Michel Mandjes

Overdispersion and multi-timescale models: exact asymptotics

Poisson processes are frequently used, e.g. to model the customer arrival process in service systems, or the claim arrival proces in insurance models. In various situations, however, the fluctuations in the arrival rate are so severe that the Poisson assumption ceases to hold. In a commonly followed approach to remedy this, the deterministic parameter λ is replaced by a stochastic process Λ(t); in this way the arrival process becomes “overdispersed”.

The first part of this talk considers the case that the Poisson rate is sampled periodically, with a focus on an infinite-server queue fed by the resulting overdispersed arrival process. After having presented a functional central limit theorem, we concentrate on tail probabilities under a particular scaling of the arrival process and the sampling frequency. We derive logarithmic tail asymptotics, and in specific cases even exact tail asymptotics.

In the second part of the talk we embed our overdispersion setting in a more general framework. The probability of interest is expressed in terms of the composition of two Lévy processes, which can alternatively be seen as a Lévy process with random time change. For this two-timescale model we present exact tail asymptotics. The proof relies on an adaptation of classical techniques developed by Bahadur and Rao, in combination with delicate Edgeworth expansion arguments. The resulting asymptotics have a remarkable form, with finitely many sublinear terms in the exponent.
[Joint work with Mariska Heemskerk and Julia Kuhn]

References:
-- the paper itself can be found on https://arxiv.org/pdf/1801.02999.pdf
-- Background material is:
R. Bahadur and R. Rao (1960). On deviations of the sample mean. Annals of Mathematical Statistics, 31, pp. 1015-1027.
A. Dembo and O. Zeitouni (1998). Large Deviations Techniques and Applications. 2nd ed. Springer, New York.
M. Heemskerk, J. Kuhn, and M. Mandjes (2017). Rare-event analysis of mixed Poisson random variables, and applications in staffing. Performance Evaluation 114, pp. 56-77.
M. Heemskerk, J. van Leeuwaarden, and M. Mandjes (2017). Scaling limits for infinite-server systems in a random environment. Stochastic Systems 7, pp. 1-31.
T. Höglund (1979). A unified formulation of the central limit theorem for small and large deviations from the mean. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 49, pp. 105-117.