Deviations of infinite-server queues and Ornstein-Uhlenbeck processes in a random environment

We will consider an infinite-server queue in a random environment and investigate deviations from its mean behavior. A random environment is represented by a stochastic background process, which modulates both the arrival process and the service process of the queue. This means that changes in the environment influence the behavior of the queue. This leads to overdispersion, in the sense that a modulated infiniteserver queue exhibits more variability than its non-modulated counterpart. We are interested to see how the random environment influences deviations of the infinite-server queue from its typical (mean) behavior on different scales. It turns out that, under a linear scaling of the arrival rates and an appropriate scaling of the background process, we may derive several limit theorems. We will present a central limit theorem, a moderate deviations principle and a large deviations principle for the number of jobs in the system. In addition, we will compare estimates for overflow probabilities using these limit theorems. Finally, we will argue that the results obtained for the modulated infinite-server queue carry over to the modulated Ornstein-Uhlenbeck process