There are two parallel servers in a system. Customers arrive this system with a Poisson process with rate . The service rates of the two

There are two parallel servers in a system. Customers arrive this system with a Poisson process with rate λ. The service rates of the two servers are μ1 and μ2,
respectively. If the system is empty, a new arrival joins server 1 with probability α, and joins to the second server with probability 1- α. Otherwise, the arrival joins the queue and will get service from the first free server. If only one of the servers is busy, the arrival automatically goes to the empty server.

a) Suppose a customer arrives to the system when there are 5 customers waiting in line. What is the probability that this customer will be served by server 1?
b) Suppose a customer arrives to the system when there are 3 customers waiting in line. What is the expected waiting time in line for this customer? What is the
expected time spent in the system?
c) Define the states of the system and draw the diagram. Set up the balance equations (no need to solve) to find the stationary probabilities.
d) In terms of the stationary probabilities what is the average number of people in the system? What is the average number of servers idle?
e) In terms of the stationary probabilities what is the probability that an arbitrary arrival gets serviced by server 1?