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