Journal of Applied Mathematics and Stochastic Analysis
The author 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. The author establishes an ergodicity criterion for both the queueing process with continuous time parameter and the imbedded process. Under this criterion, the author obtains explicit formulas for the stationary distributions of both processes by using semi-regenerative techniques.
Dshalalow, J. (1991). A single-server queue with random accumulation level. Journal of Applied Mathematics and Stochastic Analysis, 4(3), 203.