Infinite-server System with Hawkes Arrivals and Hawkes Services

TL;DR

This paper analyzes infinite-server queueing systems with self-exciting Hawkes processes for arrivals and services, deriving their Markov properties and differential equations for joint distributions.

Contribution

It introduces a novel framework for modeling infinite-server queues with Hawkes and sdHawkes processes, including Markov property derivation and differential equations for distributions.

Findings

Derived Markov property for Hawkes/sdHawkes/infty system.

Characterized joint distributions via differential equations.

Provided time-dependent results for M/sdHawkes/inite system.

Abstract

This paper is devoted to the study of the number of customers in infinite-server systems driven by Hawkes processes. In these systems, the self-exciting arrival process is assumed to be represented by a Hawkes process and the self-exciting service process by a state-dependent Hawkes process (sdHawkes process). Under some suitable conditions, for the Hawkes/sdHawkes/infty system, the Markov property of the system is derived. The joint time-dependent distribution of the number of customers in the system, the arrival intensity and the server intensity is characterised by a system of differential equations. Then, the time-dependent results are also deduced for the M/sdHawkes/\infty system.