A complete solution for the stationary queue‐length distribution of a bulk‐arrival, bulk‐service (GI
X
/M
Y
/1) queue is presented. Beginning with a known expression for the probability generating function of the stationary pre‐arrival‐epoch queue‐length distribution, the roots method is used to invert it and determine all probabilities. Next, using level crossing arguments, theoretical relationships between pre‐arrival and arbitrary‐epoch probabilities are developed. These relationships are then used to directly determine a complete set of probabilities for the arbitrary‐epoch queue‐length distribution. Finally, selected examples are presented. These demonstrate how, given arbitrary arrival time, arrival group size and service batch size probability distributions, a complete solution for the stationary queue‐length probabilities can be readily determined.
