queuing.ppt

8/16/2019 Queuing.ppt

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Queuing

CEE 320Anne Goodchild


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Outline

1. Fundamentals

2. Poisson Distribution

3. Notation4. Alications

!. Anal"sis

a. Grahicalb. Numerical

#. E$amle


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Fundamentals of Queuing Theory

• %icroscoic tra&&ic &lo'

– Di&&erent anal"sis than theor" o& tra&&ic &lo'

– (nter)als bet'een )ehicles is imortant

– *ate o& arri)als is imortant

• Arri)als

• Deartures• +er)ice rate


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Server/bottleneck

Arrivals Departures

Activated

Upstream of bottleneck/server Downstream

Direction of flow


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server

Arrivals Departures

Not Activated


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Flow Analysis

• ,ottlenec- acti)e

– +er)ice rate is caacit"

– Do'nstream &lo' is determined b" bottlenec-ser)ice rate

– Arri)al rate dearture rate

– /ueue resent


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Flow Analysis

• ,ottle nec- not acti)e

– Arri)al rate dearture rate

– No ueue resent – +er)ice rate arri)al rate

– Do'nstream &lo' euals ustream &lo'


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• htttra&&iclab.ce.5atech.edu&ree'a"a

*oadAlet.html


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Fundamentals of Queuing Theory

• Arri)als

– Arri)al rate 6)ehsec7

• ni&orm

• Poisson

– 9ime bet'een arri)als 6sec7• Constant

• Ne5ati)e e$onential

• +er)ice

– +er)ice rate

– +er)ice times

• Constant

• Ne5ati)e e$onential


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Queue Discipline

• First (n First :ut 6F(F:7

– re)alent in tra&&ic en5ineerin5

• ;ast (n First :ut 6;(F:7


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Queue Analysis – Graphical

Arrival

Rate

Departure

Rate

Time

V e h i c l e s

t1

Queue at time, t1

Maximum delay

Maximum queue

D/D/1 Queue

Delay o !th arrivi!" vehicle

Total vehicle delay

Where is capacity?


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Poisson Distriution

• Good &or modelin5 random e)ents

• Count distribution

– ses discrete )alues

– Di&&erent than a continuous distribution

( ) ( )

!n

et n P

t n λ λ −=

#$!% & pro'a'ility o exactly ! vehicles arrivi!" over time t

! & !um'er o vehicles arrivi!" over time t

( & avera"e arrival rate

t & duratio! o time over )hich vehicles are cou!ted


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Poisson !deas

• Probabilit" o& e$actl" 4 )ehicles arri)in5

– P6n47

• Probabilit" o& less than 4 )ehicles arri)in5

– P6n47 P607 < P617 < P627 < P637

• Probabilit" o& 4 or more )ehicles arri)in5

– P6n=47 1 > P6n47 1 ? P607 < P617 < P627 < P637

• Amount o& time bet'een arri)al o& successi)e )ehicles

( ) ( ) ( ) 36

!

qt t t

eeet

t h P P −−−

===≥= λ

λ λ


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"#ample Graph

*+**

*+*

*+1*

*+1

*+-*

*+-

* 1 - . 0 2 3 1* 11 1- 1. 1 1 10 1 12 13 -*

Arri)als in 1! minutes

P r o

b a b i l i t " o & : c c u r a n c e


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*+**

*+*

*+1*

*+1

*+-*

*+-

* 1 - . 0 2 3 1* 11 1- 1. 1 1 10 1 12 13 -*

Arri)als in 1! minutes

P r o

b a b i l i t " o & : c c u r a n c e

Mea! & *+- vehicles/mi!ute

Mea! & *+ vehicles/mi!ute

"#ample Graph


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"#ample$ Arrival !ntervals

*+*

*+1

*+-

*+.

*+

*+

*+0

*+

*+2

*+3

1+*

* - 0 2 1* 1- 1 10 12 -*

9ime ,et'een Arri)als 6minutes7

P r

o b a b i l i t " o & E $ c e

d a n c e

Mea! & *+- vehicles/mi!ute

Mea! & *+ vehicles/mi!ute


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Queue Notation

• Poular notations

– DD1@ %D1@ %%1@ %%N – D deterministic

– % some distribution

N Y X //

Arrival rate !ature

Departure rate !ature

4um'er o

service cha!!els


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Queuing Theory Applications

• DD1

– Deterministic arri)al rate and ser)ice times

– Not t"icall" obser)ed in real alications but

reasonable &or aro$imations• %D1

– General arri)al rate@ but ser)ice times

deterministic

– *ele)ant &or man" alications

• %%1 or %%N

– General case &or 1 or man" ser)ers


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Queue times depend onvariaility


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Queue Analysis – Numerical

• %D1

– A)era5e len5th o& ueue

– A)era5e time 'aitin5 in ueue

– A)era5e time sent in s"stem

µ

λ ρ = "


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Queue Analysis – Numerical

• %%1

– A)era5e len5th o& ueue

– A)era5e time 'aitin5 in ueue

– A)era5e time sent in s"stem

µ

λ ρ = "


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%&%&N – %ore 'tu(

– Probabilit" o& ha)in5 no )ehicles

– Probabilit" o& ha)in5 n )ehicles

– Probabilit" o& bein5 in a ueue

( )∑−

= −+

="

"!!

" N

n

N

c

n

c

c

N N n

P

ρ

ρ ρ

µ

λ ρ =

%nfor!

≤=n

P P

n

n

ρ %nfor!

≥= − N N

P P

N n

n

n

ρ

( ) N N N

P P

N

N n ρ

ρ

−=

+

>"!

"

"


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Poisson Distriution "#ample

6rom 78M -***

Vehicle arrivals at the 9lympic 4atio!al #ar: mai! "ate are assumed

#oisso! distri'uted )ith a! avera"e arrival rate o 1 vehicle every

mi!utes+ ;hat is the pro'a'ility o the ollo)i!"<

1+ =xactly - vehicles arrive i! a 1 mi!ute i!terval>

-+ ?ess tha! - vehicles arrive i! a 1 mi!ute i!terval>.+ More tha! - vehicles arrive i! a 1 mi!ute i!terval>

( )

( ) ( )

!

minveh#minveh#

n

et

n P

t n −×

=


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"#ample )alculations

( ) ( ) ( )

&'####'!#

"(##

"(##

==×

=

−e P =xactly -<

?ess tha! -<

More tha! -<

( ) ( ) ( ) "))#"# =+=< P P n P

( ) ( ) ( ) ( )( ) (*6+#""# =++−=> P P P n P

#$*%&e@+-1&*+*32, #$1%&*+13


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"#ample *

• Dearture rate 18 secondserson or 3.33

ersonsminute

• Arri)al rate B 3 ersonsminute

• 33.33 0.0

• /?bar 0.02 61?0.07 8.1 eole

• ?bar 33.3363.33?37 2.3 minutes

• 9?bar 163.33 > 37 3.03 minutes


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"#ample +

Bou are !o) i! li!e to "et i!to the Are!a+ There are . operati!"

tur!stiles )ith o!e tic:et@ta:er each+ 9! avera"e it ta:es . seco!ds

or a tic:et@ta:er to process your tic:et a!d allo) e!try+ The avera"e

arrival rate is * perso!s/mi!ute+

6i!d the avera"e le!"th o queue, avera"e )aiti!" time i! queueassumi!" M/M/4 queui!"+


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"#ample +

• N 3

• Dearture rate 3 secondserson or 20 ersonsminute

• Arri)al rate B 40 ersonsminute

• 4020 2.0• N 2.03 0.## 1 so 'e can use the other euations

• P0 1620 0 < 21 1 < 22 2 < 23 361?2377 0.1111

• /?bar 60.11117624763H37H6161 > 23727 0.88 eole

• 9?bar 62 < 0.88740 0.02 minutes 4.32 seconds• ?bar 0.02 > 120 0.022 minutes 1.32 seconds


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"#ample ,

Bou are !o) i!side the Are!a+ They are passi!" out 7arry the 7us:y

do""y 'a"s as a ree "ivea)ay+ There is o!ly o!e perso! passi!"

these out a!d a li!e has ormed 'ehi!d her+ t ta:es her exactly 0

seco!ds to ha!d out a do""y 'a" a!d the arrival rate avera"es

3 people/mi!ute+

6i!d the avera"e le!"th o queue, avera"e )aiti!" time i! queue, a!d

avera"e time spe!t i! the system assumi!" M/D/1 queui!"+


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"#ample ,

• N 1• Dearture rate # secondserson or 10

ersonsminute

• Arri)al rate B ersonsminute• 10 0.

• /?bar 60.72 6261 > 0.77 4.0! eole

• ?bar 0.62610761 > 0.77 0.4! minutes 2seconds• 9?bar 62 > 0.7662610761 > 0.7 0.!! minutes 33

seconds


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Primary -eferences

• Ma!!eri!", 6+?+E Filares:i, ;+#+ a!d ;ash'ur!, G+G+ $-**.%+ Principles of

Highway Engineering and Traffic Analysis, Third =ditio! $Drat%+ 8hapter

• Tra!sportatio! Research Coard+ $-***%+ Highway Capacity Manual

2000 + 4atio!al Research 8ou!cil, ;ashi!"to!, D+8+

queuing.ppt

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