Markov-modulated infinite-server queues driven by a common background process

ABSTRACT

This paper studies a system with multiple infinite-server queues that are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λ_{i, j}, whereas the service times of customers present in this queue are exponentially distributed with mean μ^{− 1}_{i, j}; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature).

Three types of results are presented: in the first place (i) we derive differential equations for the probability-generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large.

Keywords:

• Markov-modulation
• infinite-server queues
• central limit theorems

Mathematics Subject Classification:

• 60K25; 60F05; 60J60; 60J27

Acknowledgements

This research was partly performed when K. De Turck was a Postdoctoral Fellow of Fonds Wetenschappelijk Onderzoek / Research Foundation–Flanders. The authors thank Peter Taylor (University of Melbourne) for suggesting this problem, and for stimulating discussions.

Funding

M. Mandjes is also affiliated to Eurandom, Eindhoven University of Technology, Eindhoven, the Netherlands, and IBIS, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands. His research is partly funded by the NWO Gravitation project Networks, grant number 024.002.003.