Question: Each time a machine is repaired it remains up for an exponentially distributed time with rate \(\lambda \). It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with rate \({{\mu }_{1}}\); if it is type 2 failure, then the repair time is exponential with rate \({{\mu }_{2}}\). Each failure is, independently of the time it took the machine to fail, a type 1 failure with probability p and a type 2 failure, with probability \(1-p\). What proportion of time is the machine down due to a type 1 failure? What proportion of time is the machine down due to a type 2 failure? What proportion of the time the machine is up?