Weak Convergence of High-speed Network Traffic Models

We consider a network traac model consisting of an innnite number of sources linked to a server. Sources initiate transmissions to the server at Poisson time points. The duration of each transmission has a heavy tailed distribution. We show that suitable scalings of the traac process converge to a totally skewed stable L evy motion in Skorohod space, equipped with the Skorohod M1 topology. This allows us to prove a heavy traac theorem for a single server uid model. 1. Introduction In modern telecommunications traac data, features such as long range dependence, self-similarity, and heavy tails are prominent. Furthermore, the characteristics of the data vary greatly over measurements at diierent sites and diierent points in time. This is especially true for internet data, as described in e.g. 32]. Black box time series modeling, which has been succesful in-nite variance (low variability) settings, is inadequate for modeling teletraac datasets. See e.g. and the discussions therein. Hence, we should attempt more structural modeling. The size of recent data sets of traac network measurements, the dynamic nature of communication networks, and the complexity of the traac they carry argue strongly in favor of parsimonious and structural network traac models. As a more structural traac model, a superposition of a large number of ON/OFF type sources whose activity periods are heavy tailed, has received considerable attention. Such models are approximated by uid models with M/G/1 inputs, sometimes referred to as innnite source Poisson models. The M/G/1 input model described in 17] is of this type. Using a distributional limit theorem, Konstantopoulos and Lin ((17]) explain the suitability of an totally skewed stable L evy motion as a macroscopic traac model for a high-speed network switch. The limit process is self-similar, but somewhat surprisingly has independent increments. The marginal distribution of the L evy stable motion matches those sometimes empirically obtained in practice ((11]). Also, results on the queue length process of queueing systems with L evy inputs are reported in 17]. However, in 17] the convergence of the scaled traac model to the limit L evy motion is only shown to occur in the sense of nite dimensional distributions. To validate application of the limit model in telecom applications, it is important to establish convergence in a stronger sense of functional weak convergence in a suitable topology. Typically, when the limit process has almost surely continuous