Infinite-server queues with Hawkes input

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 equatio...

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Bibliographic Details
Published in:Journal of applied probability 2018-09, Vol.55 (3), p.920-943
Main Authors: Koops, D. T., Saxena, M., Boxma, O. J., Mandjes, M.
Format: Article
Language:English
Subjects:
Branching (mathematics)
Computer simulation
Customers
Differential equations
Fixed points (mathematics)
Markov analysis
Markov processes
Queues
Queuing theory
Research Papers
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Summary: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.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2018.58