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 $α\in (0,1)$. 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 first joint moments of the rescaled process.