Poisson Process Directed by a Cyclic Renewal Process. The state of a system is represented by a continuous-time cyclic renewal process X(t) on states 0, 1,...,K−1 as in Example 8. The sojourn times in...

Poisson Process Directed by a Cyclic Renewal Process. The state of a system is represented by a continuous-time cyclic renewal process X(t) on states 0, 1,...,K−1 as in Example 8. The sojourn times in the states are independent, and the sojourn time in state i has a continuous distribution Fi with mean μi. By Exercise 47, limt→∞ P{X(t) = i} = μi/ K−1 k=0 μk. Suppose the system fails occasionally such that, while it is in state i, failures occur according to a Poisson process with rate λi, independent of everything else. Let N(t) denote the number of failures in (0, t]. Show that Example 8