The Markov-modulated Erlang loss system

This paper focuses on a loss system in which both the arrival rate and the per-customer service rate vary according to the state of an underlying finite-state, continuous-time Markov chain. Our first contribution consists of a closed-form expression for the stationary distribution of this Markov-modulated Erlang loss queue. This, in particular, provides us with an explicit formula for the probability that the queue is full, which can be regarded as the Markov-modulated counterpart of the well-known Erlang loss formula. It facilitates the computation of the probability that an arbitrary arriving customer is blocked.
Furthermore, we consider a regime where, in a way that is common for this type of loss system, we scale the arrival rate and the number of servers, while also scaling the transition rates of the modulating Markov process. We establish convergence of the stationary distribution to a truncated Normal distribution, which leads to an approximation for the blocking probability. In this 'fast regime', the parameters of the limiting distribution critically depend on the precise scaling imposed. We also derive scaling results for a 'slow regime', in which the modulating Markov process is slow relative to the arrival process. Numerical experiments show that the resulting approximations are highly-accurate.