Hi I have some problem understanding the calculation of the following transition values for my transition matrix.
The task is the following:
A system consists of two elements that both work simultaneously, the system remains functional as long as the other element continues to work. It is known that the elements, independently of each other, fail with the probability p within one hour. Repairs are performed when the second element fails. Then both will be repaired at the same time. The repair is successful with probability q
This is the transition graph i come up with: transition graph
And this is my transition matrix so far:
$$ \left(\begin{array}{cc} (1-p)^2 & 2p(1-p)& p^2\\ 0 & -- & --\\ q & 0 & 1-q \end{array}\right) $$
Is this going in the right direction? How do I calculate the missing -- values?
A denotes active(running) and R denotes Repaired. The transition matrix is given by the states of these two machines to coexist in the system.
$$ \left(\begin{array}{cc} &A_1A_2 & A_1R_2 & R_1A_2& R_1R_2\\ A_1A_2 & (1-p)^2 & (1-p)p& p(1-p)& p^2\\ A_1R_2 &0 & (1-p)& 0& p\\ R_1A_2 &0 & 0& (1-p)& p\\ R_1R_2 &q & 0& 0& 1-q\\ \end{array}\right) $$ Edit
$$ \left(\begin{array}{cc} &0 & 1 & 2\\ 0 & (1-p)^2 & 2(1-p)p& p^2\\ 1&0 & (1-p)& p\\ 2 &q & 0& (1-q)\\ \end{array}\right) $$
It also follows that $\pi(1).p(1,2) =\pi(2).p(2,1) =0$ but p(1,2) >0 and so we would have to have $\pi(1) = 0$ and using $\pi(1)p(1, i) = \pi(i)p(i, 1)$ we conclude all the $\pi(i) = 0.$ for all i's. Thus this does not have a stationary distribution.