Denumerable semi-Markov decision chains with small interest rates

In this paper the authors investigate denumerable state semi-Markov decision chains with small interest rates. They consider average and Blackwell optimality and allow for multiple closed sets and unbounded immediate rewards. The present analysis uses the existence of a Laurent series expansion for the total discounted rewards and the continuity of its terms. The assumptions are expressed in terms of a weighted supremum norm. The present method is based on an algebraic treatment of Laurent series; it constructs an appropriate linear space with a lexicographic ordering. Using two operators and a positiveness property the authors establish the existence of bounded solutions to optimality equations. The theory is illustrated with an example of a K-dimensional queueing system. This paper is strongly based on the work of Denardo and Dekker and Hordijk.