Journal of Applied Probability

Fractional Brownian motion with H < 1/2 as a limit of scheduled traffic

Victor F. Araman and Peter W. Glynn

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 49, Number 3 (2012), 710-718.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1346955328

Digital Object Identifier
doi:10.1239/jap/1346955328

Mathematical Reviews number (MathSciNet)
MR3012094

Zentralblatt MATH identifier
1255.60062

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes [See also 58J65] 60G99: None of the above, but in this section
Secondary: 60G70: Extreme value theory; extremal processes 90B30: Production models

Keywords
Fractional Brownian motion scheduled traffic heavy-tailed distribution limit theorem

Citation

Araman, Victor F.; Glynn, Peter W. Fractional Brownian motion with H &lt; 1/2 as a limit of scheduled traffic. J. Appl. Probab. 49 (2012), no. 3, 710--718. doi:10.1239/jap/1346955328. https://projecteuclid.org/euclid.jap/1346955328


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References

  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
  • Cox, D. R. and Smith, W. L. (1961). Queues. John Wiley, New York.
  • Gurin, C. A. \et (1999). Empirical testing of the infinite source Poisson data traffic model. Tech. Rep. 1257, School of Operations Research and Information Engineering, Cornell University.
  • Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.
  • Kaj, I. (2005). Limiting fractal random processes in heavy tailed systems. In Fractals in Engineering, New Trends in Theory and Applications, eds J. Lévy-Lehel and E. Lutton, Springer, London, pp. 199–218.
  • 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 M. E. Vares and V. Sidoravicius, Birkhäuser, Basel, pp. 383–427.
  • Kurtz, T. G. (1996). Limit theorems for workload input models. In Stochastic Networks: Theory and Applications, eds S. Zachary, F. P. Kelly and I. Ziedins, Clarendon Press, Oxford, pp. 119–140.
  • Mandjes, M., Norros, I. and Glynn, P. (2009). On convergence to stationarity of fractional Brownian storage. Ann. Appl. Prob. 18, 1385–1403.
  • 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.
  • 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.
  • 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.
  • Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302. \endharvreferences