We consider the problem of maximizing the long-run average (also the long-run average expected) reward per unit time in a semi-Markov decision processes with arbitrary state and action space. Our main result states that we need only consider the set of stationary policies in that for each $\varepsilon > 0$ there is a stationary policy which is $\varepsilon$-optimal. This result is derived under the assumptions that (roughly) (i) expected rewards and expected transition times are uniformly bounded over all states and actions, and that (ii) there is a state such that the expected length of time until the system returns to this state is uniformly bounded over all policies. The existence of an optimal stationary policy is established under the additional assumption of countable state and finite action space. Applications to queueing reward systems are given.
Steven A. Lippman. "Maximal Average-Reward Policies for Semi-Markov Decision Processes With Arbitrary State and Action Space." Ann. Math. Statist. 42 (5) 1717 - 1726, October, 1971. https://doi.org/10.1214/aoms/1177693170