An Infinite-Server System with L\'evy Shot-Noise Modulation: Moments and Asymptotics

#### O.J. Boxma, M. Mandjes, M. Saxena

2020, v.26, Issue 4, 757-778

ABSTRACT

We consider an infinite-server system with as input process a non-homogeneous Poisson process
with rate function $\Lambda(t) = \vect{a}^\intercal \vect{X}(t)$. Here $\{\vect{X}(t): t \geq 0\}$ is a generalized multivariate shot-noise process fed by a L\'evy subordinator rather than by just a compound Poisson process. We study the transient behavior of the model, analyzing the joint distribution of the number of customers in the queueing system jointly with the multivariate shot-noise process. We also provide a recursive procedure that explicitly identifies transient as well as stationary moments and correlations. Various heavy-tail and heavy-traffic asymptotic results are also derived, and numerical results are presented to provide further insight
into the model behavior.

Keywords: infinite-server queue, non-homogeneous Poisson process, L\'evy subordinator, modulated shot-noise process