Estimation of the Memory Parameter of the Infinite Source Poisson Process

Gilles Fay (1) , François Roueff (2) , Philippe Soulier (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 MODAL'X - Modélisation aléatoire de Paris X

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. 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.

Domaines: Statistiques [math.ST]

Gilles Fay, François Roueff, Philippe Soulier. Estimation of the Memory Parameter of the Infinite Source Poisson Process. Bernoulli, 2007, 13 (2), pp.473--491. ⟨10.3150/07-BEJ5123⟩. ⟨hal-00008797v4⟩