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 $\alpha \in (0,1).$ 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, $\gamma <\alpha ,$ we show the convergence of the two first joint moments of the rescaled process.

AMS 2000 Subject Classifications::

• Primary 60G50
• 60K30
• 62P05
• 60K25

Keywords:

• Infinite server queues
• Incurred But Not Reported (IBNR) claims
• Markov modulation
• rescaled process