Queuing Theory Models
By
Nancy Hutchins
Agenda
•
•
•
•
•
•
•
What is queuing
Why is queuing important
How can this help our company
Explanation
How it works
Summary
Reading list
What is Queuing?
• A queue is a line of waiting people,
vehicles, products, etc.
• Queuing theory models use a mathematical
approach to study queues and make them as
efficient as possible
Video Clip
Office Space Grid Lock
Why is this important?
Inadequate queue management may lead to
• Customers leaving before completing their
transaction
• Decrease in customer satisfaction
• Reduction in number of return customers
Why is this important?
• Retaining customers much more cost
effective than finding new customers
• Many businesses depend on revenue from
repeat customers
How can this help your company?
•
•
•
•
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Decrease average customer wait time
Increase customer satisfaction
Increase number of return customers
Increase revenue
Increase positive word-of-mouth customer
advertising
Basic Ways to Manage Queues
•
•
•
•
Train employees to be friendly
Segment customers by needs
Ensure customers know what to expect
Divert the customer’s attention during wait
times
• Encourage customers to come during slack
times
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub,
2006. 112. Print.
The Queuing System
Source
Population &
Arrival Rate
Servicing
System
Condition of
Exiting
Customers
Source Population
Finite
• Limited size
• Probabilities affected by an
increase/decrease in the
population
Infinite
• Large size
• Probabilities not affected by
an increase/decrease in the
population
Distribution of Arrivals
• Arrival Rate: is the number of units per
period
– Constant
– Variable
Exponential Distribution
t (minutes)
f(t)= λe ^ (-λt)
Probability Probability
that the next that the next
arrival will arrival will
occur in t
occur in t
minutes or
minutes or
more
less
0
1.00
0
0.5
0.61
0.39
1.0
0.37
0.63
1.5
0.22
0.78
2.0
0.14
0.86
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New
York: Irwin Professional Pub, 2006. 114. Print.
Customer Arrivals in Queues
• Arrival Characteristics
–
–
–
–
Distribution
Pattern
Size of Arrival
Degree of Patience
Probability of n Arrivals in
Time T
Poisson Distribution
0.224
PT(n) =
0.224
0.149
0.16
0.102
Mean = 𝜆 = 3
0.05
1
𝜆𝑇 𝑛 𝑒 −𝜆𝑇
𝑛!
2
3
4
5
6
7
8
Number of Arrivals (n)
Variance = 𝜆 = 3
9
10
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The
Core. New York: Irwin Professional Pub, 2006. 115. Print.
11
Pattern of Arrivals
Controllable
Pattern
Uncontrollable
Size of Arrival Units
Single
Size of
Arrival Units
Batch
Degree of Patience
Patient (in line
and stay)
Degree of
Patience
Arrive, View,
and Leave
Impatient
Arrive, Wait
Awhile, then
Leave
Queuing System Factors
• Length
– Infinite potential length
– Limited capacity
• Number of Lines
– Single
– Multiple
• Queue Discipline
Queue Discipline
First Come, First
Served (FCFS)
Emergencies
First
Shortest
Processing
Time
Limited
Needs
Reservations
First
Other
Service Time Distribution
• Service rate: the capacity of the server in
number of units per time period and not as
service time.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management
The Core. New York: Irwin Professional Pub, 2006. 118. Print.
Line Structures
•
•
•
•
•
Single Channel, Single Phase
Single Channel, Multiphase
Multichannel, Single phase
Multichannel, multiphase
Mixed
Exiting the Queuing System
Low Probability
of Re-service
Exit
Return to
Source
Population
Layout
Service
Phase
Source
Population
Arrival
Pattern
Queue
Discipline
Service
Pattern
Permissible
Queue
Length
1
Single
Channel
Single
Infinite
Poisson
FCFS
Exponential
Unlimited
One-lane toll
bridge
2
Single
Channel
Single
Infinite
Poisson
FCFS
Constant
Unlimited
Roller coaster
rides in
amusement
park
3
Multichannel
Single
Infinite
Poisson
FCFS
Exponential
Unlimited
Parts counter
in auto agency
Example
Model
Properties of Some Specific Line Models
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The
Core. New York: Irwin Professional Pub, 2006. 121. Print.
Infinite Queuing Notation:
Models 1-3
• λ = arrival rate
• µ = service rate
• 1/µ = average service time
• 1/λ = average time between arrivals
• ρ = ratio of total arrival rate to service rate for a single server
(λ/µ)
• Lq = average number waiting in line
• Ls = average number in system (including and being served)
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New
York: Irwin Professional Pub, 2006. 121. Print.
Infinite Queuing Notation:
Models 1-3
• Wq = average time waiting in line
• Ws = average total time in system (including time
to be served)
• n = number of units in the system
• S = number of identical service channels
• Pn = Probability of exactly n units in system
• Pw = Probability of waiting in line
Equations for Model 1
•
𝜆2
Lq =
𝜇(𝜇−𝜆)
• Ls =
𝜆
𝜇−𝜆
• Wq =
• Ws =
Lq
𝜆
Ls
𝜆
• Pn = (1 −
• ρ=
𝜆 𝜆 𝑛
)( )
𝜇 𝜇
𝜆
𝜇
• Po = (1 −
𝜆
)
𝜇
Equations for Model 2 and 3
Model 2
•
Model 3
𝜆2
Lq =
2𝜇(𝜇−𝜆)
• Ls =
𝜆
Lq +
𝜇
• Wq =
• Ws =
Lq
𝜆
Ls
𝜆
• Ls =
𝜆
Lq +
𝜇
• Wq =
• Ws =
Lq
𝜆
Ls
𝜆
𝑆𝜇
𝜆
• Pw = Lq (
− 1)
Brainstorming Exercise
Summary
• Effective queue management may lead to
improved customer satisfaction and
increased revenue
• Many queue management methods require
little money to implement
• Software is available to help with queue
analysis
Reading List
• An Introduction to Queuing Theory: Modeling and
Analysis in Applications (Statistics for Industry and
Technology) by U. Narayan Bhat
• Introduction to Queuing Networks by Erol Gelenbe and
Guy Pujolle
• Optimal Design of Queuing Systems by Shaler Stidham
• Fundamentals of Queuing Theory by Donald Gross and
Carl M. Harris
• Operations and Supply Management The Core by
Jacobs, F. Robert, and Richard B. Chase
Reference
** Jacobs, F. Robert, and Richard B. Chase.
"5." Operations and Supply Management The
Core. New York: Irwin Professional Pub,
2006. 111-127. Print.