  In this paper we study the number of customers in infinite-server queues with
a self-exciting (Hawkes) arrival process. Initially we assume that service
requirements are exponentially distributed and that the Hawkes arrival process
is of a Markovian nature. We obtain a system of differential equations that
characterizes the joint distribution of the arrival intensity and the number of
customers. Moreover, we provide a recursive procedure that explicitly
identifies (transient and stationary) moments. Subsequently, we allow for
non-Markovian Hawkes arrival processes and non-exponential service times. By
viewing the Hawkes process as a branching process, we find that the probability
generating function of the number of customers in the system can be expressed
in terms of the solution of a fixed-point equation. We also include various
asymptotic results: we derive the tail of the distribution of the number of
customers for the case that the intensity jumps of the Hawkes process are
heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by
discussing how our results can be used computationally and by verifying the
numerical results via simulations.
