Potential customers arrive at a small bank according to a Poisson process with rate of $4$ customers per hour. The bank has only two tellers A and B to serve the customers and a waiting area for four customers to queue while the two tellers are busy. The service times for the two tellers are exponentially distributed with rates $3$ customers per hour for teller A and $2$ customer per hour for teller B$.$

Assuming that the system is in steady state and that if there is only one customer is in the bank, teller A handles the customer;

$($a$)$What type of queuing process is defined above?$1$ mark

$($b$)$What is the probability that an arriving customer is turned away? $6$ marks

$($c$)$Find the average time spent by customers in the bank $5$ marks

$($d$)$ If the two tellers are busy, on average how long does a customer wait before being served?$2$ marks

$($e$)$What is the probability than the time until the first customer arrives is less than quarter an hour?$2$ marks