Suppose that a production process changes states in accordance with an irreducible, positive recurrent Markov chain having transition proba- bilities Pij , i, j = 1, ... , n, and suppose that certain of the states are considered acceptable and the remaining unacceptable. Let A denote the acceptable states andAc the unacceptable ones. If the production process is said to be "up" when in an acceptable state and "down" when in an unacceptable state, determine
1. The rate at which the production process goes from up to down (that is, the rate of breakdowns);
2. The average length of time the process remains down when it goes down; and
3. The average length of time the process remains up when it goes up.