Infinite server queues in a random fast oscillatory environment

Abstract

In this paper, we consider a $$\text {Cox}/G_t/\infty $$ infinite server queueing model in a random environment. More specifically, the arrival rate in our server is modeled as a highly fluctuating stochastic process, which arguably takes into account some small timescale variations often observed in practice. We prove a homogenization property for this system, which yields an approximation by an $$M_t/G_t/\infty $$ queue with some effective parameters. Our limiting results include the description of the number of active servers, the total accumulated input and the solution of the storage equation. Hence, in the fast oscillatory context under consideration, we show how the queuing system in a random environment can be approximated by a more classical Markovian system.