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For a Markov decision process with countably infinite states, the optimalvalue may not be achievable in the set of stationary policies. In this paper,we study the existence conditions of an optimal stationary policy in acountable-state Markov decision process under the long-run average criterion.With a properly defined metric on the policy space of ergodic MDPs, theexistence of an optimal stationary policy can be guaranteed by the compactnessof the space and the continuity of the long-run average cost with respect tothe metric. We further extend this condition by some assumptions which can beeasily verified in control problems of specific systems, such as queueingsystems. Our results make a complementary contribution to the literature in thesense that our method is capable to handle the cost function unbounded fromboth below and above, only at the condition of continuity and ergodicity.Several examples are provided to illustrate the application of our mainresults.

On the existence of optimal stationary policies for average Markov decision processes with countable states