智慧型运输系统概论2.pdf

该【智慧型运输系统概论2 】是由【小屁孩】上传分享，文档一共【43】页，该文档可以免费在线阅读，需要了解更多关于【智慧型运输系统概论2 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..TrafficFlowAnalysis(2)-LenChang,,CollegePark1:..TimeHeadwayDistributionDistance1st2nd3rdGapTimeTOccupancy?Givenatimehorizon:T?nheadways,thedistributionofonditions?Givenafixedtimeinterval?t(),thenumberofT=k??tarrivalsduringeach?tisadistribution2:..?ounts?Numberofvehiclesperinterval?1st:T(interval)=n-1vehicles()?T=10secs?2nd:T(interval)=n2vehicles()rd?3:T(interval)=n3vehicles()?……?K-th:T(interval)=nkvehicles()?Distributionisreferredtoarrivingvehiclesperinterval:n1,n2,…,nk3:..Ifk=180intervals(10secondsperinterval)=111/180=>300180111Poissondistribution:onditionsIfTis30secs,Then:m31=×mComputeper30seconds4:..DistributionsforTrafficAnalysis?PoissonDistribution:()=Totalobservationsoftime2σ/m=1x?mP(x)=m?e/x!x=0,1,2,…m=λ??t?t:selectedtimeinterval??mP(0)=exm?exp(?m)P(x)x!m=x?1=P(x?1)m?exp(?m)x(x?1)!m∴P(x)=?P(x?1)x?Severalpoissondistributionswithmeanvalues:m1,m2,m3,…NThenm=m∑ii=1Limitations:onlyfordiscreterandomevents5:..(0)×1809716363P(1)×(2)×(3)×>360??xxePxm()=?=111/180=mx!TheprobabilityofhavingXvehiclesarriving?0eatthecountinglineduringtheintervalof10seconds0Pm(0)=?0!6:..?Theprobabilitiesthat0,1,2carsarriveateachT(10secs)intervalCanbeexpressedas:im?2xim?memePn(2)≤=Pnx()≤=∑∑i=0i!i!i=0ForthecaseofXormoreim?yim?x?1memePnx()1≥=?Pxiy()≤≤=∑∑i!ix=i!i=07:..PoissonArrival?urringinatimeintervalofis:t(=n??t)k?λt(λt)?k=0,1,2,…eP[M(t)=k]=k!?Theprobabilitythatthereareatleastknumberofvehiclesarrivingduringintervaltis:∞k?λt(λt)?eP[M(t)≥k]=∑k'=kk!∴onditions8:..PoissonArrivalInter-arrivalTimes=headwayLetLk=urrenceofthek-tharrival,k=1,2,3,…ThepdffLk(x)dx≡P[ursintheintervalxtox+dx]=P[exactlyk-1arrivalsintheinterval[0,x]andexactlyonearrivalin[x,x+dx]]1LkthkTimek?1?λx?λ?dxk?1?λxkk?1?λx?(λx)?e??(λ?dx)?e?(λx)?e?λ?dxλ?x?e=?=?[λ?dx?e]=?????(k?1)!??1!?(k?1)!(k?1)!=fL(x)dxk9:..PoissonArrivalkk?1?λxλ?x?e∴f(x)=,x≥0;k=1,2,3,…Lk(k?1)!?thekth-orderinterarrivaltimedistributionforapoissonprocessisakth-orderErlangpdfsetk=1(headway)?λxx≥f(x)=λ?e0(negativeexponentialdistribution)L1Theprobability=P(h≥x)(.)∞?λx?λx=∫λ?e?dx=ex10:..PoissonArrivalFromaPoissonperspective:If“Novehiclearrivesduringthetimelengthx”≡atimeheadway≥x0(sameasthepreviouscase)(λx)?λx?λxPx[M=0]=?e=e0!Note:=λx?Headwayisacontinuousdistribution:P(h≥x)=em?x(λx)?e?Arrivalrateisadiscretedistribution:Px(M=m)=m!11:..DistributionsforTrafficAnalysis?BinomialDistributionvariance?Forcongestedtrafficflow---<<1meannxn?xP(x)=cxP(1?p)x=0,1,2,…,n?PistheprobabilitythatonecararrivesMeanvalue:m=np2Variance:s=np(1?p)Pisunknownfromthefield,butcanbeestimatedfromthemeanandvarianceofobservedvehiclesperinterval12:..Binominaldistribution?putedfromthefielddata()2^()ms?p=m2^mn==mp/2()ms?13:..NegativeBinominalDistribution?ountswithhighvariance–extendoverbothapeakperiodandanoff-peakperiod(.,ashortcountingintervalfortrafficoveracycle,ordownstreamfromatrafficsignalx+k?1kkP(x)=ck?1Pqx=0,1,2,…2mmk?=p?=22q?=(1?p?)ss?mkp(0)=px+k?1p(x)=?q?p(x?1)x14:..Summary?ThePoissondistributionrepresentstherandomoccurrenceofdiscreteevents.?Poissondistributionfitstheeventsofmeanequaltovariance,especiallyunderlighttraffic.?Binomialdistributioncanbefittedtocongestedonditionswherethevariance/meanratiosubstantiallylessthanone.?:..ContinuousDistributionsInterval(betweenarrivingvehicles)Distribution→NegativeExponentialDistributionLetV:hourlyvolume,λ=V/3600(cars/sec)?Vt/3600V?txeP(x)=()?∴3600x!∴?Vt/3600P(0)=eIfthereisnovehiclearriveinaparticularintervaloflengtht,therewillbeaheadwayofatleasttsec.∴P(0)=theprobabilityofaheadway≥tsec∴16:..?Vt/3600P(h≥t)=eH=headwayMeanheadwayT=3600/V?t/TP(h≥t)=e?t/TP(h<t)=1?eVarianceofheadways=T217:..NegativeexponentialfrequencycurveBarindicateobserveddatatakenonsamplesizeof60918:..haracteristics19:..Dashedcurveappliesonlytoprobabilityscale20:..ShiftedExponentialDistribution?(t?τ)/(T?τ)P(h≥t)=e?(t?τ)/(T?τ)P(h<t)=1?eP(t)=0,att<τ()1andP(t)=exp[?(t?τ)/(T?τ)]T?τ21:..Shiftedexponentialdistributiontorepresenttheprobabilityofheadwayslessthentwithaprohibitionofheadwayslessthanτ.(AverageofobservedheadwaysisT)22:..Exampleoffhiftedexponentialfittedtofreewaydata23:..ErlangDistributionk?1?kt/TktieP(h≥t)=∑()t=0Ti!fork=1→Reducedtotheexponentialdistribution?kt??kt/Tfork=2→P(h≥t)=1+e???T??ktkt21??kt/Tfork=3→P(h≥t)=1+()+()e???TT2!?k:aparameterdeterminingtheshapeofthedistribution~222k=T/ST:meaninterval,S:variance*k=1,thedataappeartoberandom*kincrease,thedegreeofnonrandomnessappearstoincrease24:..CompositeHeadwayModel??t1??t?τ?P(h<t)=(1?α)?1?exp()?+α?1?exp(?)??T1??T2?τ??Constrainedflows()?Unconstrainedfreeflows()25:..SelectionofHeadwayDistribution?GeneralizedPoissondistribution(DenseTraffic)k(x+i)?1j?λλeP(x)=∑x=0,1,2,…j=k?xj!k?λxk+i?1e(λ)orP(x)=∑x=0,1,2,…i=1(xk+i?1)!λ=km+1/2(k?1)?λ?λk=2,P(0)=e+λ?e2?λ3?λλeλeP(1)=+2!3!2?λ?λ?λλek=3,P(0)=e+λ?e+2!3?λ4?λ5?λλeλeλeP(1)=++3!4!5!26:..27:..DistributionModelsforSpeeds?Normaldistributionsofspeeds?Lognormalmodelofspeeds?eptancedistributionmodel28:..Cumulative(normal)distributionsofspeedsoffourlocations29:..Samedataasabovefigurebutwitheachdistributionnormalized30:..Lognormalplotoffreewayspotspeeds31:..Comparisonofobservedandtheoreticaldistributionsofrejectedgaps32:..Lagandgapdistributionforthroughmovements33:..eptedandrejectedlagsandgapsatintersectionleftturns34:..PoissonArrival?onditionsplatoonDistancest1TimeT?Twotypesofheadways?betweenandwithinplatoonsduringthesameperiodT?T=T1+T2,eachperiodhasadifferentmeanheadwayλ1andλ235:..MultipleIndependentPoissonProcessesTwoPoissonprocesses:λandλ12binedprocess:N(t)=N(t)+N(t)isalsoapoisson12process?λ1x1pdfforλ?λ?ex≥0(time-period)111?λ2x2pdfforλ?λ?ex≥0(time-period)222Thetwoareindependent:Whatistheprobabilitythatanarrivalformprocess1(type1arrival)occursbeforeanarrivalfromprocess2(type2arrival)?36:..MultipleIndependentPoissonProcessesxandxarebothrandomvariables12∞∞P[x1<x2]=fxx(x1,x2)?dx1?dx2∫0∫x112x1≥0,x2≥0?λ1x1?λ2x2fx,x(x1,x2)=fx(x1)?fx(x2)=λ1λ2?de1212∞∞∞?λ1x1?λ2x2?λ1x1?λ2x2∴P[x<x]=dx1dx2?λ1λ2?e?e=dx1?λ1?e(e)12∫0∫x∫01λ∞λ1?u1=edu=∫0λ1+λ2λ1+λ2λ2Similarly,P[x2<x1]=λ1+λ237:..MultipleIndependentPoissonProcessesxandxarebothrandomvariables12∞∞P[x1<x2]=fxx(x1,x2)?dx1?dx2∫0∫x112x1≥0,x2≥0?λ1x1?λ2x2fx,x(x1,x2)=fx(x1)?fx(x2)=λ1λ2?de1212∞∞∞?λ1x1?λ2x2?λ1x1?λ2x2∴P[x<x]=dx1dx2?λ1λ2?e?e=dx1?λ1?e(e)12∫0∫x∫01λ∞λ1?u1=edu=∫0λ1+λ2λ1+λ2λ2Similarly,P[x2<x1]=λ1+λ238:..MultipleIndependentPoissonProcessesFortheentireprocess:T=T1+T2(λ1andλ2)Theprobabilityofatime-headwayX>xis??TotalnumberofarrivalsduringTperiod=T1λ1+T2λ2?P(X>x)duringT1periodandT2period?λ1x?λ2x=andee?Totalarrivalshavingtheirheadways>xTotalnumberofarrivals?λ1x?λ2xT1λ1?e+T2λ2?e=(weightedaverage)T1λ1+T2λ2k?λix∑Tiλi?eGeneralization,i=1λ:arrivalrateP[X>x]=k∑Tiλii=139:..ConstrainedFlow-Platoon?Headwaywithinaplatoonareexponentiallydistributedwithameanarrivalrateλandminimumheadwayz0?1,forx<z0(shiftedexponentialdistribution)P[X>x]=?λ'(z?z)?0?e,forz≥z0?Therelationbetweenλandλ’?Theexpectedvalueoftheshifteddistributionmustbeequaltotheactualmeanheadway40:..ConstrainedFlow-Platoon?Headwaywithinaplatoonareexponentiallydistributedwithameanarrivalrateλandminimumheadwayz0?1,forx<z0(shiftedexponentialdistribution)P[X>x]=?λ'(z?z)?0?e,forz≥z0?Therelationbetweenλandλ’?Theexpectedvalueoftheshifteddistributionmustbeequaltotheactualmeanheadway41:..ConstrainedFlow-Platoon?Thearrivalrateforsuchashifteddistributionλ’1λ'=wherez=1/λz?z0λ∴λ'=1?z0λ∴λ’?cannotbeobservedλ?actuallyobservedλ?()(z?z0)1?z0λ∴P[X>x]=e42:..SomeTravelFree,SomeAreinPlatoon?Combinationoftwopoissonprocesses:P[X>x]=P[X>x|occursintravelfreetraffic]+P[X>x|inplatoontraffic]=P1+P2?λ1xT1λ1?eP1=Totalnumberofarrivals(=T1λ1+T2λ2)λ2?()(z?z0)1?z0λ2T2λ2?eP2=Tλ+Tλ1122?T1:totalobservedperiodduringwhichtrafficisnotmovedinplatoon?T2:totalobservedperiodduringwhichvehiclesaremovedinplatoon43