Question: Each time a machine is repaired it remains up for an exponentially distributed time with rate 2. It then fails, and its failure is either

Each time a machine is repaired it remains up for an exponentially distributed time with rate 2.

It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with rate ; if it is a type 2 failure, then the repair time is exponential with rate . Each failure is, independently of the time it took the machine to fail, a type 1 failure with probability p and a type 2 failure with probability 1 - p. What proportion of time is the machine down due to a type 1 failure? What proportion of time is it down due to a type 2 failure? What proportion of time is it up?

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