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Markov-modulated infinite-server queues driven by a common background process
Authors:Michel Mandjes  Koen De Turck
Institution:1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands;2. CWI, Amsterdam, the Netherlands;3. TELIN, Ghent University, Gent, Belgium;4. Laboratoire Signaux et Systémes, école CentraleSupélec, Université Paris Saclay, Gif-sur-Yvette, France
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 λi, j, whereas the service times of customers present in this queue are exponentially distributed with mean μ? 1i, 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
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