AN ANALYTICAL METHOD TO CALCULATE THE ERGODIC AND DIFFERENCE MATRICES OF THE DISCOUNTED MARKOV DECISION PROCESSES

was decomposed into N component forms:

. ,Dji      ʌ, k =1, 2, 3,...,N.

det (1 - αkβ)

It can be shown, that |Dij |^k < 1 and even values express probability and α1 ≡ 1.
Next connecting so received N elements on condition equal factors (1 - αKβ),
we create N separate (N × N) matrices. The first matrix is ergodic matrix pro-
ducted by 1/ (1 - β). For irreducible ergodic Markov Chain, this matrix will be con-
structed from the same rows.

Next N - 1 matrices will be difference matrices, each different and elements will
be divided by factors (I - αkβ), k = 2, 3, 4, . . . , N.

Now the formula for total rewards can be written in the following form:

ν∞ (β ) = (ʌ D¹ii + i-½ hD2ii +... + i—½ [dn ]) ∙ q. (19)

1 - β ^j 1 - α₂β  ^j         1 - α_N β  ^j

Next we consider and solve two simple examples, which show application of pre-
sented method.

4 Examples

Example 1

Let P be one-step stochastic matrix of transition of irreducible Markov Chain
with finite set of states N . Let N = 2 and let matrix P have the following form
25]:

0, 5 0, 5

0, 4 0, 6

Matrix of rewards R is also given,

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