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

TL;DR

This paper derives exact asymptotics for the probability of large deviations in a multi-timescale Lévy process model, with applications to overdispersed customer streams, revealing complex asymptotic forms and practical accuracy.

Contribution

It provides the first precise asymptotic formulas for a two-timescale Lévy process model, including sublinear correction terms, and demonstrates their practical relevance in service system applications.

Findings

Asymptotic formulas accurately approximate large deviation probabilities.

The results include sublinear correction terms in the exponent.

Numerical experiments confirm the practical utility of the asymptotics.

Abstract

In this paper we study the probability $\xi_n(u):={\mathbb P}\left(C_n\geqslant u n \right)$, with $C_n:=A(\psi_n B(\varphi_n))$ for L\'{e}vy processes $A(\cdot)$ and $B(\cdot)$, and $\varphi_n$ and $\psi_n$ non-negative sequences such that $\varphi_n \psi_n =n$ and $\varphi_n\to\infty$ as $n\to\infty$. Two timescale regimes are distinguished: a `fast' regime in which $\varphi_n$ is superlinear and a `slow' regime in which $\varphi_n$ 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 $\varphi_n$ 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.