A Duality Relation for Busy Cycles in GI/G/1 Queues

Using a generalization of the classical ballot theorem, Niu and Cooper (1989) established a duality relation between the joint distribution of several variables associated with the busy cycle in M/G/1 (with a modified first service) and the corresponding joint distribution of several related variables in its dual GI/M/1. In this note, we generalize this duality relation to GI/G/1 queues with modified first services; this clarifies the original result, and shows that the generalized ballot theorem is superfluous for the duality relation.