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 index α∈(0,1). When arrival rate is sped up by factor nγ, transition probabilities of underlying chain divided nγ n, we identify two regimes (”fast arrivals”, when γ>α, and” equilibrium”, γ=α) in which prove that properly rescaled converges pointwise distribution to some limiting process. In third” slow arrivals” regime, γ<α, show convergence first joint moments