Queues Driven by Hawkes Processes

TL;DR

This paper models queues with arrivals driven by Hawkes processes, deriving exact formulas for moments and covariances, and demonstrates applications in internet traffic and nightclub management.

Contribution

It introduces a novel queueing model with Hawkes process arrivals, providing analytical solutions for moments, covariances, and applications in real-world scenarios.

Findings

Derived differential equations for moments and generating functions.

Obtained exact expressions for mean, variance, and auto-covariance.

Applied model to internet traffic and nightclub admission control.

Abstract

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.