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 - αN 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,
R=
3 -7