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A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime

Part of: Operations research and management science Limit theorems

Published online by Cambridge University Press:  01 July 2016

Vicky Fasen  and
Gennady Samorodnitsky
Vicky Fasen*
Affiliation:
Technische Universität München
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
∗ Postal address: Center for Mathematical Sciences, Technische Universität München, D-85747 Garching, Germany. Email address: fasen@ma.tum.de
∗∗ Postal address: School of Operations Research and Information Engineering, Cornell University, 206 Rhodes Hall, Ithaca, NY 14853, USA. Email address: gennady@orie.cornell.edu
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Abstract

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We show that, contrary to common wisdom, the cumulative input process in a fluid queue with cluster Poisson arrivals can converge, in the slow growth regime, to a fractional Brownian motion, and not to a Lévy stable motion. This emphasizes the lack of robustness of Lévy stable motions as ‘birds-eye’ descriptions of the traffic in communication networks.

Keywords

Cluster Poisson processfluid queuefractional Brownian motioninput modelslow growth regimescaling limitworkload process

MSC classification

Primary: 90B22: Queues and service
Secondary: 60F17: Functional limit theorems; invariance principles

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 393 - 427
Copyright
Copyright © Applied Probability Trust 2009 

References

Anderson, K. K. and Athreya, K. B. (1988). A strong renewal theorem for generalized renewal functions in the infinite mean case. Prob. Theory Relat. Fields 77, 471–479.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
D'Auria, B. and Samorodnitsky, G. (2005). Limit behavior of fluid queues and networks. Operat. Res. 53, 933–945.CrossRefGoogle Scholar
Delgado, R. (2007). A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic. Stoch. Process. Appl. 117, 188–201.Google Scholar
Denisov, D., Dieker, A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Prob. 36, 1946–1946.Google Scholar
Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Prob. Theory Relat. Fields 107, 451–465.CrossRefGoogle Scholar
Faÿ, G., Gonzälez-Arävalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems 54, 121–140.Google Scholar
Gaigalas, R. and Kaj, I. (2003). Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9, 671–703.CrossRefGoogle Scholar
Kaj, I., Leskelä, L., Norros, I. and Schmidt, V. (2007). Scaling limits for random fields with long-range dependence. Ann. Prob. 35, 528–550.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41, 623–638.CrossRefGoogle Scholar
Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for workload process. Operat. Res. 32, 890–918.Google Scholar
Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.CrossRefGoogle Scholar
Park, K. and Willinger, W. (2000). Self-Similar Network Traffic and Performance Evaluation. John Wiley, New York.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar