Question: After being repaired, a machine functions for an exponential time with rate \(\lambda \), and then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct phases. First a phase 1 repair must be performed, then a phase 2, and so on. The times to complete these phases are independent, with phase \(i\) taking an exponential time with rate \({{\mu }_{i}}\), with \(i=1,2,...,k\).

(a) What proportion of time is the machine undergoing a phase \(i\) repair?

(b) What proportion of time is the machine working?