Infinite-server systems with Hawkes arrivals and Hawkes services

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 $\mathrm{Hawkes}/\mathrm{sdHawkes}/\infty$ 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 characterized by a system of differential equations. Then, the time-dependent results are also deduced for the $M/\mathrm{sdHawkes}/\infty$ system.