Overflow asymptotics for an infinite-server queue in different random environments

In this talk, we will consider an infinite-server queue in two different random environments. A random environment is represented by a stochastic background process, which modulates both the arrival process and the service process of the queue. The two different background processes that we will consider are a continuous-time Markov chain and a reflected Brownian motion. We will look at a queue that has to divide its attention between allowing customers to enter and serving the customers. The background process determines how much attention goes to either task. We will study the probability that the number of jobs in the system becomes unusually large, i.e. we will study overflow. Scaling the arrival process and using large deviations techniques, we compute the rate functions that describe the exponential rate of convergence of the overflow probabilities in the respective random environments. Surprisingly, the rate functions turn out to be the same for both random environments. Apparently, two very different modulating processes may lead to the same large deviations principle.

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