On the relationship between queue lengths at a random instant and at a departure in the stationary queue with BMAP arrivals

This paper considers the queue length distributions at a random point in time and at a departure in the stationary queue with a batch Markovian arrival process (BMAP). Using the rate conservation law of Miyazawa, we prove a simple relationship between the vector generating functions of the queue length distributions at a random point in time and at a departure. An interesting feature of the proof is that we do not assume any particular service mechanism. The relationship then holds for a broad class of stationary queues with BMAP arrivals.