ABSTRACT. This paper studies an infinite-server queue in a Markov environment, that is, an infiniteserver queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates λ i by a factor N , tail probabilities are examined when letting N tend to ∞; non-standard large deviations results are obtained. An importance-sampling based estimation algorithm is proposed, that is proven to be logarithmically efficient.

We analyze a two-node tandem queue with Brownian input. We first derive an explicit expression for the joint distribution function of the workloads of the first and second queue, which also allows us to calculate their exact large-buffer asymptotics. The nature of these asymptotics depends on the model parameters, i.e., there are different regimes. By using sample-path large-deviations (Schilder's theorem) these regimes can be interpreted: we explicitly characterize the most likely way the buffers fill.

We analyze the number of zeros of det(F(α)), where F(α) is the matrix exponent of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the complex plane, where F(α) is analytic. In addition, we also consider the case of a MAP killed at an independent exponential time. The corresponding zeros can be seen as the roots of a generalized Cramér-Lundberg equation. We argue that our results are particularly useful in fluctuation theory for MAPs, which leads to numerous applications in queueing theory and finance.