The manager of a bank must determine how many tellers, out of a maximum number of $6,$ should work on Fridays. For every minute a customer is in the bank, the manager believes that a cost of $$0\mathrm{.}05$ is incurred. On average $2$ customers per minute arrive at the bank, and a teller takes an average of $2$ minutes to complete a customer's transaction.

It costs the bank $$9$ per hour to hire a teller. Assuming an M$/$M$/$c queuing system,

$($a$)(13$ points$)$ find all the feasible numbers of tellers needed to serve the customers $($cu $>1).$

Calculate the average amount of time a customer spends in the system for each case. $($You can use the approximations introduced in class$)$

$($b$)(12$ points$)$ find the number of tellers the manager should hire in order to minimize the sum of the service costs and customer waiting costs.