Arrival and departure state distributions in the general bulk-service queue

In this paper, we give an explicit relation between steady-state probability distributions of the buffer occupancy at customer entrance and departure epochs, for the classical single-server system G/G^[N]/1 with batch services and for the finite capacity case. The method relies on level-crossing arguments. For the particular case of Poisson input, we also express the loss probability in terms of state probabilities at departure epochs, yielding probabilities observed by arriving customers. This work provides the ‘bulk queue’ version of a result established by Burke, who stated the equality between probabilities at arrival and departure epochs for systems with ‘unit jumps’.