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2014MCMOutstandingWinners賈冬TemContr2014MCMOutstandingWinners賈冬TemControlur謝延ForofficeuseForofficeuseF1 校苑數(shù)25142全球首T1 T2 T3 T4 F3 F4 AMathematicalContestinModeling(MCM/ICM)SummaryFreewayTrafficModelBasedonCellularAutomataandMonte-CarloMethodBasedonCellularAutomataandMonte-Carlomethod,webuildamodeltodiscusstheinfluenceofthe“Keeprightexcepttopass”rule.Firstwebreakdowntheprocessofvehiclemovementandestablishcorrespondingsub-models,inflowmodelforcar-generation,vehicle-followingmodelforonevehiclefollowinganother,andovertakingmodelforvehiclepassingThenwedesignrulestosimulatethemovementofvehiclesinsub-models.Wefurtherdiscussrulesforourmodeltoadapttothekeep-rightsituation,theunrestrictedsituation,andthesituationwheretransportationiscontrolledbyintelligentsystem.Wealsodesignaformulatoevaluatethedangerindexoftheroad.Wesimulatethetrafficontwo-lanefreeway(twolanesperdirection,fourlanesintotal),andthree-lanefreeway(threelanesperdirection,sixlanesintotal)viacomputerandanalyzethedata.Werecordtheaveragevelocity,overtakingrate,roaddensityanddangerindexandassesstheperformanceofthekeep-rightrulebycomparisonwithunrestrictedrule.Wevarytheupperspeedlimitationstoanalyzethesensitivityofmodelandseetheimpactsofdifferentupperspeedlimits.Left-handtrafficisalsoBasedonouranalysis,wecomeupwithanewrulecombiningthetwoexisting(thekeep-rightruleandtheunrestrictedrule)foranintelligentsystemtoachievebetter2014MCMOutstandingWinnersTeam#Page2of1Introduction......2014MCMOutstandingWinnersTeam#Page2of1Introduction.................................455Terminology..............................Assumptions..............................2TheModels.................................6678DesignofCellular....................Inflow .............................Vehicle-FollowingModel.......................OvertakingModel...........................OvertakingProbability....................OvertakingCondition....................DangerIndex.........................TwoSetsofRulesforCAModel...................KeepRightExcepttoPassRule...............UnrestrictedRule.......................3SupplementaryAnalysisontheModel.................DesignoftheAccelerationandDecelerationProbabilityDistribu-tions...................................DesigntoAvoidCollision......................4ModelImplementationwithComputer.................5DataAnalysisandModelValidation.................. ...........................2014MCMOutstandingWinnersTeam#Page3of..2014MCMOutstandingWinnersTeam#Page3of...Density.................................OvertakingRate............................Danger .............................6SensitivityEvaluationoftheModelunderDifferentSpeed7DrivingontheLeft.............................8TransportationunderIntelligentSystem................NewRuleforIntelligentSystem...................AdaptionoftheModel........................ResultofIntelligentSystem.....................9Conclusions.................................Strengthsand ........................10.1Strengths................................10.2Weakness................................References....................................Appendices...................................2014MCMOutstanding2014MCMOutstandingWinnersPage4ofTeam# Today,about65%oftheworld’spopulationliveincountrieswithright-handtrafficand35%incountrieswithleft-handtraffic.[worldstandards.eu,2013]Incountrieswithright-handtraffic,likeUSAandChina,regulationsrequestdriv-ingandwalkingkeeptotherightsideoftheroad.Multi-lanefreewaysinthesecountriesoftenemployarulethatrequiresdriverstodriveintheright-mostlaneunlesstheyarepassinganothervehicle,inwhichcasetheymoveonelanetotheleft,pass,andreturntotheirformertravellane.Thisruleondrivingandovertakingisreferredtoasthe”Keeprightexcepttopass”rule,orthekeep-rightruleinourpaper.Theruleincountrieswithleft-handtrafficisexactlymirrorsymmetricaltothekeep-rightrule(”Keepleftexcepttopass”).So,what’sthepurposeofapplyingsucharule?Doesthekeep-rightruleamelioratethefreewaytrafficcondition?Transportationfreeoftherestrictionofthekeep-rightrule(Vehiclescanchooseeithersideforovertaking.)isreferredtoasobeyingtheunrestrictedrule.Howdoesthekeep-rightruleperformcomparingwiththeunrestrictedrule?BasedontheCellularAutomatamodelandtheMonteCarloalgorithm,weestablishamodeltosimulatefreewaytrafficunderdifferentconditions(underthekeep-rightruleortheunrestrictedrule,inlighttrafficorinheavytraffic,2-laneor3-laneperdirection).Ourmodelisdividedinto3sub-modelsthein-flowmodel,thevehicle-followingmodelandtheovertakingmodel.TheinflowmodelemploysthePoissonprobabilitydistributionforthesimulationofthevehicle-generationprocess.Thevehicle-followingmodelintroducesaspecialprobabilitydistributionmodelwhichmakesthesimulationoftheprocessofacarfollowinganothermorerealistic.Theovertakingmodelsimulatestheover-takingbehavioranddefinesthedangerindextoevaluatethesafetyriskofacertainfreeway.Wealsobuildanextendedmodelfortransportationunderthecontrolofanintelligentsystem.WeimplementthemodelinMATLAB,andobtainsufficientdata.Wetesttheaveragevelocity,thedensity,theovertakingrateandthedangerindex,analyzetheirpropertiesandassesstheperformanceofthekeep-rightrulebycomparisonwiththeunrestrictedrule.Inaddition,weanalyzethesensitivityofourmodelunderdifferentspeedlimits.Itturnsoutthatourmodelisrobust.Thenwecometoourconclusionswhichconsistwithcommonsense.Wealsoputforwardanewrulefortransportationunderthecontrolofanintelligent2014MCMOutstandingWinnersPage5ofTeam#2014MCMOutstandingWinnersPage5ofTeam# Table1: VcurrentvelocityofthevehiclemaximumvelocityofthevehicletheupperspeedlimitofthefreewaythevelocitybeforeovertakingthevelocityintheovertakingthedistancebetweenavehicleandthevehicleaheadofittheminimumgaprequiredforsafetyconsiderationtheminimumgapafterthevehiclestopsPIEVtime(humanreactiontime)theovertakingprobabilitytheaccelerationprobabilitythedecelerationthefrictionalforcewhenbrakingdangerindexinoneovertakingeventdangerindexoftheroadsystemtheaccelerationduringthecomponentparalleltothelaneofaccelerationduringovertakingtheavailabledecelerationTwo-laneroad:Twolanesonright-halfoftheroad,fourlanesinThree-laneroad:Threelanesonright-halfoftheroad,sixlanesinDangerindex:Anindexdesignedinourpapertoevaluatethedangeroftheroadsystem.Minimumsafetygap:Thedistancebetweentwovehiclesthatdeemedsafeenoughinourmodel.Keep-rightrule:KeeprightexcepttopassUnrestrictedrule:Vehiclesarenotrestrictedandcanovertakeothersfromeitherside.Free-drivingstyle:Whenthereisnovehiclesnearby,driverswillnotaccel-erateordeceleratedeliberately,butthespeedwillstillfluctuateslightly.???????Theroadisstraightandthereisno?2014MCMOutstandingWinnersTeam#2014MCMOutstandingWinnersTeam#Page6ofThewidthofonelaneisonlyenoughforonevehicle.Allvehicleshavethesamevolume.Thereareonlytwokindsofvehiclesontheroad(fastoneandslowone).Theenvironmentandclimatearegoodfordriving.DrivingontherightisthePedestriansare?????? TheDesignofCellularLargequantitiesofformertrafficsimulations[WagnerPetal.2005]basedonCellularAutomata(CA)indicatethatCAmodelisafeasibleandeffectivemethodtoemulatetrafficflow.Space,timeandstatusarealldiscreteinCellu-larAutomata.Forexample,themodeldividestheroadintosmallrectangularcells,andtimeisdividedintosmallunits.Thisfeaturepredigeststhesimulationprocesssignificantly.Besides,thestatusofacelliscontrolledbyitsneighboringcellsfollowingasetofrules,whichismuchsimilartoreal-lifetrafficwhereacar’smovementlargelydependsonitsneighboringcars’movement.Therefore,itisrationalforustoapplyCellularAutomatainsolvingourproblem.Inoursimulation,wedivideeachlaneinto1000cells.Eachcellis4meterslonginlengthandwidthandhastwoproperties,thecurrentvelocityVandthemaximumvelocityVm.AcellisemptywhenVis0,becauseacarwon’tstopinourcrash-freesimulation.Weconsideronlyonedirectionofthefreewayforsimplicity.Thus,afreewayofnlanesisconvertedintoann*1000matrix.Inoursimulation,weemploytwokindsofvehicles,fasteronestosimulatethecarsandsloweronestosimulatethetrucks.Foreachlane,thefirst6cellsareusedascar-generationarea,trafficflowisobservedinthelast10cellsandtrafficdensityiscalculatedonthebasisofthelast500cells.Ourmodelupdatesoncepersecond,whiletheperiodT=1sistheaveragereactiontimeofadriver.WediscussthebasicprocessesfortheCAInflowProcess:Accordingtotheinflowmodelthatwewilldiscussassignvehiclesinthevehicle-generation2014MCMOutstandingWinners上海交通大學(xué)賈冬雨，史昭陽Pof2014MCMOutstandingWinners上海交通大學(xué)賈冬雨，史昭陽PofTeam#<ac??speedisV0=V+?VDecelerationProcess:Ifthedistancebetweenavehicleandthevehicleaheadofit(Frontbumpertofrontbumperdistance,wecallitthegap,andthegapisdenotedbyGanditsunitiscell.Whenthereisnovehicleahead,Gissetto+∞.)isnomorethanV,thevehicledeceleratesV0=(G?1)/T?MovingProcess:VehiclesmoveforwardbyV?TcellsonlywhenG0G(V).(G(V)istheminimumgaprequiredforsafety0ssandistobedefinedSpecificruleswillbesetintheinflowmodel,thefollowingmodelandtheovertakingmodeltosimulatetrafficundertheKeep-Right-Except-To-Passruleandtrafficundernosuchrestriction.InflowTheinflowmodel,orthevehicle-generationmodel,simulatesthestochasticarrivalsofvehiclesattheentranceofthefreeway.Foreachlane,thefirstsixcellsinthecellularautomataaresetasvehicle-generationarea.Weassumethatthearrivalofeachvehicleobeysthebinomialprobabilitydistribution.LettsdenotethesamplingtimeintervalandNdenotethetotalofvehiclearrivalsduringts.ThenNcanbeapproximatedtoobeythePoissonprobabilitydistribution.LetPts(N)betheprobabilityofNandwehaveλNe?λN>Pts(N)NWithtsbeingonesecondinourimplementation,wecanassigntheexpecta-tionofNtoarangeofvaluesfrom0to3.6.Nbeingthetotalofvehiclearrivalsineachsecond,theexpectationofN,caneffectivelyreflectthetrafficcondition.Thesmallertheλis,thelighterthetrafficis;thegreater,theheavier.Thusweareabletosimulatedifferenttrafficconditions,lightorheavy,byassigningcor-respondingvaluestoλ.Afterthevalueofλisset,wegetthestochasticnumberofvehiclesenteringthefreewayforeverysecondinsimulation.Whichlanetoenteristhenrandomlyassigned.Ourmodelsupportstwokindsofvehiclesofdifferentvelocityranges,theinitialspeedofallvehiclesaresetto20m/s.Suchpracticebringssimplificationanddoesn’tweakentheresult.2014MCMOutstandingWinnersTeam#Page8of2014MCMOutstandingWinnersTeam#Page8ofThatisbecausethespeedofallvehiclestendstoconvergetowardavaluecontrolledbythetrafficdensityandbythedistributionofaccelerationproba-bilitywhichistobeintroducedlater.Whentrafficdensityislow,vehiclescanalwaysacceleratefreelytothemaximumspeedwithoutworryingaboutcolli-sions,sotheconvergencespeedisnearthehighestspeedallowed.Whentrafficdensityishigh,allthelaneswillbefilledwithvehicles,andthespeedofthetrafficflowisdecidedbythespeedoftheslowestvehicleonthelane,sothecon-vergencespeedisnearthelowerspeedlimit.Thepreliminaryanalysisontheconvergencespeedwillbejustifiedbylaterimplementationofthemodel.elclosetorealityandpractical.Becauseoftheconvergencetendency,thesameinitialspeedpolicycanyieldsimplificationwithoutharmtothesimulation.Vehicle-FollowingTheFederalHighwayAdministrationoftheUnitedStatesDepartmentofdriver’sreactiontimeasPIEVtime.PIEVtimeconsistsoffourparts:????Perceptionprocess:AdriverperceivesthechangeindrivingIntellectionprocess:ThedriveranalyzestheinformationabouttheEvaluationprocess:ThedriverdeterminesdrivingbehaviorbasedonVolitionprocess:ThedriverexecutesthedrivingWeapplythePIEVprocessinourvehicle-followingmodelandovertakingmodel.Ineverytimecycle,wefirstobtainthevelocityandpositionofeachvehicle,calculatethegap,andthendeterminethedrivingbehavior(whethertocontinuefollowingortochangelaneforovertaking).Accordingtodrivingbehavior,computetheaccelerationandupdatethespeedandposition.Decisionondrivingbehaviorbasedprimarilyonthecurrentgap.IftheGissafeenough,accelerationisfeasible;otherwise,thevehicleshoulddown.Here,wedefinetheminimumsafegapGtobeT?V.(TstandsforsrrPIEVtime,andVisthecurrentWeassumethatdecisionsondrivingbehaviorfollowcertain2014MCMOutstandingWinnersTeam#Page9ofWhenG>Gs,2014MCMOutstandingWinnersTeam#Page9ofWhenG>Gs,thevehicletendstoaccelerate(Laterwewillintroduceprobabilitymodeltosimulatethetendency.),untilachievingthespeedlimitoritsmaximumpossibleWhenG<Gs,whethertoovertakeortofollowisdeterminedbythetakingprobabilityPoandtheovertakingconditions(PoandconditionsaretobediscussedintheovertakingWhenfollowing,avehiclecanaccelerate,decelerateorkeeptheoriginalspeed.Weintroducetwoparameters[SUNyue2005],theaccelerationproba-bilityPaanddecelerationprobabilityPb.Thehigherthespeed,thesmallerthePaandthegreaterthePb.Vlrepresentsthehigherspeedlimitofthefreeway,andVmaxisthemaximumspeedthevehiclecanreach.Thisprobabilitymodeltakesintoaccountthefactthatspeedingcan’tbeignored.WhenV>Vl,PagetsevensmallerandPbgetsevenbigger,whichmakesitpossibleforspeedingbutthepossibilityissmall.WeuseastochasticvariableRforimplementation:???IfR<Pb,thevehicleIfR>1?Pa,thevehicleOtherwise,thevehiclemaintainsthecurrentBasedontheprobabilitymodel,wecreateseveralrulesfortheCellulartomata:(Themaximumpossiblespeedofvehiclei,thecurrentgapG,theminimumsafegapisGsanditsvelocityisdenotedasV,PaandPbarefunctionsofvelocityV,andPa+Pb≤1.)Freedrivingrule:IfG≥min(Vmax,V+ V0probabilityPb;,V??Safedecelerationrule:IfG<Gsbutwon’tcrashifmovingV0=max(Vmin,V?Vminisunder-postedthelowerspeed?Crash-freerule:Ifmovingforwardcausescrash,stopbehindtheThevaluesofPaandPbarelistedintable2forfastcars,table3forslowOutstandinPaef4Team#:OutstandinPaef4Team#:foF 6數(shù) 34V/cell·?1100Table3:AccelerationandDecelerationProbabilityforSlow 56V/cell· 0 OvertakingThedriverwilldecidewethertoovertakeanothercarwiththeprobabilityPo.ProbabilityPodependsonthevehicleAandthevehicleBahead.LetVmax1bethevelocityofvehicleA,andVmax2bethevelocityofvehicleB.TheprobabilityPosatisfies:Vmax2<,,1?0.9·Po Vmax2≥max maxItisrationaltoassumethatthelargerthedifferenceinvelocity,themorelike-lythevehicleistoaccelerate.TheprobabilitydistributionreflectsthetendencyAdrivercan’tovertakeashepleases.Overtakingcanbesometimesdanger-ous,andavehicleshouldtobeabletoovertakesuccessfully,i.e.abletoreturntotheformerlane,undertheKeeprightexcepttopassrule.Thus,restrictionsaremadeonovertaking.ConditionofthegapG0onthetargetlaneislargerthanGs,thevelocityofthevehicleislargerthanthatofthevehicle2014MCMOutstandingWinnersTeam#2014MCMOutstandingWinnersTeam#Page11ofDangerHereweredefinetheminimumsafegapGsusingadifferentmethodtocal-culatethedangerindex.ThetheoreticalrelationshipbetweenGsandthecurrentvelocityVis:12mV2=f(Gs?fisthefrictionalforcewhenbraking;G0istheminimumgapafterthevehicleConsideringnormaldrivingspeedisunder200km/hinrealityanddrivers’acceptancegapisusuallylargerthanthetheoreticalsafevalue,andforthesakeofsimplicityofcomputerimplementation,weapproximateGs,asafunctionofpresentspeedV,tobelineartoV.Figure1:Relationshipbetweensafe-gapandWesetG0tobe10meters,anduse0.7asthefrictioncoefficient.ThelinearrelationshipwegetisGs=10+2.8×Whenchanginglanesforovertaking,avehiclesparepartofitsaccelerationtochangethedirectionandtherestaccelerationistocopewiththe2014MCMOutstandingWinnersTeam#Page12ofFigure2:Velocitywhen2014MCMOutstandingWinnersTeam#Page12ofFigure2:Velocitywhenfacingemergency.Sotheassessmentofsafetyforlane-changingshouldbedifferentfromthatforvehicle-following.Asisshowninthefigure2,V0denotesthevelocitybeforeovertaking,denotesthevelocityintheovertakingprocess,andaistheaccelerationovertaking.Empirically,V1=V?4m/s(decelerationduringlane-changing0cedureforsafetyconcern).Vehiclesfinishlane-changingin1s,andWecalculatethecomponentparalleltothelaneofa:ap=V0?V1·cos(arcsin(V1Then,thedecelerationavailable 2a=6.86?a=6.86?(V?2Vdp00ThevaluesoftheavailabledecelerationadchangesslightlyasV0varies,sowesetadto5.76m/s2forsimplicity.Gschangesto10+3.4Vaccordingly.Wecreateafunctiontoassessthedangercoefficientofavehicleinunit0 ifGs?Gr<dGs?,ifGs?Gr≥WhenG≥Gs,weassumethedangerissmallenoughtoneglect,sothercoefficientissetto0.WhenGr<Gs,weusethedifferenceofGsandGrtostateofthevehicletobemoredangerous.Thedangercoefficientsofvehicle-followingaresimilarinvariousroadandruleconditions,soweconsideronlythedangercoefficientsofovertakinginfurtherdiscussion.2014MCMOutstandingWinnersTeam#Page13ofNow2014MCMOutstandingWinnersTeam#Page13ofNowwedefinethedangerindextoindicatetheriskofacertainroadunderacertainrule.LetDbethesumofdangercoefficientsofovertakingeventshappenedin300stime:D A·1,whenpassingfrom3,whenpassingfromAAccordingtoresearches,ifaleft-handdrivevehicle(thevehiclecontrolslo-catedonthelefthandside)triestopassfromtherightside,driver’ssightwillgetrestrictedandthusincreasingthedangerindex.Weassumethatthedangerindexofpassingfromtherightsideisthreetimeshigherthanthatofpassingfromtheleftside.SowesetAto1whenpassingfromtheleft,andsetAto3whenfromtheright.ThedangerindexDweintroducedhereisthebasistoevaluatethesafetyoftheroadinourmodel. TwoSetsofRulesforCAKeepRightExcepttoPassWeanalyzetheperformanceoftheKeepRightexcepttoPassrulebycom-paringtheoutcomesfromthemodelunderthisruleandthemodelwithoutsuchrestriction.ApplyingthisruledemandssomerulesfortheCellularAutomata:(Therulesarelistedindescendingpriority,thatis,ifthefirstruleissatisfied,thefollowingonesaretobeneglected.)???IfthegapGintherightlaneislargerthanGs,changetotherightIfthecurrentgapGislargerthanGs,applythefreedrivingrulesetinvehicle-followingIfthegapGintheleftlaneislargerthanGs,applytheovertakingmodelwithprobabilityPo,andapplythefollowingmodelwithprobability1?Po.2014MCMOutstandingWinnersTeam#Page14of2014MCMOutstandingWinnersTeam#Page14ofLikewise,whenweimplementthemodelwithoutsuchrestriction,anothersetofrulesisneeded:(Therulesarelistedindescendingpriority,thatis,ifthefirstruleissatisfied,thefollowingonesaretobeneglected.)IfthecurrentgapGislargerthanGs,followfreedrivingruleemployedinvehicle-followingmodel;Iftheovertakingconditionsaresatisfied,andthegapGintheleftlaneislargerthanGs,applytheovertakingmodeltopassfromtheleftsidewith??probabilityP,andapplythevehicle-followingmodelwithprobability1oIftheovertakingconditionsaresatisfied,andthegapGintherightlaneislargerthanGs,applytheovertakingmodeltopassfromtherightsidewith?probabilityP,andapplythevehicle-followingmodelwithprobability1o SupplementaryAnalysisontheDesignoftheAccelerationandDecelerationProbabilityDis-Thedesignoftheaccelerationanddecelerationprobabilitydistributionsweintroducedinthevehicle-followingmodelsimulatesthechangeinvelocitydur-ingthedrivingprocess.Thesystemcanself-adjusttheaveragevelocityaccord-ingtothedensity.Whenthedensityissmall,theaveragevelocityofthetrafficflowisclosetotheaveragespeedofafree-drivingcarwhichfollowstheprob-abilitydistribution.Whenthedensityislarge,slowcarsontheroaddeceleratethefollowingcars.Inotherwords,slowercarsinthelanedeterminetheaveragevelocity.Whenthefreewayisrelativelycrowded,theexpectationoftheslowestspeeddecreasesandthusaffectstheaveragevelocityoftheroad.DesigntoAvoidWhilesimulatingheavytraffic,wedesignrulestoavoidcarcrashes.Normal-ly,afreewaylimitsthelowerspeed,butwhenavehiclefindsitselftooclosetotheoneahead,itcanbraketoavoidcollisionregardlessofthelowerspeedlimit.2014MCMOutstandingWinners2014MCMOutstandingWinnersTeam#Page15ofWhenthefreewayiscrowded,thefrequencyofdecelerationtoavoidcollisionrises,sotheaveragevelocitygoesbelowthelowerlimitofspeed. ModelImplementationwithBasedontheCellularAutomatamodelandMonteCarloalgorithm,weim-plementourmodelsuccessfullyviaMATLAB.Startingfromasimplesituation,wefirstsimulateafreewayof2lanesundertheKeep-rightrule.Thenwithasmallchangetotherules,wehavetheunrestricted2-lane-freewaymodelforcomparison.Weworkontoextendthemodeltosimulateafreewayof3lanesandbothruleconditionsarerealized.Moreover,trafficunderakeep-leftrule,withdifferentspeedlimitsaswellastransportationdirectedbyanintelligentsystemisalsoemulated.Wetestthesemodelswithdifferentinflowratesinor-dertoseetheinfluenceoftrafficheaviness.Supportedbysufficientsimulateddata,weareabletoevaluateaccuratelytheperformanceoftheKeeprightex-cepttopassruleinlightandheavytraffic,includingthetradeoffsbetweenflowandsafety,theaveragespeed,thetrafficdensityandtheovertakingfrequency.Andwefurtherdiscusstheimpactofakeep-leftruleandthatofanintelligentsystem.Figurespatiotemporaldiagramofvehicles,expectationofinflowrate0.5veh/s,theratioofsmallercarstolargervehiclesis1,three-lane.Thedi-agramrecordspositionsofallthevehiclesineverytimecycle.Redrepresentssmallercarsandgreenrepresentslargervehicles.Everythreecolumnsstandforthefreewaystateofonetimecycle.Figure3:Tempo-SpatialDistributionof2014MCMOutstandingWinnersTeam#2014MCMOutstandingWinnersTeam#Page16of DataAnalysisandModelAveragelocityofthevehicleflowtoreflectthetrafficefficiency.Weanalyzethestatisticsfromthetwo-lanemodelandthethree-lanemodel,bothunderthekeep-rightruleandunrestricted.Therelationshipsbetweentheaveragespeedandthein-flowrateinvariousconditionsareshowninthefigure4andfigure5.Figure4:AverageVelocityunderDifferentFigure5:AverageVelocityunderRules(2Rules(3lanesIt’sclearthatinthetwo-lanemodel,thekeep-rightruleyieldsfasteraveragevelocityingeneral.Whenitcomestothefreewayofthreelanes(bystatistics)ormore(bydeduction),thekeep-rightruledoesn’timprovetheaveragevelocity.Wecanseethatwhenthevehicle-generationrateisover0.75veh/s,theunre-strictedruleoutperformthekeep-rightruleaccordingtothefigure.Highinflowratemaytriggertrafficjam,aswecanlearnfromthefigure.Whentheinflowrateishigherthan1.8veh/s,theaveragevelocityinbothmod-elsgoesbelowthelowerspeedlimitofthefreeway.Iftheinterferenceofothervehiclesisneglected(thatis,avehicledrivesonanemptyfreewayunderthefree-drivingrule),theaveragespeed,orwecallittheidealspeed,ofslowvehiclesis19.44m/s,andtheidealspeedoffastcarsis25.88m/s.(ThedatacomefromourMATLABsimulation.)Wecanseefromfigure4and5thatwhentheinflowrateislow,thekeep-rightrulecanalmostreachtheidealspeed,buttheunrestrictedrulegivesaworseperformance.Wecanconcludefromtheanalysisthatonathree-lanefreeway,theKeeprightexcepttopassrulepromotestheaveragevelocityofvehicleflowinlighttrafficbutmakesnoimprovementontrafficefficiencyinheavytraffic.However2014MCMOutstandingWinnersTeam#Page172014MCMOutstandingWinnersTeam#Page17ofonatwo-lanefreeway,theKeeprightexcepttopassrulepromotestheaveragevelocityofvehicleflowprominently.AverageVelocityofFastWecalculatedtheaveragevelocityoffastercarsinthethree-lanemodel.Wemainlyfocusonthefastercarstostudythattowhatextentthefastercarisblockedbyslowervehicles.ThegeneraltendencyisgoingdownforthefollowingLargevehicles(slower)mayblocktheroadresultinginthelimitedofsmallThemorecrowdedthefreewayis,themorewilltheaveragespeedbefectedbytheslowestThevelocitygoesupwhentheinflowrateisrelativelylow.Thatisbecauseatthebeginning,theinflowrateissolowthatcarshavealmostnocompanionstoovertake,whichmakesthemmoveinthefree-drivingstyle.Asinflowrategoeshighinthelowerrange(0-0.5veh/s),carshavemorechancetoovertakesothat2014MCMOutstandingiP2014MCMOutstandingiPofTeam#eveesctirnsiefFigure7:DensityofTrafficinEachLanes(2Figure8:DensityofTrafficinEachlanes,Figure9:DensityofTrafficinEachLanes(3Figure10:DensityofTrafficinEachlanes,thatthekeep-rightrulecausestheunbalanceduseoftheroad,whichmayresultindifferentdegreesofwearinthelanesinreality.Sotherepairfordifferentlanescanbestaggeredwhichreducestheharmduetosuspensionofthelaneunderrepair.2014MCMOutstandingWinnersTeam#2014MCMOutstandingWinnersTeam#Page19ofOvertakingmodelinfiveminutes.Figure11:PassingEventsWithDifferentRules(3lanes,Undertheunrestrictedrule,overtakingfromtheleftandfromtherightareequivalentinpriority,sotheratesonbothsidesareapproximatelythesame.Underthekeep-rightrule,mostvehiclesdriveintherightlaneifpossible,andvacatetheleftlane,thusmakingtheovertakingrequirementseasiertomeetandcausingthesumofpassingeventstobemuchmorethanthatoftheunre-strictedleftlane.Highpassingratekeepsfastercarsfromthelimitproducedbyslowervehicles,whichincarnatesthequicknessofthekeep-rightrule.Besides,toomanypassingeventsontherightlaneharmthesafetygreatly.ThesedataisveryimportantforustoevaluatethedangerindexoftheroadDangerInlighttr