Exact asymptotics for a multi-timescale model, with applications in modeling overdispersed customer streams

Abstract

In this paper we study the probability ${\xi }_n(u):=\mathbb{P}\left(C_n⩾un\right)$, with $C_n:=A({\psi }_{n}B({\phi }_n))$ for Lévy processes $A(\cdot )$ and $B(\cdot )$, and ${\phi }_n$ and ${\psi }_n$ non-negative sequences such that ${\phi }_n{\psi }_n=n$ and ${\phi }_n\to \infty$ as $n\to \infty$. Two timescale regimes are distinguished: a `fast' regime in which ${\phi }_n$ is superlinear and a `slow' regime in which ${\phi }_n$ is sublinear. We provide the exact asymptotics of ${\xi }_n(u)$ (as $n\to \infty$) for both regimes, relying on change-of-measure arguments in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms (the number of which depends on the precise form of ${\phi }_n$ and ${\psi }_n$). To showcase the power of our results we include two examples, covering both the case where $C_n$ is lattice and non-lattice. Finally we present numerical experiments that demonstrate the importance of taking into account the doubly stochastic nature of $C_n$ in a practical application related to customer streams in service systems; they show that the asymptotic results obtained yield highly accurate approximations, also in scenarios in which there is no pronounced timescale separation.

arXiv comment25 pages, no figures