D

Waiting-Line Models

PowerPoint presentation to accompany
Heizer and Render
Operations Management, 10e
Principles of Operations Management, 8e
PowerPoint slides by Jeff Heyl
Modifications and additions by M Peter Jurkat

© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-1

Outline
 Queuing Theory
 Characteristics of a Waiting-Line
System
 Arrival Characteristics
 Waiting-Line Characteristics
 Service Characteristics
 Measuring a Queue’s Performance

 Queuing Costs
© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-2

Outline – Continued
 The Variety of Queuing Models
 Model A(M/M/1): Single-Channel
Queuing Model with Poisson Arrivals and Exponential Service Times
 Model B(M/M/S): Multiple-Channel
Queuing Model
 Model C(M/D/1): Constant-ServiceTime Model
 Little’s Law
 Model D: Limited-Population Model
 Other Queuing Approaches
© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-3

Learning Objectives
When you complete this module you should be able to:
1. Describe the characteristics of arrivals, waiting lines, and service systems 2. Apply the single-channel queuing model equations
3. Conduct a cost analysis for a waiting line
© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-4

Learning Objectives
When you complete this module you should be able to:
4. Apply the multiple-channel queuing model formulas
5. Apply the constant-service-time model equations
6. Perform a limited-population model analysis

© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-5

Queuing Theory
 The study of waiting lines
 Waiting lines are common situations  Useful in both manufacturing and service areas © 2011 Pearson Education, Inc. publishing as Prentice Hall

D-6

Common Queuing
Situations
Situation

Arrivals in Queue

Service Process

Supermarket

Grocery shoppers

Checkout clerks at cash register Highway toll booth

Automobiles

Collection of tolls at booth

Doctor’s office

Patients

Treatment by doctors and nurses Computer system

Programs to be run

Computer processes jobs

Telephone company

Callers

Switching equipment to forward calls

Bank

Customer

Transactions handled by teller

Machine maintenance Broken machines

Repair people fix machines

Harbor

Ships and barges

Dock workers load and unload

© 2011 Pearson Education, Inc. publishing as Prentice Hall

Table D.1

D-7

Characteristics of WaitingLine Systems
1. Arrivals or inputs to the system
 Population size, behavior, statistical distribution 2. Queue discipline, or the waiting line itself  Limited or unlimited in length, discipline of people or items in it

3. The service facility
 Design, statistical distribution of service times © 2011 Pearson Education, Inc. publishing as Prentice Hall

D-8

Parts of a Waiting Line
Population of dirty cars

Arrivals from the general population …

Queue
(waiting line)

Service facility Dave’s
Car Wash

Enter

Arrivals to the system

Arrival Characteristics
 Size of the population
 Behavior of arrivals
 Statistical distribution of arrivals

Exit the system

In the system

Waiting Line
Characteristics
 Limited vs. unlimited  Queue discipline

Exit

Exit the system

Service Characteristics
 Service design
 Statistical distribution of service
Figure D.1

© 2011 Pearson Education, Inc. publishing as Prentice Hall

D-9

Arrival Characteristics
1. Size of the population
 Unlimited (infinite) or limited (finite)

2. Pattern of arrivals
 Scheduled or random, often a Poisson distribution 3. Behavior of arrivals
 Wait in the queue and do not switch lines
 No balking or reneging

© 2011 Pearson Education, Inc. publishing as Prentice Hall

D - 10

Poisson Distribution e- x
P(x) = x! where

for x = 0, 1, 2, 3, 4, …

P(x)
=
probability of x arrivals x = number of arrivals per unit of time
 = average arrival rate e =
2.7183 (which is the base of the natural logarithms)

© 2011 Pearson Education, Inc. publishing as Prentice Hall

D - 11

Poisson