Stochastic shortest path problems with associative accumulative criteria

We consider a stochastic shortest path problem with associative criteria in which for each node of a graph we choose a probability distribution over the set of successor nodes so as to reach a given target node optimally. We formulate such a problem as an associative Markov decision processes. We show that an optimal value function is a unique solution to an optimality equation and find an optimal stationary policy. Also we give a value iteration method and a policy improvement method.