Queueing Systems

, Volume 66, Issue 4, pp 313–350 | Cite as

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes



We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.


Cluster Poisson model Covariance function Cumulative input process Fluid queue Fractional Brownian motion Heavy tails Input model Long-range dependence Self-similarity Stable Lévy motion Regular variation Scaling Teletraffic 

Mathematics Subject Classification (2000)

90B22 60F05 60F10 60F17 60G52 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany

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