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:$\frac{1}{\mu }+\frac{1}{\lambda }$

Step-by-step explanation:

If a machine is replaced at some time t, then the expected time until next failure is $\frac{1}{\mu }$

and the time between the checks is exponentially distributed with rate \lambda, the expected time until next failure is $\frac{1}{\lambda }$  

Because of memory less property of the exponential  The answer is

$\frac{1}{\mu }+\frac{1}{\lambda }$