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Consider a machine that operates for a certain amount of time that is exponentially distributed, and then fails. Once it fails, it gets repaired.
Please help me answer these questions.
Consider a machine that operates for a certain amount of time that is exponentially
distributed, and then fails. Once it fails, it gets repaired. The repair time also is an
exponential random variable and is independent of the past. The machine is as good as
new after the repair is complete. Let X(t) be the state of the machine at time t, 1 if it is
up and 0 if it is down. The expected time until failure of the machine is 20 days, while
the expected repair time is 3 day. Suppose the machine is working at the beginning of
June.
(a) Write down the generator matrix for the Markov Process (X(t)).
(b) Find the probability that the machine is working at the start of June 11. The answer
should be correct to 3 decimal places.
(c) Compute the expected total uptime of the machine in the month of June.
