Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a machine repair time exponentially distributed with respectively rate parameters $\mu_A$ and $\mu_B$. I would like to write the machine repair time as one cumulative distribution function G(s).
I know the probability of machine A breaking down first is the minimum of two exponentially distributed variables with parameters $\lambda_A$ and $\lambda_B$, which gives another exponentially distributed variable with parameter $\lambda_A+\lambda_B$:
$\frac{\lambda_A}{\lambda_A+\lambda_B}$, therefore the probability of machine B breaking down first: $\frac{\lambda_B}{\lambda_A+\lambda_B}$.
Can i now say that the $G(s) = \frac{\lambda_A}{\lambda_A+\lambda_B}(1-\exp(-\mu_At))+ \frac{\lambda_B}{\lambda_A+\lambda_B}(1-\exp(-\mu_Bt))$
thank you for your answer in advance
You are seeking the CDF of the unconditional first machine repair time?
If so, your solution is correct if only one machine can be repaired at the same time. Otherwise there are possible events of the form "machine A breaks down first and starts repairing, but machine B breaks down and is repaired before the repairing of machine A has been completed".
*Your solution should have $s$ on the right hand side instead of $t$.
I have a feeling you are looking for something else here and need to state the question with a little more information.