Determining the optimal evolution time for Markov processes with final sequence of states

This paper describes a class of dynamical stochastic systems that represents an extension of classical Markov decision processes. The Markov stochastic systems with given final sequence of states and unitary transition time, over a finite or infinite state space, are studied. Such dynamical system stops its evolution as soon as given sequence of states in given order is reached. The evolution time of the stochastic system with fixed final sequence of states depends on initial distribution of the states and probability transition matrix. The considered class of processes represents a generalization of zero-order Markov processes, studied in [3]. We are seeking for the optimal initial distribution and optimal probability transition matrix that provide the minimal evolution time for the dynamical system. We show that this problem can be solved using the signomial and geometric programming approaches.