On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals

We consider finiteand infinite-capacity queues with discrete-time batch Markovian arrival processes (D-BMAP) under the assumption of the Late Arrival System with Delayed Access as well as the Early Arrival System. Using simple arguments such as the balance equation, “rate in = rate out,” we derive relationships among the stationary queue lengths at arrival, at departure, and at random epochs. Such relationships hold for a broad class of discrete-time queues with D-BMAP arrivals.