Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through selfexcitation 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 optimal rate to admit club-goers.

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through selfexcitation 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 optimal rate to admit club-goers.

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