Queueing Systems

, Volume 54, Issue 2, pp 121–140

Modeling teletraffic arrivals by a Poisson cluster process

  • Gilles Faÿ
  • Bárbara González-Arévalo
  • Thomas Mikosch
  • Gennady Samorodnitsky
Article

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 \(\Gamma_j+\sum_i=1^kX_ji, k\le K_j\), 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 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 Kj, Xji, and of the Poisson process.

Keywords

Teletraffic Poisson cluster model Long-range dependence Palm distribution Regular variation 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Gilles Faÿ
    • 1
  • Bárbara González-Arévalo
    • 2
  • Thomas Mikosch
    • 3
  • Gennady Samorodnitsky
    • 4
  1. 1.Laboratoire Paul-PainlevéUniversité Lille 1Villeneuve d’Ascq cedexFrance
  2. 2.Department of Mathematics and Actuarial SciencesRoosevelt UniversityChicagoUSA
  3. 3.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark
  4. 4.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

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