$1.$ An interior design company provides home and office decorating assistance to its customers. In normal operation, an average of $2\mathrm{.}5$ customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages $10$ minutes with each customer.

a$.$ Find the average number of customers waiting in the system.

b$.$ Find the average time a customer spends in the system.

c$.$ Service goals dictate that an arriving customer should not wait for service more than an average of $5$ minutes. Is this goal being met? Justify.

d$.$ If the consultant can reduce the average time spent per customer to $8$ minutes, what is the mean service rate? Will the service goal be met?

e$.$ Suppose the company has the following alternatives:

$($i$)$ Use one consultant with an average service time of $8$ minutes per customer.

$($ii$)$ Expand to $2$ consultants, each of whom has an average service time of $10$ minutes per customer. If the consultants are paid $$3$ per hour and the customer waiting time is valued at $$25$ per hour for waiting time prior to service, should the company expand to the two$-$consultant system? Justify.