交通灯英文论文

Eur.Phys.J.B67,133–138(2009)DOI:10.1140/epjb/e2009-00010-2RegularArticleTHEEUROPEANPHYSICALJOURNALBTASEPrelatedmodelswithtrafficlightboundaryT.NeumannaGermanAerospaceCenter(DLR),InstituteofTransportationSystems,Rutherfordstr.2,12489Berlin,GermanyReceived2October2008Publishedonline13January2009cEDPSciences,Societ`aItalianadiFisica,Springer-Verlag2009Abstract.Cellularautomatamodelsplayanimportantroleintrafficmodeling.ForsomevariantsoftheNagel-Schreckenbergmodel,theeffectsoftrafficlightboundaryconditionsareconsidered.Basedonpreviousresults,theexactdensityprofilescanbederivedeasilyfordeterministicdynamics.Additionally,theexactaverageoutflowpertrafficlightcycleispresentednotonlyinthedeterministiccase,butalsoforanimportantsemi-stochasticvariantwithslow-to-startbehaviour.Thereby,themodelsarestronglyrelatedtothewell-knowntotallyasymmetricsimpleexclusionprocess(TASEP)whichcanberegardedasagenericmodelformanydrivenparticlesystems.PACS.45.70.VnGranularmodelsofcomplexsystems;trafficflow–45.70.-nGranularsystems–89.40.-aTransportation–05.60.-kTransportprocesses1IntroductionFormanyyears,trafficflowmodelingmostlyfocusedonfreewaytraffic.Manymodelsweredevelopedwhereatthecellularautomataapproachwithitsmostfamousrep-resentative,theNagel-Schreckenberg(NaSch)model[1],playsamajorrole.Sincethesemodelsoftencanalsobeusedtodescribeurbantraffic[2],meanwhilethereisanincreasinginterestintheoreticalresultsabouttheirbe-haviourwhentrafficlightsareintroduced.Forexample,HuangandHuang[3,4]analyzedtheim-pactofdifferenttrafficlightoffsetsintheframeworkoftheNaSchmodelwithperiodicboundaries.Furthermore,afterthedensityprofilesforthetotallyasymmetricsim-pleexclusionprocess1(TASEP)withopenboundariesandconstantinflowandoutflowrateshavebeendescribed[5],Popkovetal.[6]nowconsideredthesamemodelwithran-domsequentialupdatewheretheoutflowisperiodicallyblocked,i.e.withtrafficlightboundaryconditions.Indo-ingso,theyfoundacharacteristicsawtoothstructureforthetrafficlightqueue.However,inthecontextofcellulartrafficflowmod-elsmoreoftenparallelupdateproceduresareused[7]asitisusualespeciallyfortheNaSchmodel.Furthermore,theTASEPonlyallowsfortwodifferentspeedlevels(0and1)whileintheNaSchmodelthisnumberisafreelyse-lectableparameter.So,letp∈[0,1]beacertainslowdownae-mail:[e-mail]1Whilethegeneralasymmetricsimpleexclusionprocess(ASEP)allowsmovementsinbothdirectionsonaone-dimen-sionallattice,usingtheTASEPthereisonlyoneflowdirection.Accordingly,theTASEPtypicallyismuchmoreappropriatetotrafficflowmodelingthanthegeneralASEP.probabilityandvmax∈Nthemaximumspeed(cellspertimestep).Then,therulesoftheclassicalNaSchmodel(appliedtoallparticlessimultaneously)aregivenbyvi(t+12)=min{vi(t)+1,xi+1(t)−xi(t)−1,vmax}(1)vi(t+1)=max{vi(t+12)−1,0}withprob.pvi(t+12)withprob.1−p(2)xi(t+1)=xi(t)+vi(t+1)(3)wherexi(t)andvi(t)arethepositionandthespeedoftheithparticle(vehicle)attimet.Inotherwords,vehiclesmoveonadiscretizedsingle-laneroadwheretheycanhavedifferentspeedsrangingfrom0tovmaxonadiscretescale.Asiswellknown[8],theTASEPwithparallelupdateisrecoveredwhenvmax=1.AnothergeneralizationoftheNaSchmodelisgivenwhensomeslow-to-startbehaviourisintroduced[9].Thisisdonebyanadditionalrulewhichisappliedtoallparti-clespriortotherules(1)–(3):p:=p0p1ifvi(t)=0vi(t)0(4)wherep0∈[0,1]istheslow-to-startprobabilityforstand-ingvehicles,andp1∈[0,1]istheslowdownprobabilityfordrivingvehicles.Whenp0=p1,theclassicalNaSchmodelisrecovered.Inthefollowing,thefocuswillbeonthecaseofde-terministicdrivingbehaviourwherep1=0.So,p0=0yieldsthedeterministicNaSchmodel,andp00resultsinasemi-stochasticvariantwithdeterministicdriving,butstochasticslow-to-startbehaviour.134TheEuropeanPhysicalJournalBFurthermore,thegeneralsetupisgivenbyadiscretizedsingle-laneroadwithfinitelengthandopenboundarieswheretheinflowisdrivenstochasticallywithaveragein-flowQ,andtheexitisperiodicallyblockedtomimictheeffectoftrafficlights.Inthisprocess,thegreen(red)phaselastsg(r)timestepswhereg,r∈N.Accordingly,thecycletimeisgivenbyc=g+rtimesteps.2OutflowThenumbers(g;p0)ofvehicleswhichcanleavethecon-sideredroadsectionduringasingletrafficlightcycle,i.e.duringasinglegreenphase,typicallydeterminestheca-pacityoftheroad.So,itisacrucialvaluewhichvitallydependsontheusedmodelparameters.Letp0=0,thens(g):=s(g;0)isgivenby(see[10])s(g)=max{n∈N0|d(n,vmax)≤g}(5)whered(n,vmax)isthetimeneededthatnvehiclespassthetrafficlightsoutofacompacttrafficlightqueueundertheassumptionofaninfinitegreentime.Furthermore[10],thereisevenanexactclosedformulafors(g):s(g)=5+2g2−2g+174(6a)ifgg∗,ands(g)=5+2g∗2−2g∗+174+vmax(g−g∗+1)vmax+1(6b)ifg≥g∗,whereg∗:=vmax(vmax+1)/2−1.Similarto(5),s(g;p0)inthemoregeneralcasep0≥0canbewrittenass(g;p0)=max{n∈N0|d(n,vmax;p0)≤g}(7)whered(n,vmax;p0)isthecorrespondingstochasticver-sionofd(n,vmax).Bysimplecombinatoricalarguments,itisshownthattheprobabilitydistributionofd(n,vmax;p0)isdefinedasfollows2:Pd(n,vmax;p0)d(n,vmax) =0(8a)Pd(n,vmax;p0)=d(n,vmax)+k =n+k−1kpk0(1−p0)n(8b)2Thesimplecasep0=1canbedisregarded.Inthissit-uation,onejustgetsd(n,vmax;1)=∞foralln∈Nandd(0,vmax;1)=0.Accordingly,s(g,1)=0holdsforallg∈N.