Customers arrive according to a Poisson process with rate λ at a bank where two clerks work. Clerk 1 (respectively, 2) serves at an exponential rate 𝜇1 (resp., 𝜇2)' We suppose that the customers form a single queue and that. when the system is empty, an arriving customer will go to clerk 1 (resp., 2) with probability pi (resp., 1-p1). On the other hand, when a customer must wait, she will eventually be served by the first available clerk. We also suppose that an arbitrary customer can enter the bank only if there are no more than 10 customers waiting in fine. We say that the system is in state n = 0,2,... if there are n customers in the bank, and in state 11 (resp., 12) if there is exactly one customer in the bank and if this customer is being served by clerk 1 (resp., 2).
(a) Write the balance equations of the system.
(b) In terms of the limiting probabilities, what is the probability that an entering customer will be served by clerk 1?