Infinite-server queues with Hawkes input. (16th November 2018)

• Record Type: Journal Article
• Authors: Koops, D. T.; Saxena, M.; Boxma, O. J.; Mandjes, M.
• Is Part Of: Journal of applied probability. Volume 55:Number 3(2018)
• Page Start: 920
• Page End: 943
• Publication Date: 2018-11-16
• Subjects: Self-exciting process, -- Hawkes process, -- infinite-server queue, -- branching process, -- heavy-tailed distribution, -- heavy traffic; Primary 60K25, -- 60G55, -- 60J80; 519.2
• DOI: 10.1017/jpr.2018.58
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.