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)ehicle 'o)e'ent characteristics #ro' upstrea' signal to do&nstrea' signal need to be

considered and si'ulated* +ra##ic Progression !odels 'odel the )ehicle 'o)e'ent

characteristics and help in the lin%ing o# signals* First, the concept o# platoon, platoon )ariablesis discussed and then platoon ratio is de#ined &hich is reuired #or deter'ination o# arri)al type*

+hen, the pheno'enon o# platoon dispersion and platoon dispersion 'odel is introduced #or

understanding dispersion beha)ior o# the )ehicles* Finally, one o# the platoon dispersion 'odelsi*e*, Roberson"s platoon dispersion 'odel is discussed, &hich esti'ates the )ehicle arri)als at

do&nstrea' locations based on an upstrea' departure pro#ile*

Characterizing Platoon

$ )ehicle Platoon is de#ined as a group o# )ehicles tra)elling together* $ )ehicle Platoon is

sho&n in Fig* 1*

Variables describing platoon

+he )arious )ehicle platoon characteristics or )ariables include platoon size, platoon head&ay,platoon speed and inter-arri)al head&ay* Platoon beha)iour and distribution patterns could be

identi#ied &ith respect to these para'eters* +he )arious platoon characteristics are illustrated in

Fig* 2*

Figure 1$ )ehicle Platoon

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Figure !Illustration o# Platoon Variables

Platoon "ize #$p%It is the nu'ber o# )ehicles in a platoon*

Platoon &ead'a( #hp%It is the a)erage )alue o# head&ays &ithin a platoon*

Platoon "peed #Vp%It is the a)erage speed o# all the )ehicles &ithin a platoon*

Inter)Arri*al #IA%It is the head&ay bet&een the last )ehicle o# the preceding platoon

and the #irst )ehicle o# the #ollo&ing platoon*

Various )alues o# platoon head&ay and inter-arri)al bet&een consecuti)e platoons can be used todeter'ine appropriate critical head&ay #or platoon identi#ication and detection* .nce the criticalhead&ay is deter'ined, platoon size and platoon speed can be detected to calculate the signal

ti'ing ad(ust'ent to acco''odate the approaching )ehicle platoon* It is o# great i'portance to

select a proper )alue o# the critical head&ay since a s'all change in the critical head&ay &illgenerate tre'endous changes in the resultant platoon characteristics* /se o# a large critical

head&ay &ill result in a large a)erage platoon size and reuire a large detection area in order to

detect large )ehicle platoons* Conseuently, a large detection area leads to an increase o# detector

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installation and 'aintenance costs* .n the other hand, use o# a s'all critical head&ay &ill result

in a s'all a)erage platoon size, but 'ay not pro)ide su##icient )ehicle platoon in#or'ation*

+here#ore, it is desired to #ind an appropriate critical head&ay so that su##icient platoonin#or'ation can be obtained &ithin a proper detection area* Research has sho&n that head&ays

are rarely less than 0* seconds or o)er 10 seconds at di##erent tra##ic )olu'es* !any

in)estigations ha)e been done on #inding the e##ects o# critical head&ays o# 1*2, 1*, 2*1 and 2*seconds on platoon beha)iour and these in)estigations ha)e sho&n that a critical head&ay o# 2*1

seconds corresponding to a tra##ic )olu'e o# 100 )ehicles per hour per lane )phpl3 can be

ta%en #or data collection*

Platoon +atio

+he platoon ratio denoted as Rp, is a nu'erical )alue used to uanti#y the uality o# progressionon an approach* +he platoon ratio represents the ratio o# the nu'ber o# )ehicles arri)ing during

the green phase to the proportion o# the green inter)al o# the total cycle* +his is gi)en by

1

3

&here, P Proportion o# all )ehicles during green ti'e, C Cycle length, g 5##ecti)e green

ti'e* Its )alue ranges #ro' 0* to 2*0* It is used in the calculation o# delays, capacity o# an

approach* +he arri)al types range #ro' 1 &orst platoon condition3 to 6 the best platooncondition3* +he platoon ratio appro7i'ates the arri)al type and the progression uality* For

e7a'ple 8C! 20003 has suggested the #ollo&ing relationship bet&een platoon ratio and arri)al

&hich is as sho&n in +able 1*

Table 1Relationship bet&een $rri)al +ype and Platoon Ratio

$rri)al Range o# platoone#ault )alue

Progression uality

typeratio

1 0* Very poor

2 0*66 /n#a)orable

1*000 Rando' arri)als

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4 1* Fa)orable

1*66 8ighly #a)orable

6 2*000 57ceptional

Platoon ,ispersion

$s a platoon 'o)es do&nstrea' #ro' an upstrea' intersection, the )ehicles disperse i*e*, thedistance bet&een the )ehicles increase &hich 'ay be due to the di##erences in the )ehicle speeds,

)ehicle interactions lane changing and 'erging3 and other inter#erences par%ing, pedestrians,

etc*,3* +his pheno'enon is called asPlatoon Dispersion*

ispersion has been #ound to be a #unction o# the tra)el ti'e #ro' a signal to a do&nstrea'

signal or other do&nstrea' location3 and the length o# the platoon* +he longer the tra)el ti'e

bet&een signals, the greater the dispersion* +his is intuiti)ely logical since the longer the tra)el

ti'e, the 'ore ti'e opportunity3 there is #or di##erent dri)ers to de)iate #ro' the a)erage tra)el

ti'e* $ si'ple case o# Platoon ispersion is as sho&n in Fig* * Fro' the #igure, it can be

obser)ed that, initially the pea% o# the platoon is high and the length o# the platoon is

co'parati)ely s'all, but as the platoon progresses do&nstrea', the pea% o# the platoon

decreases and the length increases*

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Figure -$ si'ple case o# Platoon ispersion

Various tra##ic engineering so#t&are li%e +R$:;

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Figure .$ +R$:;

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)arious tra##ic si'ulation so#t&are* Research has already been conducted on the applicability o#

platoon dispersion as a reliable tra##ic 'o)e'ent 'odel in urban street net&or%s* !ost o# the

research has sho&n that Robertson"s 'odel o# platoon dispersion is reliable, accurate, and robust*

+obertson/s Platoon ,ispersion Model

+he basic Robertson"s recursi)e platoon dispersion 'odel ta%es the #ollo&ing 'athe'atical #or'

2

3

&here, arri)al #lo& rate at the do&nstrea' signal at ti'e t, departure #lo& rate at

the upstrea' signal at ti'e t-+, + 'ini'u' tra)el ti'e on the lin% 'easured in ter's o# unit

steps + 3, a)erage lin% tra)el ti'e, n 'odeling ti'e step duration, is the

s'oothing #actor gi)en by=

3

platoon dispersion #actor unit less3 tra)el ti'e #actor unit less3 5uation sho&sthat the arri)al #lo&s in each ti'e period at each intersection are dependent on the departure

#lo&s #ro' other intersections* :ote that the Robertson"s platoon dispersion euation 'eans that

the tra##ic #lo& , &hich arri)es during a gi)en ti'e step at the do&nstrea' end o# a lin%, is a

&eighted co'bination o# the arri)al pattern at the do&nstrea' end o# the lin% during the

pre)ious ti'e step and the departure pattern #ro' the upstrea' tra##ic signal + seconds

ago *

Fig* gi)es the graphical representation o# the 'odel* It clearly sho&s that predicated #lo& rate

at any ti'e step is a linear co'bination o# the original platoon #lo& rate in the corresponding

ti'e step &ith a lag ti'e o# t3 and the #lo& rate o# the predicted platoon in the step i''ediately

preceding it* ;ince the dispersion 'odel gi)es the do&nstrea' #lo& at a gi)en ti'e inter)al, the

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'odel needs to be applied recursi)ely to predict the #lo&* ;eddon de)eloped a nu'erical

procedure #or platoon dispersion* 8e re&rote Robertson"s euation as,

43

Figure 0>raphical Representation o# Robertson"s Platoon ispersion

!odel

+his euation de'onstrates that the do&nstrea' tra##ic #lo& co'puted using the Robertson"splatoon dispersion 'odel #ollo&s a shi#ted geo'etric series, &hich esti'ates the contribution o#

an upstrea' #lo& in the inter)al to the do&nstrea' #lo& in the inter)al* $success#ul application o# Robertson"s platoon dispersion 'odel relies on the appropriate

calibration o# the 'odel para'eters* Research has sho&n that the tra)el-ti'e #actor is

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dependent on the platoon dispersion #actor * /sing the basic properties o# the geo'etricdistribution o# 5uation , the #ollo&ing euations ha)e been deri)ed #or calibrating the

para'eters o# the Robertson platoon dispersion 'odel*

3

5uationde'onstrates that the )alue o# the tra)el ti'e #actor is dependent on the )alue o#

the platoon dispersion #actor and thus a )alue o# 0*? as assu'ed by Robertson results in

inconsistencies in the #or'ulation* Further, the 'odel reuires calibration o# only one o# the'

and the other #actors can be obtained subseuently*

6

3

&here, is the standard de)iation o# lin% tra)el ti'es and is the a)erage tra)el ti'e

bet&een upstrea' and do&nstrea' intersections* 5uation de'onstrates that tra)el ti'e #actor

can be obtained %no&ing the a)erage tra)el ti'e, ti'e step #or 'odeling and standard de)iation

o# the tra)el ti'e on the road stretch*

3

5uation#urther per'its the calculation o# the s'oothing #actor directly #ro' the standard

de)iation o# the lin% tra)el ti'e and ti'e step o# 'odeling* +hus, both and can be

'athe'atically deter'ined as long as the a)erage lin% tra)el ti'e, ti'e step #or 'odeling and itsstandard de)iation are gi)en*

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$umerical Eample 1

In a case study, the a)erage tra)el ti'e #or a particular stretch &as #ound out to be 22*? seconds,

standard de)iation is *91 and 'odel ti'e step duration is 10 sec* Find out the Robertson"s

'odel para'eters and also the #lo& at do&nstrea' at di##erent ti'e steps &here the upstrea'#lo&s are as gi)en as=

*

"olution

>i)en, +he 'odel ti'e step duration n10sec, a)erage tra)el ti'e 322*?sec, standard

de)iation 3*91* Fro' euations abo)e*

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2pstream Flo's

;ince the 'odelling ti'e step duration is gi)en as n10 sec, the gi)en upstrea' #lo&s can be

&ritten as #ollo&s=

.n si'ilar lines , , , can be &ritten*

,o'nstream Flo's

.n the do&nstrea', at 10 sec the #lo& &ill be zero since the 'odelling step duration is 10 sec*

8ence the do&nstrea' #lo&s can be &ritten as #ollo&s*

;i'ilarly, do&nstrea' #lo&s can be &ritten till ?0 sec* :ote that since n10 sec, + is ta%en in

units o# n* +he 'ini'u' tra)el ti'e +3 is gi)en as

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Calculating on si'ilar lines, &e get

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+he total upstrea' )ehicles in 60 sec is ?9* $nd total do&nstrea' )ehicles in ?0 sec is ?9* +hat

is, all ?9 )ehicles co'ing #ro' upstrea' in 6 inter)als too% inter)als to pass the do&nstrea'*

$umerical Eample !

In a case study, the a)erage tra)el ti'e #ro' the upstrea' point to 1st do&n&ard point point in

bet&een upstrea' and do&nstrea'3 &as #ound out to be 22*? seconds and #ro' upstrea' pointto do&n&ard point end point3 &as #ound out to be 2*? seconds , standard de)iation is *91 and

'odel ti'e step duration is 10 sec* Find out the Robertson"s 'odel para'eters and also the #lo&

at do&nstrea' at di##erent ti'e steps &here the upstrea' #lo&s are as gi)en belo&*

*

"olution

+his proble' is si'ilar to the earlier proble'* .nly there are 2 do&nstrea' points gi)en in this*For the #irst do&nstrea' point, upstrea' )alues o# #lo& gi)en in the proble' &ill be used,

&hereas #or the do&nstrea' point, the #lo& #ro' the do&nstrea' point is to be used*

8ence at do&nstrea' point, #lo& in the #irst inter)al is zero and at the do&nstrea')alue, #lo& is zero #or #irst 2 inter)als* +he calculations ha)e been done in e7cel and the

#ollo&ing sho&s the results*

/pstrea' Vol* #or in sec*3 :o* o# Vehicles

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10 20

20 10

0 1

40 1?

0 14

60 12

0

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0

0

0

34

;'oothing Factor F 0*?

@ag +i'eFor In Aet&een Point3 20 sec

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@ag +i'eFor 5nd Point3 0 sec

o&nstrea' Volu'e o&nstrea' Volu'e

At in bet'een Point At End Point

in seconds3 in seconds3

10 0 10 0*00

20 1*66 20 0*00

0 11*2 0 12*26

40 14*1? 40 11*4

0 15615 0 1*9

60 14*69 60 176-4

0 12*? 0 1*06

?0 2* ?0 1*12

90 0*9 90 4*99

100 0*1 100 1*

0*00 110 0*44

33647 120 0*09

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3363.

Four graphs are plotted belo&* +he #irst graph sho&s the upstrea' pro#ile, the second sho&s thedo&nstrea' pro#ile at in bet&een point, the third sho&s the do&nstrea' pro#ile at the end point*

+he last graph sho&s the co'parison o# all the three*

Conclusion

Initially, the concept o# platoon and platoon )ariables &as discussed* +he platoon )ariables are

reuired #or the deter'ination o# critical head&ay &hich #urther helps in platoon identi#ication*

+hen, the platoon ratio &as de#ined &hich helps us in identi#ying the arri)al type* @ater, platoondispersion 'odel &as discussed &hich 'odel the departure pro#ile o# the do&nstrea' )ehicles

based on the upstrea' departure pro#ile* Finally, Robertson"s platoon dispersion 'odel is

discussed &ith the help o# nu'erical e7a'ples* +he Robertson"s platoon dispersion 'odel

esti'ates the do&nstrea' )olu'e at di##erent ti'e inter)als &hich can be used #or the lin%ing o#the signals and opti'ization o# signal ti'ings*

+eferences

1* 8igh&ay Capacity !anual* Transportation Research Board* :ational Research Council,

Bashington, *C*, 2000*

2* < iang, @ ;hou, and 5 aniel*A Platoon-based Traffic Signal Timing Algorithm for

Major-Minor Intersection Types* +ransportation Research Part A 40, 2006*

* $ !anar and D > Aaass* Traffic lo! Theory and Traffic flo! sim"lation models*

+ransportation Research Record= 166, 1996*

4* F Eiao, 8

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* 8 Ra%ha and ! Farzaneh* #alibration of TRA$S%T Traffic Dispersion Model: Iss"es and

Proposed Sol"tions* Virginia +ech +ransportation Institute, 2004*

6* R 8 ;ho&ers*In&estigation and 'nhancement of models that describe the flo! of traffic

on arterial streets* $ Ph** +hesis sub'itted to the /ni)ersity o# Florida,4,,9, 2002*

* B Bey*Model form"lation and sol"tion algorithm of traffic signal control in an "rban

net!or(* Co'puters, 5n)iron'ent and /rban ;yste's, 24, 2000*

Ac8no'ledgments

I &ish to than% 'y student !r* Chetan Du'ar and !s* D ;ra)ya #or their assistance inde)eloping the lecture note, and 'y sta## !r* Rayan in typesetting the 'aterials* I also &ish tothan% se)eral o# 'y students and sta## o# :P+5@ #or their contribution in this lecture*

Prof) Tom *) Mathe! +,.-,+-/