智慧型运输系统概论2

TrafficFlowAnalysis(2)StatisticalProperties.Flowratedistributions.Headwaydistributions.SpeeddistributionsbyDr.Gang-LenChang,ProfessorDirectorofTrafficsafetyandOperationsLab.UniversityofMaryland,CollegePark1TimeHeadwayDistributionGivenatimehorizon:T⇒nheadways,thedistributionofsuchheadwaysdependsontrafficconditionsGivenafixedtimeinterval∆t(),thenumberofarrivalsduringeach∆tisadistribution2TimeTDistance1st3rd2ndGapOccupancytkT∆⋅=TrafficcountsNumberofvehiclesperinterval1st:T(interval)=n-1vehicles(e.g.3)T=10secs2nd:T(interval)=n2vehicles(e.g.2)3rd:T(interval)=n3vehicles(e.g.0)……K-th:T(interval)=nkvehicles(e.g.3)Distributionisreferredtoarrivingvehiclesperinterval:n1,n2,…,nk3No.ofVeh/TObservedFrequencyObsVehicles094016363221423263001801114Ifk=180intervals(10secondsperinterval)Ave.vehiclesperinterval=111/180=mPoissondistribution:Lighttrafficconditions1IfTis30secs,Then:m3Computeper30secondsm=×DistributionsforTrafficAnalysisPoissonDistribution:lighttrafficconditionse.g.Severalpoissondistributionswithmeanvalues:m1,m2,m3,…Then5timeofnsobservatioTotalsoccurrenceTotalvalueavem=).(1/2=mσ!/)(xemxPmx−⋅=tm∆⋅=λx=0,1,2,…∆t:selectedtimeintervalmeP−=)0(xmmxmmxmxPxPxx=−⋅−−⋅=−−)exp()!1()exp(!)1()(1)1()(−⋅=xPxmxP∴∑==Niimm1Limitations:onlyfordiscreterandomeventsNo.ofVeh/TObsFrequencyTotalProbTheFre0940P(0)×1809716363P(1)×18059.922142P(2)×18018.5326P(3)×1803.8360?0.8180111180.06Testthedistribution()!xxePxmx−=00(0)0!ePm−=Ave.vehiclesperinterval=111/180=mTheprobabilityofhavingXvehiclesarrivingatthecountinglineduringtheintervalof10secondsTheprobabilitiesthat0,1,2carsarriveateachT(10secs)intervalCanbeexpressedas:70()!imximePnxi−=≤=∑20(2)!imimePni−=≤=∑ForthecaseofXormore10()1!imximePnxi−−=≥=−∑()!imyixmePxiyi−=≤≤=∑PoissonArrivalThenumberofPoissonarrivalsoccurringinatimeintervalofis:k=0,1,2,…Theprobabilitythatthereareatleastknumberofvehiclesarrivingduringintervaltis:∴Poissonisonlyapplicableinlighttrafficconditions8)(tnt∆⋅=!)(])([ketktMPtkλλ−⋅==∑∞=−⋅=≥kktkketktMP'!)(])([λλPoissonArrivalInter-arrivalTimes=headwayLetLk=timeforoccurrenceofthek-tharrival,k=1,2,3,…ThepdffLk(x)dx≡P[ktharrivaloccursintheintervalxtox+dx]=P[exactlyk-1arrivalsintheinterval[0,x]andexactlyonearrivalin[x,x+dx]]9LkTimekth1⋅⋅⋅−⋅=⋅−−−!1)()!1()(1dxxkedxkexλλλλ[])!1()!1()(11−⋅⋅=⋅⋅⋅−⋅=−−⋅−−−kexedxkexxkkdxxkλλλλλλdxxfkL)(=PoissonArrival∴,x≥0;k=1,2,3,…⇒thekth-orderinterarrivaltimedistributionforapoissonprocessisakth-orderErlangpdfsetk=1(headway)x≥0(negativeexponentialdistribution)Theprobability(C.D.F.)10)!1()(1−⋅⋅=−−kexxfxkkLkλλxLexfλλ−⋅=)(1)(xhP≥=xxxedxeλλλ−∞−=⋅⋅=∫PoissonArrivalFromaPoissonperspective:If“Novehiclearrivesduringthetimelengthx”≡atimeheadway≥x(sameasthepreviouscase)Note:Headwayisacontinuousdistribution:Arrivalrateisadiscretedistribution:11xxxeexMPλλλ−−=⋅==!0)(]0[0xexhPλ==≥)(!)()(mexmMPxmx−⋅==λBinomialDistributionForcongestedtrafficflow---PistheprobabilitythatonecararrivesMeanvalue:Variance:12DistributionsforTrafficAnalysis1meanvariancexnxnxpPcxP−−=)1()(npm=)1(2pnps−=x=0,1,2,…,nPisunknownfromthefield,butcanbeestimatedfromthemeanandvarianceofobservedvehiclesperintervalBinominaldistributionmandscanbecomputedfromthefielddata(no.Vehiclesperinterval)132^2^2()/()mspmmmpnms−===−Trafficcountswithhighvariance–extendoverbothapeakperiodandanoff-peakperiod(e.g.,ashortcountingintervalfortrafficoveracycle,ordownstreamfromatrafficsignal14NegativeBinominalDistributionkkkxkqPcxP11)(−+−=2ˆsmp=msmk−=22ˆ)ˆ1(ˆpq−=kpp=)0()1(1)(−⋅⋅−+=xpqxkxxpx=0,1,2,…SummaryThePoissondistributionrepresentstherandomoccurrenceofdiscreteevents.Poissondistributionfitstheeventsofmeanequaltovariance,especiallyunderlighttraffic.Binomialdistributioncanbefittedtocongestedtrafficconditionswherethevariance/meanratiosubstantiallylessthanone.Negativebinomialdistributioncanbefittedtotrafficwherethereisacyclicvariationintheflowandmeanflowischangingduringthecountingperiod.1516ContinuousDistributionsInterval(betweenarrivingvehicles)Distribution→NegativeExponentialDistributionLetV:hourlyvolume,λ=V/3600(cars/sec)∴∴Ifthereisnovehiclearriveinaparticularintervaloflengtht,therewillbeaheadwayofatleasttsec.∴P(0)=theprobabilityofaheadway≥tsec∴!)3600()(3600/xetVxPVtx−⋅∆=3600/)0(VteP−=17H=headwayMeanheadwayT=3600/VTtethP/)(−=≥TtethP/1)(−−=Varianceofheadways=T23600/)(VtethP−=≥18NegativeexponentialfrequencycurveBarindicateobserveddatatakenonsamplesizeof60919Statisticaldistributionsoftrafficcharacteristics20DashedcurveappliesonlytoprobabilityscaleShiftedExponentialDistribution21)/()()(ττ−−−=≥TtethP)/()(1)(ττ−−−−=TtethP,0)(=tP)]/()(exp[1)(τττ−−−−=TtTtPattτ(min.headway)and22Shiftedexponentialdistributiontorepresenttheprobabilityofheadwayslessthentwithaprohibitionofheadwayslessthanτ.(AverageofobservedheadwaysisT)23ExampleoffhiftedexponentialfittedtofreewaydataErlangDistribution24∑−=−=≥10/!)()(ktTktiieTktthPTkteTktthP/1)(−+=≥TkteTktTktthP/2!21)()(1)(−++=≥22/~STk=fork=1→k:aparameterdeterminingtheshapeofthedistributionfork=2→fork=3→ReducedtotheexponentialdistributionT:meaninterval,S