Consider a system of n components such that the working times of component i, i = 1,..., n, are exponentially distributed with rate λi . When a component fails, however, the repair rate of component i depends on how many other components are down. Specifically, suppose that the instantaneous repair rate of component i, i = 1,..., n, when there are a total of k failed components, is αkμi .

(a) Explain how we can analyze the preceding as a continuous-time Markov chain. Define the states and give the parameters of the chain.

(b) Show that, in steady state, the chain is time reversible and compute the limiting probabilities.