Infinite-server queues with Hawkes input

TL;DR

This paper analyzes infinite-server queues driven by Hawkes self-exciting processes, deriving differential equations, moments, and distributional properties, including asymptotics and heavy-traffic limits, with computational methods and simulations.

Contribution

It extends queue analysis to Hawkes input, providing explicit equations, moments, and distributional insights for both Markovian and non-Markovian cases.

Findings

Derived differential equations for joint distribution

Provided recursive moments calculation method

Established asymptotic tail behavior and heavy-traffic results

Abstract

In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and non-exponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by discussing how our results can be used computationally and by verifying the numerical results via simulations.