1. Reid & Sanders, Operations Management © Wiley 2002 Waiting Line Models A SUPPLEMENT

2. Reid & Sanders, Operations Management © Wiley 2002 Page 2 Learning Objectives Describe the elements of a waiting line problem Use waiting line models to estimate system performance Use waiting line models to make managerial decisions

3. Reid & Sanders, Operations Management © Wiley 2002 Page 3 Waiting Line System A waiting line system consists of two components: –The customer population (people or objects to be processed) –The process or service system Whenever demand exceeds available capacity, a waiting line or queue forms

4. Reid & Sanders, Operations Management © Wiley 2002 Page 4 Terminology Finite versus Infinite populations: –Is the number of potential new customers affected by the number of customers already in queue? Balking –When an arriving customer chooses not to enter a queue because it’s already too long Reneging –When a customer already in queue gives up and exits without being serviced Jockeying –When a customer switches back and forth between alternate queues in an effort to reduce waiting time

5. Reid & Sanders, Operations Management © Wiley 2002 Page 5 Service System The service system is defined by: –The number of waiting lines –The number of servers –The arrangement of servers –The arrival and service patterns –The service priority rules

6. Reid & Sanders, Operations Management © Wiley 2002 Page 6 Number of Lines Waiting lines systems can have single or multiple queues. –Single queues avoid jockeying behavior & all customers are served on a first-come, first-served fashion (perceived fairness is high) –Multiple queues are often used when arriving customers have differing characteristics (e.g.: paying with cash, less than 10 items, etc.) and can be readily segmented

7. Reid & Sanders, Operations Management © Wiley 2002 Page 7 Servers Single servers or multiple, parallel servers providing multiple channels Arrangement of servers (phases) –Multiple phase systems require customers to visit more than one server –Example of a multi-phase, multi-server system: CCCCC DepartArrivals 1 2 36 5 4 Phase 1Phase 2

8. Reid & Sanders, Operations Management © Wiley 2002 Page 8 Arrival & Service Patterns Arrival rate: –The average number of customers arriving per time period –Modeled using the Poisson distribution Service rate: –The average number of customers that can be serviced during the same period of time –Modeled using the exponential distribution

9. Reid & Sanders, Operations Management © Wiley 2002 Page 9 Priority Rules First come, first served Best customers first (reward loyalty) Highest profit customers first Quickest service requirements first Largest service requirements first Earliest reservation first Emergencies first

10. Reid & Sanders, Operations Management © Wiley 2002 Page 10 Common Performance Measures The average number of customers waiting in queue The average number of customers in the system (multiphase systems) The average waiting time in queue The average time in the system The system utilization rate (% of time servers are busy)

11. Reid & Sanders, Operations Management © Wiley 2002 Page 11 Infinite Population, Single-Server, Single Line, Single Phase Formulae

12. Reid & Sanders, Operations Management © Wiley 2002 Page 12 Infinite Population, Single-Server, Single Line, Single Phase Formulae

13. Reid & Sanders, Operations Management © Wiley 2002 Page 13 Example A help desk in the computer lab serves students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour. Based on this description, we know: –Mu = 20 (exponential distribution) –Lambda = 15 (Poisson distribution)

14. Reid & Sanders, Operations Management © Wiley 2002 Page 14 Average Utilization

15. Reid & Sanders, Operations Management © Wiley 2002 Page 15 Average Number of Students in the System

16. Reid & Sanders, Operations Management © Wiley 2002 Page 16 Average Number of Students Waiting in Line

17. Reid & Sanders, Operations Management © Wiley 2002 Page 17 Average Time a Student Spends in the System

18. Reid & Sanders, Operations Management © Wiley 2002 Page 18 Average Time a Student Spends Waiting (Before Service)

19. Reid & Sanders, Operations Management © Wiley 2002 Page 19 Probability of n Students in the Line

20. Reid & Sanders, Operations Management © Wiley 2002 Page 20 Changing System Performance Demand management: –change customer arrival rates through non-peak discounts or price promotions Modify the number of servers Division of labor: –change the number of phases in the system – change the number of workers at each station (e.g.: add a bagger to assist each cashier)

21. Reid & Sanders, Operations Management © Wiley 2002 Page 21 Changing System Performance Apply technology to improve efficiency –e.g.: price scanners Change priority rules –e.g.: implement a reservation protocol Change the number of lines: –Reduce multiple lines to single queue to avoid jockeying –Dedicate specific servers to specific transactions

22. Reid & Sanders, Operations Management © Wiley 2002 Page 22 The End Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.