The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Mult

The E ect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer Predrag R. Jelenkovi c The main theme of this dissertation is the evaluation of the capacity of broadband multimedia network multiplexers. This problem calls for the modeling of network tra c streams and the analysis of a network multiplexer that is loaded with the corresponding models. For modeling we focus on MPEG video tra c streams that are expected to be predominant in the tra c mixture of future multimedia networks. We experimentally demonstrate that real-time MPEG video tra c exhibits multiple time scale characteristics, as well as subexponential rst and second order statistics. Then we construct a model of MPEG video that captures both of these characteristics and accurately predicts queueing behavior for a broad range of bu er and capacity sizes. Depending on whether a network multiplexer (loaded with MPEG) is strictly or weakly stable the dominant e ect on the queueing behavior arises from themultiple time scale or subexponential structure, respectively. Correspondingly, we identify two general classes of stochastic processes, to which the constructed MPEG model belongs, and advance mathematical tools for the associated queueing analysis. First, we consider a class of multiple time scale processes. When these processes are loaded into a strictly stable queue, bu er occupancy probabilities exhibit a distinctive functional behavior that, as we explain, results from their multiple time scale structure. In this case, we demonstrate that the Dominant Root queueing approximation (also called Equivalent Bandwidth (EB)) may be o by orders of magnitude, and that an EB-based admission control policy might be too conservative. Furthermore, we show that the EB approximation does not depend on the slow time scale statistics and is equal to the case when the arrival processes stay in the highest activity states all of the time. This explains the observed mismatch between the EB approximation and experimental results. In order to alleviate this problem, we develop an asymptotic expansion (Perturbation Theory) technique for approximating all of the bu er probabilities. Second, for a class of subexponential (non Cram er type) processes, we extend a well known probability/queueing result by proving that the asymptotic behavior of a subexponential GI/GI/1 queue is invariant with respect to Markov modulation. Then we construct a class of subexponentially correlated processes for modeling MPEG video sources. When these processes are fed into a uid ow queue, we prove that the queue length distribution is directly asymptotically proportional to their autocorrelation function. For the problem of multiplexing a large number of On-O sources with subexponential (long-tailed) On periods, we derive precise and bounding asymptotic results for approximating large bu er occupancy probabilities. Based on these asymptotic results, we suggest a practical approximation method for the case of nitely many multiplexed sources. The e cacy of the method is demonstrated on extensive numerical examples. In addition, when subexponential On-O sources are multiplexed with exponentially bounded sources, the dominant e ect for the large bu er probabilities is due to subexponential sources. This might have a practical implication for multiplexing voice (exponential) and video (subexponential) tra c sources. Finally, we propose both multiple time scale and subexponential approximation techniques for e cient capacity evaluation and admission control in broadband multimedia networks. Table of