Case $4(33$ p$.)$ A company is considering opening a fast$-$food service operation $($drive through$).$ Prior research shows that customer arrivals would follow a Poisson process, with an arrival rate of $30$ cars per hour, and that service times would follow a negative exponential probability distribution. Arriving customers would place orders at an intercom station at the back of the parking lot and then drive to the service area to pay for and receive their orders. The following service alternatives are being considered: $($show all time$-$related answers in minutes$)$

Q$1.(10$ p$.)$ A single$-$server operation in one service window in which one employee fills the order and takes the money from the customer. The average service time for this alternative is $1\mathrm{.}5$ minutes.

Information given to answer Q$1$:

X $=30$ cars $/$hour

Y $=1\mathrm{.}5$ minutes $/$car

$-$ What is the average number of cars waiting for service in the line?

$-$ What is the average number of cars in the system?

$-$ What is the average time $($in minutes$)$ a car waits for service in the line?

$-$ What is the average time $($in minutes$)$ in the system?

$-$ What is the utilization factor of the system?

Q$2.(10$ p$.)$ A two$-$server operation with two service windows and two employees $($one employee per window$).$ The employee stationed at each window fills the order and takes the money from the customer. The average service time for this alternative is also $1\mathrm{.}5$ minutes for each server. This service operation assumes that the two service windows are located at the opposite sides of the restaurant building with two separate lanes leading to them, i$.$e$.,$ the arriving customers must choose from the very beginning $($after placing their order$)$ which lane they will follow $($with no chance for backing up or switching to the other lane$).$

$-$ What is the average number of cars waiting for service in the line?

$-$ What is the average number of cars in the system?

$-$ What is the average time $($in minutes$)$ a car waits for service in the line?

$-$ What is the average time $($in minutes$)$ in the system?

$-$ What is the utilization factor of the system?

Q$3.(10$ p$.)$ A two$-$server operation with two service windows and two employees $($one employee per window$).$ The employee stationed at each window fills the order and takes the money from the customer. The average service time for this alternative is also $1\mathrm{.}5$ minutes for each server. This service operation assumes that although the two service windows are located at the opposite sides of the restaurant building, there would be only one common lane leading to the building from the parking lot, i$.$e$.,$ the arriving customers would follow the same, common, lane and only at the very end they would move to the first window available $($on the left or on the right side of the building$).$

$-$ What is the average number of cars waiting for service in the line?

$-$ What is the average number of cars in the system?

$-$ What is the average time $($in minutes$)$ a car waits for service in the line?

$-$ What is the average time $($in minutes$)$ in the system?

$-$ What is the utilization factor of the system?

Q$4.(3$ p$.)$ Which option would you recommend and why? Fully justify and interpret your answer.