Bernoulli

  • Bernoulli
  • Volume 13, Number 2 (2007), 473-491.

Estimation of the memory parameter of the infinite-source Poisson process

Gilles Faÿ, François Roueff, and Philippe Soulier

Full-text: Open access

Abstract

Long-range dependence induced by heavy tails is a widely reported feature of internet traffic. Long-range dependence can be defined as the regular variation of the variance of the integrated process, and half the index of regular variation is then referred to as the Hurst index. The infinite-source Poisson process (a particular case of which is the $M/G/∞$ queue) is a simple and popular model with this property, when the tail of the service time distribution is regularly varying. The Hurst index of the infinite-source Poisson process is then related to the index of regular variation of the service times. In this paper, we present a wavelet-based estimator of the Hurst index of this process, when it is observed either continuously or discretely over an increasing time interval. Our estimator is shown to be consistent and robust to some form of non-stationarity. Its rate of convergence is investigated.

Article information

Source
Bernoulli, Volume 13, Number 2 (2007), 473-491.

Dates
First available in Project Euclid: 18 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1179498757

Digital Object Identifier
doi:10.3150/07-BEJ5123

Mathematical Reviews number (MathSciNet)
MR2331260

Zentralblatt MATH identifier
1127.62070

Keywords
heavy tails internet traffic long-range dependence Poisson point processes semiparametric estimation wavelets

Citation

Faÿ, Gilles; Roueff, François; Soulier, Philippe. Estimation of the memory parameter of the infinite-source Poisson process. Bernoulli 13 (2007), no. 2, 473--491. doi:10.3150/07-BEJ5123. https://projecteuclid.org/euclid.bj/1179498757


Export citation

References

  • Barakat, C., Thiran, P., Iannaccone, G., Diot, C. and Owezarski, P. (2002). A flow-based model for internet backbone traffic. In, Proceedings of the 2nd ACM SIGCOMM Workshop on Internet Measurement, pp. 35–47. New York: ACM Press.
  • Cohen, A. (2003)., Numerical Analysis of Wavelet Methods. Amsterdam: North-Holland.
  • Duffield, N.G., Lund, C. and Thorup, M. (2002). Properties and prediction of flow statistics from sampled packet streams. In, Proceedings of the 2nd ACM SIGCOMM Workshop on Internet Measurement, pp. 159–171. New York, ACM Press.
  • Fa\"y, G., Roueff, F. and Soulier, P. (2005). Estimation of the memory parameter of the infinite source Poisson process., arXiv:math.ST/0509371v2.
  • Hall, P. and Welsh, A.H. (1984). Best attainable rates of convergence for estimates of regular variation., Ann. Statist. 3 1079–1084.
  • Künsch, H.R. (1987). Statistical aspects of self-similar processes. In Yu.A. Prohorov and V.V. Sazonov (eds), Proceedings of the First World Congres of the Bernoulli Society, Vol. 1, pp. 67–74. Utrecht: VNU Science Press.
  • Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model., J. Appl. Probab. 39 671–699.
  • Meyer, Y. (1992)., Wavelets and Operators. Cambridge: Cambridge University Press.
  • Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab. 12 23–68.
  • Parulekar, M. and Makowski, A.M. (1997). M/G/infinity input processes: A versatile class of models for network traffic. In, Proceedings of INFOCOM '97, pp. 419–426. Los Alamitos, CA: IEEE Computer Society Press.
  • Resnick, S. (1987)., Extreme Values, Regular Variation and Point Processes. New York: Springer-Verlag.
  • Resnick, S. and Rootzén, H. (2000). Self-similar communication models and very heavy tails., Ann. Appl. Probab. 10 753–778.
  • Wornell, G.A. and Oppenheim, A.V. (1992). Wavelet-based representations for a class of self-similar signals with application to fractal modulation., IEEE Trans. Inform. Theory 38 785–800.