Traffic Flow Theory

arXiv:physics/0507126v1[physics.soc-ph]15Jul2005TrafficFlowTheorySvenMaerivoet∗andBartDeMoorDepartmentofElectricalEngineeringESAT-SCD(SISTA)†,KatholiekeUniversiteitLeuvenKasteelparkArenberg10,3001Leuven,Belgium(Dated:February2,2008)Thescientificfieldoftrafficengineeringencompassesarichsetofmathematicaltechniques,aswellasre-searcherswithentirelydifferentbackgrounds.Thispaperprovidesanoverviewofwhatiscurrentlythestate-of-the-artwithrespecttotrafficflowtheory.Startingwithabriefhistory,weintroducethemicroscopicandmacroscopiccharacteristicsofvehiculartrafficflows.Movingon,wereviewsomeperformanceindicatorsthatallowustoassessthequalityoftrafficoperations.Afinalpartofthispaperdiscussessomeoftherelationsbetweentrafficflowcharacteristics,i.e.,thefundamentaldiagrams,andshedssomelightonthedifferentpointsofviewadoptedbythetrafficengineeringcommunity.PACSnumbers:02.50.-r,45.70.Vn,89.40.-aKeywords:xxxContentsI.Abriefhistoryoftrafficflowtheory2II.Microscopictrafficflowcharacteristics3A.Vehiclerelatedvariables3B.Trafficflowcharacteristics3III.Macroscopictrafficflowcharacteristics4A.Density51.Mathematicalformulation52.Passengercarunits6B.Flow61.Mathematicalformulation62.Obliquecumulativeplots7C.Occupancy8D.Meanspeed91.Mathematicalformulation92.Fundamentalrelationoftrafficflowtheory10E.Movingobservermethodandfloatingcardata11IV.Performanceindicators12A.Peakhourfactor12B.Traveltimesandtheirreliability121.Traveltimedefinitions122.Queueingdelays133.Anexampleoftraveltimeestimationusingcumulativeplots134.Reliabilityandrobustnessproperties14C.Levelofservice14D.Efficiency15V.Fundamentaldiagrams16A.Trafficflowregimes161.Free-flowtraffic162.Capacity-flowtraffic163.Congested,stop-and-go,andjammedtraffic16†Phone:[phone]Fax:[phone]URL:∗Electronicaddress:[e-mail]ficflowcharacteristics171.Thehistoricoriginofthefundamentaldiagram172.Thegeneralshapeofafundamentaldiagram183.Empiricalmeasurements21C.Capacitydropandthehysteresisphenomenon22D.Kerner’sthree-phasetheory231.Freeflow,synchronisedflow,andwide-movingjam232.Fundamentalhypothesisofthree-phasetraffictheory243.Transitionstowardsawide-movingjam244.Fromdescriptionstosimulations25E.Theoriesoftrafficbreakdown25VI.Conclusions27A.Glossaryofterms271.Acronymsandabbreviations272.Listofsymbols28Acknowledgements30References30Thescientificfieldoftrafficengineeringencompassesarichsetofmathematicaltechniques,aswellasre-searcherswithentirelydifferentbackgrounds.Thispaperprovidesanoverviewofwhatiscurrentlythestate-of-the-artwithrespecttotrafficflowtheory.Startingwithabriefhistory,weintroducethemicroscopicandmacro-scopiccharacteristicsofvehiculartrafficflows.Movingon,wereviewsomeperformanceindicatorsthatallowustoassessthequalityoftrafficoperations.Afinalpartofthispaperdiscussessomeoftherelationsbetweentrafficflowcharacteristics,i.e.,thefundamentaldiagrams,andshedssomelightonthedifferentpointsofviewadoptedbythetrafficengineeringcommunity.Becauseofthelargediversityofthescientificfield(en-gineers,physicists,mathematicians,...alllackaunifiedstandardorconvention),oneoftheprincipalaimsofthispaperistodefinebothalogicalandconsistentterminol-ogyandnotation.Itisourstrongbeliefthatsuchaconsis-tentnotationisanecessitywhenitcomestocreatingor-derinthe‘zooofnotations’thatinouropinioncurrentlyexists.Foraconcisebutcompleteoverviewofallabbre-viationsandnotationsproposedandadoptedthroughoutthispaper,wereferthereadertoappendixA.I.ABRIEFHISTORYOFTRAFFICFLOWTHEORYHistorically,trafficengineeringgotitsrootsasaratherpracticaldiscipline,entailingmostofthetimeacommonsenseofitspractitionerstosolveparticulartrafficprob-lems.However,allthischangedatthedawnofthe1950s,whenthescientificfieldbegantomature,attractingengi-neersfromallsortsoftrades.Mostnotably,JohnGlenWardropinstigatedtheevolvingdisciplinenowknownastrafficflowtheory,bydescribingtrafficflowsusingmath-ematicalandstatisticalideas[115].Duringthishighlyactiveperiod,mathematicsestablisheditselfasasolidbasisfortheoreticalanalyses,aphe-nomenonthatwasentirelynewtotheprevious,more‘rule-of-thumb’oriented,lineofreasoning.Twoex-amplesoftheprogressduringthisdecade,includethefluid-dynamicmodelofMichaelJamesLighthill,Ger-aldBeresfordWhitham,andPaulRichards(ortheLWRmodelforshort)fordescribingtrafficflows[72,102],andthecar-followingexperimentsandtheoriesoftheclubofpeopleworkingatGeneralMotors’researchlaboratory[17,40,41,50].Simultaneousprogresswasalsomadeonthefrontofeconomictheoryappliedtotransportation,mostnotablybythepublicationofthe‘BMWtrio’,Mar-tinJ.Beckmann,CharlesBartlettMcGuire,andChristo-pherB.Winsten[6].Fromthe1960son,thefieldevolvedevenfurtherwiththeadventoftheearlypersonalcomputers(althoughatthattime,theycouldonlybeconsideredasmerecomputingunits).Morecontrol-orientedmethodswerepursuedbyengineers,asameansforalleviatingcongestionattun-nelsandintersections,bye.g.,adaptivelysteeringtraf-ficsignaltimings.Nowadays,thefieldhasbeenkindlyembracedbytheindustry,resultinginwhatiscalledin-telligenttransportationsystems(ITS)