  Many stochastic systems have arrival processes that exhibit clustering
behavior. In these systems, arriving entities influence additional arrivals to
occur through self-excitation of the arrival process. In this paper, we analyze
an infinite server queueing system in which the arrivals are driven by the
self-exciting Hawkes process and where service follows a phase-type
distribution or is deterministic. In the phase-type setting, we derive
differential equations for the moments and a partial differential equation for
the moment generating function; we also derive exact expressions for the
transient and steady-state mean, variance, and covariances. Furthermore, we
also derive exact expressions for the auto-covariance of the queue and provide
an expression for the cumulant moment generating function in terms of a single
ordinary differential equation. In the deterministic service setting, we
provide exact expressions for the first and second moments and the queue
auto-covariance. As motivation for our Hawkes queueing model, we demonstrate
its usefulness through two novel applications. These applications are trending
internet traffic and arrivals to nightclubs. In the web traffic setting, we
investigate the impact of a click. In the nightclub or "Club Queue" setting, we
design an optimal control problem for the rate to admit club-goers.
