Modeling teletraffic arrivals by a Poisson cluster process |
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Authors: | Gilles Faÿ Bárbara González-Arévalo Thomas Mikosch Gennady Samorodnitsky |
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Institution: | (1) Laboratoire Paul-Painlevé, Université Lille 1, 59655 Villeneuve d’Ascq cedex, France;(2) Department of Mathematics and Actuarial Sciences, Roosevelt University, 430 S Michigan Avenue, Chicago, IL 60605, USA;(3) Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark;(4) School of Operations Research and Industrial Engineering, Cornell University, 220 Rhodes Hall, Ithaca, NY 14853, USA |
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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
, 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
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.
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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Keywords: | Teletraffic Poisson cluster model Long-range dependence Palm distribution Regular variation |
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