A machine either works or does not work. The time for which it does work is exponentially distributed with mean _ (hours). When the machine does not work, the time it takes to repair is exponentially distributed with mean / (hours). Denote by X, the state (1 = works and 2 = does not work) of the machine at the time t. It is well-known that { Xt, t > 0} is a homogeneous continuous Markov Chain with state space S = {1, 2}.. Show that the transition probability pgh) satises the basic assumption: \"1(a) : 1 MA + 0(a), mun] : hf). + o[h}, Pad?!) = his + DOLL mm} = 1 - his + 0131.]. . Find Q-matrix and the transition matrix 1" of the jlmlp chain. . Find stationary distribution and limit distribution of the MC. . When the machine does not work, it incurs a wet of $100 per hour. Find the minimum ratio of Afp. so that the total cost is not more than $51] on average in a working day with 8 hours