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Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic

Part of: Limit theorems Operations research and management science Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Victor F. Araman  and
Peter W. Glynn
Victor F. Araman*
Affiliation:
American University of Beirut
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Olayan School of Business, American University of Beirut, Beirut 1107-2020, Lebanon. Email address: va03@aub.edu.lb
∗∗ Postal address: Management Science and Engineering, Stanford University, Stanford, CA 94305-4121, USA. Email address: glynn@stanford.edu
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Abstract

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In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

Keywords

Fractional Brownian motionscheduled trafficheavy-tailed distributionlimit theorem

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes 60G99: None of the above, but in this section
Secondary: 60G70: Extreme value theory; extremal processes 90B30: Production models
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 710 - 718
Copyright
© Applied Probability Trust 

References

Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Cox, D. R. and Smith, W. L. (1961). Queues. John Wiley, New York.Google Scholar
Gurin, C. A. et al. (1999). Empirical testing of the infinite source Poisson data traffic model. Tech. Rep. 1257, School of Operations Research and Information Engineering, Cornell University.Google Scholar
Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.CrossRefGoogle Scholar
Kaj, I. (2005). Limiting fractal random processes in heavy tailed systems. In Fractals in Engineering, New Trends in Theory and Applications, eds Lévy-Lehel, J. and Lutton, E., Springer, London, pp. 199–218.Google Scholar
Kaj, I. and Taqqu, M. S. (2008). Convergence to fractional Brownian motion and to the telecom process: the integral representation approach. In In and Out of Equilibrium 2 (Progress Prob. 60), eds Vares, M. E. and Sidoravicius, V., Birkhäuser, Basel, pp. 383–427.Google Scholar
Kurtz, T. G. (1996). Limit theorems for workload input models. In Stochastic Networks: Theory and Applications, eds Zachary, S., Kelly, F. P. and Ziedins, I., Clarendon Press, Oxford, pp. 119–140.CrossRefGoogle Scholar
Mandjes, M., Norros, I. and Glynn, P. (2009). On convergence to stationarity of fractional Brownian storage. Ann. Appl. Prob. 18, 1385–1403.Google Scholar
Mandjes, M., Mannersalo, P., Norros, I. and van Uitert, M. (2006). Large deviations of infinite intersections of events in Gaussian processes. Stoch. Process. Appl. 116, 1269–1293.CrossRefGoogle 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.Google Scholar
Pipiras, V., Taqqu, M. S. and Levy, J. B. (2004). Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10, 121–163.Google Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.Google Scholar