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. We focus on the probability that the number of jobs in the system attains an unusually high value. Scaling the arrival rates λ i by a factor N and the transition rates ν ij of the background process as well, a large-deviations based approach is used to examine such tail probabilities (where N tends to ∞). The paper also presents qualitative properties of the system's behavior conditional on the rare event under consideration happening.