A SIMULATION STUDY OF THE IMPACT OF SWITCHING SYSTEMS ON SELF-SIMILAR PROPERTIES OF TRAFFIC. Yunkai Zhou and Harish Sethu

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1 Proceedings of the IEEE Workshop on Statistical Signal and Array Processing Pocono Manor, Pennsylvania, USA, August 14 16, 2000 A SIMULATION STUDY OF THE IMPACT OF SWITCHING SYSTEMS ON SELF-SIMILAR PROPERTIES OF TRAFFIC Yunkai Zhou and Harish Sethu Department of ECE, Drexel University 3141 Chestnut Street Philadelphia, PA {kenty, sethu}@ece.drexel.edu ABSTRACT Recent research has shown that traffic in Ethernet and other networks tends to exhibit properties of self-similarity such as long-range dependence and a high degree of correlation between arrivals. This paper investigates the impact of the switching network on the self-similar properties of the traffic. This simulation study reveals that switching networks tend to reduce the self-similarity of highly self-similar traffic. This is because of the truncation of long bursts due to packet discards, and also because of aggregation of flows through concatenated rather than superposed bursts. On the other hand, switching systems have the opposite effect of increasing the self-similarity of input traffic that has no selfsimilar properties such as traffic with Poisson or uniformly random distributions. This paper also presents simulationbased evidence of the causes behind these phenomena. 1. INTRODUCTION Recent work by many researchers has shown that traffic in Ethernet and other networks tends to be bursty at many or all time-scales [2,6], and that this phenomenon can be mathematically described using the notion of self-similarity. Extensive research has been done on the impact of the selfsimilar properties of traffic on network design issues, such as queueing performance [10], switch performance [3], congestion control [5] and scheduling algorithms [4]. While it is clear that traffic characteristics have an impact on network design issues, it is also true that the properties of a network have an impact on the characteristics of the traffic as it progresses through the network. Very few studies, however, have addressed this issue of changes in traffic characteristics caused by the network [1, 8, 12, 14]. These studies have focused on only the impact of individual components of a network such as the traffic shaper [12], the packet scheduler [1, 14] or a single-server queue [8], as opposed This work was supported in part by U.S. Air Force Contract F to the impact of the entire network as a whole. In addition, studies such as [8] have obtained insightful theoretical results which, however, cannot be readily applied to realistic network environments to solve problems in network engineering. Further, studies such as in [12, 14] only consider short-range burstiness, which does not capture all of the features of self-similar traffic, especially long-range burstiness as observed in [2, 6]. This paper presents a simulation study of the impact of a switching network on the self-similar properties of the traffic, and investigates the causes underlying the observed phenomena. We use self-similar traffic generated using the fractional ARIMA model [7], and a baseline Banyan topology for the switching network. Section 2 discusses the network and the traffic model in greater detail. Our simulation study reveals that switching networks tend to reduce the self-similarity of highly self-similar traffic. This is because of the truncation of long bursts due to packet discards, and also because of the aggregation of flows through concatenated rather than superposed bursts. On the other hand, switching systems also increase the selfsimilarity of input traffic that has no self-similar properties such as traffic with Poisson or uniformly random distributions. Section 3 presents these simulation results and the related analysis with simulation-based evidence of the causes behind the phenomena that yield these results. This section also explains our results in relation to those obtained in [8] and [11]. Section 4 concludes the paper. 2. NETWORK AND TRAFFIC MODEL 2.1. Network Model This study uses an N N baseline Banyan multistage network, with N source nodes and N destination nodes. The switching network consists of log m N stages of m m switching elements. In Banyan topologies, the path between a source end-point and a destination end-point is unique. This property of Banyan networks helps our study of the

2 Inlet 0 Inlet 15 Outlet 0 Outlet 15 Figure 1: Banyan network with N = 16 and m = 4. impact of switching systems, since it eliminates other secondary effects such as due to the choice of a routing algorithm. The popularity of Banyan topologies in real implementations is an additional motivation behind our use of this network model. Figure 1 shows a baseline Banyan network topology with N = 16 and m = 4. Each source node consists of N traffic generators, each of which generates traffic intended for a distinct destination node. Thus, the system consists of a total of N 2 traffic generators. In our simulations, traffic generators are all independent, and generate no more than one packet per cycle. We assume that packet lengths are constant, and that exactly one packet can be transmitted during each cycle across any port. If more than one packet are created in a source node during the same cycle, only one of these is allowed to be transmitted while all the others are buffered in a queue. We assume that the queue sizes at the source nodes are large enough that no packet is ever dropped before it enters the network. Destination nodes drain packets from the output ports of the last stage of switching elements, at the maximum rate of one packet per cycle. In the switching elements, each input port is associated with an input buffer of a fixed small capacity of 4 packet lengths. Each output port contains a dedicated output buffer. In addition, our simulations also use a shared output buffer of capacity equivalent to 4 packets per output port for additional space for the output queues. Under most traffic conditions, the shared buffer improves performance through better buffer utilization. During each cycle in our simulations, switching elements can accept no more than one packet at each input port into the input queue. Each non-empty output queue transmits exactly one packet to the output port in each cycle. We use the round-robin scheduling algorithm to transfer packets to and from the shared queue. A packet arriving at an input port first enters the associated input buffer, then the shared output buffer, and finally the output buffer corresponding to the destination port. Our model ensures that the maximum bandwidth with which the shared buffer can be written into or read from, is equal to the maximum aggregate input or output bandwidth of the switch. Packets arriving at a full input buffer are dropped. No packets, however, are dropped at any other point within the switching element, i.e., packets are forwarded to the shared buffer, or to an output buffer only if there is room available Traffic Model We use the fractional autoregressive integrated moving average (FARIMA) model [7] to synthesize self-similar traffic. FARIMA(p, d, q) is defined as Φ(B)X n = Θ(B) d ɛ n, where B is the backward operator, i.e., Bx n = x n 1. The definition above can be also expressed as X i = d ɛ i θ 1 d ɛ i 1 θ q d ɛ i q +φ 1 X i φ p X i p. (1) In equation (1), d is defined as d = i=0 b i( d)b i, where b 0 ( d) = 1 and b i ( d) = Γ(i + d), i = 1, 2,... Γ(d)Γ(i + 1) When the innovation ɛ i is a stable process with index α, i.e., ɛ i S α (σ, β, µ), the Hurst parameter, H, and the quantities α and d are related by d = H 1/α. In this paper, we use the Hurst parameter as the measure of the degree of self-similarity. The Hurst parameter has a range of 0.5 H 1, and a larger value of H implies a higher degree of self-similarity. Throughout our work, we use α = 1.2, σ = 1, β = 0, µ = 0, p = 50, and q = 400. Θ(B) is generated by selecting θ i in [0, 0.05] randomly and independently. Unlike Θ(B), Φ(B) is generated by selecting p/2 complex roots and their conjugates, since X i converges only if all roots of Φ(B) are in the unit circle. The real and imaginary components of each root are uniform in [0, 0.05]. Finally, we normalize X i to a series of 1 s or 0 s indicating whether or not a packet is generated during a given cycle. The variance-time plot [9] is used to estimate the Hurst parameter of observed network traffic. For a self-similar time series X(k), X m (k) is defined as and X m (k) = 1 m (m+1)k 1 i=mk varx m = varx/m β, X(i), where H = 1 β/2. Taking the logarithm of the equation above, we get, log varx m = β log m + log varx, From the above equation, the Hurst parameter is determined by the slope of the plot of log varx m vs. log m.

3 Hurst Parameter self similar traffic random traffic Number of Bursts (a) Input Traffic Number of Hops Traversed Figure 2: Per-hop changes in self-similarity. 3. SIMULATION RESULTS AND ANALYSIS In a switching element with buffers, a flow typically consumes more space during a bursty period. Under such conditions, depending on the buffer sharing policy and the buffer sizes, either other flows suffer from less empty space, or the bursty flow suffers a higher packet loss rate. In either of these cases, the traffic characteristics change due to delays or losses or both. If two or more flows are bursty at the same time, these effects are further magnified. This section presents our study of these effects on the self-similar properties of traffic. Our study includes two kinds of traffic sources, selfsimilar and uniform random traffic. A uniform random traffic source generates a packet during each cycle with a certain probability p, with uniformly distributed packet destinations. Like Poisson traffic, uniform random traffic has a Hurst parameter of 0.5, indicating that it has no self-similar properties. Our simulation study shows that the impact of a switching system on the self-similarity of the traffic depends on the self-similarity of the input traffic itself. A switching system reduces the self-similarity of highly self-similar traffic, while it increases that of non-self-similar traffic such as uniform random traffic. Figure 2 illustrates this phenomenon of the opposite nature of the effects observed depending on the self-similarity of the input traffic itself. When the traffic is uniformly random, the Hurst parameter increases from 0.5 to 0.64 after the first stage and stays around 0.65 thereafter. When the input traffic has a high level of self-similarity, the Hurst parameter drops from 0.86 to 0.78 after the first stage, and further to 0.76 after the second stage. This interesting phenomenon shows us that, switching networks have the effect of shaping the traffic characteristics to a moderate level of self-similarity. In the following, we investigate Number of Bursts Burst Length (b) Output Traffic Burst Length Figure 3: Distribution of short burst length, (a) input traffic and (b) output traffic. and present simulation-based evidence of the causes of this phenomenon. In the case of uniform random traffic, the probability of packet arrivals during each cycle is independent of the packet arrival pattern during the previous cycles. As the traffic progresses through a switching network with buffers in the switching elements, this independence assumption progressively becomes less valid. Packets for the same output port that independently arrive at different times, due to congestion, end up waiting in the buffers for transmission, and get transmitted in a burst at the output port. This phenomenon adds burstiness to the traffic at each new hop in the path of the traffic, changing the output traffic characteristics to something other than random uniform traffic. Packet arrivals at subsequent hops of the network are now correlated, as reflected in the increased Hurst parameter of the traffic. The distribution of burst lengths in highly self-similar traffic is heavy-tailed, i.e., the probability distribution is given by P [X > x] x α. Such a distribution decays

4 Number H (Input H (Output Percentage of Nodes Traffic) Traffic) Decrease % % % Table 1: Self-similarity of traffic vs. number of nodes. Hurst Parameter m=16 m=4 m=2 more slowly than exponentially, causing a high likelihood of long bursts in self-similar traffic. However, because of congestion and limited buffering capacities in the switching elements, long bursts do not easily survive in switching networks. In fact, long bursts self-destruct through causing congestion, triggering discarding of packets and thus breaking the long burst into smaller ones. For example, in our simulations, the longest burst observed at the traffic source had more than 6,000 consecutive packets, while the output traffic had no bursts longer than 900 packets. This is also illustrated in Figure 3, which shows that the distribution of short burst lengths of the input and the output traffic of the network. The output traffic has a larger percentage of shorter bursts, and with a significantly smaller average burst length. Note that the relative increase in shorter bursts increases the value of the index α in a heavy-tailed distribution given by P [X > x] x α. This, in turn, has the effect of reducing the Hurst parameter since, as shown in [13], H = (3 α)/2. In addition to the reduction in burst lengths, the other reason for this phenomenon is that aggregation of flows in networks typically has the effect of reducing variation over larger scales. It should be understood that this phenomenon is quite different from that shown in [11]. Willinger et al. show that a superposition of many ON/OFF traffic sources exhibits properties of self-similarity when the lengths of the ON and OFF periods are independent and follow a heavytailed distribution. Because of the limited bandwidth of output links, a true superposition is never possible in switching networks. Bursts are actually concatenated rather than superposed on top of each other on the output links. A superposition increases variation across scales, but a concatenation actually has the effect of spreading out the peaks and thus smoothening out the variations. The phenomenon discussed above can be verified through simulation using a single m m switching element and varying m. In this model, each output link is fed by m selfsimilar traffic sources at the inputs. Table 1 shows the impact on the self-similarity of the output traffic for different values of m. As can be observed from Table 1, the selfsimilarity of traffic decreases as the level of aggregation increases. This reduction in the self-similarity of traffic with aggregation, is also the reason that highly self-similar traffic Stage Number Figure 4: Per-hop changes for different switch sizes. reduces in self-similarity as it progresses through each hop in the switching network. This is easily understood from noting that the output links further hops away from the traffic sources carry more of an aggregated traffic than the ones closer to the sources. The same phenomenon is apparent in the impact of the size of switching elements used in the topology of a switching network. A network designed using 2 2 switching elements, as compared to 4 4 switching elements, will contribute to a smaller decrease in the observed Hurst parameter after the first hop. A network with 4 4 switching elements achieves the same level of aggregation in fewer hops than one using 2 2 switching elements. Figure 4 shows the perhop changes in the self-similarity of traffic for switching networks with different sizes of switching elements. It is worthwhile to discuss our results in relation to those obtained by Song et al. [8] in their study of self-similarity of output traffic at a single server with an infinite buffer. It is proved in [8] that if the queue length has finite variance, the self-similar properties of input and output traffic remain the same. In fact, it is shown that both the input and output traffic have the same Hurst parameter. Noting that it is unrealistic to assume an infinite buffer, the authors in [8] argue that in real switches, the condition that queue length has finite variance is always satisfied. However, another important impact of finite buffers should be considered traffic can be accepted into the buffer only if there is available space. In the absence of a feedback mechanism, packet discarding becomes inevitable which changes the traffic characteristics; in the presence of a feedback mechanism such as credit-based flow control, the characteristics of arriving traffic itself changes. A second important reason for the apparent discrepancy between our results and that in [8] is that our results use the self-similar properties of aggregate traffic at each output link, while the results in [8] compare

5 the properties of individual flows before and after service by the server. Our approach to only analyze aggregate traffic is motivated by the fact that most switches and routers do not maintain per-flow queueing, and therefore, only the characteristics of the aggregate traffic at each input or output link is important to the performance and related issues in the design of switches and routers. All of the results presented in this paper were obtained at moderate or heavy traffic loads. As one might expect, the impact of the switching network on the traffic characteristics is minimal when the traffic load is small. 4. CONCLUSION In this paper, we have presented simulation studies that show that highly self-similar input traffic reduces in its selfsimilarity as it progresses through a switching network. Our analysis indicates that this phenomenon is caused by the truncation of long bursts due to packet discarding, and by the aggregation of flows through concatenation of bursts. On the other hand, during periods of congestion in networks with buffers, input traffic with no self-similar properties increases in self-similarity as it progresses through the network. These results have important implications relevant to the design of routers and switches. For example, our results suggest that core Internet routers receive traffic that is much less self-similar than traffic that emerges out of border routers directly connected to Ethernet LANs. Acknowledgments The simulation code to generate the self-similar traffic was partly written by Haiguang Cheng. The authors would also like to gratefully acknowledge his contribution through discussions that further improved the traffic model. REFERENCES [1] S. Borst, O. Boxma and P. Jelenković. Asymptotic Behavior of Generalized Processor Sharing with Long-Tailed Traffic Sources. Proceedings of IEEE INFOCOMM, Tel Aviv, Israel, Mar [2] M. E. Crovella and A. Bestavros. Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes. IEEE/ACM Transactions on Networking, vol 5, no. 6, Dec [3] S. Fong and S. Singh. Performance Evaluation of Shared-Buffer ATM Switches Under Self-Similar Traffic. Proceedings of IEEE International Performance, Computing, and Communications Conference, Arizona, [4] R. G. Garroppo, S. Giordano, S. Miduri, M. Pagano and F. Russo. Statistical Multiplexing of Self-Similar VBR Video-conferencing Traffic. Proceedings of GLOBECOM, [5] A. Gersht, G. Pathak and A. Shulman. Burst Level Congestion Control in ATM Networks. Proceedings of IEEE Symposium on Computer and Communications, Athens, Greece, July [6] W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson. On the Self-Similar Nature of Ethernet Traffic (Extended Version). IEEE/ACM Transactions on Networking, vol. 2, no. 1, Feb [7] G. Samorodnitsky and M. S. Taqqu. Stable Non- Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, NY, [8] S. Song, J. K.-Y. Ng and B. Tang. On the Self- Similarity Property of the Output Process from a Network Server with Self-Similar Input Traffic. Proceedings of Sixth International Conference on Real-Time Computing Systems and Applications, pp , Dec [9] W. Stallings. High-Speed Networks: TCP/IP and ATM Design Principles. Prentice Hall, Upper Saddle River, NJ [10] B. Tsybakov and N. D. Georganas. Overflow Probability in an ATM Queue with Self-Similar Input Traffic. Proceedings of IEEE International Conference on Communications, Montreal, Canada, June [11] W. Willinger, M. S. Taqqu, R. Sherman and D. V. Wilson. Self-Similarity Through High-Variability: Statistical Analysis of Ethernet LAN Traffic at the Source Level. IEEE/ACM Transactions on Networking, vol.5, no.1, Feb [12] S. Wittevrongel and H. Bruneel. Output Traffic Analysis of a Leaky Bucket Traffic Shaper Fed by a Bursty Source. Proceedings of IEEE International Conference on Communications, pp , [13] X. Yang, A. P. Petropulu and V. Adams. The Extended ON/OFF Model for High-Speed Data Network. Workshop on Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics, Washington, D.C., Jun [14] H. Zhang. Service Disciplines for Guaranteed Performance Service in Packet-Switching Networks. Proceedings of the IEEE, vol. 83, no. 10, Oct