Systems Methodology and Design II

7/30/2019 Systems Methodology and Design II

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Department of Systems Engineering

George Mason University

Kuo-Chu ChangFairfax, Virginia

SYST 302: Systems Methodologyand Design II #6

SYST 302: Systems Methodologyand Design II #6


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Queuing Theory and Analysis

Queuing Theory and Analysis

Concepts and Introduction

Monte Carlo Analysis of Queuing

Single-Channel Queuing Models

Multiple-Channel Queuing Models

Finite Population Queuing Models


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Introduction

Introduction

The Queuing System (Waiting-Line) A facility or a group of facilities for service

A population of individuals or units which form a line tobe served with a pre-determined waiting-line discipline

Single-channel or multiple-channel

Single-stage or multiple-stage

Common Examples The public forms waiting-lines at grocery stores,

theaters, doctors office, etc. Items in process produces a waiting-line at each machine

center in a production system

Automobiles form waiting-lines at toll-gates, traffic

signals, docks in a transportation system


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The Queuing SystemThe Queuing System

Objective: determine the capacity of the service facility

in the light of the relevant costs and the characteristics ofthe arrival patterns so that the overall cost associated withthe queuing system is minimized

The arrival mechanism: finite or infinite population,arrival pattern may be time-dependent

The waiting line: form a queue with a certain disciplinesuch as first come/first serve, relative urgency, or firstcome/last serve. Waiting cost is incurred.

The service mechanism: discrete process provides

service for units in line, can be single channel or multiplechannel with a certain capacity. The cost is facilitydependent.

The decision model: decide a policy of service capacity

to meet the demand at a minimum cost


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Some ExamplesSome Examples

Population o o o o o o o

waiting line

o

o

o

Service facility

Population o o o o o o o

waiting line

o

Service facility

o. . .


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Monte Carlo AnalysisMonte Carlo Analysis The Mechanism

Uncertain Waiting-line decision models usually based on

assumptions of the arrival and service-time distributions Formal mathematic solutions may be difficult or

impossible in general

Monte Carlo analysis does not require these distributionsto obey certain theoretical forms

Provides analysis tool through simulation

5 6 7 8 9 10 11

0

0.05

0.1

0.15

0.2

0.25

0.3

Arrival Time

Probability

5 6 7 8 9 10 11

0

0.2

0.4

0.6

0.8

1

Arrival Time

Cumulativ

eProb

6 7 8 9 10 11 120

0.2

0.4

0.6

0.8

1

Service Time

CumulativeProb

6 7 8 9 10 11 120

0.05

0.1

0.15

0.2

0.25

0.3

Service Time

Probability


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Single Channel Queuing MC AnalysisSingle Channel Queuing MC Analysis

Economic Analysis:

Total cost = unit-waiting-cost-per-time-period * (waiting-time-in-queue + waiting-time-in-service)

+ service-cost-per-time-period * service-timeEx: $9.6 * (23 + 337) + $16.10 * 400 = $9,896


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Queuing ModelsQueuing Models

Queuing Models A/B/C: Arrival/Service/Number of Servers

M/M/1: Markov (Poisson) / Markov (Exponential) /

one server M/M/S: Multiple servers

M/G/1: General service time distribution

M/D/1: Deterministic service time


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Single-Channel Queuing ModelsSingle-Channel Queuing Models Model Assumptions

Infinite population pool

Arrival per period is a Poisson distribution

Service time is an exponential distribution

Provides theoretical analysis tool

System Parameters

ttedt

tttt

tttt

tntP

tn

tt

t

n

===

+

+

)1(1e:Note

andbetweenoccurscompletionserviceay thatprobabilit:

andmebetween tioccursarrivalany thatprobabilit:

at timesystemin theunitsofyprobabilit:)(

at timesystemin theunitsofnumber:

periodper timescompletionserviceofnumberexpected:

periodper timearrivalofnumberexpected:

00t-


10/2210

Probability of n Units in the SystemProbability of n Units in the System[ ][ ]{ } [ ][ ]{ } [ ]{ }

[ ]{ } [ ][ ]{ }

n

n

nnnn

t

nnnnnn

t

nn

nnnnn

nnnn

P

tPtPtPtPtPdt

d

tPtPtPtPdt

d

tPtPtPdt

d

t

tPttP

tttPttPttP

tPtPtPtPdt

d

t

tPttP

t

ttPttP

ttPttPttPttPtP

tttPtttPtttPttP

=

=+==

+++==

+==

+

+=+

+++==

+

+

+++=

++=+

+

+

++

+

1

)()()()(0)(

)()()()(0)(

state,steadyIn

)()()()()(

lim

1)(1)()(but

)()()()()()()(

lim

haveweterms,ignoring

)()(

)()()()()()(

1)(1)(11)()(

01100

11

10000

0

100

110

2

211

2

11

2

11


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Probability of n Units in the SystemProbability of n Units in the System

( )

1 1 1 2

1 0

1 2 0 1 2 1 0 1 2 0 0

0 0 0

0 0

Let average number of units being served

( ) (1 ) ( ) ( ) 0

But, 0,

1But 1 1

1

(1 ) 1

n

n n n n

n

n

n

n

n n

n

n

P t P t P t P c c

P Pc c P c c P P c c P P P

P P P P

P

+

= =

=

+ + = = +

= + = + = = = = =

= = = =

= =

n

0 1 2 3 4 5 6 7 8 9 10

0

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0. 7Ex:

4.0

25.0

1.0

==

==

8.0=

Ex:

0 1 2 3 4 5 6 7 8 9 10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2


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Other Performance MeasuresOther Performance Measures

( ) ( ) ( )

( ) ( )

Expected number of units in the system:

n nP n nnn

nn

n

n

n

n

= = = =

=

=

=

=

=

=

=

=

1 1 1

11

11

1

1 1

00 0 0

2

( )

Note: is the average number of units being serviced =

Average length of the queue:

avg. number of units in the system - avg. numbers of units being serviced

=

m =

=

2

( )

[ ] kn

nk

kn

n

kn

nPkP

k

=

===

=

=

= 0'

')1()1(systemin

:systemin theunitsthanmoreofyProbabilit


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ExampleExample25.0

4

1,1.0

10

1,)1(1 =====

=

nn

nP

Expected number of units in the system:

0.6671

n

= = =

267.0)(

:queuetheoflengthAverage2

=

=

m

Average length of the nonempty queue:

in the system]m

m

P m

m

Pm> = >

=

= = =0 20 20 267 0 267

0161667

( ) [

. .

..


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Waiting TimeWaiting Time

( ) 67.6101667.0example,previousIn the1

111fact,In

11

)(

=

timeserviceaverage+timewaitingaverage

:systemin thespenttimetotalAverage

)()()1()(

10.3)(Sec.ICBST0,>for w

1=system}in theunit0{)0(

:timewaitingofonDistributi

0

)(

0

==

=

=

===

=+

=

===

===

T

nTTn

T

dwwwfwewf

PPwP

w

w


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System Performance CurvesSystem Performance Curves

n =

1

T =

1

1

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

TotalTim

einSystem

0 0.2 0.4 0.6 0.8 10

5

10

15

20

Average

#inSystem


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ExampleExample

=

==

1

1][

bygivenisdelaymeanThe.nutilizatiotheand

,rateservice,artearrivalwithqueueM/M/1anisprocessorlargeThe:

systems.proposedtheandexistingtheofeperformancdelaymean

theCompare.ratearrivalandrateprocessingwitheachprocessorswith

processorlargethereplacetomadeisproposalathatSupposetime.processing

ofamountddistributellyexponentiaanrequiresactionsthat tranandsecond,per

ionstransactrateofprocessa PoissontoaccordingarrivalonstransactiSupposesecond.perionstransactofrateaatonstransactihandlesprocessorlargeA:

KTE

KK

KK

K

KK

A

Q

t.improvemeneperformancdelaytsignificaninresultssystemsingleaintodemandcustomerofionconcentratThe

][1

1]'[

isdelaymeanThe.nutilizatioand,rateservice

,ratearrivalwithsystemM/M/1anisprocessorssmalltheofEach

=

=

=

TKETE


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Cost Analysisper unit time

Cost Analysisper unit time

Expect total cost = expected waiting cost + expected facility cost

=

Minimize cost:

0

where : waiting cost per unit per tim

w f w f

w

f

w

TC C n C C C

CTC

C

C

+ = +

= = +

e period

: facility cost per unit serviced

Example:

1 8 , $0.10, $0.165

0.4, 0.1115

f

w f

C

C C

TC

= = =

= =


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Cost CurvesCost Curves

0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

Service Rate

TotalCost

FCWC

TC


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Multiple-Channel Queuing ModelsMultiple-Channel Queuing Models

Model Assumptions Same as single channel

Multiple service facility, # of channel = c

( ) ( ) ( ) ( )

, 0,0 ,0 0,0

0,0 1

0

,

Let , then

1 1,

! !1

1 ! [1/(1 )] 1 !

other measures such , , and can also be obtained (see Sec. 10.4),

where is the probability that ther

c m

n

c n m

r cc r

r

m n

c

P P P P

c m

P

c r

n m w

P

=

=

=

= =

= +

,

e are n units waiting in queue

and m channels are busy

Note: 0 m c and 0 for and 0m nP m c n =


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Queuing with Nonexponential ServiceQueuing with Nonexponential Service

Model Assumptions Poisson arrival

Nonexponential service time

( )

( )[ ]

( )

( )[ ]

( )

( )[ ]

( )

( )[ ]( )

For any service time distribution, if is its variance, then

and

For constant service time,

and

For exponential service time,

and

2

2 2 2 2 2

2

2

2 2

2 1 2 1

1

0

2 1 2 1

1

1

1

n T

n T

n T

=+

+ =+

+

=

= + = +

=

= = = w fT C C n C +


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Example*Example*

( )( )

( )( )

( )

2

22

For constant service time, 0.1, 0.25

1=0.5333 and 5.332 1 2 1

For exponential service time, 1

1=0.667 and 6.67

n T

n T

= =

= + = + =

=

= = =

*See MATLAB code lec61.m and lec62.m


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SummarySummary

Queuing

Monte Carlo Analysis of Queuing

Single-Channel Queuing Models

Multiple-Channel Queuing Models

Finite Population Queuing Models Control Concepts and Techniques

Statistical Process Control

Optimal Policy Control

Project Control

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Systems Methodology and Design II

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