1. 1-1 McGraw-Hill/Irwin ©2009 The McGraw-Hill Companies, All Rights Reserved 1 Chapter 8A Waiting Line Management

2. 1-2 2 Suggestions for Managing Queues 1. Determine an acceptable waiting time for your customers 2. Try to divert your customer’s attention when waiting 3. Inform your customers of what to expect 4. Keep employees not serving the customers out of sight 5. Segment customers

3. 1-3 3 Suggestions for Managing Queues (Continued) 6. Train your servers to be friendly 7. Encourage customers to come during the slack periods 8. Take a long-term perspective toward getting rid of the queues

4. 1-4 4 Components of the Queuing System Customer Arrivals Servers Waiting Line Servicing System Exit Queue or

5. 1-5 5 Customer Service Population Sources Population Source FiniteInfinite Example: Number of machines needing repair when a company only has three machines. Example: The number of people who could wait in a line for gasoline.

6. 1-6 6 Service Pattern Service Pattern ConstantVariable Example: Items coming down an automated assembly line. Example: People spending time shopping.

7. 1-7 7 The Queuing System Queue Discipline Length Number of Lines & Line Structures Service Time Distribution Queuing System

8. 1-8 8 Examples of Line Structures Single Channel Multichannel Single Phase Multiphase One-person barber shop Car wash Hospital admissions Bank tellers’ windows

9. 1-9 9 Degree of Patience No Way! BALK No Way! RENEG

10. 1-10 10 Waiting Line Models ModelLayout Source PopulationService Pattern 1Single channelInfiniteExponential 2Single channelInfiniteConstant 3MultichannelInfiniteExponential 4Single or MultiFiniteExponential These four models share the following characteristics:  Single phase  Poisson arrival  FCFS  Unlimited queue length

11. 1-11 11 Notation: Infinite Queuing: Models 1-3

12. 1-12 12 Infinite Queuing Models 1-3 (Continued)

13. 1-13 13 Assume a drive-up window at a fast food restaurant. Customers arrive at the rate of 25 per hour. The employee can serve one customer every two minutes. Assume Poisson arrival and exponential service rates. Determine: A) What is the average utilization of the employee? B) What is the average number of customers in line? C) What is the average number of customers in the system? D) What is the average waiting time in line? E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be in the system? Determine: A) What is the average utilization of the employee? B) What is the average number of customers in line? C) What is the average number of customers in the system? D) What is the average waiting time in line? E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be in the system? Example: Model 1

14. 1-14 14 Example: Model 1 A) What is the average utilization of the employee?

15. 1-15 15 Example: Model 1 B) What is the average number of customers in line? C) What is the average number of customers in the system?

16. 1-16 16 Example: Model 1 D) What is the average waiting time in line? E) What is the average waiting time in the system?

17. 1-17 17 Example: Model 1 F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)?

18. 1-18 18 Example: Model 2 An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution. Determine: A) The average number of customers in line. B) The average total waiting time in the system. Determine: A) The average number of customers in line. B) The average total waiting time in the system.

19. 1-19 19 Example: Model 2 A) The average number of customers in line. B) The average total waiting time in the system.

20. 1-20 20 Example: Model 3 Recall the Model 1 example: Drive-up window at a fast food restaurant. Customers arrive at the rate of 25 per hour. The employee can serve one customer every two minutes. Assume Poisson arrival and exponential service rates. If an identical window (and an identically trained server) were added, what would the effects be on the average number of cars in the system and the total time customers wait before being served?

21. 1-21 21 Example: Model 3 Average number of cars in the system Total time customers wait before being served