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

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 alpha is an element of(0,1). When the arrival rate is sped up by a factor n gamma, the transition probabilities of the underlying Markov chain are divided by n gamma and the service times are divided by n, we identify two regimes (''fast arrivals'', when gamma>alpha, and'' equilibrium'', when gamma=alpha) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third'' slow arrivals'' regime, gamma