51. In a semi-Markov process, letI mage denote the conditional expected time that the process spends in state i given that the next state is j.

(a) Present an equation relating to the Image.

(b) Show that the proportion of time the process is in i and will next enter j is equal to Image.

Hint: Say that a cycle begins each time state i is entered. Imagine that you receive a reward at a rate of 1 per unit time whenever the process is in i and heading for j. What is the average reward per unit time?