Duality between the M/ G/ s/ s and GI/ M/1/ s/ s queues

Abstract

We provide a duality relation between the joint distribution of several variables associated with busy servers in an M/ G/ s/ s queue and the corresponding joint distribution associated with idle input sources in a GI/ M/1/ s/ s queue. The M/ G/ s/ s queue, so called Erlang's loss system, has been extensively studied as a model of telephone exchanges. Using the duality relation and the previously established results for the M/ G/ s/ s queue, we can derive some well-known and some new results for the GI/ M/1/ s/ s queue that is a useful model of multiple access systems such as time-sharing computer systems or multiple access communication channels.