I am asked to solve the following problem with the following hint: use Renewal Equation. I get stuck on the hint… it seems a lot easier to solve it without the hint.
Exercise: a machine is working for an exponential time with rate $\lambda_1$ and when it breaks it is being repaired for an exponential time with rate $\lambda_2$. a) what is the rate of breakdown of the machine b) what’s the probability that the machine is working at a time uniformly taken between 0 and t?
It seems intuitive to look at a 2 state birth rate process / Markov chain where the mean time spent in working state is 1/$\lambda_1$ and the mean time spent in repair state is $1/\lambda_2$.
Then the breakdown rate is the inverse of $1/\lambda_1 + 1/\lambda_2 $.
And P(Machine works at time uni(0,t))= $\lambda_1/(\lambda_1+\lambda_2)*e^-\lambda_1t$
Am I missing something? How can I do this more thoroughly using renewal equations?