  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.
