What is the probability that an arriving car will drive away because the line is full?

A branch of Drugs-R-Us pharmacy in Middletown, USA, is evaluating the effectiveness of its drive thru service. Open from 8 a.m. to 5 p.m., the branch has a one-window drive thru service where patients pick up their prescriptions that have been pre-ordered via an automated phone call-in system. The window is currently staffed by one server who takes each customer’s order (asks for name and birthdate), collects the pre-filled prescriptions, hands the prescriptions to the customer, and asks the customer to complete the electronic signature device, including payment for the prescription, if required. As needed, a pharmacist gives the customer additional instruction and information about the prescription(s). Once the transaction is complete, the customer drives away. The window serves the next car (next customer in line).
In studying the effectiveness (or lack thereof) of the drive-thru service, the pharmacy wants to have a solid idea of certain line characteristics (such as average length of the queue, the system utilization, average time that a car will be in line and in the system, etc. You were hired by the pharmacy to carry out the study. As part of your empirical study of the line, you observed and collected data on the arrival times and service times during a typical 2-business-hour period (from 10:00 a.m. to noon) on a typical day. Below is the complete data.
CarArrival TimeService Time
110:013.00
210:081.50
310:123.00
410:182.00
510:192.00
610:221.70
710:232.00
810:261.20
910:273.00
1010:290.80
1110:311.40
1210:330.80
1310:353.00
1410:392.00
1510:412.00
1610:421.70
1710:442.20
1810:473.00
1910:512.00
2010:532.00
2110:561.30
2210:581.00
2310:594.00
2411:052.50
2511:095.00
2611:151.40
2711:174.00
2811:212.00
2911:232.10
3011:251.50
3111:251.80
3211:272.50
3311:301.10
3411:341.80
3511:351.50
3611:371.10
3711:381.60
3811:392.20
3911:402.00
4011:432.00
4111:480.90
4211:501.50
4311:532.00
4411:571.00
4511:581.90
Using the complete data in this link, and using the appropriate Waiting Line model Excel template in Moodle (Do not use the formulas shown in the textbook), answer the following questions.
1.What are the hourly arrival (number of cars arriving each hour) and hourly service (number of cars served per hour) rates? Show and explain how you obtained them. (4 pts)
2. Enter the information in #1 into the appropriate Excel waiting line model template. Copy and paste below an image of the output produced by the waiting line model template you used. Use the results in this output in answering items #3 to #9. (5 points)
3.Assume that the pattern in #1 above applies to the 8 a.m. to 5 p.m. periods from Monday to Friday. On a typical weekday, how many hours is the window idle (i.e. no cars or customers in the system) over the 9-hour period? (4 pts)
4.During a typical day for the drive-thru window, what is the probability that an arriving customer/car will have to wait for service? (4 pts)
5.On average, how many cars will be waiting to be served at the window? (4 pts)
6.If a customer has to wait for service, how long (express your answer in minutes) will the wait be, on average? (4 pts)
7. What is the expected amount of time (in minutes) a customer will be at the drive thru (in line plus being served? (4 pts)
8. The location where the drive thru can accommodate only 5 cars, i.e. one at the window and four waiting for service. What is the probability that an arriving car will drive away because the line is full? Show proof. (4 pts)
9. Analyze and comment on the performance measures (e.g. time of waiting time of service, length of queue, etc.) given the arrival and service rates in #1 and your answers to #3 to #8. If the line measures need to be improved, suggest ways you can implement to improve the line performance measures. Of these suggested ways, which one will you recommend and why? Be specific, clear, and complete in your explanation. (6 pts)
10. Assume that you want the customers to spend 3 minutes or less in the drive thru (in line plus being served). What service rate per hour and average service time in minutes will meet this criterion? Support your answer with data or output (paste image below) from the Waiting Lines Excel template. (6 pts)

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