Answer:
(a)0.16
(b)0.588
(c)
Step-by-step explanation:
The matrix below shows the transition probabilities of the state of the system.
(a)To determine the probability of the system being down or running after any k hours, we determine the kth state matrix .
(a)
If the system is initially running, the probability of the system being down in the next hour of operation is the
The probability of the system being down in the next hour of operation = 0.16
(b)After two(periods) hours, the transition matrix is:
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,
Using a calculator to raise matrix P to large numbers, we find that the value of approaches [0.75 0.25]:
Furthermore,
The steady-state probabilities of the system being in the running state and in the down-state is therefore:
