# Duality and Other Results for *M/G/1* and *GI/M/1* Queues,
via a New Ballot Theorem

We generalize the classical ballot theorem and use it to obtain direct
probabilistic derivations of some well-known and some new results relating
to busy and idle periods and waiting times in *M/G/1* and *GI/M/1* queues.
In particular, we uncover a duality relation between the joint distribution
of several variables associated with the busy cycle in *M/G/1* and the
corresponding joint distribution in *GI/M/1*. In contrast with the classical
derivations of queueing theory, our arguments avoid the use of transforms,
and thereby provide insight and term-by-term "explanations" for the
remarkable forms of some of these results.