申論題

(b) Consider a two-server system in which customer inter-arrival times are exponentially distributed with rate λ at server 1. After being served by server 1, customers then join the queue for server 2. We assume that there is infinite waiting space at both servers. Each server serves one customer at a time with server i taking an exponential time with rate μi for a service, i = 1,2. To analyze this system, we need to keep track of the number of customers at server 1 and the number of customers at server 2. Let define the state by pair (n,m)- meaning that there are n customers at server 1 and m customers at server 2 and denote the probability of being in that state.
(b-1) (10 points) What are the balance equations for this system?
(b-2) (S points) If the number of customers at server 1 and server 2 were independent random variables, what would be the expression for  ?
 (b-3) (S points) Verify that your solution satisfies the balance equations from (b-1).