Optimal threshold probability and expectation in semi-Markov decision processes

We consider undiscounted semi-Markov decision process with a target set and our main concern is a problem minimizing threshold probability. We formulate the problem as an infinite horizon case with a recurrent class. We show that an optimal value function is a unique solution to an optimality equation and there exists a stationary optimal policy. Also several value iteration methods and a policy improvement method are given in our model. Furthermore, we investigate a relationship between threshold probabilities and expectations for total rewards.