACK (TD). Consider a Binomial flow starting at time t =0.For any given time t>0,denen t as the number of ackets transmitted in the interval [0;t], andt t = N t =t, the sending rate on that interval. Note that T t is the number of ackets sent er unit of time regardless of their eventual fate (i.e. whether they are received or not). Thus, T t reresents the sending rate of the connection. We dene the long-term steady-state rate T to be T = t!1 lim T N t t = t!1 lim t Dene a TD eriod (TDP) to be the interval of time between two TD congestion indications. For the ith TD eriod we dene random variable Y i as the number of ackets send in the eriod, A i the duration of the eriod, and W i the window size at the end of the eriod. Consider fw i g to be a Markov regenerative rocess with rewards fy i g. From renewal theory [3, 4], we know that (7) T = E[Y ] E[A] (8) 5

6 In order to derive an exression for T, the long-term steady-state Binomial sending rate, we next derive exressions for the means of Y and A. Consider a TD eriod as in Figure 2. Packets sent W i ACKed Packt Lost acket W i-1 βw i η i -1 η i θ i TD occurs TDP ends X i Number of rounds TDP i - l Figure 2: A trile-dulicate eriod (TDP) A TD eriod starts immediately after a TD congestion indication, and thus the congestion window size at the start of the ith TD eriod is eual to W i 1 W l. i 1 At the end of each round, the window is incremented by ff=w k,wherew is the window size at the beginning of the round. We denote by i the rst acket lost in TDP i,andx i the round where this loss occurs. After acket i, W i 1 more ackets are sent in an additional round before a TD congestion indication occurs (and the current TD eriod ends). Thus a total of Y i = i + W i 1 ackets are sent in X i +1rounds. It follows that: E[Y ]=E[ ] +E[W ] 1 (9) To derive E[ ], consider a random rocess f i g,where i is the number of ackets sent in a TD eriod u to and including the rst acket that is lost. Based on the assumtion that ackets are lost in a round indeendently of any ackets lost in other rounds, f i g is a seuence of indeendent and identically distributed (i.i.d.) random variables. Given the loss model, the robability of i = k is eual to the robability that exactly k 1 ackets are successfully acknowledged before a loss occurs P [ = k] =(1 ) k 1 ; k =1; 2;::: (10) 6

7 The mean of is thus E[ ] = 1X k=1 (1 ) k 1 k = 1 (11) Plugging (11) into (9), we have E[Y ]= 1 + E[W ] (12) To derive E[W ] and E[A], consider again TDP i. Dene r ij to be P the duration of the jth round of TDP i. Then, the duration of TDP i is A i = i +1 X j=1 r ij. Consider the round-tri time r ij to be random variables that are assumed to be indeendent of congestion window size, and thus indeendent of the round number, j. It follows that E[A] =(E[X] +1)E[r] (13) Henceforth, let R = E[r] denote the average value of the round-tri time. Finally, to derive an exression for E[X], consider the evolution of W i as a function of the number of rounds. First we observe that during the ith TD eriod, the window size increases between W i 1 W l and W i 1 i (see Figure 2). Consider the differential euation: dw dt = ff RW k (14) Solve the differential euation, we have that for t 2 [0;RX i ] W (t) = ff(k +1) R )k+1 1 t + W k+1 k+1 l 1 (1 W i 1 i 1 (15) From (15), and lug in W (RX i )=W i, we solve the exression for X i as 1 X i = W k+1 i W k+1 l 1 i 1 (1 Wi 1 (k +1)ff )k+1 (16) The fact that Y i ackets are transmitted in TDP i is exressed by Y i = Z RXi 0 W (t)dt + i (17) W k+2 i W k+2 l 1 (1 W i 1 = ff(k +2) 7 i 1 )k+2 + i (18)

8 where i is the number of ackets sent in the last round. Consider that i,the number of ackets in the last round, is uniformly distributed between 1 and W i, and thus E[ ] =E[W ]=2 (19) fw i g is a Markov rocess for which a stationary distribution can be obtained numerically. However, a simler aroximate solution can be obtained. Next, we consider two secial cases. The rst case is called IIAD (inverseincrease/additive decrease); the second, SQRT (because the increase and decrease are roortional to the suare-root of the current window). A.1.1 IIAD (k = 1, l= 0) First, lug in k =1, l =0into (16), we have X i = 1 2ff Take exectation on (20), and we have W 2 i (W i 1 ) 2 (20) 2E[W ] 2 E[X] = (21) 2ff Plug in k = 1, l = 0 into (18), take exectations on both sides, comare to (12), we have 1 + E[W ] 2» W 3 (W ) 3 ) = E 3ff = 3E[W 2 ] 3 2 E[W ]+ 3 (23) 3ff Since Var[W ]=E[W 2 ] E[W ] 2, and we assume the variance of W is small, therefore, we can aroximate E[W 2 ] by E[W ] 2. We solve the Euation (23) and derive the exression for E[W ] as E[W ] ß ff Simlify, and we have + s E[W ]= (22) ff + 3ff2 48ff +12ff (24) r ff + o(1= ) (25) 8

9 Therefore, for small value of,wehave E[W ] ß r ff (26) According to (21), and lug in the exression for E[W ], we can derive the exression for E[X], simlify, and we have E[X] = s ff + o(1= ) (27) Next, consider the derivation for E[A]. Plugging the exression for E[X] into (13), we have E[A] = R(E[X] +1) (28) = R s ff + o(1= ) (29) Then, according to (8) for T, (12) for E[Y ], (24) for E[W ], (29) for E[A], we have T = ß 1 1 R + E[W ] E[A] + ff ff (30) (31) Simlify, and we have T ß 1 R r ff + o(1= ) (32) A.1.2 SQRT (k = l = 0.5) First, lug in k = l = 0:5 into (16), we have X i = 1 1:5ff W 1:5 i (W i 1 W 0:5 i 1 )1:5 Assume E[W t ] ß E[W ] t, take exectations on (33), we have (33) E[X] = E[W ] 1:5 E[W ] 1:5 (1 9 1:5ff E[W ] )1:5 (34)

10 Plug in k = l = 0:5 into (18), take exectations on both sides, assume E[W t ] ß E[W ] t, and comare to (12), we have 1 + E[W ] 2 E[W ] 2:5 (1 (1 = 2:5ff E[W ] )2:5 ) To get an analytical exression for E[W ], aroximate (1 2:5, we solve the euation to get E[W ] E[W ]= ff s ff ff 1 (35) E[W ] )2:5 as 1 (36) Simlify, and we have r ff E[W ]= + o(1= ) (37) Therefore, for small value of,wehave r ff E[W ] ß (38) Plug in (36) into (34), simlify, and we have E[X] = s ff + o(1= ) (39) Next, consider the derivation for E[A]. Plug in the exression for E[X] into (13), we have have E[A] = R(E[X] +1) (40) = R s ff + o(1= ) (41) Then, according to (8) for T, (12) for E[Y ], (36) for E[W ], (41) for E[A], we have T = ß 1 1 R 10 + E[W ] E[A] + ff ff (42) (43)

11 Simlify, and we have T ß 1 R r ff + o(1= ) (44) Summarize the result for IIAD and SQRT, we found that for both cases, and E[W ] ß E[X] ß r ff s ff (45) (46) A.2 Congestion indications are trile-dulicate ACKs and timeouts Next, we extend the analysis to include timeouts. The derivation in this section is the same as in [7] excet for ^Q(E(W )). However, we include it here for comleteness. In the revious section, we considered Binomial flows where all congestion indications are due to trile-dulicate ACKs. However, under certain circumstances the majority of window decreases can be due to timeouts. Therefore, a good model should also cature timeout congestion indications. Timeout occurs when ackets (or ACKs) are lost, and less than three dulicate ACKs are received. The sender waits for a eriod of time denoted by T 0,andthen retransmits the rst unacknowledged acket. Following a timeout, the congestion window is reduced to one, and one acket is resent in the rst round after a timeout. If this retransmission is unsuccessful, the eriod of timeout doubles to 2T 0 ;this doubling is reeated for each unsuccessful retransmission until 64T 0 is reached, after which the timeout eriod remains constant at 64T 0. Figure 3 shows a trace with both TDP and timeouts. W i2 W i3 W W i1 R =2 i t i A i1 A i2 A i3 T 0 2T 0 TD Z i Z i TO 4T 0 t S i Figure 3: A trace with both TDP and timeouts 11

12 Let Zi TO denote the duration of a seuence of timeouts and Zi TD interval between two consecutive timeout seuences. Dene S i to be the time S i = Z TD i + Z TO i (47) Also, dene M i to be the number of ackets sent during S i.thenf(s i ;M i )g is an i.i.d seuence of random variables, and we have T = E[M ] E[S] (48) Extend the denition of TD eriod dened reviously to include eriods starting after, or ending in, a TO congestion indication (besides eriods between two TD congestion indications). Let n i be the number of TD eriods in interval Zi TD. For the jth TD eriod of interval Zi TD we dene Y ij to be the number of ackets sent in the eriod, A ij to be the duration of the eriod, X ij to be the number of rounds in the eriod, and W ij to be the window size at the end of the eriod. Also, R i denotes the number of ackets sent during timeout seuence Zi TO.Wehave M i = S i = Xn i j=1 Xn i j=1 Y ij + R i A ij + Z TO i And thus, E[M ] = E[ E[S] = E[ Xn i j=1 Xn i j=1 Y ij ]+E[R] (49) A ij ]+E[Z TO ] (50) If n i is an i.i.d. seuence of random variables, indeendent of fy ij g and fa ij g, then for any i we have E[( E[( Xn i j=1 Xn i j=1 Y ij ) i ] = E[n]E[Y ] (51) A ij ) i ] = E[n]E[A] (52) 12

13 To derive E[n], observe that, during Zi TD, the time between two consecutive timeout seuences, there are n i TDPs, where each of the rst n i 1 end in a TD, and the last TDP ends in a TO. It follows that in Zi TD there is one TO out of n i loss indications. Therefore, if we denote by Q the robability that a congestion indication ending a TDP is a TO, we have Q =1=E[n]. Conseuently, T = E[Y ]+QE[R] E[A] +QE[Z TO ] (53) Since Y ij and A ij do not deend on timeouts, their means are those derived before. However, we still need to derive exressions for Q, E[R], E[Z TO ]. seuence number s k LEGEND received acket lost acket ACK k s m+1 m f w s 1 TD occurs, TDP ends f k+1 w f k k f 1 RTT enultimate round RTT last round time Figure 4: Packet and ACK transmissions receding a loss indication First consider Q. Consider the round of ackets where a loss indication occurs; this round will be referred to as the enultimate round (see Figure 4). We choose the ACK such that ACKs acknowledge individual ackets (i.e. ACKs are not delayed). We will see that the analysis does not deend on whether ACKs are delayed or not. Let w be the current window size. Thus acket f 1 ;::: ;f w are sent in the enultimate round. Packets f 1 ;::: ;f k are acknowledged, and ackets f k+1 is the rst acket to be lost (or not ACKed). We again assume acket losses are correlated within a round: if a acket is lost, so too are all ackets that follow, until the end of the round. Thus all ackets following f k+1 are also lost. However, since ackets f 1 ;::: ;f k are ACKed, another k ackets, s 1 ;::: ;s k are sent in the next round, which we refer to the last round. This round of acket may have another loss, say acket s m+1. Again, our assumtion on acket loss correlation mandates that ackets s m+2 ;::: ;s k are also lost in the last round. The m ackets 13

14 successfully sent in the last round are resonded to by ACKs for acket f k,which are counted as dulicate ACKs. These ACKs are not delayed, so the number of dulicate ACKs is eual to the number of successfully received ackets in the last round. If the number of such ACKs is higher than three, then a TD indication occurs, otherwise a TO occurs. In both cases the current eriod ends. We denote by A(w; k) the robability that rst k ackets are ACKed in a round of w ackets, given there is a seuence of one or more losses in the round. Then A(w; k) = (1 )k 1 (1 ) w (54) Also, let C(n; m) denote the robability that m ackets are ACKed in seuence in the last round (where n ackets are sent) and the rest of the ackets in the round, if any are lost. Then C(n; m) = ρ (1 ) m if m» n-1 (1 ) n otherwise (55) by Then, ^Q(m), the robability that a loss in a window of size w isato,isgiven ^Q(w) =ρ 1 P if w» 3 2 A(w; k=0 k) +P w A(w; k)p 2 k=3 After some algebraic maniulation, we have ^Q(w) = min Observe that m=0 C(k; m) otherwise (56) 1; (1 (1 )3 )(1 + (1 ) 3 (1 (1 ) w 3 )) 1 (1 ) w (57) lim ^Q(w) = 3!0 w (58) A numerical aroximation of ^Q(w) then is ^Q(w) ß min(1; 3 w ) (59) Q, the robability that a congestion indication is a TO, is Q = 1X w=1 ^Q(w)P [W = w] =E[ ^Q] (60) 14

15 We aroximate Q ß ^Q(E[W ]) (61) where E[W ] is from (45). Next, consider the derivations of E[R] and E[Z TO ]. A seuence of k TOs occurs when there are k 1 consecutive losses (the rst loss is given) followed by a successfully transmitted acket. Conseuently, the number of TOs in a TO seuence has a geometric distribution, and thus P [R = k] = k 1 (1 ) (62) Then we calculate the mean of R as E[R] = 1X k=1 kp [R = k] = 1 1 (63) Next, consider E[Z TO ], the average duration of a timeout seuence excluding retransmissions, which can be calculated in a similar way. We know that the rst six timeouts in one seuence have length 2 i 1 T 0, with all immediately following timeouts having length 64T 0. Then the duration of a seuence with k timeout is L k = ρ (2 k 1)T 0 for k» 6 ( (k 6))T 0 for k 7 (64) And the mean of Z TO is E[Z TO ] = 1X k=1 L k P [R = k] (65) = T Now we can lug (12) for E[Y ], (63) for E[R], (13) for E[A], (66) for E[Z TO ], and (61) for Q into (53), and have (66) T = 1 + E[W ]+ ^Q(E[W ]) 1 1 R(E[X] + 1) + ^Q(E[W ])T 0 f () 1 (67) where f () = (68) 15

16 Now we can lug the common exression (45) of IIAD and SQRT for E[W ], the common exression (46) of IIAD and SQRT for E[X], and (59) for ^Q into (67), simlify, and we have T ff; (; R; T 0 ;b) ß R ff + T 0 min 1 1; 3 ff ( ) (69) References [1] M. Allman, V. Paxson, and W. Stevens. TCP Congestion Control, RFC 2581, Ar [2] D. Bansal and H. Balakrishnan. TCP-friendly congestion control for real-time streaming alications. Technical Reort MIT LCS TR 806, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A., May [3] W. Feller. An Introduction to Probability Theory and Its Alication, Vol. 1, volume 1. John Wiley and Sons, 3rd edition, [4] W. Feller. An Introduction to Probability Theory and Its Alication, Vol. 2, volume 2. John Wiley and Sons, 2nd edition, [5] V. Jacobson. Congestion avoidance and control. In Proceedings of ACM SIGCOMM 88, Aug [6] V. Jacobson. Modied TCP congestion avoidance algorithm. Note sent to end2endinterest mailing list, [7] J. Padhye, V. Firoiu, D. Towsley, and J. Kurose. Modeling TCP throughut: A simle model and its emirical validation. In Proceedings of ACM SIGCOMM 98, Set [8] W. Stevens. TCP/IP Illustrated, Volume 1, The Protocols. Addison-Wesley,