Convergence to Fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach

Abstract

This is an author-produced version of a chapter published in In and Out of Equilibrium 2. It does not include the final publisher proof-corrections or pagination.Citation for the published chapter: Kaj, I.; Taqqu, M. S. "Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach" In: In and Out of Equilibrium 2. Ed. V. Sidoravicius and M. E. Vares. Basel:Abstract. It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.3. In Section 4, the convergence in finite-dimensional distributions of the continuous flow model is extended to weak convergence in function space. The infinite source Poisson modelInfinite source Poisson models are arrival processes with M/G/∞ input obtained by integrating the standard M/G/∞ queueing system size. The resulting class of Poisson shot noise processes are widely used traffic models which describe the amount of workload accumulating over time. Such models have been suggested as realistic workload processes for Internet traffic, where is is natural to assume that while web sessions are inititated according to a Poisson process, duration lengths and transmission rates could vary considerably. More exactly, the aggregated traffic consists of sessions with starting points distributed according to a Poisson process on the real time line. Each session lasts a random length of time and involves workload arriving at a random transmission rate. There are two slightly different sets of assumptions that are natural to make regarding the precise traffic pattern during a session. The first is that the workload arrives continuously at a randomly chosen transmission rate, which is fixed throughout the session and independent of the session length. The second type of model assumes that the workload arrives in discrete entities, packets, according to a Poisson process throughout the session, and such that the size of each packet is chosen independently from a given packet size distribution. The duration and the continuous or discrete rate of traffic in one session is independent of the traffic in any other session, although in general the sessions overlap. One novelty in this work is that we point out how these two types of models differ in their asymptotic behavior and that we explain the origin of the qualitative differencies.We are going to introduce the workload models using directly an integral representation with respect to Poisson measures, as in Kurtz (1996) andÇ aglar (2004), rather than working with a more traditional Poisson shot...

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