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

Germany Herty

Description

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.

• trucks having


• sociated riemann problem


• shortest path


• ??n ??


• single commodity


• ??n


• arc e?


• unknown state


• riemann problem

Sujets

• Antipolis
• Shortest path problem
• Riemann problem

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

MULTICOMMODITYFLOWSONROADNETWORKS M.HERTY ∗ ,C.KIRCHNER ∗ ,S.MOUTARI † , AND M.RASCLE † Abstract.Inthispaper,wediscussthemulticommodityflowforvehiculartrafficonroadnet- works.Tomodelthetraffic,weusethe“Aw-Rascle”multiclassmacroscopicmodel[3].Wedescribe asolutiontotheRiemannproblematjunctionswithacriterionofmaximizationofthetotalflux, takingintoaccountthedestinationpathofthevehicles.Atsuchajunction,theactualdistribution dependsonthedemandsandthesuppliesontheincomingandoutgoingroads,respectively.Further- more,thisnewdistributionschemecapturesefficientlykeymergingcharacteristicsofthetrafficand contraryto[15]leadstoaneasycomputationalmodeltosolveapproximatelythehomogenization problemdescribedin[15],[16].Furthermore,wededucetheequivalentdistributionschemeforthe LWRmulticlassmodelin[9]andwecomparetheresultswiththoseobtainedwiththe“Aw-Rascle” multiclassmodelforthesameinitialconditions. Keywords.Aw–Rasclemodel,multicommodityflowmodels,trafficnetworks subjectclassifications.35LXX,35L6 1.IntroductionAtypicalvehiculartrafficsystemconsistsofthevehicle- driverpairsandtheinfrastructures,i.e.,acollectionofhighwayssystemsandall theiroperationalelements.Inavehiculartrafficsystem,anumberoftrips-defined bytheirdestinationpath,thetravelroute,etc.-interactontheroadnetworkand generatevariousdynamicsandphenomena.Tostudythesetrafficphenomenaand thecorrespondingapplications,manytrafficmodels[8,9,14,18,20,21,24,29]and simulationpackages[19]havebeensuggestedintheliterature.However,mostofthe modelsofmulticommoditytrafficsystemshavebeenmadeundertheframeworkof theLWRmodel[25,9].Inthisstudy,weareparticularlyinterestedintheimpactof thedestinationpathofthevehiclesontrafficdynamicsatroadsintersectionswhen modellingwithaclassof“secondorder”modelsoftrafficflow.Inthepresentwork, weconsideramulticlassmacroscopicmodel[3]derivedfromthe“Aw-Rascle”(AR) secondordermodeloftrafficflowandweproposeadefinitionofthesolutiontotheas- sociatedRiemannproblematajunction.Thissolutionisbasedonthemaximization ofthemassfluxatthejunctionandtheconservationofthepseudo-momentum(see below).Thevehicleswiththesamedestinationareconsideredtobelongtothesame commodity.EachcommodityisattachedtoeachvehicleandisthereforeaLagrangian variablethatwemustkeeptrackofinthewholeflowandinparticularthroughjunc- tions(seealso[9]foranotherdefinitionofcommodity).Givensomeinitialboundary conditions,wemakeuseofthesupplyanddemandmethods[6,23]tocomputethe fluxesthroughajunctionandthenwesolvetheassociatedRiemannproblem.We thenusehomogenizationtechniques[16,15]toobtaincouplingconditionsforthesec- ondmomentumoftheAw–Rascleequation.Otherapproachesconservingthesecond momentarerecentlyintroducedin[12,27]. 2.Preliminaries Wemodelaroadnetworkasafinite,directedgraphG=(E,N)wherewedenoteby EandNthesetofarcsandverticesofthegraphG,respectively.Welabeleach arcinEbye=1,...,Eandassumethateacharccorrespondstoaroad.Similarly, eachvertexinNlabeledn=1,...,Ncorrespondstojunction.Foranarbitrarygiven ∗ FachbereichMathematik,TUKaiserslautern,D-67653Kaiserslautern,Germany. [e-mail],[e-mail] † LaboratoireJ.A.Dieudonne´,Universite´deNice-SophiaAntipolis,ParcValrose,06108Nice Cedex2,France.{salissou,rascle}@math.unice.fr 1