2. A mini supermarket has two checkouts: one is staff-operated (SO), and the other is self-service (SS). When either the SO or SS checkout breaks, it requires an exponentially distributed number of days to fix with parameter 1 and 2, respectively. Let {X,;t > 0) be a continuous time Markov chain denoting the status of both checkouts at time t. Its state-space is S = {(1, 1), (1, 0), (0, 1), (0,0)} where (i, j) denotes the status of the SO checkout (i=1 if working, i=0 if broken) and SS checkout (j=1 if working, j=0 if broken). A single repair person can fix either checkout. The repair to the SO checkout always takes priority, so that if the SO checkout breaks while the SS checkout is being repaired, the repair person will switch immediately to repairing the SO checkout. The generator matrix for the process { X,;t > 0} is given by: -0.3 0.2 0.1 0 19 -2.1 0 0.1 Q = 0 -1.2 0.2 ? (a) Complete the generator matrix of the process { X,; t 2 0}. (b) Briefly explain the notation o(h) used during the course. (c) Both checkouts are currently working. Using the o(h) notation, state the approximate probability that in the next h days at least one checkout will break, assuming that h is very small. (d) Show that the equilibrium distribution for this process is given by 1463 (1200, 130, 100, 33). (e) State the long-run proportion of time that at least one checkout is not working