A machine works for an exponentially distributed time with rate μ and then fails. 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.
What I have so far:
I believe that if M represents the failure time of the machine and T represents the time when the machine is checked, then the probability of the machine being replaced is:
P(M < T) = μ/(μ+λ)
Since both M and T are exponentially distributed. So, that got me into thinking that this as Exponential(μ+λ) distributed, and so the expected time for the machine to be fixed is just (μ+λ)/μ.
Another thought was to find the expected time of the failure of the machine (1/μ) and then the expected time of checking the machine conditioned on the machine having failed, which I think is just 1/λ by the no memory property.
However, I feel like I do not understand what I am doing and am just aimlessly trying to solve the question. I can't seem to find a similar problem or example in my textbook (I'm on Ch 5 of "Introduction to Probability Models" by Ross, 10ed), and if anyone can help me with the problem or refer me to some good resources, I would be extremely grateful.
Quick edit: It looks like the question asks for the expected time between replacements, so now I feel like I did things completely wrong :(