Question

Amachineworksforanexponentiallydistributedtimewithrateμandthenfails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the machine is found to have failed then it is immediately replaced. Find the expected time between replacements of machines.

Answer 1

Answer:
`(1)/(\mu )+(1)/(\lambda )`

Explanation:

If a machine is replaced at some time t, then the expected time until next failure is
`(1)/(\mu )`

and the time between the checks is exponentially distributed with rate \lambda, the expected time until next failure is
`(1)/(\lambda )`

Because of memory less property of the exponential The answer is

`(1)/(\mu )+(1)/(\lambda )`