The computer center at Dong-A University has been experiencing computer down time. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities: From Running down Running 0.90 0.10 Down 0.30 0.70 If the system is initially running, what is the probability of the system being down in next hour of operation? At a current period, the system is in a down state. After 2 periods of time, what is the probability that the system will be in the state of running? c. What are the steady-state probabilities of system being in the running state and in the down state?

(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: