1. Supplement C Waiting Line Models Operations Management by R. Dan Reid & Nada R. Sanders 4th Edition © Wiley 2010

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. Elements of Waiting Lines “Queue” is another name for a waiting line. 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 There is a tradeoff between cost and service level.

4. Customer Population Characteristics Finite versus Infinite populations: Is the number of potential new customers materially 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 between alternate queues in an effort to reduce waiting time.

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 waiting line priority rules

6. Number of Lines Waiting lines systems can have single or multiple queues. Single queues avoid jockeying behavior and perceived fairness is usually 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. 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. Example Queuing Systems

9. Arrival & Service Patterns Arrival rate: The average number of customers arriving per time period Modeled using the Poisson distribution Arrival rate usually denoted by lambda ( ) Example: =50 customers/hour; 1/ =0.02 hours between customer arrivals (1.2 minutes between customers)

10. Arrival & Service Patterns con’t Service rate: The average number of customers that can be served during the period of time Service times are usually modeled using the exponential distribution Service rate usually denoted by mu (µ) Example: µ=70 customers/hour; 1/µ=0.014 hours per customer (0.857 minutes per customer). Even if the service rate is larger than the arrival rate, waiting lines form! Reason is the variation in specific customer arrival and service times.

11. Waiting Line 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 Etc.

12. Waiting Line Performance Measures L q = The average number of customers waiting in queue L = The average number of customers in the system W q = The average waiting time in queue W = The average time in the system p = The system utilization rate (% of time servers are busy)

13. Single-Server Waiting Line Assumptions Customers are patient (no balking, reneging, or jockeying) Arrivals follow a Poisson distribution with a mean arrival rate of. This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/ The service rate is described by a Poisson distribution with a mean service rate of µ. This means that the service time for one customer follows an exponential distribution with an average of 1/µ The waiting line priority rule is first-come, first-served Infinite population

14. Formulas: Single-Server Case

15. Formulas: Single-Server Case con’t

16. State Univ Computer Lab 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: µ = 20 students/hour (average service time is 3 minutes) = 15 students/hour (average time between student arrivals is 4 minutes)

17. Average Utilization

18. Average Number of Students in the System, and in Line

19. Average Time in the System & in Line

20. Probability of n Students in the Line

21. Single Server: Spreadsheet Approach Key Formulas B9: =1/B5 B10: =1/B6 B13: =B5/B6 B14: =1-B13 B15: =B5/(B6-B5) B16: =B13*B15 B17: =1/(B6-B5) B18: =B13*B17 B22: =(1-B$13)*(B13^B21) Use Data Table (tracking B22) to easily compute the probability of n customers in the system.

22. Single Server: Probability of n Students in the System

23. Multiple Server Case Assumptions Same as Single-Server, except here we have multiple, parallel servers Single Line When server finishes with customer, first person in line goes to the idle server All servers are identical

24. Multiple Server Formulas

25. Multiple Server Formulas con’t

27. Example: Multiple Server Computer Lab Help Desk Now 45 students/hour need help. 3 servers, each with service rate of 18 students/hour Based on this, we know: µ = 18 students/hour s = 3 servers = 45 students/hour

28. Flexible Spreadsheet Approach Formulas are somewhat complex to set up initially, but you only need to do it once! For other multiple-server problems, can just change the input values. This approach also makes sensitivity analysis possible.

29. Key Formulas for Spreadsheet F10: =F$5^E10 (copied down) G10: =E10*G9 (copied down) H10: =H9+(F10/G10) (copied down) F5: =B5/B6 F6: =INDEX(G9:G109,B7+1) B10: =1/B5 B11: =1/B6 B12: =B7*B6 B15: =B5/B12 B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1) B17: =B5*B19 B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2) B19: =B20+(1/B6) B20: =B18/B5 B24: =IF(B23<=B7, ((F5^B23)*B16)/INDEX(G9:G109,B23+1), ((F5^B23)*B16)/ (INDEX(G9:G109,B7+1)*(B7^(B23-B7))))

30. Probability of n students in the system

31. Changing System Performance Customer Arrival Rates Try to smooth demand through non-peak discounts or price promotions Number and type of service facilities Increase or decrease number of servers, or dedicate specific servers for certain tasks (e.g., express line for under 10 items) Change Number of Phases Can use multi-phase system instead of single phase. This spreads the workload among more servers and may result in better flow (e.g., fast food restaurants having an order phase, pay phase, and pick-up phase during busy hours)

32. Changing System Performance Server efficiency Add resources to each phase (e.g., bagger helping a checker at the grocery store) Use technology (e.g. price scanners) to improve efficiency Change priority rules Example: implement a reservation protocol Change the number of lines Reduce multiple lines to single queue to avoid jockeying Dedicate specific servers to specific transactions

33. Waiting Lines Models within OM: How it all fits together Although it is unlikely that you calculate performance measures for the lines you wait in on a day-to-day basis, you should now be aware of the potential for mathematical analysis of these systems. More importantly, management has a tool by which it can evaluate system performance and make decisions as to how to improve the performance while weighing performance against the costs to achieve that performance. Waiting line models are important to a company because they directly affect customer service perception and the costs of providing a service.

34. Supplement C Highlights The elements of a waiting line system include the customer population source, the patience of the customer, the service system, arrival and service distributions, waiting line priority rules, and system performance measures. Understanding these elements is critical when analyzing waiting line systems. Waiting line models allow us to estimate system performance by predicting average system utilization, average number of customers in the service system, average number of customers waiting in line, average time a customer waits in line, and the probability of n customers in the service system. The benefit of calculating operational characteristics is to provide management with information as to whether system changes are needed. Management can change the operational performance of the waiting line system by altering any or all of the following: the customer arrival rates, the number of service facilities, the number of phases, server efficiency, the priority rule, and the number of lines in the system. Based on proposed changes, management can then evaluate the expected performance of the system.

35. Homework Hints Problems C.3 and C.4: these are based on the single-server model, the “additional server” is one who works within the single server system. C.3 asks for the utilization rate and average number of customers waiting in the system and in line. C.4 asks for the average time in the system and in line and the probability of more than 3 and 4 customers in the system