In the paper, a theorem about the existence of the ergodic and difference matrices of the finite state discounted Markov decision processes had been formulated and proved. On the basis of this theorem an analytical method to calculate these matrices is presented. The theorem allows the distribution of the overall value into two parts: the so-called "constant" part, which represents a part of the value related to the ergodic matrix and the "variable" part which represents a sum of the difference matrices. On the basis of the mentioned analytical method a new performance index to the discounted optimal control Markov problem is proposed and some interpretation of the received results is given. The proposed new performance index is formulated as a quotient of the distinguished parts of the overall value. The method is illustrated by two simple examples.
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