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We deal with infinite state Markov decision processes with unbounded costs. Three simple conditions, based on the optimal discounted value function, guarantee the existence of an expected average cost optimal stationary policy. Sufficient conditions are the existence of a distinguished state of smallest discounted value and a single stationary policy inducing an irreducible, ergodic Markov chain for which the average cost of a first passage from any state to the distinguished state is finite. A result to verify this is also given. Two examples illustrate the ease of applying the criteria.

A Markov decision chain with denumerable state space incurs two types of costs — for example, an operating cost and a holding cost. The objective is to minimize the expected average operating cost, subject to a constraint on the expected average holding cost. We prove the existence of an optimal constrained randomized stationary policy, for which the two stationary policies differ on at most one state. The examples treated are a packet communication system with reject option and a single-server queue with service rate control.

Semi-Markov decision processes underlie the control of many queueing systems. In this paper, we deal with infinite state semi-Markov decision processes with nonnegative, unbounded costs and finite action sets. Axioms for the existence of an expected average cost optimal stationary policy are presented. These conditions generalize the work in Sennott [22] for Markov decision processes. Verifiable conditions for the axioms to hold are obtained. The theory is applied to control of the M/G/l queue with variable service parameter, with on-off server, and with batch processing, and to control of the G/M/m queue with variable arrival parameter and customer rejection. It is applied to a timesharing network of queues with a single server and finally to optimal routing of Poisson arrivals to parallel exponential servers. The final section extends the existence result to compact action spaces.

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