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Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-1 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
OperationsOperationsManagementManagement
Waiting-Line ModelsWaiting-Line ModelsModule DModule D
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-2 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
OutlineOutline♦ 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
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-3 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
When you complete this chapter, you should beable 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
Learning ObjectivesLearning Objectives
2
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-4 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
‘The other linealways moves faster.’
‘If you change lines, theone you left will start tomove faster than the oneyou’re in.’
© 1995 Corel Corp.
Thank you for holding.Hello...are you there?
You’ve Been There Before!You’ve Been There Before!
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-5 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Bank Customers Teller Deposit etc.
Doctor’s Patient Doctor Treatmentoffice
Traffic Cars Light Controlledintersection passage
Assembly line Parts Workers Assembly
Tool crib Workers Clerks Check out/intools
Situation Arrivals Servers Service Process
Waiting Line ExamplesWaiting Line Examples
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-6 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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
Waiting LinesWaiting Lines
3
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-7 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Level of serviceLevel of service
CostCost
Service costService costTotal waiting line cost
Total waiting line cost
Waiting time costWaiting time cost
Optimal
Waiting Line CostsWaiting Line Costs
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-8 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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
Waiting Line TerminologyWaiting Line Terminology
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-9 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ Size of the source population♦ limited♦ unlimited
♦ Behavior of the arrivals♦ join the queue, and wait until
served♦ balk; refuse to join the line♦ renege; leave the line
ServiceFacilityWaiting LinePopulation
♦ Arrival rate distribution♦ Poisson♦ Other
♦ Pattern of arrivals♦ random♦ scheduled
ArrivalArrival CharacteristicsCharacteristics
Characteristics of aCharacteristics of a Waiting Line System Waiting Line System
4
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-10 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
ServiceFacilityWaiting LinePopulation
Waiting Line Characteristics♦ Length of thequeue
♦ limited♦ unlimited
♦ Service priority♦ FIFO♦ other
Characteristics of aCharacteristics of a Waiting Line System - continued Waiting Line System - continued
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-11 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
ServiceFacilityWaiting LinePopulation
Service Facility Characteristics
♦ Number of channels♦ single♦ multiple
♦ Number of phasesin service system
♦ single♦ multiple
♦ Service time distribution♦ negative exponential♦ other
Characteristics of aCharacteristics of a Waiting Line System - continued Waiting Line System - continued
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-12 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Service system
Waitingline
Servicefacility
Inputsource
© 1995 Corel Corp.
Waiting Line SystemWaiting Line System
5
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-13 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size
Infinite
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-14 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size
FiniteInfinite
© 1995 Corel Corp.
Fixed number ofaircraft to service
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-15 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size ArrivalPattern
FiniteInfinite Random Non-Random
Input CharacteristicsInput Characteristics
6
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-16 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size ArrivalPattern
FiniteInfinite Random Non-Random
Poisson Other
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-17 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ Number of events thatoccur in an interval oftime♦ Example: Number of
customers that arrivein 15 min.
♦ Mean = λλλλ (e.g., 5/hr.)♦ Probability:
.0
.3
.6
0 1 2 3 4 5X
P(X)
.0
.3
.6
0 2 4 6 8 10X
P(X)
λλλλλλλλ = 0 = 0..55
λλλλλλλλ = 6 = 6..00
Poisson DistributionPoisson Distribution
e)x(P
xλ=
λ−
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Poisson Distributions for ArrivalPoisson Distributions for ArrivalTimesTimes
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1 2 3 4 5 6 7 8 9 10 11 12x
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1 2 3 4 5 6 7 8 9 10 11 12x
Prob
abilit
y
Prob
abilit
y
λ=2 λ=4
7
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-19 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size BehaviorArrivalPattern
FiniteInfinite Random Non-Random Patient Impatient
Poisson Other
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-20 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size BehaviorArrivalPattern
FiniteInfinite Random Non-Random Patient Impatient
BalkPoisson Other
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-21 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Inputsource Service
facilityWaiting
line
Service system
© 1995 Corel Corp.
Line wastoo long!
BalkingBalking
8
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-22 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Input Source(Population)
Size BehaviorArrivalPattern
FiniteInfinite Random Non-Random Patient Impatient
Balk RenegePoisson Other
Input CharacteristicsInput Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-23 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
RenegingRenegingInput
source Servicefacility
Waitingline
Service system
© 1995 Corel Corp.
I give up!
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-24 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Waiting Line
Length
Unlimited
© 1995 Corel Corp.
Waiting Line CharacteristicsWaiting Line Characteristics
9
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-25 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Waiting Line
Length
LimitedUnlimited
© 1995 Corel Corp.
© 1995 Corel Corp.
Waiting Line CharacteristicsWaiting Line Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-26 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Waiting Line
Length QueueDiscipline
LimitedUnlimited FIFO(FCFS) Random Priority
Waiting Line CharacteristicsWaiting Line Characteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-27 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
ServiceFacility
Configuration
Multi-Channel
SingleChannel
SinglePhase
Service CharacteristicsService Characteristics
10
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-28 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ Service time, &time between arrivals♦ Example: Service time
is 20 min.♦ Mean service rate = µµµµ
♦ e.g., customers/hr.♦ Mean service time = 1/µµµµ♦ Equation:
Negative Exponential DistributionNegative Exponential Distribution
xìe)xt(f −=>
0.
.1
.2
.3
.4
0 2 4 6 8 10 x
Prob
abilit
y t>x
µµµµ=1µµµµ=2µµµµ=3µµµµ=4
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-29 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
0
0.01
0.02
0.03
0.04
0.05
0.06
Pro
babi
lity
0 30 60 90 120 150 180 Service time (minutes)
Negative Exponential DistributionNegative Exponential Distribution
Average serviceAverage servicetime = 1 hourtime = 1 hour
Average service time = 20Average service time = 20minutesminutes
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-30 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
ArrivalsServedunits
Servicefacility
Queue
Service system
Dock
Waiting ship lineShips atsea
Ship unloading system Emptyships
Single-Channel, Single-PhaseSingle-Channel, Single-PhaseSystemSystem
11
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-31 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Cars& food
Single-Channel, Multi-PhaseSingle-Channel, Multi-PhaseSystemSystem
ArrivalsServedunits
Servicefacility
Queue
Service system
Pick-upWaiting cars
Carsin area
McDonald’s drive-through
Pay
Servicefacility
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-32 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Arrivals
Servedunits
ServicefacilityQueue
Service system
Servicefacility
Example: Bank customers wait in single line for oneof several tellers.
Multi-Channel, Single PhaseMulti-Channel, Single PhaseSystemSystem
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-33 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Servicefacility
Arrivals
Servedunits
ServicefacilityQueue
Service system
Servicefacility
Example: At a laundromat, customers use one of severalwashers, then one of several dryers.
Servicefacility
Multi-Channel, Multi-PhaseMulti-Channel, Multi-PhaseSystemSystem
12
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-34 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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, ρρρρ♦ Probability of k units in system, Pn > k
Waiting-Line PerformanceWaiting-Line PerformanceMeasuresMeasures
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-35 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Assumptions of the BasicAssumptions of the BasicQueuing ModelQueuing Model
♦ Arrivals are served on a first come, first servedbasis
♦ Arrivals are independent of preceding arrivals♦ Arrival rates are described by the Poisson
probability distribution, and customers come from avery large population
♦ Service times vary from one customer to another,and are independent of one and other; the averageservice time is known
♦ Service times are described by the negativeexponential probability distribution
♦ The service rate is greater than the arrival rate
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-36 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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
Types of Queuing ModelsTypes of Queuing Models
13
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-37 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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
Simple (M/M/1) ModelSimple (M/M/1) ModelCharacteristicsCharacteristics
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Simple (M/M/1) Model EquationsSimple (M/M/1) Model Equations
Average number ofunits in queue
Average time insystem
Average number of unitsin queue
Average time in queue
System utilization
L
W
L
W
s
s
q
q
=
=
=
=
λλλλ
µµµµ - λλλλ
λλλλ
1
2
λλλλ
µµµµ - λλλλ
µµµµ (µµµµ - λλλλ )
µµµµ (µµµµ - λλλλ )
=ρρρρλλλλ µµµµ
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-39 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Probability of 0 units in system, i.e., system idle:
Probability of more than k units in system:
Where n is the number of units in the system
P
k+1
0 1 1= - = -
=Pn>k
ρρρρ λλλλµµµµ
Simple (M/M/1) ProbabilitySimple (M/M/1) ProbabilityEquationsEquations
λλλλµµµµ( )
14
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-40 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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♦ ΣΣΣΣ Service rates > arrival rate
Multichannel (M/M/S) ModelMultichannel (M/M/S) ModelCharacteristicsCharacteristics
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-41 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Model B (M/M/S) EquationsModel B (M/M/S) Equations
ëMì
Mì
ì
ë
M!
1
ì
ë
n!
1
1P
M1M
0n
n0
−
+
∑
=
−
=
Probability of zeropeople or units in thesystem:
λλ
µ
λλµ= 0P
M!ML
M
s
Average number ofpeople or units in thesystem:
1+λ
µ
λµ= 0P
M!MW
M
s
Average time a unitspends in the system:
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-42 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Model B (M/M/S) EquationsModel B (M/M/S) Equations
1−=
µλ−=
sq
sq
WW
LL
Average number ofpeople or units waitingfor service:
Average time a personor unit spends in thequeue
15
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-43 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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♦ ΣΣΣΣ Service rates > arrival rate
Constant Service Rate (M/D/1)Constant Service Rate (M/D/1)Model CharacteristicsModel Characteristics
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Model C (M/D/1) EquationsModel C (M/D/1) Equations
Average number of people orunits in the system:
Average time a unit spends inthe system:
( )
( )λ2λ=
λ−µµ2λ=
2
q
q
W
L
Average number of peopleor units waiting for service:
Average time a person orunit spends in the queue
1+=
µλ+=
qs
qs
WW
LL
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-45 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ 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♦ ΣΣΣΣ Service rates > arrival rate
Limited Population ModelLimited Population ModelCharacteristicsCharacteristics
16
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Model D (Limited Population)Model D (Limited Population)EquationsEquations
XF
)F(T
)LN(
)UT(LW
)F(NL
UT
TX
−1=−+=
−1=
+=Service Factor:
Average number of peopleor units waiting for service:
Average time a person orunit spends in the queue
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-47 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Model D (Limited Population)Model D (Limited Population)Equations - continuedEquations - continued
HLJN
FNXH
)X(NFJ
=
−1=Average numberrunningAverage numberbeing served:
Number in thepopulation:
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-48 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Model D (Limited Population)Model D (Limited Population)Equations - continuedEquations - 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
17
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-49 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Model D (Limited Population)Model D (Limited Population)Equations - continuedEquations - 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
Transparency Masters to accompany OperationsManagement, 6E (Heizer & Render) D-50 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
♦ λλλλ = Mean number ofarrivals per time period♦ e.g., 3 units/hour
♦ µµµµ = Mean number ofpeople or items servedper time period♦ e.g., 4 units/hour
♦ 1/µµµµ = 15 minutes/unit
Remember: Remember: λλλλλλλλ & & µµµµµµµµ Are Rates Are Rates
© 1984-1994 T/Maker Co.
If average service time is15 minutes, then μ is 4customers/hour