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Modeling teletraffic arrivals by a Poisson cluster process

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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 Γ j , a flow of packets is initiated which is modeled as a partial iid sum process \(\Gamma_j+\sum_i=1^kX_ji, k\le K_j\), with a random limit K j which is independent of (X ji ) and the underlying Poisson points (Γ j ). 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(K j > x) is regularly varying with index α∊ (1, 2), the distribution of the X ji ’s being irrelevant. This means that the increment process exhibits long-range dependence. If var(K j ) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N(t)) t≥ 0 and give conditions on the distribution of K j and X ji under which the random sums \(\sum_{i=1}^{K_j}X_{ji}\) 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 distribution al characteristics of K j , X ji , and of the Poisson process.

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Authors and Affiliations

  1. Laboratoire Paul-Painlevé, Université Lille 1, 59655, Villeneuve d’Ascq cedex, France

    Gilles Faÿ

  2. Department of Mathematics and Actuarial Sciences, Roosevelt University, 430 S Michigan Avenue, Chicago, IL, 60605, USA

    Bárbara González-Arévalo

  3. Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen, Denmark

    Thomas Mikosch

  4. School of Operations Research and Industrial Engineering, Cornell University, 220 Rhodes Hall, Ithaca, NY, 14853, USA

    Gennady Samorodnitsky

Authors
  1. Gilles Faÿ
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  2. Bárbara González-Arévalo
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  3. Thomas Mikosch
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  4. Gennady Samorodnitsky
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Correspondence to Thomas Mikosch.

Additional information

AMS Subject Classifications Primary—60K30; Secondary—60K25

A large part of this research was done with support of Institut Mittag-Leffler of the Royal Swedish Academy of Sciences when the authors participated in the Fall 2004 program on Queuing Theory and Teletraffic Theory. Mikosch’s research is also partially supported by MaPhySto, the Danish research network for mathematical physics and stochastics and the Danish Research Council (SNF) Grant No 21-04-0400. Samorodnitsky’s research is also partially supported by NSF grant DMS-0303493 and NSA grant MSPF-02G-183 at Cornell University. González-Arévalo’s research is partially supported by BoRSF grant LEQSF(2004-2007)-RD-A-31 at the University of Louisiana at Lafayette.

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Faÿ, G., González-Arévalo, B., Mikosch, T. et al. Modeling teletraffic arrivals by a Poisson cluster process. Queueing Syst 54, 121–140 (2006). https://doi.org/10.1007/s11134-006-9348-z

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  • DOI: https://doi.org/10.1007/s11134-006-9348-z

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