The Robotics Manufacturing Company operates an equipment repair business where emergency jobs arrive randomly at the rate of two jobs per $8-$hour day. The company's repair facility is a single$-$server system operated by a repair technician. The service time varies, with a mean repair time of $3\mathrm{.}0$ hours and a standard deviation of $1\mathrm{.}9$ hours. The company's cost of the repair operation is $$27$ per hour. In the economic analysis of the waiting line system, Robotics uses $$36$ per hour cost for customers waiting during the repair process.

$($a$)$ What are the arrival rate and service rate in jobs per hour? $($Round your answers to four decimal places.$)$

$=$

$=$

$($b$)$ Show the operating characteristics. $($Round your answers to four decimal places. Report time in hours.$)$

$L_q=$

$L=$

$w_q=$

$w=$

$h$ h

Show the total cost per hour. $($Express the total cost per hour in dollars. Round your answer to the nearest cent.$)$

$TC=$$

$$

$($c$)$ The company is considering purchasing a computer$-$based equipment repair system that would enable a constant repair time of $3\mathrm{.}0$ hours. For practical purposes, the standard deviation is $0.$ Because of the computer$-$based system, the company's cost of the new operation would be $$30$ per hour. What effect will the new system have on the waiting line characteristics of the repair service? $($Round your answers to four decimal places. Report time in hours.$)$

$w_q=,h$

$w=,h$

Show the total cost per hour. $($Express the total cost per hour in dollars. Round your answer to the nearest cent.$)$

$TC=$$

$$

$($d$)$ Dces paying for the computer$-$based system to reduce the variation in service time make economic sense? The firm's director of operations rejected the request for the new system because the hourly cost is $$3$ higher and the mean repair time is the same. Do you agree? How much $($in dollars$)$ will the new system save the company during a $40$ hour work week? $($Round your answer to the nearest cent. Enter $0$ if there are no savings.$)$

The average savings over a $40-$hour work week amount to $

$|.$ Based on this, the director's argument should be