# 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.