A Markov-modulated \(M/M/1\) queue with group arrivals (Q1175220)

The author considers a Markov-modulated \(M/M/1\) queue with group arrivals described as follows. Let \(\{J_ t, t\geq 0\}\) be the underlying Markov process with finite state space. If \(J_ t=j\), then groups of customers arrive according to a Poisson process with intensity \(\lambda_ j\), the group sizes are characterized by random variables \(\xi_ j\) and the service times are distributed exponentially with parameter \(\mu_ j\). The author gives a representation of the queue length and departure process in terms of Laplace transforms. Using a Tauberian theorem he gets the limiting distribution of the queue length process. For some special cases explicit results are obtained, furthermore time reversibility is investigated.