ASE SCENARIO: DRIVER$$S DONUTS

Driver$$s Donuts is a bakery and coffee shop that depends on morning commuters for $80\%$ of its

daily business. During the morning rush from $5$:$30$ a$.$m$.$ to $8$:$30$ a$.$m$.,$ an average of $35$ cars arrive

at the Driver$$s Donuts drive$-$ through lane every hour, to purchase donuts and coffee through a

single store window. The employee who staffs that window can serve each car in an average of

$90$ seconds, although this speed is partially due to the fact that a second employee talks to the car

behind the one being served, inputting that car$$s order so that the window employee knows the

order and its cost when the second car arrives at the window. Without that assistance, it takes the

window employee an average of $2$ minutes to serve any car, which is the practice outside of the

morning rush hours, as customer traffic in the drive$-$through lane drops to an average of four cars

an hour. Assuming that Driver$$s Donuts drive$-$through lane meets the assumptions of the M$/$M$/1$

model, answer the following questions.

Questions

$1.$ On average, how many cars are in the drive$-$through lane during the morning rush? $($This

includes any car being served$).$ During the morning rush, how much time does the average

Driver$$s Donuts customer spend in the drive$-$through lane?

$2.$ If a car arrived between $1$:$00$ p$.$m$.$ and $2$:$00$ p$.$m$.,$ how much time could the customer expect to

spend in the drive$-$through lane? What is the probability that the customer would find no other

cars in the lane, and be able to pull up to the window immediately?

$3.$ If a customer arrives at the drive$-$through lane during the morning rush, what is the probability

that the customer will have to wait until a Driver$$s Donuts employee takes their order