Ergodicity of Stochastic Processes Describing the Operation of Open Queueing Networks

We consider stochastic processes that model the operation of open queueing networks with calls of different types. Each call type has its own route. A new ergodicity criterion is proposed for continuous-time countable Markov chains. This criterion is applied to reduce the problem of finding the ergodicity conditions of the Markov process that describes the operation of the queueing network to the analysis of the behavior of a special (limiting) deterministic process obtained from the original process by normalization and a change of time scale. For the simplest nontrivial network of this class – a two-node network with two types of calls moving in opposite directions, the natural condition of “less than unit load at each node” is sufficient for ergodicity of the modeling Markov process under the FIFO discipline. An example of a simple priority discipline is considered for which the corresponding Markov process is nonrecurrent under this condition.