A High-Order Model for Congestion Control Using Exponential Increase

Transcription

1 A High-Order Model for Congestion Control Using Exponential Increase D-H. HOANG Department of Computer Science and Automation Technical University of Ilmenau D Ilmenau GERMANY Abstract: Due to the dynamic behavior of the network traffic, a suitable congestion control remains a critical issue especially for multimedia communication. In contrast to the typical increase-by-one decrease-to-half strategy used in various schemes, we investigate a general approach in which the increase and decrease values are parameters. We observe the exponential behavior of the round-trip delay. Therefore, we propose an exponential increase in order to consider the influence of the round-trip time on the throughput. Thus, a general model for higher order control is achieved that overcomes the problem of using constant additive increase while improving good smoothness and convergence of the control. Furthermore, we derive the relationship between the throughput and the increase/decrease parameters and show how to choose these parameters to obtain high throughput. The performance analyze shows that the exponential control model provides better protection against rate fluctuations compared to constant additive increase in other schemes. Key-Words: Network performance, Congestion control, Multimedia communication, Quality of Service. 1 Introduction Congestion control is becoming an important issue in a shared network due to the growth of the network and the network load. In generality, congestion control is concerned with allocating the network resources and is heavily influenced by the traffic dynamic. It is not easy to get a satisfactory solution due to the complexity of the problem. There is a number of network policies that affect the congestion control and there are several requirements for the design of a congestion control scheme [1]. Among the existing congestion control algorithms (see [2] for details), the additive increase multiplicative decrease (AIMD) algorithm is widely deployed. This algorithm is the basis of congestion control for Transmission Control Protocol (TCP)[23]. The key principle of AIMD is that it raises load by a constant if no loss occurs and responds to packet losses, i.e. congestion indication, by reducing load to a fraction of the current value. TCP is not well-suited for new applications such as streaming multimedia. To address this issue, one approach is to deploy numerous mechanisms at the network and the link layer in order to ensure the application's quality of service. A second approach is to promote the use of end-to-end congestion control schemes [6]. Since TCP is the dominant traffic, the new congestion control schemes should be TCP-friendly. That means, the throughput of a non-tcp flow should be approximately the same as that of a TCP flow under the same conditions of round trip time and packet loss rate. However, it was recently shown that the additive increase nature of AIMD appears inferior [20]. Two desired properties of any congestion control scheme are fairness and convergence. But additive increase does not guarantee the quickest convergence and AIMD is unfair under certain conditions, as pointed out in [20,3,10,11]. Floyd et.al. [14] also suggested that a better smoothness can be achieved with a decrease parameter less than ½. They showed that large rate variations can be observed when loss rate is high. Other works [17,15,13,12] argued that increase-by-one decrease-to-half of AIMD is not a fundamental requirement of congestion control. Other values have been suggested for increase and decrease. One question arises, how to choose the increase and decrease parameters so that high throughput can be achieved while obtaining the reasonable response time according to the dynamic behavior of the network. We propose therefore in this paper a general version for congestion control with a high order model. Instead of using concrete increase-by-one decrease-to-half such as in existing schemes, we use general parameters a for increase and b for decrease, respectively. Moreover, we consider the exponential behavior of the round-trip delay using an exponential increase function. With the model, we derive the throughput as a function of a and b, and of loss rate and round-trip delay. In addition, we obtain the relationship between a and b and show how to choose these parameters to achieve high throughput. The paper is organized as follows. In section 2 we review the basic principle of congestion control and some related works. We describe our general version of congestion control in section 3. Section 4 shows our simulation results and finally, section 5 concludes the paper.

2 2 Basic Principle of Congestion Control The existing congestion control schemes can be roughly classified into three approaches: window-based [2,3,18], rate-based [24,7,4] and formula-based schemes [5,13,17,22]. The common objective of these schemes is to find a suitable way for controlling traffic load according to the state of the network. Nevertheless, most schemes are based on the basic principle of AIMD, i.e. increase by one and decrease to half, similar to TCP. The analytical model of AIMD was first investigated by Chiu and Jain [1,2]. The analysis of this classical AIMD is based on the binary feedback scheme (1=overloaded, 0=underloaded) depending upon the network load. The authors of [1,2] suggested a congestion control based on the state equation as follows: x i (t+1) = x i (t) + f(x i (t), y(t)) (1) Where x i (t) and x i (t+1) are the i th user s load at the time instance t and t+1, respectively. The load change due to i th user s control is interpreted by a function f, which is a function of x i (t) and the system feedback y(t). The control function f(.) can be linear or nonlinear. However, the authors have focused on the linear case. Figure 1 [1] shows general pattern of throughput and response time of a network as the network load increases. Throughput knee cliff Response time Load Load Fig. 1. Throughput and response time upon the network load The knee describes the point after which the increase in the throughput is smaller when the response time significantly increases as the load increases. The cliff is the point at which the packets start getting lost. Any congestion control schemes should operate in the zone to the left of the cliff [1]. According to this model, the authors suggested an increase/decrease algorithm as follows: Additive increase: w w + a (2) Multiplicative decrease: w b * w Where w is the window size, a=1, and b= The value b=0.875 (or 7/8) is chosen due to the fact that if the network is congested, the multiplicative decrease makes users with higher windows go down more than those with lower windows [1,2]. Parallel to the works of Jain and Chiu, Jacobson analyzed the congestion avoidance strategy in [3] and suggested using the first term of a Taylor series of the load, resulting in the same algorithm (2). However, as opposed to the b=7/8 suggested in [1,2], he suggested using decrease parameter b=1/2. Despite of this large performance penalty, he interpreted it by assuming that at the time of congestion a half of the current window size was successfully exchanged with no drops. This algorithm is implemented in the popular TCP protocols, (for instance [23]). A TCP session begins or restarts in the slow-state [3]. In the begin of this state, the congestion window is set to the size of one packet. The congestion window w is then doubled as each acknowledgement arrives. The network capacity will be reached at some time. After receiving the first congestion indication, the window is halved and the session will enter the congestion avoidance phase. In this state, the congestion window is increased by a/w for each acknowledgement received. That is, the congestion window size increases by a (one packet by TCP) each round trip time (RTT) as long as no congestion is detected. Whenever congestion is detected, the window size is changed to b*w. However, it was recently shown e.g. in [20,10,11] that the additive increase feature of TCP and AIMD control does not guarantee the quickest convergence of fairness and efficiency. Using simulation and the same analytical method as that in [2], Gorinsky and Vin [20] showed that the TCP congestion control algorithm is unfair. TCP congestion control can show chaotic behavior under certain conditions [11]. Veres and Boda showed in [11] that the congestion window process at a certain time instance does not reveal the underlying state of the system in all details. Although the self similarity behavior is a property of the low level system, it is heavily influenced by the congestion control mechanisms. The existing works on congestion control schemes have been mainly focused on the stationary behavior. On the other hand, one can be observed that the increase by one decrease to half is not the fundamental requirement of congestion control. In [3], Jacobson also showed that a high order model is required as the Internet increases. There is an incentive to derive new schemes which can better describe the traffic dynamic of the network. Several schemes have been proposed in last few years to improve AIMD (see e.g. [9]). A number of rate-based congestion control schemes have been recently proposed for supporting multimedia communications, for instance [19,24,5,22]. In [6], Floyd et.al. defined the TCPcompatibility of congestion control schemes and introduced the term "TCP-friendliness (see also [14] for references). Sisalem et.al. [4] tried to use an equation for rate control using exponential function. Padhye et.al. analyzed a probabilistic model for TCP and derived an

3 equation-based control with respect to time-out, loss event rate, etc [5,22]. Bansal et.al. suggested a binomial congestion control [17]. Jin et.al [13] proposed a squareincrease scheme (SIMD). Our work differs from previous works that we propose a general version for congestion control with a high order model. We use general parameters a for increase and b for decrease, respectively. Due to the exponential behavior of the round-trip delay we propose using an exponential increase function. With this model, we derive the relationship between a and b and show the criteria for choosing these parameters. 3 High-Order Model for Congestion Control Using Exponential Increase Motivated by the idea of [3], we propose using a higher order model to develop a general version for congestion control. Furthermore, we also observe that the exponential function can be described by a time series. The idea is to replace the function f(.) of equation (1) with an exponential function of time and load. Therefore, other models such as the zero model (no congestion) and the first order model (on congestion) of the existing schemes can be modeled as the approximations of an exponential function. We investigate now a more general version of the AIMD algorithms. Instead of using concrete increase-byone decrease-to-half such as in existing schemes, we use general parameters a and b for increase and decrease, respectively. Moreover, it was argued that the round-trip delay has a considerable impact on the control. The exponential behavior of the round-trip delay was first observed in [3]. An approximation was suggested for the calculation of this delay by RTT new =(1-g)*RTT old + g*rtt old, where g is the weight. However, the influence of the round-trip delay was not considered explicitly in the existing congestion control schemes. Therefore, we consider in this paper the exponential behavior of the round-trip delay using an exponential increase function. In contrast to the existing schemes that use a constant increase parameter, we propose an exponential increase parameter as follows: Increase: w t+r = w t + a.e -Kt (3) Decrease: w t+δt = w t (1-b) With a, b and K are constants. W denotes the window size (or congestion window) at a time instant (w t at t, w t+r at t+r, w t+δt at t+δt). R denotes the time interval of window change (usually one round-trip time). For the sake of convenience, we call our general AIMD version the exponential increase multiplicative decrease (EIMD) due to the use of the exponential function in the increase. The reasons for using this function are as follows: 1) The exponential function helps us to describe the dynamic of the system. 2) The congestion control algorithm should converge while keeping small responsiveness and smoothness. Thus, an exponential increase is a suitable. It was also suggested in [3] that an unstable system can be stabilized by adding some exponential damping to its primary excitation. By using the function exp(-kt), the smoothness and responsiveness converges by the factor 1/K. 3) The updating interval used in the typical control algorithms is the round trip time. The mean round trip time is estimated by an exponential evolution [23][3]. Thus a conservation at the equilibrium is achieved. However, only a constant RTT is used by deriving the formula for the mean throughput in the existing schemes (for instance [4][5][21][17]). We claim that it is better to cover the effect of RTT using the exponential increase to describe the evolution of the congestion window. We can compare this modification with some modifications of TCP, e.g. Slow-start. 4) Finally, the number of packets (the window size) in the network can be presented by an accumulative queue building process. This process has intuitively an exponential behavior. Network Load Goal Responsiveness Smoothness Time Fig. 2. Responsiveness and Smoothness One key issue is the choice of the factor K. The smaller the factor K is, the better convergence is achieved. The factor K should reflect the influence of the increase, the round-trip delay and the maximum window size. For a these reasons, we choose K = (4) RW m where W m is the maximum window size. We now try to analyze the throughput of our proposed algorithm. We consider a deterministic model as same as the model described in [16,8,17,12], rather than the stochastic TCP model from [5,21,22]. For the deterministic model, a congestion epoch is defined as a time period beginning with a congestion window of (1-b)*W m packets. W m is the maximum window size, at which a packet drop occurs. The congestion window w(t) is increased with the formula (3) up to the maximum congestion window W m. At this time, the sender is notified about the packet loss and is required to reduce its window size to the value of (1-b)*W m. Note that b is the decrease parameter, which is 1/2 by TCP[3,23] and other TCP-friendly schemes [4,22,17,13,24]. In this model, we also assume that the window size reduces in

4 response to a loss event rather than to a single packet drop. Furthermore, we do not consider the effect of TCP retransmission timeouts in this paper. However, the model can be extended to take into account the timeouts in order to achieve a more accuracy under high loss rates, but it requires the further investigation. The window evolution under periodic loss is depicted in Figure 3 for exponential increase. Window size evolution Fig. 3. The window size evolution of EIMD Because of the monotone and the continuity of the exponential function we can use a fluid approximation and the linear interpolation to get the following differential equation for the window w(t) during each time interval (t, t+r) as follows: dw ( t ) a Kt = e dt R Thus, by integral we get: a Kt W ( t ) = e + C (5) RK where C is an integration constant. From (4) and (5) we have: Kt W ( t) = Wm e + C We are interested in the evolution of one congestion window. Therefore, we can shift the time origin to t o for a simple calculation. At time t o =0 we have W(t=0) = (1-b)W m. Thus, we get C=(2-b)W m and: Kt W ( t) = Wm (2 b e ) (6) The duration of a congestion epoch (t 0,t 1 ) is the time interval between two successive packet drops. During this time interval, we assume that 1/p packets are delivered with a packet loss probability of p followed by one drop packet. By this assumption, one packet is dropped at time t 1 (more exactly will be one loss event) and the congestion window is reduced to (1-b)W m. Let N denote the number of packet sent between two drops. N is then the shaded area under the curve in Figure 3. We can calculate N as: 1 t1 Wm 1 Kt t N = 2t bt e 1 W ( t) dt = R t + 0 R K t 0 For t 0 =0, we have: N = W m 1 Kt 1 2 t bt + e R K K (7) At the time t 1, we have W(t 1 ) = W m. Substitute this in (6) we have: Kt W = W ( 2 b e 1 ) m W m (1-b)W m m Time Solving this equation for t 1, we get: 1 1 t1 = log (8) K 1 b From (7) and (8) we conclude that: 2 ( 2 b) Wm 1 N = log 1 (9) a 1 b Due to the assumption of packet drops, the packet loss probability p is: p=1/n. Thus, from (9) we get: a (10) W m = 1 p. ( 2 b). log 1 1 b The average throughput is the number of packets delivered in each congestion epoch N divided by the duration of this epoch, i.e. t 1 -t 0. Let R ave denote the average throughput, we have: N R ave = t 1 t 0 1 a. ( 2 b). log 1 1 b R ave = 1 R p. log 1 b a R ave =. F( b) (11) R. p This result shows an exact relationship between the mean throughput R ave, the round trip delay R, the loss rate p and the increase/decrease parameters a and b. We emphasize here that the result of other works [5,17,16] has been obtained using approximations. By choosing a. F( b) = 3/ 2, we get exactly the same result of the simple TCP model in [8], i.e.: 3 / 2 R ave = (12) R. p Now it is interesting to examine the relationship between the throughput and the parameters a and b. This relationship is depicted in figure 4. Fig. 4. The relationship of throughput and decrease parameter b As showed in the figure, we can choose the decrease parameter b either around 0.875, i.e. 7/8 (as same as in

5 DECbit scheme[1]) or less than 0.6 for good throughput. If we choose b=0.5, we should choose a=1.56 in order to obtain the result (12) as same as the one indicated in the simple TCP model [8]. The smaller the parameter b is, the the throughput increases more. But it means also more penalty for the flow. Figure 5 shows the throughput as a function of b and the round-trip delay. The larger the round-trip delay (RTT) is, the the influence of the parameter b on the throughput is less. Figure 6 shows a comparison of our model and the simple TCP model under various roundtrip delay and loss rates. The figure shows the results of two models in roughly similar throughput under b=1/2. The bottleneck link has a capacity of 2Mbps and a link delay of 10ms. The figures 8-10 show the simulation in which EIMD flows share the bottleneck link with AIMD flows (TCP-SACK[9,23] flows). Fig. 5. Influences of the parameter b and the round-trip delay on throughput Fig. 8. Sending Rate Figure 8 shows the sending rate of the EIMD scheme with different parameters b. Each flow begins with a slow start phase. As shown in the figure, the less the parameter b is, the the congestion control is more aggressive. Although a higher rate is gained with b=0.2, the loss rate is higher than with larger b. That is, the control is more aggressive, more penalty is given to the flow. Fig. 6. Comparison of EIMD model and the simple TCP model From (11) and (12) it is obviously that in order to obtain TCP-friendliness we must have the following relationship of a and b: 3 a = (12) 2.( F ( b ) ) 2 where F(b) is a function of the parameter b (see (11)). 4 Performance Simulation For performance simulation we use the typical bottleneck topology as shown in figure 7. Bottleneck link Fig. 7. Simulation topology Fig. 9. Throughput Figure 9 shows the throughput of a EIMD flow which shares the bottleneck link with a TCP-SACK flow. Since the bottleneck link is 2Mbps, a fair share rate should be around 1Mbps for each of two flows sharing the link. In the figure we see that the smaller parameter b is, the the flow is more aggressive. The unfairness with TCP flow is increases with the smaller parameter b. Furthermore, the packet loss rate increases with the decrease of b. That is the penalty for the aggressive flow. Table 1 shows the results of loss rate according to the parameter b. Note that we consider a constant parameter a in all experiments. The results show that a small loss rate is obtained with b around 0.7. b Loss rate Table 1. Loss Rate

6 Figure 10. Throughput of EIMD and AIMD Figure 10 shows the throughput of EIMD and AIMD flows both with b=0.5. The AIMD used in the experiment is TCP SACK [9,23]. The result indicates that the inter-protocol unfairness remains since TCP flows frequently experience retransmission timeout while the time-out issue is still not considered in our scheme. This outlines our further development to extend the scheme with timeout handling. 5 Conclusion The paper presents a general version for congestion control with a high order model. Instead of using concrete increase-by-one decrease-to-half such as in existing schemes, we use general parameters a for increase and b for decrease, respectively. Moreover, we consider the exponential behavior of the round-trip delay using an exponential increase function. With the model, we derive the throughput as a function of a and b, of loss rate and round-trip delay. In addition, we obtain the relationship between a and b and show how to choose these parameters to achieve high throughput. Further work is going on the investigation of timeout effects on the scheme. Although it is difficult to describe the network dynamic especially in the presence of new multimedia services, the proposed model is a new trial approach which first tries to include the effect of the round-trip delay within congestion control using a higher order model. References: [1] R.Jain, K.K.Ramakrishnan, D.M.Chiu. Congestion Avoidance in Computer Networks With a Connectionless Network Layer, DEC-TR-506, Aug [2] D.M.Chiu, R.Jain. Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks, DEC-TR-509, Aug [3] V.Jaconson. Congestion Avoidance and Control, ACM SIGCOMM 88, p [4] D.Sisalem, H.Schulzrinne, The Loss-based Adjustment Algorithm: A TCP-friendly Adaptation Schemes", in Proc. of NOSSDAV'98, Cambridge, England, July [5] J.Padhye, V.Firoiu, D.Towsley, J.Kurosee, Modeling TCP Throughput: A Simple Model and Its Empirical Validation, In ACM SIGCOMM 98, Vancouver, Sept [6] S.Floyd, K.Fall. Promoting the Use of End-to-End Congestion Control in the Internet. IEEE/ACM Transactions on Networking, Vol.7, No.4, pp , Aug [7] R.Jain, A Delay-Based Approach for Congestion Avoidance in Interconnected Heterogeneous Computer Networks, DEC-TR-566, Apr [8] M.Mathis, J.Semke, J.Mahdavi. The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm, in Computer Communication Review, Vol.27, pp.67-82, July [9] S.Floyd. A Report on Some Recent Developments in TCP Congestion Control, in IEEE Communication Magazine, Apr [10] R.Morris. Scalable TCP Congestion Control, in Proc. of IEEE INFOCOM 2000, [11] A.Veres, M.Boda. The Chaotic Nature of TCP Congestion Control. IEEE INFOCOM 2000, pp , Tel-Aviv, Israel, Mar [12] K.W.Lee, T.Kim, V.Bharghavan. A Comparison of Two Popular End-to-End Congestion Control Algorithms: The Case of AIMD and AIPD, in Proc. of IEEE INFOCOM [13] S.Jin, L.Guo, I.Matta, A.Bestavros. TCP-friendly SIMD Congestion Control and Its Convergence Behavior. in Proc of ICNP'01, The 9th IEEE International Conference on Network Protocols, Riverside, CA, November [14] S.Floyd, M.Handley, J.Padhye. A Comparison of Equation-Based and AIMD Congestion Control. May, [15] Y.R.Yang, S.S.Lam. General AIMD Congestion Control, Techn. Rep. at the Uni. of Texas at Austin, TR , May [16] S.Floyd. Connections With Multiple Congested Gateways in Packet-Switched Networks- Part 1: One-Way Traffic. IEEE ACM Comp. Commun. Review, Vol.21, No.5, pp.30-47, Oct [17] D.Bansal, H.Balakrishnan. Binomial Congestion Control Algorithms. IEEE INFOCOM 2001, Apr [18] M.Handley, J.Padhye, S.Floyd. TCP Congestion Window Validation. UMass CMPSCI Tech. Rep.99-77, Sept [19] D.Sisalem, A.Wolisz. Towards TCP-friendly Adaptive Multimedia Applications Based on RTP", In 4th IEEE Symposium on Computers and Communications (ISCC'99), Egypt, July [20] S.Gorinsky, H.Vin. Additive Increase Appears Inferior. Technical Report TR , Dept. of CS, Uni. of Texas at Austin, May [21] T.J.Ott, J.H.B.Kemperman, M.Mathis. The Stationary Behavior of Ideal TCP Congestion Avoidance. Aug [22] M.Handley, J.Padhye, S.Floyd, J.Widmer. TCP Friendly Rate Control (TFRC): Protocol Specification. draft-ietftsvwg-tfrc-03.ps, July 2001, exp. Jan [23] W.Steven. TCP/IP Illustration, Volume I: The Protocols, Addison-Wesley [24] R.Rejaie, M.Handley, D.Estrin, RAP: An end-to-end rate based congestion control mechanism for real-time streams in the Internet, in Proc. of IEEE INFOCOM'99, Vol.3, pp , Mar.1999.