Modeling teletraffic arrivals by a Poisson cluster process
by
Barbara Gonzalez
University of Louisiana at Lafayette
Coauthors: Gilles Fay, Thomas Mikosch, and Gennady Samorodnitsky

In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point G a flow of packets is initiated which is modeled as a partial iid sum process G+X1+...+Xk, k<=K, with a random limit K which is independent of the X's and the underlying Poisson points. We study the covariance structure of the increment process of N. In particular, the covariance function of the increment process is not summable if the right tail of the distribution of K is regularly varying with index between 1 and 2, the distribution of the X's being irrelevant. This means that the increment process exhibits long-range dependence. If the variance of K is finite, long-range dependence is excluded. We study the asymptotic behavior of the process N, and give conditions on the distributions of K and X under which the random sums have a regularly varying tail. Using the form of the distribution functions of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distributional characteristics of K, X, and of the Poisson process.

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Date received: September 28, 2005