17.2 A GENERAL SOURCE MODEL

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obtain models compatible with traf c measurements cited earlier, we allow the holding-time cdf's Fjk 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

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of cells or bursts, but admission control and congestion control are suf ciently shortterm 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 withinlevel 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 s 13, 16, and 18 in this volume. 17.3 OUTLINE OF THE CHAPTER

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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-

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