Queues Driven by Hawkes Processes

Abstract

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.

An elementary derivation of moments of Hawkes processes

“…In the following, a numerical example is given for a simple Hawkes process. From (10) and (11) As shown in Figure 1, we see that correlation coefficient ρ(t) of N(t) and λ(t) is a decreasing function of time t and lim t↓0 ρ(t) = 1, lim t↑∞ ρ(t) = 0, as we intuitively expect.…”

“…In fact these are not so easy to describe in general, but results for low-order moments are easy to obtain. Results for all first-and second-order moments can be found: from (10) and (11) for first-order moments, and from (8), (12), (13), and (14) for second-order moments; see Example 1 below. All expectations of order m can be found as follows.…”

Infinite-server systems with Coxian arrivals

“…While the arrival process of visitors to the Web site may well be a Poisson process, its rate may jump up due to some (external) event, then decay gradually, only to jump up again because of another event. Such an example formed one of the motivations for [2,7,8], which all study infinite-server queues with an overdispersed arrival process.…”

“…Infinite-server queues are also studied in [2] and [8], but the arrival process there is a Hawkes process, the so-called self-exciting process-jumps in the shot-noise process in turn increase the intensity of the occurrence of the jumps. Daw and Pender [2] present several interesting motivating examples. They consider deterministic jump sizes in the shot-noise process and study in particular the Hawkes/Ph/∞ and Hawkes/D/∞ queues, obtaining detailed expressions for moments and autocovariances.…”

New perspectives on the Erlang-A queue

“…Because the relationships between the true quantities and their approximations are conditioned only on the service and abandonment rate, it may be possible for this to be extended to stochastic-intensity, non-Poisson arrival processes, such as the Hawkes process or shot noise driven queues studied in [23], [24], and [6]. Finally, in a similar manner it would be interesting to extend this to networks of Erlang-A queues; however, we would have to keep track of the routeing probabilities carefully to keep track of the convexity/concavity of the rate functions.…”