On the joint distribution of runs in the sequence of Markov-dependent multi-state trials

Under different popular counting schemes of runs we have obtained probability-generating function (pgf) of the joint distribution of runs in a sequence of Markov-dependent multi-state trials. A method of conditional pgf's is used to obtain these joint distributions. Expanding the matrix polynomial of degree n, involved in the pgf, a recurrent formula for evaluation of exact probability distribution is obtained. To show the feasibility and compatibility of the results obtained, the joint distributions are numerically evaluated.