Show all your work otherwise no credit given $($attach printouts from queuing templates if you used ones$)$

$1.$ A small suburban post office is frequented by customers at the rate of $10$ per hour. The post office has a single employee who typically processes each customer$$s request at an average rate of $4$ minutes per transaction. Assume that arrival rate follows a Poisson distribution and service time follows an exponential distribution. To determine the efficiency of operations, the postmaster wishes to examine several queue operating characteristics.

a$.$ What is the utilization rate of this service system?

b$.$ What is the average number of customers in line?

c$.$ What is the average time that each customer spends in the queue?

d$.$ What is the average time that each customer spends in the queue and being serviced?

e$.$ What is the probability that the post office employee will be idle?

$2.$ Refer to problem $1.$ The postmaster has been receiving complaints from customers about long waiting times. The postmaster is considering hiring an additional employee who will also be able to process each customer$$s request at the same rate of $4$ minutes per transaction. Assume that the post office pays its employees $$9$ per hour and that the cost of customer waiting time, in terms of customer dissatisfaction and goodwill, is $$11$ per hour. Assume that arrival rate follows a Poisson distribution and service time follows an exponential distribution. Should the postmaster hire an additional employee? Compare the total costs of this $2-$employee system with the old single$-$server queuing system.