Queueing Systems

, 69:29 | Cite as

The on–off network traffic model under intermediate scaling

  • Clément Dombry
  • Ingemar Kaj


The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on–off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the traffic fluctuations are described by a stable Lévy motion, while the limiting fluctuations for the intermediate scaling regime are given by fractional Poisson motion. In this paper, we prove an invariance principle for the normalized cumulative workload of a network with m on–off sources and time rescaled by a factor a. When both the number of sources m and the time scale a tend to infinity with a relative growth given by the so-called ’intermediate connection rate’ condition, the limit process is the fractional Poisson motion. The proof is based on a coupling between the on–off model and the renewal type model.


On–off process Workload process Renewal process Intermediate scaling Fractional Poisson motion Fractional Brownian motion Lévy motion Heavy tails Long-range dependence 

Mathematics Subject Classification (2000)

60G22 60F05 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Laboratoire LMAUniversité de PoitiersFuturoscope-Chasseneuil cedexFrance
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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