×

Estimation of the memory parameter of the infinite-source Poisson process. (English) Zbl 1127.62070

Summary: 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 \(M/G/\infty\) 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.
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.

MSC:

62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
65T60 Numerical methods for wavelets
60K25 Queueing theory (aspects of probability theory)
62G20 Asymptotic properties of nonparametric inference
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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.
[2] Cohen, A. (2003)., Numerical Analysis of Wavelet Methods . Amsterdam: North-Holland. · Zbl 1038.65151
[3] 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.
[4] Faÿ, G., Roueff, F. and Soulier, P. (2005). Estimation of the memory parameter of the infinite source Poisson process., · Zbl 1127.62070 · doi:10.3150/07-BEJ5123
[5] Hall, P. and Welsh, A.H. (1984). Best attainable rates of convergence for estimates of regular variation., Ann. Statist. 3 1079–1084. · Zbl 0539.62048 · doi:10.1214/aos/1176346723
[6] 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. · Zbl 0673.62073
[7] Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model., J. Appl. Probab. 39 671–699. · Zbl 1090.90017 · doi:10.1239/jap/1037816012
[8] Meyer, Y. (1992)., Wavelets and Operators . Cambridge: Cambridge University Press. · Zbl 0776.42019
[9] 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. · Zbl 1021.60076 · doi:10.1214/aoap/1015961155
[10] 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.
[11] Resnick, S. (1987)., Extreme Values, Regular Variation and Point Processes . New York: Springer-Verlag. · Zbl 0633.60001
[12] Resnick, S. and Rootzén, H. (2000). Self-similar communication models and very heavy tails., Ann. Appl. Probab. 10 753–778. · Zbl 1083.60521 · doi:10.1214/aoap/1019487509
[13] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.