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

Part of: Limit theorems Special processes Stochastic processes

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.

References

Crovella, M. E. and Bestavros, A. (1996). Self-similarity in world wide web traffic: evidence and possible causes. Perf. Evaluation Rev. 24, 160–169.CrossRefGoogle Scholar
Crovella, M. E., Taqqu, M. S. and Bestavros, A. (1996). Heavy-tailed probability distributions in the world wide web. Preprint. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications, eds Adler, R., Feldman, R. and Taqqu, M. S., Birkhäuser, Boston, pp. 3–25.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Fowler, H. J. and Leland, W. E. (1991). Local area network traffic characteristics with implications for broadband network congestion management. IEEE J. Sel. Areas Commun. 9, 1139–1149.CrossRefGoogle Scholar
Gut, A. (1988). Stopped Random Walks: Limit Theorems and Applications. Springer, New York.CrossRefGoogle Scholar
Heath, D., Resnick, S. I. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Operat. Res. 23, 145–165.CrossRefGoogle Scholar
Kallenberg, O. (1986). Random Measures, 4th edn. Akademie-Verlag, Berlin.Google Scholar
Kallenberg, O. (1996). Improved criteria for distributional convergence of point processes. Stoch. Process. Appl. 64, 93–102.CrossRefGoogle Scholar
Leland, W. E., Willinger, W., Taqqu, M. S. and Wilson, D. V. (1993). On the self-similar nature of Ethernet traffic. Comput. Commun. Rev. 23, 183–193.CrossRefGoogle Scholar
Mikosch, T., Resnick, S. I., Rootzén, H., and Stegeman, A. W., (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? To appear in Ann. Appl. Prob.Google Scholar
Paxton, V. and Floyd, S. (1995). Wide-area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Networking 3, 226–244.Google Scholar
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23.CrossRefGoogle Scholar
Whitt, W. (1970). Weak convergence of probability measures on the function space C[0,∞). Ann. Math. Statist. 41, 939–944.CrossRefGoogle Scholar
Willinger, W., Taqqu, M. S., Sherman, R. and Wilson, D. V. (1995). Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level (extended version). Comput. Commun. Rev. 25, 100–113.CrossRefGoogle Scholar