Stationary queue length in a first-in-first-out single server queue with service interruptions and multiple batch Markovian arrival streams

This paper considers a FIFO single-server queue with service interruptions and multiple batch Markovian arrival streams. The server state (on and off), the type of arriving customers and their batch size are assumed to be governed by a continuous-time Markov chain with finite states. To put it more concretely, the marginal process of the server state is a phase-type alternating Markov renewal process, the marginal arrival process is a batch marked Markovian arrival process, and they may be correlated. Further, service times of arriving customers are allowed to depend on both their arrival stream and the server state on arrival. For such a queue, we derive the vector joint generating function of the numbers of customers from respective arrival streams. Further assuming discrete phase-type batch size distributions, we establish a numerical algorithm to compute the joint queue length distribution at a random point in time. Finally, we show some numerical examples and examine the impact of system parameters on the queue length distribution.