This paper establishes a rather complete optimality theory for the average cost semi-Markov decision model with a denumerable state space, compact metric action sets and unbounded one-step costs for the case where the underlying Markov chains have a single ergotic set. Under a condition which, roughly speaking, requires the existence of a finite set such that the supremum over all stationary policies of the expected time and the total expected absolute cost incurred until the first return to this set are finite for any starting state, we shall verify the existence of a finite solution to the average costs optimality equation and the existence of an average cost optimal stationary policy.
Federgruen, Awi, A. Hordijk, and H. C. Tijms. "Denumerable state semi-Markov decision processes with unbounded costs, average cost criterion." Stochastic Processes and their Applications 9, no. 2 (November 1979): 223-235.
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