3MCMBM.docx

3MCMBM.docx

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3MCMBM

Introduction/Summary

Wesetupamodel,inwhichtheoptimalnumberofthetollboothsinthetollplazacanbecalculatedundertheconditionthatlanenumberisgiven.

Firstofall,wedefined'theoptimal'asthemaximalnumberofvehiclespassingthroughwithinadefiniteperiodoftime,whichcanbeattained,i.e.,theoutflowrateofatollplaza.Consideringdrivers'complaint,weaddedaconstraintconditionthatthetotalwaitingtimeofavehicleshouldbelessthan5minuteswhentrafficcomesstable.

Secondly,undertheextremeconditionassumingthatthenumberoftollboothsequalsthatoflanes,wegraduallyincreasedthenumberoftollboothsusingcomputersimulationandfinallyobtainedtheoptimal.Wehavediscoveredthatwhentrafficisblockedup,theformercongestionwillgraduallydecreaseandthelattercongestionwillgraduallyincreasealongwiththeincreaseoftollbooths，whilethelattercongestionwillaggravatetheformercongestion,whichisconsistentwithfact.Theoptimalnumberoftollboothswillemergeinthephasewhenthelattercongestionaggravatestheformercongestion,fromthereon,theincreaseofthetollboothsshouldmakethewholehighwaysectionblockedmoreheavily.

Therearequiteafewfactorswhichcanaffecttheoptimalnumber,amongwhichwetaketwoasrelativelymajor:

chargingtimeatthetollboothandthespeedatwhichvehiclesleaveout-poolbackforhighway.Theresultofsensitivityanalysisisasbelow:

fluctuationofchargingtimeatthetollboothaffectstargetlittle,whilethatofthelatteraffectsmuch.

Finally,usingourmodel,weanalyzedtherealhighwaysection'svehicleflowrate（of4lanes）andgot7tollbooths,whichisthesameasthereality.

Content

Content2

ProblemRestatement3

TermsandDefinition3

Assumptions4

AboutInterstate-954

Abouttheweather4

Aboutthedriver4

Aboutthevehicles4

Aboutthetollbooths5

Aboutthein-pool&out-pool5

SomeFactsDerivedfromAssumptions5

AnalysisandModelDesign6

Overview6

Analysis6

Modeling7

ModelResultAnalysis10

SingleLaneCase11

Multilanecase13

Special（n-n）case14

Theanswertothequestionraisedintheproblem16

ModelVerification17

DataAcquisition17

Simulation19

ModelSensitivity20

SensitivityofminLeavingIntervalInSingleLane20

Sensitivityofservingtimeoftollbooths21

FurtherWork23

ModelStrengthandWeakness23

References24

TheOptimalNumberofTollbooths

ProblemRestatement

Somebusytollroadsaremulti-lanehighwayswithsometollplazasscatteredallover.Sincenooneiswillingtoexperiencecontinuoustrafficdisruption,itisdesirabletolimittheamountoftimespentontollplazas.

Commonlyatollplazaismadeoftollbooths,amuchlargernumberofwhichislocatedthantheenteringtravellanessoastofitwellintotheroaringtrafficinflow.Vehiclesfanouttotollboothstobeserved,andthentaketroubleinsqueezingoutofthebottleneckatthedepartureoftollplaza,finallygobacktoanormalnumberoflanes.Chancesarethatcongestionhappensinthebottleneckwhentrafficisheavy.Congestionattheentryoftollplazaisalsopossibleaslongastheinflowrateisoverwhelming,thoughthepossibilityisnotsohigh.

Makeamodeltodeterminetheoptimalnumberoftollboothsinabarrier-tollplaza,giventhefreedomtodefinewhattheword“optimal”meansinthecontextofpaper.Payspecialattentiontothesituationwherethenumberoftollboothsisthesameasthatoftravellanesandcomparethatsituationoneffectivenesswithcurrentpractice.

TermsandDefinition

Weassumethetollplazaas[Figure1]:

Figure1:

Thetopviewofatollplaza.

Alistofrelevantvariables,constants,andparametersisin[Table1].

Terms

Definitions

In-pool

Spaceforthevehiclestopassthetollbeforegoingtothetollbooths

Out-pool

Spaceforthevehiclestoleavethetollplazawhenpaidforthetoll

Inflow

Thetrafficflowintothein-pool

Outflow

Thetrafficflowintotheout-pool

Interval

Theword“Interval”canbeeitherseparationintimeorinspace.Herewejustuseitintimedelay,whiletheword“Distance”isusedforspace.

Distance

numOfLanes

ThenumberofLanesinthehighway.

numOfBooths

Thenumberoftollboothsintheplaza.

Theformercongestion

The50%filledstateinin-pool.

Thelattercongestion

The50%filledstateinout-pool.

numOfWaitingInFormer

Thenumberofvehicleswaitingintheformercongestion.

numOfWaitingInLatter

Thenumberofvehicleswaitinginthelattercongestion

minLeavingIntervalInSingleLane

Theminimalintervalbetweentwovehiclesgettingoutofthedepartureinonelane.

Thetotalwaitingtime

ThetotaltimeforonevehiclefromenteringIn-pooltogettingoutofOut-pool.

maxDriverTolerance

Themaximaltotalwaitingtimethatadrivercantolerate.

（n-m）deployment

[abbr.（n-m）]

AtollplazaconfigurationwithnumOfLanes=n&numOfBooths=m

Ci

Thetimeconstantusedintheformercongestion

Co

Thetimeconstantusedinthelattercongestion

α

Thesecondterm’scoefficientoftimedelayinout-pool

η

Thethirdterm’scoefficientoftimedelayinout-pool

U-Vdiagram

Adiagramwithx-axislabeledUandy-axislabeledV

Table1:

TermsandDefinitions

Assumptions

AboutInterstate-95

1.Accordingtoreference[1],theAADT（AnnualAverageDailyTraffic）ofInterstate-95inyear2001is60,414.

2.Accordingtoreference[1],Interstate-95has8lanesand4lanesineachdirection.

Abouttheweather

1.Weassumetheweatherisfineandhasnobadeffectoncharacteristicsofthevehicles.

Aboutthedriver

1.Usually,therangeofreactiontimeofthehumanbeingsisfrom200msto300ms.Weassumeitis250ms.

2.maxDriverTolerance=5min.

Aboutthevehicles

1.Itisassumedthatthearrivalofvehiclesisabitstochastic,butstationaryinthelonghurl.Asusual,wemodelthearrivaltimeofvehiclesasPoissondistributedrandomvariables,withtheintervalsfollowingtheprobabilitydensityfunction:

With1/λtheaveragetrafficflowrate.

2.Vehiclesarehomogeneous.

a）Allvehiclesareinthesamelengthof4.5m.

b）Thedistancebetweentwovehicleswaitingpayingforthetollisthe1.5m.

c）Empirically,ittakesavehicleabout25sec~30sec（27seconaverage）tospeedupfrom0km/hto80km/h.

d）Empirically,thevehiclehastoslide40muntilitcanbrakeataspeedof80km/h

e）Thedistancebetweentwovehiclesinthesamelanegettingoutofthedepartureofthetollplazaisatleast10msoastoavoidcollision.

Aboutthetollbooths

1.Allthetollboothsareindependent.Theyprovidetheserviceforthevehiclesindependently,thustheyhavenoideaofcorrelatingtheservicetimetoavoidthecongestionintheoutpollwhichastheempiricalparametershowsis60s.

Aboutthein-pool&out-pool

1.Empirically,thedistancebetweentheentryoftollplazaandeachtollboothis200m.Sodoeseachtollboothtothedeparture.（shownin[Figure1]）

2.Thecapacityofthein-pool&out-poolisbasedontheirshapes.

3.Ci=16sec.

4.Co=20sec

5.η=5sec.

6.α=10sec.

SomeFactsDerivedfromAssumptions

1.Theaccelerationanddecelerationofavehiclecanbeevaluated:

（mentionedinAssumption:

vehicles.2.c））

（mentionedinAssumption:

vehicles.2.d））

2.minLeavingIntervalInSingleLanecanbeevaluated:

Interval=5sec（mentionedinAssumption:

vehicles.2.e）andaccelerationvalue）

Notes.Ifweassumeavehiclewon’tstarttoleaveuntilitsprecedentisonitswaywith10mdistance,thenaccordingtoaccelerationdata,wecanderive:

3.Thepoolsizeofbothin-poolandout-poolsatisfiesthefollowingfunction:

Note:

Wecanderivetheexpressionsfromincrementanalysis.

IfnumOfBoothsincreasebyone,thepoolwillincreaseby10slots.Ifdoingso,theareaofpoolwillgrowbyonetriangle,asshowninthefollowingfigure.

Howmanyvehiclesthetrianglecanhold?

Thetrianglehasaheightof200meterslongandabottom,abletoholdonevehicle（atollbooththere）.Weassumeeachvehicleis4.5meterslongandsharesafreespaceof5.5meterswithitsneighbors.Bydivision,20vehiclesareabletopileinarectanglewithheightof200metersandwidthof1vehicle,whoseareaistwiceasmuchasthatoftriangle.Thussuchatrianglecanhold10vehicles,whichisjustthecoefficientassociatedwithnumOfBooths.

Ontheotherhand,ifnumOfLanesincreasebyone,theprocessofanalysisisalmostthesameexceptthatanothertollboothhastobeaddedsoastopreventcongestion.ThusthecoefficientassociatedwithnumOfLanesis20.

Figure2:

Illustrationonincrementanalysis

AnalysisandModelDesign

Overview

Ourgoalistosetupamodeltogettoknowthebehavioroftollplazaserving,andthentrytousethatmodeltoworkouttheoptimalnumberoftollbooths,withrespecttothenumberoflanesandthedefinitionof“optimal”weraised.

Analysis

Twoplacesarehotspotsofcongestion,theplacebeforethetollboothswheretherecouldbeaverylongline（“theformercongestion”）andtheplacebeforethedepartureoftollplaza（“thelattercongestion”）.

Theformercongestionisrelatedtotheservingtimeoftollbooths,thesmallertheservingtimeis,thelessprobablewillthiscongestionhappen.Thelattercongestionisrelatedtotheleavingrateofthedepartureandthenumberofvehiclesstayinginthatposition.

Thesetwokindsofcongestionarenotuncorrelated.Oncethelattercongestionhappens,theformeronewillhappensoonerorlaterastheonlydrainofthetrafficflowisblocked.However,theformercongestionwillnothavesucheffects.

Wecanimagethattheformercongestionwillhappenifthearrivalofthevehicleshasexceededthecapacityofthetoll