1. McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Chapter 8A
Waiting Line Analysis

3. OBJECTIVES Waiting Line Characteristics
Suggestions for Managing Queues
Examples (Models 1, 2, 3, and 4) 2

4. Components of the Queuing System
Servers
Waiting Line
Servicing System
Queue or
Customer
Arrivals
Exit 3

5. Customer Service Population Sources
Finite
Infinite
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. 4

6. Service Pattern Constant Variable
Example: Items coming down an automated assembly line.
Example: People spending time shopping. 5

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

8. Examples of Line Structures
Single
Phase
Multiphase
One-person
barber shop
Car wash
Hospital
admissions
Bank tellers’
windows
Single Channel
Multichannel 6

9. Degree of Patience
No Way!
No Way!
BALK
RENEG 7

10. 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 8

11. 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 9

12. These four models share the following characteristics: · Single phase
Waiting Line Models
Source
Model
Layout
Population
Service Pattern 1
Single channel
Infinite
Exponential 2
Single channel
Infinite
Constant 3
Multichannel
Infinite
Exponential 4
Single or Multi
Finite
Exponential
These four models share the following characteristics:
Single phase
Poisson arrival
FCFS
Unlimited queue length 10

13. Notation: Infinite Queuing: Models 1-3

14. Infinite Queuing Models 1-3 (Continued)

15. Assume a drive-up window at a fast food restaurant.
Example: Model 1
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? 11

16. A) What is the average utilization of the employee?
Example: Model 1
A) What is the average utilization of the employee? 12

17. B) What is the average number of customers in line?
Example: Model 1
B) What is the average number of customers in line?
C) What is the average number of customers in the system? 13

18. D) What is the average waiting time in line?
Example: Model 1
D) What is the average waiting time in line?
E) What is the average waiting time in the system? 14

19. 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)? 15

20. An automated pizza vending machine heats and
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. 16

21. A) The average number of customers in line.
Example: Model 2
A) The average number of customers in line.
B) The average total waiting time in the system. 17

22. Recall the Model 1 example: Drive-up window at a fast food restaurant.
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? 18

23. Average number of cars in the system
Example: Model 3
Average number of cars in the system
Total time customers wait before being served 19

24. Notation: Finite Queuing: Model 4

25. Finite Queuing: Model 4 (Continued)

26. The copy center of an electronics firm has four copy
Example: Model 4
The copy center of an electronics firm has four copy
machines that are all serviced by a single technician.
Every two hours, on average, the machines require
adjustment. The technician spends an average of 10
minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service,
how many machines are “down” (on average)? 20

27. Number of machines down = L + H = .382 machines
Example: Model 4
N, the number of machines in the population = 4
M, the number of repair people = 1
T, the time required to service a machine = 10 minutes
U, the average time between service = 2 hours
From Table TN7.11, F = .980 (Interpolation)
L, the number of machines waiting to be
serviced = N(1-F) = 4(1-.980) = .08 machines
H, the number of machines being
serviced = FNX = .980(4)(.077) = .302 machines
Number of machines down = L + H = .382 machines 21

28. Queuing Approximation
This approximation is quick way to analyze a queuing situation. Now, both interarrival time and service time distributions are allowed to be general.
In general, average performance measures (waiting time in queue, number in queue, etc) can be very well approximated by mean and variance of the distribution (distribution shape not very important).
This is very good news for managers: all you need is mean and standard deviation, to compute average waiting time

29. Inputs: S, , , Queue Approximation
(Alternatively: S, , , variances of interarrival and service time distributions)

30. Approximation Example
Consider a manufacturing process (for example making plastic parts) consisting of a single stage with five machines. Processing times have a mean of 5.4 days and standard deviation of 4 days. The firm operates make-to-order. Management has collected date on customer orders, and verified that the time between orders has a mean of 1.2 days and variance of 0.72 days. What is the average time that an order waits before being worked on?
Using our “Waiting Line Approximation” spreadsheet we get:
Lq = Expected number of orders waiting to be completed.
Wq = 3.78 Expected number of days order waits.
Ρ = 0.9 Expected machine utilization.

31. Question Bowl
The central problem for virtually all queuing problems is which of the following?
Balancing labor costs and equipment costs
Balancing costs of providing service with the costs of waiting
Minimizing all service costs in the use of equipment
All of the above
None of the above
Answer: b. Balancing costs of providing service with the costs of waiting 7

32. Question Bowl
Customer Arrival “populations” in a queuing system can be characterized by which of the following?
Poisson
Finite
Patient
FCFS
None of the above
Answer: b. Finite 7

33. Question Bowl
Customer Arrival “rates” in a queuing system can be characterized by which of the following?
Constant
Infinite
Finite
All of the above
None of the above
Answer: a. Constant 7

34. Single channel, multiphase Single channel, single phase
Question Bowl
An example of a “queue discipline” in a queuing system is which of the following?
Single channel, multiphase
Single channel, single phase
Multichannel, single phase
Multichannel, multiphase
None of the above
Answer: e. None of the above (These are the rules for determining the order of service to customers, which include FCFS, reservation first, highest-profit customer first, etc.) 7

35. Single channel, multiphase Single channel, single phase
Question Bowl
Withdrawing funds from an automated teller machine is an example in a queuing system of which of the following “line structures”?
Single channel, multiphase
Single channel, single phase
Multichannel, single phase
Multichannel, multiphase
None of the above
Answer: b. Single channel, single phase 7

36. Can not be computed from data above
Question Bowl
Refer to Model 1 in the textbook. If the service rate is 15 per hour, what is the “average service time” for this queuing situation?
16.00 minutes
hours
hours
16% of an hour
Can not be computed from data above
Answer: c hours (1/15=0.0667) 7

37. Question Bowl
Refer to Model 1 in the textbook. If the arrival rate is 15 per hour, what is the “average time between arrivals” for this queuing situation?
16.00 minutes
hours
hours
16% of an hour
Can not be computed from data above
Answer: c hours (1/15=0.0667) 7

38. Question Bowl
Refer to Model 4 in the textbook. If the “average time to perform a service” is 10 minutes and the “average time between customer service requirements” is 2 minutes, which of the following is the “service factor” for this queuing situation?
0.833
0.800
0.750
0.500
None of the above
Answer: a (10/(10+2)=0.833) 7

39. 8A-39
End of Chapter 8A