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

In this same way, we decompose the other three elements of matrix and we rece-
ive:

4          5         5         _ 5

9 I__9                 9      _|__9

(1-β) ⁺ (1-0,1β)   (1-β) ⁺ (1-0,1β)

4         _4        5         4

9                 9            9               9

(1-β) ⁺ (1-0,1β)   (1-β) ⁺ (1-0,1β)

Finally formula (15) becomes the following form:

We can check that the first matrix with factor 1/ (1 - β) is ergodic matrix of Mar-
kov Process for given stochastic matrix of transition P . The second matrix is so
named difference matrix. The sum of elements is equal zero in rows of this matrix.
Taking formula (19) into consideration, we receive total finite expected reward:

Now we can find value ν_∞ (β) for two different β, β₁ = 0, 5 and β₂ = 0, 99.

After providing of values and simple calculations we receive:

ν∞ (0, 5) =

5
9
5

9

Hence we obtain for the starting state and n → ∞

ν₁,∞ (0, 5) = 2 • 1 + 1, 052 • 5 =

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