Renewal Phenomena 371 7.4.5 A Markov chain X 0 , X 1 , X 2 , . . . has the transition probability matrix 0 1 2 0.3 0.7 0 0.6 0 0.4 . 0 0.5 0.5 0 P = 1 2 A sojourn in a state is an uninterrupted sequence of consecutive visits to that state. (a) Determine the mean duration of a typical sojourn in state 0. (b) Using renewal theory, determine the long run fraction of time that the pro- cess is in state 1. 7.5 7.5.1 Generalizations and Variations on Renewal Processes Delayed Renewal Processes We continue to assume that {X k } are all independent positive random variables, but only X 2 , X 3 , . . . (from the second on) are identically distributed with distribution func- tion F, while X 1 has possibly a different distribution function G. Such a process is