A Cellular Automata Model for Urban Traffic and it

AcellularautomatamodelforurbantracanditsapplicationtothecityofGenevaBastienChopard,AlexandreDupuis,andPascalLuthiUniversityofGeneva,CUI,24rueGeneral-Dufour,CH-1211Geneva4,SwitzerlandAbstract.Wepresentasimplecellularautomatamodeloftracinurbanenvironmentsinwhichroadjunctionsareimplementedasrotaries.Westudyitsgenericpropertiesandcomparethenumericalsimulationswithanalyticalresults.Weshowthatthedynamicsiswelldescribedintermsofthequeuesthatformatthecrossings.WeapplyourmodeltothecaseofthecityofGenevaandstudythetimeneededforatestcartojointwopointsinthenetwork.Weintroducetheconceptofrisktodescribetripswithalargetimeuctuation.Finallyweshowthattheowdiagramexhibitsapronouncedhysteresis.1IntroductionSimulationofroadtraciscurrentlyreceivingagreatdealofattention.Ononehand,themanagementoftracinurbanenvironmentiscrucialfortrans-portationpolicyandroadplanning.Ontheotherhand,thecollectivebehaviorofalargenumberofvehiclesmutuallyinteractingisquiteinterestingfromthephysicalpointofview.Cellularautomata(CA),inwhichthecardynamicsissimpliedtokeeponlytheessentialfeaturesareverypowerfulmodelstodescribetracsystems.Whereasalargebodyofworkhasbeendevotedtothestudyofhighways(seeforinstancetheseproceedingsandref.[1]),thesituationofaroadnetwork(typicallyaManhattan-likecity)hasnotbeeninvestigatedindetailfromthepointofviewofitsphysicalbehavior.Two-dimensionalCAtracmodelssuchasthosedescribedinrefs.[2,3]consideraratherunrealisticsituationinwhichnostreetexits;carscanturnatanysiteofa2Dgrid,andinteractinaprimitivewayinregardstowhathappensatrealroadjunctions.InthispaperweconsidertheCAruleofref.[4]todealwithroadintersections.Ourapproachistorepresentanyjunctionasarotary.Thisisagenericwaytorepresentanytypeofcrossings,withanynumberofbranchesandmakesthenumericalimplementationquitesimpleandsuitableforaparallelmachine.ArotarycanbethoughtofasaCAringallowingseveralone-dimensionalCAstobeinterconnected.Althoughreal-lifejunctionsmayhavemanydierentstructures,weexpectthatrotariescapturethefundamentaleectofacrossing,namelytoallowseveraltraclanestomergeandsplitagain,withagivenmaximalow.Thus,whenstudyingtheentiretracsystem(overascalecomprisingalargenumberofjunctions),theprecisetopologyofacrossingisnotexpectedtoplayacrucialroleontheglobalbehavior.Thisassumptionisconrmedbythesimulationsweperformed[5,6]withatwo-dimensional\followtheleadermodel(similartothePARAMICS[7]simulator),showingthatthedetailsofthedynamicsareoftenirrelevantatthelevelofthewholestreetnetwork.Notethatourrotaryparadigmcanbeeasilycombinedwithtraclights.Itisenoughtopreventthecarsfromenteringtherotaryduringsomespecictimeintervals.Similarly,otherprioritymodelscanbeimplemented.Insections2to4westudythephysicalpropertiesoftraconaManhattan-likestreetnetwork.Weproposeananalyticaldescriptionwhichprovidesafairlygoodpictureofthephenomena.Theconceptofcarqueuesappearsasthenaturalwaytodescribethesystem.Numericalsimulationsshowthatlargeuctuationsofthequeueslengthmustbetakenintoaccountandproduceeectsthatshowupintheowdiagram.Insections5to7,weshowthatourmodelcanbesatisfactorilyappliedtoarealcitysuchasGenevaanditssuburbs.Duetothelackoftracmeasurements,wefocusoursimulationontwoaspects:(i)thetimeneededbyadrivertoreachitsdestinationdependingonthetimeitleavesand(ii)theeectontheowdiagramduetoanon-stationarytracloadduringtherushhour.2ThecellularautomatamodelWestartwiththesimplerCAmodelinwhichcarscanhavetwopossiblespeeds:0or1.Multispeedmodels[8]arecrucialwhenmodelingafreewaybutmaybeunnecessarilyrichtosimplycapturetheformationofaqueueatajunction[5].Thebasicideaistoconsiderasetofadjacentcellsrepresentingthestreetalongwhichacarcanmove.Theruleofmotionisthatacarjumpstoitsnearestneighborcell,unlessthiscellisalreadyoccupiedbyanothercar.Inthiscase,thecarstaysatrest.Accordingly,themotioncanbeexpressedbyni(t+1)=nini(t)(1ni(t))+ni(t)nouti(t)(1)whereni(t)isthecaroccupationnumber(ni=0meansafreesite,ni=1meansthatavehicleispresentatsitei).Thequantitynini(t)representsthestateofthecellfromwhichacarcouldcomeandnouti(t)indicatesthestateofthedestinationcell.Rule(1)meansthatthenextstateofcelliis1ifacariscurrentlypresentandthenextcellisoccupied,orifnocariscurrentlypresentandacarisarriving.Inordertodiscussfurthertracrule(1),wehavetospecifyhowniniandnoutiaredened.Forasimpleone-dimensionalperiodicroad,wesimplyhavenini=ni1nouti=ni+1(2)andthemicrodynamicsreducesexactlytorule184ofWolfram,usedbyseveralauthorstomodel1Dtrac(e.g[9]).Foratwo-dimensionalroadnetwork,thesequantitiescanbegeneralizedeas-ilyprovidedthatintersectionsarerepresentedasarotarybecause,then,priorityisdealtwithinasimpleway(avehicleinarotaryhasthepriorityoveranyen-teringcar).Arotaryisaringofcellsonwhichseverallanesareattachedto,asshowningure1.Foracellinsidetherotary,niniis,withpriority,thestateofthepreviouscellontherotaryor,otherwise,thestateofthelastcellofanenteringlane(ifsuchalaneisenteringatthisrotaryposition).Forthelastcellofanenteringlane,nouticorrespondstothestateoftheconnectingcellintherotaryorthestateofitspredecessor.Similarly,noutiandninicanbedenedforarot