ERGODICITY OF QUEUING-NETWORKS

We consider an open Jackson-type service network with single-channel stations. It is shown that if the load on each station is less than 1, the process defined by the length of the queue satisfies an ergodic theorem in discrete time. If it is additionally supposed that the lengths of the intervals between the arrivals of calls possesses a nonlattice distribution, an ergodic theorem will also hold for the queue length process in continuous time. These results are carried over to the case of multichannel stations. For the special case of "acyclic networks," it is proved that an ergodic theorem will also hold in the most general situation in which the elements of the control sequences form a stationary metrically transitive sequence. We also consider closed networks for which an ergodic theorem is proved under the condition that the distributions of service times is nonlattice.