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The lifetime of a certain machine is a random variable having an exponential distribution with parameter λ. When the machine breaks down, there is a probability equal to p (respectively, 1 - p) that the failure is of type I (resp., II). In the case of a type I failure, the machine is out of use for an exponential time, with mean equal to 1/μ time unit(s). To repair a type II failure, two independent operations must be performed. Each operation takes an exponential time with mean equal to 1/ μ.

(a) Use the results on regenerative processes to calculate the probability that the machine will be in working state at a (large enough) given time instant.

(b) What is the average age of the machine at time t? That is, what is the average time elapsed at time t since the most recent failure has been repaired? Assume that λ = μ.

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Category:- Statistics and Probability
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