The question asks: A physical device can be in three states: A,B,C. The device operates as follows (all time units are in hours): The device spends an exponentially distributed amount of time in stateAA (with mean of 12minutes) and then with probability 0.6 goes to state B, and with prob.0.4goes to state C. When in state B, the device moves to state C after an Exp(3) amount of time. When in state C, the device goes to state A at rate 1/hour, and to state B at rate 2/hour. Let Xt represent the device state at time tt, and suppose X0=′A0′. What is the long-run proportion of time the device spends in state ′A′. So I am thinking about maybe I should calculate the probability the device is in state A after 30 minutes and the probability the device is in state A after 30 minutes given that it was in state B after 5 minutes and in state C after 10 minutes. But what not sure how to do. Any help is appreciated!