$3.$ M$.$ M$.$ Sprout $-$ Case$$M$.$ M$.$ Sprout, a catalog mail$-$order retailer, has one customer$-$service representative $($CSR$)$ to take orders at an $800$ telephone number. If the CSR is busy, the next caller is put on hold. For simplicity, assume that any number of incoming calls can be put on hold and the nobody hangs up in frustration over a long wait. Suppose that, on average, one call comes in every $4$ minutes and that it takes the CSR an average of $3$ minutes to take an order. Both interarrival and activity times are exponentially distributed. The CSR is paid $ $20$ an hour, and the telephone company charges $ $5$ an hour for the $800$ line. The company estimates that each minute a customer is kept on hold costs $ $2$ in customer dissatisfaction and loss of future business.

$1)$ Estimate the following:

a$)$ Proportion of time that the CSR will be busy,

b$)$ Average time that a customer will be on hold,

c$)$ Average number of customers on line, and

d$)$ Total hourly cost of service and waiting.

$2)$ More realistically, suppose that M$.$ M$.$ Sprout has four telephone lines. At most, therefore, three callers can be kept on hold. Assume, too, that any caller who gets a busy signal because all four lines are occupied simply hangs up and calls a competitor. M$.$ M$.$ Sprout$$s average loss, in terms of current and potential future business, is $ $100$ per blocked caller. Estimate the total cost of the following: $($Hint: use the performance.xls spreadsheet$)$

a$)$ Providing service,

b$)$ Waiting, and

c$)$ Average hourly loss incurred because customers are blocked and cannot get through.

$3)$ Suppose that M$.$ M$.$ Sprout is considering adding another line to reduce the amount of lost business. If the installation cost is negligible, can the addition of one line be justified on economic grounds? How would it affect customer waiting time? $($Hint: use the performance.xls spreadsheet$)$

$4)$ In addition to adding another line, suppose M$.$ M$.$ Sprout wants to hire one more CSR to reduce waiting time. Should the firm hire another CSR$?($Hint: use the performance.xls spreadsheet$)$

Do not forget to interpret all the answers in detail, i$.$e$.,$ to explain and justify! Enter all the intermediate steps in the calculations and specify the units of the results! Pay attention to meaningful rounding!