Please answer and explain the solution to this problem.

1. Customers arrive at a bank according to a Poisson process with rate A per hour. The bank lobby has enough space for six customers. \"llhen the lobby is fullJ an incoming customer leaves and is lost. The bank manager assigns one teller to customer service as long as the number of customers in the lobby is two or less. He a$igns two tellers if the number is more than two but less than veJ and he assigns three tellers if it is ve or more. The service times of the customers are iid Exp(p] random variables. Let X {t} be the number of customers in the bank at time t. (a) Model {Xft} : t .3 D} as a birth and death process? and derive the birth and death rates. {b} Find the limiting distribution of the process and express it in terms of the trac intensity or = if. From Part{c} onwards, assume that A = 3 and p. = 2. (c) Find the numerical values {5 decimals]I of the limiting probabilities lirnHmJ P [X13] 2 i] for 1' E S {d} Assume that there was one customer waiting outside when the bank opened. To an accuracyr of 3 decimals? compute the probability that there are four customers in the system at time i. t : 1, ii. t = 5. Compare this probability with your answer in [c]