31. A total of N customers move about among r servers in the following manner. When a customer is served by server i, he then goes over to server j, , with probability . If the server he goes to is free, then the customer enters service; otherwise he joins the queue. The service times are all independent, with the service times at server i being exponential with rateI mage. Let the state at any time be the vector , where is the number of customers presently at server .

(a) Argue that if is the state at time t, then is a continuous-time Markov chain.

(b) Give the infinitesimal rates of this chain.

(c) Show that this chain is time reversible, and find the limiting probabilities.