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1. A fast-food franchise is considering operating a drive-up window food-service operation. Assume that customer arrivals follow a Poisson probability distribution, with an arrival rate of 24 cars per hour, and that service times follow an exponential probability distribution. Arriving customers place orders at an intercom station at the back of the parking lot and then drive to the service window to pay for and receive their orders. The following three service alternatives are being considered:

A single-server operation in which one employee fills the order and takes the money from the customer. The average service time for this alternative is 2 minutes.

A single-server operation in which one employee fills the order while a second employee takes the money from the customer. The average service time for this alternative is 1.25 minutes.

A two-server operation with two service windows and two employees. The employee stationed at each window fills the order and takes the money for customers arriving at the window. The average service time for this alternative is 2 minutes for each server.

For each of the three design alternatives, answer the following questions. Then, recommend one of the design options.

a. What is the probability that no cars are in the system?

b. What is the average number of cars waiting for service?

c. What is the average number of cars in the system?

d. What is the average time a car waits for service?

e. What is the average time in the system?

f. What is the probability that an arriving car will have to wait for service?

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Step 1Waiting model is a model which helps to know how much company is able to serve their customer and how much customer has to wait for services in line and while in the system. This model is very helpful for checking the company’s efficiency and to utilize resources to the fullest.Step 2Answer:There are three models we have in this question as follows:In System 1 where there is only one server to provide service, arrival ate is 24 cars per hour and service time is 2 minutes.So we have:λ=24 cars per hourμ=602=30 cars per hourAverage utilization of server (p):p=λμ   =2430    =0.8Average number of cars in service system (L):L=λμ-λ   =2430-24    =4Average number of cars waiting in line for service (Lq):LQ=pL     =0.8×4     =3.2In System 2, Two Server case where there are two servers and two service providers:λ=24 cars per hourμ=602=30 cars per hourWe will calculate Average utilization:p=λsμ   =242×30    =0.4In System 3, Single server but 2 employees so the arrival rate is the same as 24 but the service rate is changed:λ=24 cars per hourμ=601.5=40 cars per hourAverage utilization of server (p):p=λμ   =2440    =0.6Average number of cars in service system (L):L=λμ-λ   =2440-24    =1.5Average number of cars waiting in line for service (Lq):LQ=pL     =0.6×1.5     =0.9   Step 3Sorry for the inconvenience we have solved as much as possible but this question is too lengthy and you have asked multi parts in multiple models. ...