An airline reservation system has two computers, only one of which is in operation at any given time. a computer may break down on any given day with probability p. there is a single repair facility that takes 2 days to restore a computer to normal. the facilities are such that only one computer at a time can be dealt with. form a markov chain by taking as states the pairs (x, y) where x is the number of machines in operating condition at the end of a day and y is 1 if a day's labor has been expended on a machine not yet repaired and 0 otherwise. the transition matrix is pi_0 = q^2/1 + p^2 pi_2 = qp/1 + p^2 pi_1 = p/1 + p^2 pi_3 = p^2/1 + p^2 in order to increase the system availability, it is proposed to add a duplicate repair facility so that both computers can be repaired simultaneously. the corresponding transition matrix is now and the limiting distribution is pi_0 = q/1 + p + p^2 pi_2 = p/1 + p + p^2 pi_1 = p/1 + p + p^2 pi_3 = p^2/1 + p + p^2 when there is only one repair facility, what fraction of time is the repair facility idle? when a second repair facility is added, what fraction of time is each facility idle?