Infinite-server System with Hawkes Arrivals and Hawkes Services
Dharmaraja Selvamuthu, Paola Tardelli

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.
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