At the beginning of each day, a machine is inspected to determine its working condition, which is classified as state 1 = new, 2, 3, or 4 = broken. We assume that a machine in state 1 remains in state 1 with probability 0.95 and otherwise is degraded to state 2, that a machine in state 2 remains in state 2 with probability 0.9 and otherwise is degraded to state 3, and that a machine in state 3 remains in state 3 with probability 0.875 and otherwise is degraded to state 4.

(i) Suppose that a broken machine requires three days to fix it. To incorporate that into the Markov chain, we add states 5 and 6 and say that P(45) = P(56) = P(61) = 1. What is the long-run time fraction that the machine is working?

(ii) Suppose now that we have the option of performing preventative maintenance when the machine is in state 3 and that this maintenance takes one day and returns to state 1. Find the time fraction that the machine is working under this new policy.