6. (Problem 6.6.4 in Pinsky and Karlin) A system consists of two units, both of which may operate simultaneously, and a single repair facility. The probability that an operating system will fail in a short time interval of length t is (t) + o(t). Repair times are exponentially distributed, but the parameter depends on whether the failure was regular or severe. The fraction of regular failures is p, and the corresponding exponential parameter is . The fraction of severe failure is q = 1 p and the exponential parameter is < .

Model the system as a continuous time Markov chain by taking as states the pairs (x, y), where x = 0, 1, 2 is the number of units operating and y = 0, 1, 2 is the number of units undergoing repair for a severe failure. The possible states are (2, 0), (1, 0), (1, 1), (0, 0), (0, 1) and (0, 2). Specify the infinitesimal matrix. Assume that the repair facility can work on two computers at the same time.