$($a$)$ A crew of mechanics at the Highway Department garage repair vehicles that break down at

an average of $12$ vehicles per hour $($approximately Poisson in nature$).$ The mechanic crew can

service an average of $4$ vehicles every $10$ minutes with a repair time distribution that

approximates an exponential distribution. The crew cost is approximately $$50$ per hour. The

cost associated with lost productivity from the breakdown is estimated at $$80$ per vehicle per

hour $($or any fraction thereof$).$ Which is cheaper, the existing system with one service crew, or

a revised system with two service crews? The average waiting time for adding the second crew

is $0\mathrm{.}02.[5$ points

$($b$)$ Peter's International Barbershop is a popular haircutting and styling saloon near the

campus of the Brooklyn College. One barber is available to work full time and spend an

average of $6$ minutes on each customer. Customers arrive all day long at an average rate

of $10$ minutes Arrivals tend to follow the Poisson distribution, and service times are

exponentially distributed. Explain your results. $[5$ points$]$

$($a$)$ What is the probability that the shop is empty?

$($b$)$ What is the average number of customers in the barbershop?

$($c$)$ What is the average time spent in the shop?

$($d$)$ What percentage of time the barber is busy?

$($e$)$ What is the average number of customers waiting to be served?