A-continuum-theory-for-the-flow-of-pedestrians

AcontinuumtheoryfortheflowofpedestriansRogerL.Hughes*DepartmentofCivilandEnvironmentalEngineering,UniversityofMelbourne,Parkville,Vic.3010,AustraliaReceived21June1999;receivedinrevisedform8February2001;accepted26February2001AbstractTheequationsofmotiongoverningthetwo-dimensionalflowofpedestriansarederivedforflowsofbothsingleandmultiplepedestriantypes.Tworegimesofflow,ahigh-density(subcritical)andalow-density(supercritical)flowregimes,arepossible,ratherthantwoflowregimesforeachtypeofpedestrian.Asubcriticalflowalwaysfillsthespaceavailable.However,asupercriticalflowmayeitherfillthespaceavailableorbeself-confiningforeachtypeofpedestrian,dependingontheboundarylocation.Although,theequationsgoverningtheseflowsaresimultaneous,time-dependent,non-linear,partialdifferentialequations,remarkablytheymaybemadeconformallymappable.Thesolutionoftheseequationsbecomestrivialinmanysituations.Freestreamlinecalculations,utilizingthisproperty,revealbothupstreamanddownstreamseparationoftheflowofpedestriansaroundanobstacle.Suchanalysistellsmuchaboutthenatureoftheassumptionsusedinvariousmodelsfortheflowofpedestrians.Thepresenttheoryisdesignedforthedevelopmentofgeneraltechniquestounderstandthemotionoflargecrowds.However,itisalsousefulasapredictivetool.ThebehaviorpredictedbytheseequationsofmotioniscomparedwithaerialobservationsfortheJamaratBridgenearMecca,SaudiArabia.Itisshownthat,forthisimportantcase,pedestrians,thatispilgrims,aimatachievingeachimmediategoalinminimumtimeratherthanachievingallgoalsinoverallminimumtime.Typicalofmanyexamples,thiscaseillustratedthestrongdependenceofpathonthepsychologicalstateofthepedestriansinvolved.ItisproposedthattheflowofpedestriansovertheJamaratBridgebeimprovedbyappropriatebarrierplacement,thatforceaneffectiveglobalviewofthegoals.
2002ElsevierScienceLtd.Allrightsreserved.1.IntroductionTheflowoflargecrowdsofpedestriansislikelytobecomeincreasinglyimportantasthepopulationsofourlargecitiesgrow.ManystudiesofpedestrianflowshavebeenundertakenTransportationResearchPartB36(2002)507–535*Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](R.L.Hughes).0191-2615/02/$-seefrontmatter
2002ElsevierScienceLtd.Allrightsreserved.PII:S0191-2615(01)00015-7particularlyoverthepastthreedecades,withparticularlystronginterestinthetopicsincetheearly1990s,testifyingtoitscurrentlyperceivedimportance,(SmithandDickie,1993).Never-theless,asWigan(1993)hasnoted,ourknowledgeoftheflowofcrowdsisinadequateandbehindthatofothertransportmodes.Interestincrowdmodelinghasmanysources.Examplesincludecrowdsassociatedwithtransportsystems(Dalyetal.,1991;Toshiyuki,1993;Smith,1993),sportingandgeneralspectatoroccasions(Bradley,1993),holysites(AlGadhiandMahmassani,1991;SelimandAl-Rabeh,1991),politicaldemonstrations(SurtiandBurke,1971)andfirees-capes(Tanaka,1991).Noattemptismadeheretopresentacompletelistofsituations.However,theabovestudiesillustratethevariedbehaviorassociatedwithdifferentsituations.ThebehaviorofpedestriansvariesnotonlywiththeirphysicalcharacteristicsbutalsowiththeirpurposeasshownbyPolusetal.(1983),andPushkarevandZupan(1975).Therearetwofundamentallydistinctphilosophiesformodelingcrowdmotion.Thefirstphi-losophyinvolvestreatingpedestriansasdiscreteindividualsandwalksthemthroughthedomaingenerallyinacomputersimulation.Pedestriansaremodeledby(a)usingagranularmaterialanalogue(rare),(b)modelingthepathtakenassumingpedestriansoptimizetheirimmediatelocalbehavior,or(c)assumingtheyattempttomovealongpredefinedgloballydeterminedpaths.BothLagrangiansimulation,wherebyindividualsarefollowedthroughthedomain,andEuleriansimulation,wherebyanaccountiskeptofthenumberofindividualsineachgridboxinthedomain,areused.Thesecondphilosophy,applicableonlyinlargecrowds,involvestreatingthecrowdasawhole.Crowdsaretreatedas(a)afluid(nowrare),(b)acontinuumrespondingtolocalinfluences,or(c),asinthisstudy,byassumingindividualsinthecontinuummovesoastooptimizetheirbehaviortoreachnon-localobjectives.Totheauthor’sknowledgewhenusingthesecondphilosophyattentionhasbeenrestrictedtoEulerianmodeling.ThealternativeLagrangiantheorymaybedevelopedquitesimplybyfollowingtheprocedureinLamb(1932).Theformermodelingphilosophyprovidesflexibility.Itisparticularlywellsuitedtobeusedforsmallcrowds.Thelatterphilosophyisprobablybetterinunderstandingtherulesgoverningtheoverallbehavioroftheseflows.However,asamodelingtoolthislatterphilosophyexcelswithextremelylargecrowdsespeciallyinstudyingthoseaspectsofthemotionforwhichindividualdifferencesarenotimportant.Thus,acontinuummodelalleviatesamajorprobleminunder-standingcrowdsasnotedbyGaskellandBenewick(1987),namely,thatacrowdneedstobetreatedasanidentity.Bothphilosophieshavetheirplace.Theyshouldbeseenascomplementingeachother.Theapplicabilityofthevarioustechniquesisalsovaried.Theuseof(granularand)fluidanalogiesisoflimitedvalueexceptatdangerouslyhigh-crowddensities(4m
2)where(com-pressibility)pressurewavescanoccur.Unlikeamaterialsubstance,atlowerdensities,acrowddoesnotacceleratebutwalksatconstantspeedwhensubjectedtoamotivationalforce.Con