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

Infinite-server System with Hawkes Arrivals and Hawkes Services