Respuesta :
Answer:
(a)0.16
(b)0.588
(c)[tex][s_1$ s_2]=[0.75,$ 0.25][/tex]
Step-by-step explanation:
The matrix below shows the transition probabilities of the state of the system.
[tex]\left(\begin{array}{c|cc}&$Running&$Down\\---&---&---\\$Running&0.90&0.10\\$Down&0.30&0.70\end{array}\right)[/tex]
(a)To determine the probability of the system being down or running after any k hours, we determine the kth state matrix [tex]P^k[/tex].
(a)
[tex]P^1=\left(\begin{array}{c|cc}&$Running&$Down\\---&---&---\\$Running&0.90&0.10\\$Down&0.30&0.70\end{array}\right)[/tex]
[tex]P^2=\begin{pmatrix}0.84&0.16\\ 0.48&0.52\end{pmatrix}[/tex]
If the system is initially running, the probability of the system being down in the next hour of operation is the [tex](a_{12})th$ entry of the P^2$ matrix.[/tex]
The probability of the system being down in the next hour of operation = 0.16
(b)After two(periods) hours, the transition matrix is:
[tex]P^3=\begin{pmatrix}0.804&0.196\\ 0.588&0.412\end{pmatrix}[/tex]
Therefore, the probability that a system initially in the down-state is running
is 0.588.
(c)The steady-state probability of a Markov Chain is a matrix S such that SP=S.
Since we have two states, [tex]S=[s_1$ s_2][/tex]
[tex][s_1$ s_2]\left(\begin{array}{ccc}0.90&0.10\\0.30&0.70\end{array}\right)=[s_1$ s_2][/tex]
Using a calculator to raise matrix P to large numbers, we find that the value of [tex]P^k[/tex] approaches [0.75 0.25]:
Furthermore,
[tex][0.75$ 0.25]\left(\begin{array}{ccc}0.90&0.10\\0.30&0.70\end{array}\right)=[0.75$ 0.25][/tex]
The steady-state probabilities of the system being in the running state and in the down-state is therefore:
[tex][s_1$ s_2]=[0.75$ 0.25][/tex]
