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Extremal behavior of heavy-tailed ON-periods in a superposition of ON/OFF processes

Part of: Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Alwin Stegeman
Alwin Stegeman*
Affiliation:
University of Groningen
*
∗ Postal address: University of Groningen, Department of Mathematics, PO Box 800, NL-9700 AV Groningen, Netherlands. Email address: [email protected]
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Abstract

Empirical studies of data traffic in high-speed networks suggest that network traffic exhibits self-similarity and long-range dependence. Cumulative network traffic has been modeled using the so-called ON/OFF model. It was shown that cumulative network traffic can be approximated by either fractional Brownian motion or stable Lévy motion, depending on how many sources are active in the model. In this paper we consider exceedances of a high threshold by the sequence of lengths of ON-periods. If the cumulative network traffic converges to stable Lévy motion, the number of exceedances converges to a Poisson limit. The same holds in the fractional Brownian motion case, provided a very high threshold is used. Finally, we show that the number of exceedances obeys the central limit theorem.

Keywords

Exceedancesextreme value theorypoint processpoint process of exceedancesPoisson random measurePoisson processmartingalestopping timeheavy tailsregular variationPareto tailsON/OFF processON/OFF model

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 179 - 204
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported by a Dutch Science Foundation (NWO) grant.

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