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

Using a generalization of the classical ballot theorem, Niu and Cooper [7] 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. AMS 1980 subject classification. Primary: 90B22; Secondary: 60K25. IAOR 1973 subject classification. Main: Queues. OR/MS Index 1978 subject classification. Primary: 681 Queues.