Operations
Management
Waiting-Line Models
Module D
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Outline
♦Characteristics of a Waiting-Line System
♦
♦
♦
♦
♦
Arrival Characteristics
Waiting-Line Characteristics
Service Facility Characteristics
Measuring the Queue’s Performance
Queuing Costs
♦The Variety of Queuing Models
Model A: Single-Channel Queuing Model with
Poisson Arrivals and Exponential Service Times
♦ Model B: Multiple-Channel Queuing Model
♦ Model C: Constant Service Time Model
♦ Model D: Limited Population Model
♦
♦Other Queuing Approaches
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Learning Objectives
When you complete this chapter, you should be
able to :
♦ Identify or Define:
♦
The assumptions of the four basic waiting-line
models
♦ Explain or be able to use:
♦
♦
How to apply waiting-line models
How to conduct an economic analysis of queues
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You’ve Been There Before!
Thank you for holding.
Hello...are you there?
‘The other line
always moves faster.’
‘If you change lines, the
one you left will start to
move faster than the one
you’re in.’
© 1995 Corel Corp.
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Waiting Line Examples
Situation
Arrivals
Servers
Service Process
Bank
Customers
Teller
Deposit etc.
Doctor’s
office
Patient
Doctor
Treatment
Traffic
intersection
Cars
Light
Controlled
passage
Assembly line
Parts
Workers
Assembly
Tool crib
tools
Workers
Clerks
Check out/in
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Waiting Lines
♦ First studied by A. K. Erlang in 1913
♦
Analyzed telephone facilities
♦ Body of knowledge called queuing theory
♦
Queue is another name for waiting line
♦ Decision problem
♦
Balance cost of providing good service with cost
of customers waiting
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Waiting Line Costs
Cost
cost
g line
in
it
a
lw
Tota
ost
ic e c
Serv
Waiting time cost
Level of service
Optimal
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Waiting Line Terminology
♦ Queue: Waiting line
♦ Arrival: 1 person, machine, part, etc. that
arrives and demands service
♦ Queue discipline: Rules for determining the
order that arrivals receive service
♦ Channel: Number of waiting lines
♦ Phase: Number of steps in service
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Characteristics of a
Waiting Line System
tion
Popula
Waiting Line
Service
Facility
Arrival Characteristics
♦ Arrival rate distribution ♦ Size of the source population
♦ Poisson
♦ Other
♦ Pattern of arrivals
♦ limited
♦ unlimited
♦ Behavior of the arrivals
♦ join the queue, and wait until
♦random
♦scheduled
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served
♦ balk; refuse to join the line
♦ renege; leave the line
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Characteristics of a
Waiting Line System - continued
tion
Popula
Service
Facility
Waiting Line
Waiting Line Characteristics
♦Length of the
queue
♦ limited
♦ unlimited
♦ Service priority
♦ FIFO
♦ other
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Characteristics of a
Waiting Line System - continued
Service
Facility
Service Facility Characteristics
tion
Popula
Waiting Line
♦ Number of channels
♦ Service time distribution
♦ single
♦ multiple
♦ negative exponential
♦ other
♦ Number of phases
in service system
♦ single
♦ multiple
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Waiting Line System
Input
source
Service system
Waiting
line
Service
facility
© 1995 Corel Corp.
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Input Characteristics
Input Source
(Population)
Size
Infinite
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Input Characteristics
Input Source
(Population)
Fixed number of
aircraft to service
Size
Infinite
Finite
© 1995 Corel Corp.
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Input Characteristics
Input Source
(Population)
Size
Infinite
Finite
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Arrival
Pattern
Random
NonRandom
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Input Characteristics
Input Source
(Population)
Arrival
Pattern
Size
Infinite
Finite
Random
Poisson
NonRandom
Other
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Poisson Distribution
♦ Number of events that
occur in an interval of
time
.6
.3
.0
♦ Example: Number of
customers that arrive
in 15 min.
.6
.3
.0
−λ x
P( x ) =
1
2
3
4
5
λ = 6.0
P(X)
X
0
e λ
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X
0
♦ Mean = λ (e.g., 5/hr.)
♦ Probability:
λ = 0.5
P(X)
2
4
6
8
10
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0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
Probability
Probability
Poisson Distributions for Arrival
Times
0.10
0.05
0.00
0.10
0.05
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12
x
λ=2
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0 1 2 3 4 5 6 7 8 9 10 11 12
x
λ=4
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Input Characteristics
Input Source
(Population)
Arrival
Pattern
Size
Infinite
Finite
Random
Poisson
NonRandom
Behavior
Patient
Impatient
Other
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Input Characteristics
Input Source
(Population)
Arrival
Pattern
Size
Infinite
Finite
Random
Poisson
NonRandom
Other
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Behavior
Patient
Impatient
Balk
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Balking
Input
source
Line was
too long!
Service system
Service
facility
Waiting
line
© 1995 Corel Corp.
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Input Characteristics
Input Source
(Population)
Arrival
Pattern
Size
Infinite
Finite
Random
Poisson
NonRandom
Other
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Behavior
Patient
Impatient
Balk
Renege
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Reneging
Service system
Input
source
Waiting
line
Service
facility
I give up!
© 1995 Corel Corp.
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Waiting Line Characteristics
Waiting Line
Length
Unlimited
© 1995 Corel Corp.
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Waiting Line Characteristics
Waiting Line
Length
© 1995 Corel Corp.
Unlimited
Limited
© 1995 Corel Corp.
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Waiting Line Characteristics
Waiting Line
Queue
Discipline
Length
Unlimited
Limited
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FIFO
(FCFS)
D-26
Random
Priority
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Service Characteristics
Service
Facility
Configuration
Single
Channel
MultiChannel
Single
Phase
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Negative Exponential Distribution
♦ Service time, &
time between arrivals
.4
Probability t>x
µ=1
µ=2
µ=3
µ=4
.3
♦ Example: Service time
is 20 min.
.2
♦ Mean service rate = µ
♦ e.g., customers/hr.
.1
♦ Mean service time = 1/µ
♦ Equation:
0.
0
f ( t > x ) = e− ìx
2
4
6
8
10
x
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Negative Exponential Distribution
0.06
Average service
time = 1 hour
0.05
Probability
0.04
0.03
Average service time = 20
minutes
0.02
0.01
0
0
30
60
90
120
Service time (minutes)
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150
180
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Single-Channel, Single-Phase
System
Service system
Arrivals
Ships at
sea
Queue
Service
facility
Ship unloading system
Served
units
Empty
ships
Waiting ship line
Dock
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Single-Channel, Multi-Phase
System
Service system
Arrivals
Cars
in area
Queue
Service
facility
Service
facility
Served
units
McDonald’s drive-through
Cars
& food
Waiting cars
Pay
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Pick-up
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Multi-Channel, Single Phase
System
Service system
Arrivals
Service
facility
Queue
Served
units
Service
facility
Example: Bank customers wait in single line for one
of several tellers.
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Multi-Channel, Multi-Phase
System
Service system
Arrivals
Queue
Service
facility
Service
facility
Service
facility
Service
facility
Served
units
Example: At a laundromat, customers use one of several
washers, then one of several dryers.
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Waiting-Line Performance
Measures
♦ Average queue time, Wq
♦ Average queue length, Lq
♦ Average time in system, Ws
♦ Average number in system, Ls
♦ Probability of idle service facility, P0
♦ System utilization, ρ
♦ Probability of k units in system, Pn > k
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Assumptions of the Basic
Queuing Model
♦ Arrivals are served on a first come, first served
basis
♦ Arrivals are independent of preceding arrivals
♦ Arrival rates are described by the Poisson
probability distribution, and customers come from a
very large population
♦ Service times vary from one customer to another,
and are independent of one and other; the average
service time is known
♦ Service times are described by the negative
exponential probability distribution
♦ The service rate is greater than the arrival rate
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Types of Queuing Models
♦ Simple (M/M/1)
♦
Example: Information booth at mall
♦ Multi-channel (M/M/S)
♦
Example: Airline ticket counter
♦ Constant Service (M/D/1)
♦
Example: Automated car wash
♦ Limited Population
♦
Example: Department with only 7 drills
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Simple (M/M/1) Model
Characteristics
♦ Type: Single-channel, single-phase system
♦ Input source: Infinite; no balks, no reneging
♦ Arrival distribution: Poisson
♦ Queue: Unlimited; single line
♦ Queue discipline: FIFO (FCFS)
♦ Service distribution: Negative exponential
♦ Relationship: Independent service & arrival
♦ Service rate > arrival rate
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Simple (M/M/1) Model Equations
λ
Ls =
µ -λ
Average number of
units in queue
Ws =
Average time in
system
Average number of units
in queue
Lq =
2
λ
µ (µ - λ )
λ
µ (µ - λ )
λ
ρ=
µ
Wq =
Average time in queue
System utilization
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1
µ -λ
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Simple (M/M/1) Probability
Equations
Probability of 0 units in system, i.e., system idle:
λ
P = 1- ρ = 10
µ
Probability of more than k units in system:
P =
n>k
(µλ )
k+1
Where n is the number of units in the system
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Multichannel (M/M/S) Model
Characteristics
♦ Type: Multichannel system
♦ Input source: Infinite; no balks, no reneging
♦ Arrival distribution: Poisson
♦ Queue: Unlimited; multiple lines
♦ Queue discipline: FIFO (FCFS)
♦ Service distribution: Negative exponential
♦ Relationship: Independent service & arrival
♦ Σ Service rates > arrival rate
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Model B (M/M/S) Equations
Probability of zero
people or units in the
system:
Average number of
people or units in the
system:
Average time a unit
spends in the system:
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P0 =
1
M−1 1  ë n  1  ëM Mì
 ∑    +  
 n! ì  M! ì  Mì − ë
 n=0   
λµ  λ 
 µ
Ls =
M !M
µ λ 
 µ
Ws =
M !M
D-41
M
λ
P0
λ
M
λ
P0 +
1
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Model B (M/M/S) Equations
Average number of
people or units waiting
for service:
L q = Ls − λ
µ
Average time a person
or unit spends in the
queue
Wq = Ws −
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Constant Service Rate (M/D/1)
Model Characteristics
♦ Type: Single-channel, single-phase system
♦ Input source: Infinite; no balks, no reneging
♦ Arrival distribution: Poisson
♦ Queue: Unlimited; single line
♦ Queue discipline: FIFO (FCFS)
♦ Service distribution: Negative exponential
♦ Relationship: Independent service & arrival
♦ Σ Service rates > arrival rate
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Model C (M/D/1) Equations
Average number of people
or units waiting for service:
Lq =
Average time a person or
unit spends in the queue
Wq =
Average number of people or
units in the system:
Average time a unit spends in
the system:
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2
λ
2µ (µ − λ )
2
(
λ
λ)
Ls = Lq + λ
µ
Ws = Wq +
1
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Limited Population Model
Characteristics
♦ Type: Single-channel, single-phase system
♦ Input source: Limited; no balks, no reneging
♦ Arrival distribution: Poisson
♦ Queue: Unlimited; single line
♦ Queue discipline: FIFO (FCFS)
♦ Service distribution: Negative exponential
♦ Relationship: Independent service & arrival
♦ Σ Service rates > arrival rate
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Model D (Limited Population)
Equations
T
T+U
Service Factor:
X=
Average number of people
or units waiting for service:
L = N (1 − F)
Average time a person or
unit spends in the queue
W=
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L(T + U ) T(1 − F)
=
( N − L)
XF
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Model D (Limited Population)
Equations - continued
Average number
running
J = NF(1 − X )
Average number
being served:
H= FNX
Number in the
population:
N J L H
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Model D (Limited Population)
Equations - continued
Where:
★D = probability that a unit will have to wait in
the queue
★F = efficiency factor
♦H = average number of units being serviced
♦J = average number of units not in the queue
or service bay
♦ L = average number of units waiting for
service
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Model D (Limited Population)
Equations - continued
♦M = number of service channels
♦N = number of potential customers
♦T = average service time
♦U = average time between unit service
requirements
♦W = average time a unit waits in line
♦X = service factor
★to be obtained from finite queuing tables
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Remember: λ & µ Are Rates
♦ λ = Mean number of
arrivals per time period
♦
e.g., 3 units/hour
If average service time is
15 minutes, then μ is 4
customers/hour
♦ µ = Mean number of
people or items served
per time period
♦
e.g., 4 units/hour
♦
1/µ
µ = 15 minutes/unit
© 1984-1994 T/Maker Co.
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