Nonergodicity of a Queueing Network under Nonstability of Its Fluid Model

We study ergodicity properties of open queueing networks for which the associated fluid models have trajectories that go to infinity. It is proved that if a trajectory is stable in a certain sense and grows to infinity linearly, then the underlying stochastic process is nonergodic. The result applies to the basic nontrivial examples of nonergodic networks found by Bramson, and Rybko and Stolyar. The proof employs some general results from the large deviation theory.