Asymptotics for infinite server queues with fast/slow Markov switching and fat tailed service times

Abstract

We study a general k dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index When the arrival rate is sped up by a factor the transition probabilities of the underlying Markov chain are divided by and the service times are divided by n, we identify two regimes (”fast arrivals”, when and” equilibrium”, when ) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third” slow arrivals” regime, we show the convergence of the two first joint moments of the rescaled process.