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, j = i, with probability 1/(r−1). 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 rate μ, i = 1, . . . , r. Let the state at any time be the vector (n1, . . . , nr), where ni is the number of customers presently at server i, i = 1, . . . , r,



ini = N.

(a) Argue that if X(t) is the state at time t, then {X(t), t  0} 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.