25. Poisson () arrivals join a queue in front of two parallel servers A and B, having...

25. Poisson (λ) arrivals join a queue in front of two parallel servers A and B, having exponential service rates μA and μB (see Figure 8.4). When the system is empty, arrivals go into server A with probability α and into B with probability 1 − α. Otherwise, the head of the queue takes the first free server.

(a) Define states and set up the balance equations. Do not solve.

(b) In terms of the probabilities in part (a), what is the average number in the system? Average number of servers idle?

(c) In terms of the probabilities in part (a), what is the probability that an arbitrary arrival will get serviced in A?