Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach

In this paper we consider an infinite-server queue in a random environment. The distinguishing feature of the model is the presence of two irreducible Markov chains: one Markov chain modulates the arrival rates, while the other modulates the service times. We are interested in the probability that the number of jobs in the system becomes unusually large, i.e. we are interested in overflow. Because arrival rates and service times are stochastically varying over time, the number of jobs in the system has a Poisson distribution with random parameters rather than a 'classical' Poisson distribution. In this case we cannot use a CLT-type result to analyze the system. However, basic large-deviations techniques provide an alternative approach. Scaling the arrival rates linearly, we prove a large-deviations principle by conditioning on paths of the background processes that are very likely to lead to overflow. We show that, conditional on these paths, we are in a situation in which Cramér's Theorem may be applied. This gives us the rate function for the number of jobs in the system and we use it to describe overflow probabilities. A nice observation is that we do not need to know the transition probabilities of the Markov chains to say something about overflow probabilities.

ISBN:9789461972095

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