Exact Asymptotics for a Multitimescale Model with Applications in Modeling Overdispersed Customer Streams

In this paper, we study the asymptotics of the (scaled) exceedance probability of a multitimescale doubly-stochastic Lévy process. Two timescale regimes are distinguished: a fast regime in which one of the timescales is superlinear and a slow regime in which it is sublinear. We provide the exact asymptotics of the exceedance probability 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 the timescales chosen). To showcase the power of our results, we include two examples covering the cases where the multitimescale Lévy process is lattice and nonlattice. Finally, we present numerical experiments that demonstrate the importance of taking into account the doubly stochastic nature 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.