MULTICOMMODITY FLOWS ON ROAD NETWORKS M HERTY C KIRCHNER S MOUTARI† AND M RASCLE†

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MULTICOMMODITY FLOWS ON ROAD NETWORKS M. HERTY?, C. KIRCHNER?, S. MOUTARI† , AND M. RASCLE† Abstract. In this paper, we discuss the multicommodity flow for vehicular traffic on road net- works. To model the traffic, we use the “Aw-Rascle” multiclass macroscopic model [3]. We describe a solution to the Riemann problem at junctions with a criterion of maximization of the total flux, taking into account the destination path of the vehicles. At such a junction, the actual distribution depends on the demands and the supplies on the incoming and outgoing roads, respectively. Further- more, this new distribution scheme captures efficiently key merging characteristics of the traffic and contrary to [15] leads to an easy computational model to solve approximately the homogenization problem described in [15], [16]. Furthermore, we deduce the equivalent distribution scheme for the LWR multiclass model in [9] and we compare the results with those obtained with the “Aw-Rascle” multiclass model for the same initial conditions. Key words. Aw–Rascle model, multicommodity flow models, traffic networks subject classifications. 35LXX, 35L6 1. Introduction A typical vehicular traffic system consists of the vehicle- driver pairs and the infrastructures, i.e., a collection of highways systems and all their operational elements. In a vehicular traffic system, a number of trips - defined by their destination path, the travel route, etc. - interact on the road network and generate various dynamics and phenomena.

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MULTICOMMODITYFLOWSONROADNETWORKSM.HERTY∗,C.KIRCHNER∗,S.MOUTARI†,ANDM.RASCLE†Abstract.Inthispaper,wediscussthemulticommodityﬂowforvehiculartraﬃconroadnet-works.Tomodelthetraﬃc,weusethe“Aw-Rascle”multiclassmacroscopicmodel[3].WedescribeasolutiontotheRiemannproblematjunctionswithacriterionofmaximizationofthetotalﬂux,takingintoaccountthedestinationpathofthevehicles.Atsuchajunction,theactualdistributiondependsonthedemandsandthesuppliesontheincomingandoutgoingroads,respectively.Further-more,thisnewdistributionschemecaptureseﬃcientlykeymergingcharacteristicsofthetraﬃcandcontraryto[15]leadstoaneasycomputationalmodeltosolveapproximatelythehomogenizationproblemdescribedin[15],[16].Furthermore,wededucetheequivalentdistributionschemefortheLWRmulticlassmodelin[9]andwecomparetheresultswiththoseobtainedwiththe“Aw-Rascle”multiclassmodelforthesameinitialconditions.Keywords.Aw–Rasclemodel,multicommodityﬂowmodels,traﬃcnetworkssubjectclassiﬁcations.35LXX,35L61.IntroductionAtypicalvehiculartraﬃcsystemconsistsofthevehicle-driverpairsandtheinfrastructures,i.e.,acollectionofhighwayssystemsandalltheiroperationalelements.Inavehiculartraﬃcsystem,anumberoftrips-deﬁnedbytheirdestinationpath,thetravelroute,etc.-interactontheroadnetworkandgeneratevariousdynamicsandphenomena.Tostudythesetraﬃcphenomenaandthecorrespondingapplications,manytraﬃcmodels[8,9,14,18,20,21,24,29]andsimulationpackages[19]havebeensuggestedintheliterature.However,mostofthemodelsofmulticommoditytraﬃcsystemshavebeenmadeundertheframeworkoftheLWRmodel[25,9].Inthisstudy,weareparticularlyinterestedintheimpactofthedestinationpathofthevehiclesontraﬃcdynamicsatroadsintersectionswhenmodellingwithaclassof“secondorder”modelsoftraﬃcﬂow.Inthepresentwork,weconsideramulticlassmacroscopicmodel[3]derivedfromthe“Aw-Rascle”(AR)secondordermodeloftraﬃcﬂowandweproposeadeﬁnitionofthesolutiontotheas-sociatedRiemannproblematajunction.Thissolutionisbasedonthemaximizationofthemassﬂuxatthejunctionandtheconservationofthepseudo-momentum(seebelow).Thevehicleswiththesamedestinationareconsideredtobelongtothesamecommodity.EachcommodityisattachedtoeachvehicleandisthereforeaLagrangianvariablethatwemustkeeptrackofinthewholeﬂowandinparticularthroughjunc-tions(seealso[9]foranotherdeﬁnitionofcommodity).Givensomeinitialboundaryconditions,wemakeuseofthesupplyanddemandmethods[6,23]tocomputetheﬂuxesthroughajunctionandthenwesolvetheassociatedRiemannproblem.Wethenusehomogenizationtechniques[16,15]toobtaincouplingconditionsforthesec-ondmomentumoftheAw–Rascleequation.Otherapproachesconservingthesecondmomentarerecentlyintroducedin[12,27].2.PreliminariesWemodelaroadnetworkasaﬁnite,directedgraphG=(E,N)wherewedenotebyEandNthesetofarcsandverticesofthegraphG,respectively.WelabeleacharcinEbye=1,...,Eandassumethateacharccorrespondstoaroad.Similarly,eachvertexinNlabeledn=1,...,Ncorrespondstojunction.Foranarbitrarygiven∗FachbereichMathematik,TUKaiserslautern,D-67653Kaiserslautern,Germany.herty@rhrk.uni-kl.de,kirchner@mathematik.uni-kl.de†LaboratoireJ.A.Dieudonne´,Universite´deNice-SophiaAntipolis,ParcValrose,06108NiceCedex2,France.{salissou,rascle}@math.unice.fr1

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