Abstract. 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 Γj, a flow of packets is initiated which is modeled as a partial iid sum process Γj + P k i=1 Xji, k ≤ Kj , with a random limit Kj which is independent of (Xji) and the underlying Poisson points (Γj). 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 P (Kj > x) is regularly varying with index α ∈ (1, 2), the distribution of the Xji's being irrelevant. This means that the increment process exhibits long-range dependence. If var(Kj) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N (t)) t≥0 and give conditions on the distribution of Kj and Xji under which the random sumsXji have a regularly varying tail. Using the form of the distribution 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 Kj , Xji, and of the Poisson process.

PS with an ED/UC center visit when the patient has autism depends mostly on the quality of staff interactions with the patient and family. It is important for ED/UC center administrators to ensure that staff understand how to interact and communicate effectively with patients with ASD and their families.

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