There are three machines, all of which are needed for a system to work. Machine i functions for an exponential time with rate λi before it fails, i = 1, 2, 3.

When a machine fails, the system is shut down and repair begins on the failed machine. The time to fix machine 1 is exponential with rate 5; the time to fix machine 2 is uniform on (0, 4); and the time to fix machine 3 is a gamma random variable with parameters n = 3 and λ = 2. Once a failed machine is repaired, it is as good as new and all machines are restarted.

(a) What proportion of time is the system working?

(b) What proportion of time is machine 1 being repaired?

(c) What proportion of time is machine 2 in a state of suspended animation

(that is, neither working nor being repaired)?