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

Abstract

We study a general k dimensional innite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index α ∈ (0, 1).When the arrival rate is sped up by a factor n γ , the transition probabilities of the underlying Markov chain are divided by n γ and the service times are divided by n, we identify two regimes ("fast arrivals", when γ > α, and "equilibrium", when γ = α) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third "slow arrivals" regime, γ < α, we show the convergence of the two rst joint moments of the rescaled process.