31. A total of customers move about among r servers in the following manner. When a...

31. A total of Í customers move about among r servers in the following manner. When a customer is served by server /, he then goes over to server j9j ?± /, with probability \/(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 / being exponential with rate

/ / , · , /= 1 , . . . , r. Let the state at any time be the vector (nx,..., nr), where nt is the number of customers presently at server /, / = 1, ...,/*, Óßni = N.

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