1. Supplement D Waiting Line Models
Operations Management by
R. Dan Reid & Nada R. Sanders
2nd Edition © Wiley 2005
PowerPoint Presentation by
Roger B. Grinde, University of New Hampshire

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. Waiting Line System “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 service 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: C
Depart
Arrivals 1 2 3 6 5 4
Phase 1
Phase 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
Service rate usually denoted by lambda ()
Example: =50 customers/hour; 1/=0.02 hours between customer arrivals (1.2 minutes between customers)
Service rate:
The average number of customers that can be served during the same 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.

10. Example 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.

11. Common Performance Measures
Lq = The average number of customers waiting in queue
L = The average number of customers in the system
Wq = The average waiting time in queue
W = The average time in the system
p = The system utilization rate (% of time servers are busy)

12. Single-Server Waiting Line
Assumptions
Infinite population
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.

13. Formulas: Single-Server Case

14. Formulas: Single-Server Case (continued)

15. 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:
µ = 20 students/hour (average service time is 3 minutes)
 = 15 students/hour (average time between student arrivals is 4 minutes)

16. Average Utilization

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

18. Average Time in the System, and in Line

19. Probability of n Students in the Line

20. 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.

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

22. 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)

23. 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

24. 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.

25. Multiple Server Formulas

26. Multiple Server Formulas (continued)

27. Multiple Server Formulas (continued)

28. 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

29. 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.

30. 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))))

31. Probability of n students in the system

32. Supplement D 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.

33. The End
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