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

Abstract

In this paper we study the probability $ξ_n(u):={\mathbb P}\left(C_n\geqslant u n \right)$, with $C_n:=A(ψ_n B(φ_n))$ for Lévy processes $A(\cdot)$ and $B(\cdot)$, and $φ_n$ and $ψ_n$ non-negative sequences such that $φ_n ψ_n =n$ and $φ_n\to\infty$ as $n\to\infty$. Two timescale regimes are distinguished: a `fast' regime in which $φ_n$ is superlinear and a `slow' regime in which $φ_n$ is sublinear. We provide the exact asymptotics of $ξ_n(u)$ (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 $φ_n$ and $ψ_n$). 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.