A system can be in one of the three states 1, 2, and 3: In state 1 it operates most efficiently. In state 2 it is still working but its efficiency is lower than state 1. State 3 is the down state, the system no longer operating and has to be maintained. State changes can only occur after a fixed time unit of length 1. Transitions into the same state are allowed. If X denotes the state at , time n, then {X0, Xt,...} is assumed to be a Markov chain with transition matrix , (1-7v ) 1 2 3 V0.8 0.1 0.1\ 2 0 0.6 0.4 3 0.8 0 0.2 (a) Find the stationary state probabilities. _ (b) Suppose the system makes per unit time profits of $1000 in state 1, $600 in state 2, and loses $100 at state 3. Find long term mean profit per unit time.