4.1.11 Suppose that a production process changes state according to a Markov process whose transition probability matrix is given by

It is known that 1 D 119 379 D 0:3140 and 2 D 81 379 D 0:2137.

(a) Determine the limiting probabilities 0 and 3.

(b) Suppose that states 0 and 1 are “In-Control” while states 2 and 3 are deemed “Out-of-Control.” In the long run, what fraction of time is the process Out-of-Control?

(c) In the long run, what fraction of transitions are from an In-Control state to an Out-of-Control state?

0 1 2 3 0 0.3 0.5 0 0.2 1 0.5 0.2 0.2 0.1 P = 2 0.2 0.3 3 0.1 0.2 0.4 0.3 0.4 0.1