A single-server queue with random accumulation level

The paper studies the queueing process in a single-server bulk queueing system. Upon completion of a previous service, the server can take a group of random size from customers that are available. Or, the server can wait until the queue attains a desired level. An ergodicity criterion for both the queueing process with continuous time parameter and the imbedded process us established. Under this criterion, explicit formulas for the stationary distributions of both processes by using semi-regenerative techniques are obtained.