Semi-Markov decision processes with a reachable state-subset

The paper considers the problem of minimizing the long-run average expected cost per unit time in a semi-Markov decision process with arbitrary state and action space. Assuming the existence of a Borel subset of state space called a reachable state-subset, it derives the optimality equation for the unbounded costs. The contraction property for the average case is used, so that the assumptions of both continuity of the one-step cost function and compactness of state and action space are excluded.