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.