Chapter 14
Waiting Lines and
Queuing Theory
Models
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to
• Describe the trade-off curves for
cost-of-waiting time and cost-ofservice.
• Understand the three parts of a
queuing system: the calling
population, the queue itself, and
the service facility.
• Describe the basic queuing
system configurations.
• Understand the assumptions of
the common models dealt with in
this chapter
• Analyze a variety of operating
characteristics14-2of waiting lines.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline
14.1 Introduction
14.2 Waiting Line Costs
14.3 Characteristics of a Queuing System
14.4 Single-Channel Queuing Model with
Poisson Arrivals and Exponential
Service Times
14.5 Multiple-Channel Queuing
Model with Poisson Arrivals and
Exponential service Times
14.6 Constant Service Time Model
14.7 Finite Population Model
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline – cont.
14.8 Some General Operating
Characteristics Relationships
14.9 More Complex Queuing Models and
the Use of Simulation
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Queuing Costs and
Service Levels
Total
Expected
Cost
Optimal
Service
Level
Cost of Operating
Service Facility
Cost of
Providing
Service
Cost of
Waiting
Time
Service Level
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Waiting Line Cost Analysis
Three Rivers Shipping
Number of Stevedore Teams
1
Avg. number of
ships arriving
per shift
Average waiting
time per ship
2
3
4
5
5
5
5
7
4
3
2
Total ship
35
20
15
hours lost
Est. cost per hour $1,000 $1,000 $1,000
of idle ship time
Value of ships'
lost time
Stevedore
teams salary
Total Expected
Cost
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10
$1,000
35,000 29,000 $15,000 $10,000
$6,000 $12,000 18,000 $24,000
$41,000$32,000 $33,000 $34,000
14-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Characteristics of a
Waiting Line System
• Calling Population
• Unlimited
• Limited
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Characteristics of a
Waiting Line System - cont.
• Arrival Characteristics
• Arrival rate distribution
• Poisson
• other
• Pattern of arrivals
• random
• scheduled
• Behavior of arrivals
• join the queue, and wait till served
• balk; refuse to join the line
• renege; leave the line
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Characteristics of a
Waiting Line System- cont.
• Waiting Line Characteristics
• Length of the queue
• limited
• unlimited
• Service priority/Queue discipline
• FIFO
• other
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Characteristics of a
Waiting Line System - cont.
• Service Facility Characteristics
• Number of channels
• single
• multiple
• Number of phases in service
system
• single
• multiple
• Service time distribution
• negative exponential
• other
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-10
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Poisson Distribution for
Arrival Times
.35
.30
.25
.20
.15
.10
.05
.00
0 1 2 3 45 6 7 8 910 1
1
X
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-11
.30
.25
.20
.15
.10
.05
.00
P(X),  = 4
P(X)
P(X)
P(X),  = 2
e  x
P(X) 
X!
012345678910 1
1
X
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Basic Queuing System
Configurations
Queue
Service
facility
Single Channel, Single Phase
Service Facility
Queue
Facility
1
Facility
2
Single Channel, Multi-Phase
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Basic Queuing System
Configurations
Service
facility
1
Queue
Service
facility
2
Service
facility
3
Multi-Channel,
Single Phase
Queue
Multi-Channel,
Multiphase Phase
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-13
Type 1
Service
Facility
Type 2
Service
Facility
Type 1
Service
Facility
Type 2
Service
Facility
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Exponential Distribution
for Service Times
f(x)  μe  μx for x  0, μ  0
Probability
(for Intervals of 1 Minute)
μ  Average Number Served Per Minute
Average Service Time of 20 Minutes
Average Service Time of 1 Hour
30
60
90
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
120 150 180 X
14-14
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Assumptions: M/M/1
Model
1. Queue discipline: FIFO
2. No balking or reneging
3. Independent arrivals; constant
rate over time
4. Arrivals: Poisson distributed
5. Service times: negative
exponential
6. Average service rate > average
arrival rate
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-15
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Performance Measures
of Queuing Systems
• Average time each customer spends
in the queue
• Average length of the queue
• Average time each customer spends
in the system
• Average number of customers in the
system
• Probability that the service facility
will be idle
• Utilization factor for the system
• Probability of a specific number of
customers in the system
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-16
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Equations: M/M/1
Average number in system, L 

 -
1
Average time in system, W 
 -
2
Average number in queue, L q 
  -  

Average time waiting, Wq 
  -  

Utilizatio n Factor,  


Percent Idle, P0  1 


Pn  k   

k 1
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-17
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Equations: M/M/m
P0 
1
 n  M 1 1    n  1    M M
   
 
 
 n 0 n!     M !    M  
    
M



L
P0 
2

M  1!M   
L

W
Lq  L 

Wq  W 

1

To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna

Lq

14-18


M
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Equations: M/D/1

Lq 
2     

Wq 
2     

L  Lq 

2
W  Wq 
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-19
1

© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Equations
Finite Population
Model
P0 
1
N!   
 

n  0 ( N  n)!   
N
n
  
Lq  N  
1  P0 
  
L  Lq  1  P0 
Wq 
Lq
N  L 
W  Wq 
1

n
N!   
  P0
P(n, n  N)  Pn 
N  n !   
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-20
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
General Operating
Characteristics
Little' s Flow Equations :
L
(or W  )
λ
Lq
(or Wq  )
λ
L  λW
L q  λWq
1
W  Wq 

To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
14-21
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458