6.6.4 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 1t is .1t/Co.1t/. 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 D 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 D 0;1;2 is the number of units operating and y D 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 A.

Assume that the units enter the repair shop on a first come, first served basis.