operations management -...

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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


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


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

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‘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!


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


♦ 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

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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


♦ 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


♦ 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

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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



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


Service system

Waitingline

Servicefacility

Inputsource

© 1995 Corel Corp.

Waiting Line SystemWaiting Line System

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Input Source(Population)

Size

Infinite

Input CharacteristicsInput Characteristics



Size

FiniteInfinite

© 1995 Corel Corp.

Fixed number ofaircraft to service




Size ArrivalPattern

FiniteInfinite Random Non-Random


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Size ArrivalPattern

FiniteInfinite Random Non-Random

Poisson Other



♦ 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−


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



Size BehaviorArrivalPattern

FiniteInfinite Random Non-Random Patient Impatient

Poisson Other






BalkPoisson Other



Inputsource Service

facilityWaiting

line

Service system

© 1995 Corel Corp.

Line wastoo long!

BalkingBalking

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Balk RenegePoisson Other



RenegingRenegingInput

source Servicefacility

Waitingline

Service system

© 1995 Corel Corp.

I give up!


Waiting Line

Length

Unlimited

© 1995 Corel Corp.

Waiting Line CharacteristicsWaiting Line Characteristics

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Waiting Line

Length

LimitedUnlimited

© 1995 Corel Corp.

© 1995 Corel Corp.



Waiting Line

Length QueueDiscipline

LimitedUnlimited FIFO(FCFS) Random Priority



ServiceFacility

Configuration

Multi-Channel

SingleChannel

SinglePhase

Service CharacteristicsService Characteristics

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♦ 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


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


ArrivalsServedunits

Servicefacility

Queue

Service system

Dock

Waiting ship lineShips atsea

Ship unloading system Emptyships

Single-Channel, Single-PhaseSingle-Channel, Single-PhaseSystemSystem

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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


Arrivals

Servedunits

ServicefacilityQueue

Service system

Servicefacility

Example: Bank customers wait in single line for oneof several tellers.

Multi-Channel, Single PhaseMulti-Channel, Single PhaseSystemSystem


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

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♦ 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


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


♦ 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

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♦ 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


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


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( )

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♦ 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


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:


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

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♦ 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


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


♦ 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

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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


Model D (Limited Population)Model D (Limited Population)Equations - continuedEquations - continued

HLJN

FNXH

)X(NFJ

=

−1=Average numberrunningAverage numberbeing served:

Number in thepopulation:



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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♦ 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


♦ 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

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