Computing semi-stationary optimal policies for multichain semi-Markov decision processes

We consider semi-Markov decision processes with finite state and action spaces and a general multichain structure. A form of limiting ratio average (undiscounted) reward is the criterion for comparing different policies. The main result is that the value vector and a pure optimal semi-stationary policy (i.e., a policy which depends only on the initial state and the current state) for such an SMDP can be computed directly from an optimal solution of a finite set (whose cardinality equals the number of states) of linear programming (LP) problems. To be more precise, we prove that the single LP associated with a fixed initial state provides the value and an optimal pure stationary policy of the corresponding SMDP. The relation between the set of feasible solutions of each LP and the set of stationary policies is also analyzed. Examples are worked out to describe the algorithm.