Queues Driven by Hawkes Processes

Andrew Daw, Jamol Pender

Published 2017-07-17Version 1

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. In this 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 mean, variance, and covariances. Furthermore, we compare our analytical results with simulation, and compare the simulated limiting distributions to queues that are driven by Poisson processes instead of Hawkes processes. As motivation for our Hawkes queueing model, we demonstrate its usefulness through three applications. These applications are calls to airline help centers, trending internet data traffic, and even arrivals to night clubs. Lastly, in the night club or "Club Queue" setting, we design an optimal control problem that determines the optimal rate at which a bouncer should allow club-goers to enter the club.