Abstract A complete solution for the stationary queue-length distribution of a bulk-arrival, bulk-service (GIX/MY/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.