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Infinite-server queues with Hawkes input

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  16 November 2018

D. T. Koops ,
M. Saxena ,
O. J. Boxma  and
M. Mandjes
D. T. Koops*
Affiliation:
University of Amsterdam
M. Saxena*
Affiliation:
Eindhoven University of Technology
O. J. Boxma*
Affiliation:
Eindhoven University of Technology
M. Mandjes*
Affiliation:
University of Amsterdam
*
* Postal address: Korteweg-de Vries Institute, University of Amsterdam, PO Box 94248, 1090GE, Amsterdam, The Netherlands.
*** Postal address: Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600MB, Eindhoven, The Netherlands.
*** Postal address: Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600MB, Eindhoven, The Netherlands.
* Postal address: Korteweg-de Vries Institute, University of Amsterdam, PO Box 94248, 1090GE, Amsterdam, The Netherlands.
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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.

Keywords

Self-exciting processHawkes processinfinite-server queuebranching processheavy-tailed distributionheavy traffic

MSC classification

Primary: 60K25: Queueing theory 60G55: Point processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 920 - 943
Copyright
Copyright © Applied Probability Trust 2018 

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