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.

Infinite-server queues with Hawkes input

“…[3] or [20] for references. Very recently, Hawkes models have also been used to study trending social media [9,26,27]. Interestingly, Daw and Pender [9] add a queueing aspect: they model the arrival process of visitors of a website as a Hawkes process, thus trying to capture the viral behavior of such an arrival process, taking into account that customers leave after a time which has a phase-type distribution.…”

“…Very recently, Hawkes models have also been used to study trending social media [9,26,27]. Interestingly, Daw and Pender [9] add a queueing aspect: they model the arrival process of visitors of a website as a Hawkes process, thus trying to capture the viral behavior of such an arrival process, taking into account that customers leave after a time which has a phase-type distribution. They study the number of visitors on the website at any time t. Another queueing application with Hawkes processes is high-frequency transaction processes in limit order books, cf.…”

“…Scaling limits for infinite-server queues designed specifically for Hawkes input are derived in [10]; it states that exact and numerical analysis of this model is 'challenging'. To the best of our knowledge, only [9] pays attention to exact (i.e., nonasymptotic) analysis of queues driven by Hawkes processes. In [9], the focus is on exact analysis of an infinite-server queue driven by an unmarked Markovian Hawkes process (i.e., the jump sizes in the arrival intensity are deterministic).…”

“…To the best of our knowledge, only [9] pays attention to exact (i.e., nonasymptotic) analysis of queues driven by Hawkes processes. In [9], the focus is on exact analysis of an infinite-server queue driven by an unmarked Markovian Hawkes process (i.e., the jump sizes in the arrival intensity are deterministic). The model we consider is a general version of the one studied in [8]: in our case the driving process is a marked Hawkes process, i.e.…”

On the Distributions of Infinite Server Queues With Batch Arrivals

“…Additionally, one could explore findings of this work, like the steady-state distribution representation or the batch scaling, in contexts where there is dependence among the service durations within each batch of arrivals, such as those studied in Pang and Whitt [28], Falin [11]. Finally, we are particularly interested in studying the impact of batch arrivals in the context of selfexciting arrival processes such as Hawkes processes like in the work of Gao and Zhu [13], Koops et al [18], Daw and Pender [5]. We intend to pursue the ideas described here as well as other related concepts in our future work.…”