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17 NETWORK DESIGN AND CONTROL USING ON=OFF AND MULTILEVEL SOURCE TRAFFIC MODELS WITH HEAVY-TAILED DISTRIBUTIONS N. G. DUFFIELD AND W. W HITT AT&T Labs±Research, Florham Park, NJ 07392 17.1 INTRODUCTION In order to help design and control the emerging high-speed communication networks, we want source traf®c models (also called offered load models or bandwidth demand models) that can be both realistically ®t to data and successfully analyzed. Many recent traf®c measurements have shown that network traf®c is quite complex, exhibiting phenomena such as heavy-tailed probability distributions, long- range dependence, and self similarity; for example, see Ca  ceres et al. [7], Leland et al. [23], Paxson and Floyd [24], and Crovella and Bestavros [10]. In fact, the heavy-tailed distributions may be the cause of all these phenomena, because they tend to cause long-range dependence and (asymptotic) self-similarity. For example, the input and buffer content processes associated with an on=off source exhibit long-range dependence when the on and off times have heavy-tailed probability distributions; for example, see Section 17.9. Heavy-tailed distributions are known to cause self-similarity in models of (asymptotically) aggregated traf®c; see Willinger et al. [27]. In this chapter we propose a way to analyze the performance of a network with multiple on= off sources and more general multilevel sources in which the on-time, off-time, and level-holding-time distributions are allowed to have heavy tails. To do Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc. 421 Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger Copyright # 2000 by John Wiley & Sons, Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X so we must go be beyond the familiar Markovian analysis. To achieve the required analyzability with this added model complexity, we propose a simpli®ed kind of analysis. In particular, we avoid the customary queueing detail (and its focus on buffer content and over¯ow) and instead concentrate on the instantaneous offered load. We describe the probability that aggregate demand (the input rate from a collection of sources) exceeds capacity (the maximum possible output rate) at any time. Focusing on the probability that aggregate demand exceeds capacity is tantamount to considering a bufferless model, which we believe is often justi®ed. By also considering the probability that aggregate demand exceeds other levels, we provide a quite ¯exible performance characterization. This approach also can generate approximations describing loss and delay with ®nite capacity; for example, see Duf®eld and Whitt [14], Section 5. To a large extent, the present chapter is a review of our recent work [14, 15], to which we refer the reader for additional discussion. In Duf®eld et al. [16] the model is extended to include a nonhomoge- neous Poisson connection arrival process. Then each active connection may generate traf®c according to one of the source traf®c models presented here. It is signi®cant that we are able to obtain useful descriptions of the offered load in the nonstationary context. 17.2 A GENERAL SOURCE MODEL Motivation for considering on=off and multilevel models as source models comes from traces of frame sizes generated by certain video encoders; for example, see Grasse et al. [19]. Shifts between levels in mean frame size appear to arise from scene changes in the video, with the distribution of scene durations heavy-tailed. Indeed, the expectation that scene durations will have heavy-tailed distributions is one of the motivations behind the renegotiated constant bit rate (RCBR) proposal of Grossglauser et al. [20]. Our approach is interesting for on=off and multilevel source models, but with little extra effort we can treat a wider class. The general model we consider has two components. The bandwidth demand for each source as a function of time, fBt: t ! 0g, is represented as the sum of two stochastic processes: (1) a macro- scopic (longer-time-scale) level process fLt: t ! 0g and (2) a microscopic (shorter- time-scale) within-level variation process fW t: t ! 0g, that is, BtLtW t; t ! 0: 17:1 We let the macroscopic level process fLt: t ! 0g be a semi-Markov process (SMP) as in CË inlar [9, Chap. 10]; that is, the level process is constant except for jumps, with the jump transitions governed by a Markov process, while the level holding times (times between jumps) are allowed to have general distributions depending on the originating level and the next level. Given a transition from level j to level k, the holding time in level j has cumulative distribution function (cdf) F jk . Conditional on the sequence of successive levels, the holding times are mutually independent. To 422 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS obtain models compatible with traf®c measurements cited earlier, we allow the holding-time cdf 's F jk to have heavy tails. We assume that the within-level variation process fW t: t ! 0g is a zero-mean piecewise-stationary process. During each holding-time interval in a level, the within-level variation process is an independent segment of a zero-mean stationary process, with the distribution of each segment being allowed to depend on the level. We allow the distribution of the stationary process segment to depend on the level, because it is natural for the variation about any level to vary from level to level. We will require only a limited characterization of the within-level variation process; it turns out that the ®ne structure of the within-level variation process plays no role in our analysis. Indeed, that is one of our main conclusions. In several examples of processes that we envisage modeling by these methods, there will only be the level process. First, the level process may be some smoothed functional of a raw bandwidth process. This is the case with algorithms for smoothing stored video by converting into piecewise constant rate segments in some optimal manner subject to buffering and delay constraints; see Salehi et al. [25]. With such smoothing, the input rate will directly be a level process as we have de®ned it. Alternatively, the level process may stem from rate reservation over the period between level-shifts, rather than the bandwidth actually used. This would be the case for RCBR previously mentioned. In this situation we act as if the reservation level is the actual demand, and thus again have a level process. A key to being able to analyze the system with such complex sources represented by our traf®c model is exploiting asymptotics associated with multiplexing a large number of sources. The ever-increasing network bandwidth implies that more and more sources will be able to be multiplexed. This gain is generally possible, even in the presence of heavy-tailed distributions and more general long-range dependence; for example, see Duf®eld [12, 13] for demonstration of the multiplexing gains available for long-range dependent traf®c in shared buffers. As the scale increases, describing the detailed behavior of all sources become prohibitively dif®cult, but fortunately it becomes easier to describe the aggregate, because the large numbers produce statistical regularity. As the size increases, the aggregate demand can be well described by laws of large numbers, central limit theorems, and large deviation principles. We have in mind two problems: ®rst, we want to do capacity planning and, second, we want to do real-time connection-admission control and congestion control. In both cases, we want to determine whether any candidate capacity is adequate to meet the aggregate demand associated with a set of sources. In both cases, we represent the aggregate demand simply as the sum of the bandwidth requirements of all sources. In forming this sum, we regard the bandwidth processes of the different sources as probabilistically independent. The performance analysis for capacity planning is coarser, involving a longer time scale, so that it may be appropriate to do a steady-state analysis. However, when we consider connection-admission control and congestion control, we suggest focusing on a shorter time scale. We are still concerned with the relatively long time scale of connections, or scene times in video, instead of the shorter time scales 17.2 A GENERAL SOURCE MODEL 423 of cells or bursts, but admission control and congestion control are suf®ciently short- term that we propose focusing on the transient behavior of the aggregate demand process. In fact, even for capacity planning the transient analysis plays an important role. The transient analysis determines how long it takes to recover from rare congestion events. One application we have in mind is that of networks carrying rate-adaptive traf®c. In this case the bandwidth process could represent the ideal demand of a source, even though it is able to function when allocated somewhat less bandwidth. So from the point of view of quality, excursion of aggregate bandwidth demand above available supply may be acceptable in the short-term, but one would want to dimension the link so that such excursions are suf®ciently short-lived. In this or other contexts, if the recovery time from overload is relatively long, then we may elect to provide extra capacity (or reduce demand) so that overload becomes less likely. However, we do not focus speci®cally on actual design and control here; see Duf®eld and Whitt [14] for some speci®c examples. Our main contribution here is to show how the transient analysis for design and control can be done. The remainder of this chapter is devoted to showing how to do transient analysis with the source traf®c model. We suggest focusing on the future time-dependent mean conditional on the present state. The present state of each level process consists of the level and age (elapsed holding time in that level). Because of the anticipated large number of sources, the actual bandwidth process should be closely approximated by its mean, by the law of large numbers (LLN). As in Duf®eld and Whitt [14], the conditional mean can be thought of as a deterministic ¯uid approximation; for example, see Chen and Mandelbaum [8]. Since the within- level variation process has mean zero, the within-level variation process has no effect on this conditional mean. Hence, the conditional mean of the aggregate bandwidth process is just the sum of the conditional means of the component level processes. Unlike the more elementary M =G=I model considered in Duf®eld and Whitt [14], however, the conditional mean here is not available in closed form. In order to rapidly compute the time-dependent conditional mean aggregate demand, we exploit numerical inversion of Laplace transforms. It follows quite directly from the classical theory of semi-Markov process that explicit expressions can be given for the Laplace transform of the conditional mean. More recently, it has been shown that numerical inversion can be an effective algorithm; see Abate et al. [1]. For related discussions of transient analysis, design and control, see Chapters 13, 16, and 18 in this volume. 17.3 OUTLINE OF THE CHAPTER The rest of this chapter is organized as follows. In Section 17.4, we show that the Laplace transform of the mean of the transient conditional aggregate demand can be expressed concisely. This is the main enabling result for the remainder of the chapter. The conditional mean itself can be very ef®ciently computed by numerically inverting its Laplace transform. To carry out the inversion, we use the Fourier- 424 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS series method in Abate and Whitt [2] (the algorithm Euler exploiting Euler summation), although alternative methods could be used. The inversion algorithm is remarkably fast; computation for each time point corresponds simply to a sum of 50 terms. We provide numerical examples in Examples 17.6.2 and 17.8.1. Example 17.8.1 is of special interest, because the level-holding-time distribution there is Pareto. In Section 17.5 we show that in some cases we can avoid the inversion entirely and treat much larger models. We can avoid the inversion if we can assume that the level holding times are relatively long compared to the times of interest for control. Then we can apply a single-transition approximation, which amounts to assuming that the Markov chain is absorbing after one transition. Then the conditional mean is directly expressible in terms of the level-holding-time distributions. Alternatively, we can perform a two-transition approximation, which only involves one-dimensional convolution integrals. In Section 17.6 we describe the value of having more detailed state information, speci®cally the current ages of levels. With heavy-tailed distributions, a large elapsed holding time means that a large remaining holding time is very likely; for examples see Duf®eld and Whitt [14, Section 8] for background, and Harchol-Balter and Downey [21] for an application in another setting. In Section 17.7 we turn to applications to capacity planning. The idea is to approximate the probability of an excursion in demand using Chernoff bounds and other large deviation approximations, then chart its recovery to a target acceptable level using the results on transience. Interestingly, the time to recover from excursions suf®ciently close to the target level depends on the level durations essentially only through their mean. Correspondingly, the conditional mean demand relaxes linearly from its excursion, at least approximately so, for suf®ciently small times. If the chance for a larger excursion is negligible (as determined by the large deviation approximation mentioned) then this simple description may suf®ce. An example is given in Section 17.8. In Section 17.9 we show how long-range dependence in the level process arises through heavy-tailed level-holding-time distributions. Finally, we draw conclusions in Section 17.10. 17.4 TRANSIENT ANALYSIS 17.4.1 Approximation by the Conditional Mean Bandwidth Throughout this chapter, the state information on which we condition will be either the current level of each source or the current level and age (current time) in that level of each source. No state from the within-level variation process is assumed. Conditional on that state information, we can compute the probability that each source will be in each possible level at any time in the future, from which we can calculate the conditional mean and variance of the aggregate required bandwidth by adding. 17.4 TRANSIENT ANALYSIS 425 The Lindberg±Feller central limit theorem (CLT) for non-identically-distributed summands can be applied to generate a normal approximation characterized by the conditional mean and conditional variance; see Feller [18, p. 262]. For the normal approximation to be appropriate, we should check that the aggregate is not dominated by only a few sources. Let Bt denote the (random) aggregate required bandwidth at time t, and let I 0 denote the (known deterministic) state information at time 0. Let BtjI 0 represent a random variable with the conditional distribution of Bt given the information I0. By the CLT, the normalized random variable BtjI0À EBtjI0  VarBtjI0 p 17:2 is approximately normally distributed with mean 0 and variance 1 when the number of sources is suitably large. Since the conditional mean alone tends to be very descriptive, we use the approximation BtjI0 % EBtjI0; 17:3 which can be justi®ed by a (weaker) law of large numbers instead of the CLT. We will show that the conditional mean in Eq. (17.3) can be ef®ciently computed, so that it can be used for real-time control. From Eq. (17.2), we see that the error in the approximation (17.3) is approximately characterized by the conditional standard deviation  VarBtjI0 p . We also will show how to compute this conditional standard deviation, although the required computation is more dif®cult. If there are n sources that have roughly equal rates, then the conditional standard deviation will be O  n p , while the conditional mean is On. Given that our approximation is the conditional mean, and given that our state information does not include the state of the within-level variation process, the within-level variation process plays no role because it has zero mean. Let i index the source. Since the required bandwidths need not have integer values, we index the level by the integer j; 1 j J i , and indicate the associated required bandwidths in the level by b i j . Hence, instead of Eq. (17.1), the required bandwidth for source i can be expressed as B i tb i L i t  W L i t t; t ! 0: 17:4 Let P i jk tjx be the probability that the source-i level process is in level k at time t given that time 0 it was in level j and had been so for a period x (i.e., the age or elapsed level holding time at time 0 is x). If j j 1 ; ; j n  and x x 1 ; ; x n  are the vectors of levels and ages of the n source level processes at time 0, then the state 426 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS information is I0j; xj 1 ; ; j n ; x 1 ; ; x n  and the conditional aggregate mean is EBtjI0  Mtjj; x P n i1 P J i k i 1 P i j i k i tjx i b i k i : 17:5 From Eq. (17.5), we see that we need to compute the conditional distribution of the level, that is, the probabilities P i jk tjx, for each source i. However, we can ®nd relatively simple expressions for the Laplace transform of P i jk tjx with respect to time because the level process of each source has been assumed to be a semi-Markov process. We now consider a single source and assume that its required bandwidth process is a semi-Markov process (SMP). (We now have no within-level variation process.) We now omit the superscript i. Let Lt and Bt be the level and required bandwidth, respectively, at time t as in Eq. (17.4). The process fLt: t ! 0g is assumed to be an SMP, while the process fBt: t ! 0g is a function of an SMP, that is Btb Lt , where b j is the required bandwidth in level j.Ifb j T b k for j T k, then fBt: t ! 0g itself is an SMP, but if b j  b k for some j T k, then in general fBt: t ! 0g is not an SMP. 17.4.2 Laplace Transform Analysis Let At be the age of the level holding time at time t. We are interested in calculating P jk tjxPLtkjL0j; A0x17:6 as a function of j; k; x, and t. The state information at time 0 is the pair  j; x. Let P be the transition matrix of the discrete-time Markov chain governing level transitions and let F jk t be the holding-time cdf given that there is a transition from level j to level k. For simplicity, we assume that F c jk t1 À F jk t > 0 for all j; k, and t,so that all positive x can be level holding times. Let Ptjx be the matrix with elements P jk tjx and let ^ Psjx be the Laplace transform (LT) of Ptjx, that is, the matrix with elements that are the Laplace transforms of P jk tjx with respect to time: ^ P jk sjx  I 0 e Àst P jk tjx dt: 17:7 We can obtain a convenient explicit expression for ^ Psjx. For this purpose, let G j be the holding-time cdf in level j, unconditional on the next level, that is, G j x P k P jk F jk x: 17:8 17.4 TRANSIENT ANALYSIS 427 For any cdf G, let G c be the complementary cdf, that is, G c x1 À Gx. Also, let H jk tjx P jk F jk t  x G c j x and G j tjx P k H jk tjx17:9 for G j in Eq. (17.8). Then let ^ h jk sjx and ^ g j sjx be the associated Laplace±Stieltjes transforms (LSTs): ^ h jk sjx  I 0 e Àst dH jk tjx and ^ g j sjx  I 0 e Àst dG j tjx: 17:10 Let ^ hsjx be the matrix with elements ^ h jk sjx. Let ^ qs be the matrix with elements ^ q jk s, where Q jk tP jk F jk t and ^ q jk s  I 0 e Àst dQ jk t: 17:11 Let ^ Dsjx and ^ Ds be the diagonal matrices with diagonal elements ^ D jj sjx1 À ^ g j sjx=s; ^ D jj s1 À ^ g j s=s; 17:12 where ^ g j s is the LST of the cdf G j in Eq. (17.8). Theorem 17.4.1. The transient probabilities for a single SMP source have the matrix of Laplace transforms ^ Psjx ^ Dsjx ^ hsjx ^ Psj0; 17:13 where ^ Psj0I À ^ qs À1 ^ Ds: 17:14 Proof. In the time domain, condition on the ®rst transition. For j T k, P jk tjx P l  t 0 dH jl ujxP lk t À uj0; so that ^ P jk sjx P l ^ h jl sjx ^ P lk sj0; 428 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS while P jj tjxG c j tjx P l  t 0 dH jl ujxP lj t À uj0; so that ^ P jj sjx 1 À ^ g j sjx s  P l h jl sjx ^ P lj sj0: Hence, Eq. (17.13) holds. However, Ptj0 satis®es a Markov renewal equation, as in CË inlar [9, Section 10.3]; that is, for j T k, P jk tj0 P l  t 0 dQ jl uP lk t À uj0; and P jj tj0G c j t P l  I 0 dQ jl uP lj t Àuj0; so that Ptj0DtQtÃPtj0 where à denotes convolution, and Eq. (17.14) holds. j To compute the LT ^ Psj0, we only need the LSTs ^ f jk s and ^ g j s associated with the basic holding-time cdf 's F jk and G j . Abate and Whitt [3±5] give special attention to heavy-tail probability densities whose Laplace transforms can be computed and, thus, inverted. However, to compute ^ Psjx, we also need to compute ^ Dsjx and ^ hsjx, which require computing the LSTs of the conditional cdf's H jk tjx and G j tjx in Eq. (17.9). In general, even if we know the LST of a cdf, we do not necesssarily know the LST of an associated conditional cdf. However, in special cases, the LSTs of conditional cdf 's are easy to obtain because the cdf 's inherit their structure upon conditioning. For example, this is true for phase-type, hyperexpo- nential and Pareto distributions; Duf®eld and Whitt [15, Section 4]. Moreover, other cdf's can be approximated by hyperexponential or phase-type cdf's; see Asmussen et al. [6] and Feldman and Whitt [17]. If the number of levels is not too large, then it will not be dif®cult to compute the required matrix inverse I À qs À1 for all required s. Note that, because of the probability structure, the inverse is well de®ned for all complex s with Res > 0. To illustrate with an important simple example, we next give the explicit formula for an on=off source. Suppose that there are two states with transition probabilities 17.4 TRANSIENT ANALYSIS 429 P 12  P 21  1 and holding time cdf's G 1 and G 2 . From Eq. (17.9) or by direct calculation, ^ Psj0 ^ P 11 sj0 ^ P 12 sj0 ^ P 21 sj0 ^ P 22 sj0 ! I À ^ qs À1 ^ Ds  1 s1 À ^ g 1 s ^ g 2 s 1 À ^ g 1 s ^ g 1 s1 À ^ g 2 s ^ g 2 s1 À ^ g 1 s 1 À ^ g 2 s  : 17:15 Suppose that the levels are labeled so that the initial level is 1. Note that all transitions from level 1 are to level 2. Hence, when considering the matrix ^ hsjx in Eq. (17.10), it suf®ces to consider only the element ^ h 12 sjx. Since H c 12 tjxG c 1 tjx G c 1 t  x G c 1 x ; 17:16 then ^ h 12 sjx ^ g 1 sjx  I 0 e Àst dG 1 tjx: 17:17 Since P 11 tjx1 À P 12 tjx, it suf®ces to calculate only P 12 tjx. Hence, in this context ^ P 12 sjx ^ g 1 sjx1 À ^ g 2 s s1 À ^ g 1 s ^ g 2 s : 17:18 We now determine the mean, second moment, and variance of the bandwidth process of a general multilevel source as a function of time, ignoring the within-level variation process. It is elementary that m j tjxEBtjL0j; A0x P k P jk tjxb k ; 17:19 s j tjxEBt 2 jL0j; A0x P k P jk tjxb 2 k ; 17:20 v j tjxVarBtjL0j; A0xs j tjxÀm j tjx 2 : 17:21 We can calculate m j tjx and s j tjx by single inversions of their Laplace transforms, using ^ m j sjx  I 0 e Àst m j tjx dt  P k P jk sjxb k 430 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS