29. Consider a set of n machines and a single repair facility to service these machines.

Suppose that when machine i, i = 1, . . . , n, fails it requires an exponentially distributed amount of work with rate μi to repair it. The repair facility divides its efforts equally among all failed machines in the sense that whenever there are k failed machines each one receives work at a rate of 1/k per unit time. If there are a total of r working machines, including machine i, then i fails at an instantaneous rate λi/r.

(a) Define an appropriate state space so as to be able to analyze the preceding system as a continuous-time Markov chain.

(b) Give the instantaneous transition rates (that is, give the qij ).

(c) Write the time reversibility equations.

(d) Find the limiting probabilities and show that the process is time reversible.