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 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 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.
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 1According to guidelines, for more than three subpart only first  three subpart should be answered.Given that, the cars arrive at Poisson probability distribution rate  of 24 per hour.λ=24/hourAlso given that, the average service time is 2 minutes for a customer.μ=1 customer2 minutes (1 hour/ 60 minutes)=30 customer/hour.Part (a)Express the probability of n cars in the system.Pn=1-λμλμnCalculate the probability of 0 cars in the system.P0=1-243024300=0.2Therefore, the probability is 0.2.Step 2Part (b)Express the formula for the average number of cars in the queue.Lq=λ2μμ-λCalculate the average number of cars waiting for the service.Lq=2423030-24=3.2Therefore, the answer is 3.2Step 3Part (c)Express the formula for the average number of cars in the system.Ls=λμ-λ=2430-24=4Therefore, the number of cars in the system is 4. ...