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Examples of Queueing Theory Problems

## Problem 1: Petroco Service Station

1. The Petroco Service Station has one pump for regular unleaded gas, which (with an attendant) can service 10 customers per hour. Cars arrive at the regular unleaded pump at a rate of 6 per hour. Determine the average queue length, the average time a car is in the system, and the average time a car must wait. If the arrival rate increases to 12 cars per hour, what will be the effect on the average queue length?

2. The First American Bank of Rapid City has one outside drive-up teller. It takes the teller an average of 4 minutes to serve a bank customer. Customers arrive at the drive-up window at a rate of 12 per hour. The bank operations officer is currently analyzing the possibility of adding a second drive-up window, at an annual cost of \$20,000. It is assumed that arriving cars would be equally divided between both windows. The operations officer estimates that each minute’s reduction in customer waiting time would increase the bank’s revenue by \$2,000 annually. Should the second drive-up window be installed?

3. All trucks traveling on Interstate 40 between Albuquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 200 per 8-hour day, and the station can weigh, on the average, 220 trucks per day.

a. Determine the average number of trucks waiting, the average time spent waiting and being weighed at the weigh station by each truck, and the average waiting time before being weighed for each truck.

b. If the truck drivers find out they must remain at the weigh station longer than 15 minutes, on average, they will start taking a different route or traveling at night, thus depriving the state of taxes. The state of New Mexico estimates that it loses \$10,000 in taxes per year for each extra minute trucks must remain at the weigh station. A new set of scales would have the same service capacity as the present set of scales, and it is assumed that arriving trucks would line up equally behind the two sets of scales. It would cost \$50,000 per year to operate the new scales. Should the state install the new set of scales?

4. During registration at State University every semester, students in the college of business must have their courses approved by the college advisor. It takes an average of 2 minutes to approve each schedule, and students arrive at the adviser’s office at the rate of 28 per hour. The dean of the college of business is considering the addition of a second adviser in the college advising office to serve students waiting to have their schedules approved. This new adviser could serve the same number of students per hour as the present adviser. Determine L, Lq, W, and Wq for this altered advising system. As a student, would you recommend adding the adviser?

5. The Quick Wash 24-hour Laundromat has 16 washing machines. A machine breaks down every 20 days (exponentially distributed). The repair service with which the Laundromat contracts takes an average of 1 day to repair a machine (exponentially distributed). A washing machine averages \$5 per hour in revenue. The Laundromat is considering a new repair service that guarantees repairs in 0.50 day, but it charges \$10 more per hour than the current repair service. Should the Laundromat switch to the new repair service?