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Trafficoptimizationintransport networks basedon localrouting SalvatoreScellato,1 LuigiFortuna,2,1 Mattia Frasca,2,1 Jesu´sGo´mez-Garden˜es,1,3 andVito Latora4,1 1LaboratoriosuiSistemiComplessi, ScuolaSuperiorediCatania, ViaSanNullo5/i,95123Catania, Italy 2Dipartimento di Ingegneria Elettrica, Elettronica e dei Sistemi, Universita` degli Studi di Catania, viale A. Doria 6, 95125 Catania, Italy 3DepartamentodeMatema´ticaAplicada,ESCET,UniversidadReyJuanCarlos,E-28933Mo´stoles(Madrid),Spain 4DipartimentodiFisicaeAstronomia,Universita` diCatania,andINFN,ViaS.Sofia64,95123Catania,Italy (Dated:January8,2009) Congestionintransportnetworksisatopicoftheoreticalinterestandpracticalimportance. Inthispaperwe studytheflowofvehiclesinurbanstreetnetworks. Inparticular,weuseacellularautomatamodeltosimulate 9 themotionofvehiclesalongstreets,coupledwithacongestion-awareroutingatstreetcrossings. Suchrouting 0 makes use of the knowledge of agents about traffic in nearby roads and allows the vehicles to dynamically 0 2 updatetheroutestowardstheirdestinations.Byimplementingthemodelinrealurbanstreetpatternsofvarious cities,weshowthatitispossibletoachieveaglobaltrafficoptimizationbasedonlocalagentdecisions. n a PACSnumbers:89.20.-a,89.75.-k,89.40.Bb J 8 Introduction.-Traffic optimization has always been a cru- of the system. Therefore, to study congestion-awarerouting ] h cialissueinthecontextofcommunicationandtransportation ofvehiculartrafficintransportnetworksitisnecessarytoin- p systems [1, 2, 3, 4, 5]. A transport network is a network corporatetheabovetwoingredients. - of roads, streets, pipes, power lines, or nearly any structure In this paper, we focus on the realistic scenario in which c o whichpermitseithervehicularmovementortheflowofsome onlyalocalknowledgeofcongestionisavailableandusedby s commodity.Inmostofthedevelopedcountries,transportation theagentstomodifytheirroutes. Ourmodelisimplemented . s infrastructuresoriginallydesignedto carrya definedamount intermsofvehiculartrafficandweassumethatdriversknow c oftrafficare oftencongestedbyanoverwhelmingrequestof the shortest paths to their destinations and, simultaneously, i s resources: this is the case of railroads, airplane connections they are aware about the congestion of nearby roads. Both y and,ofcourse,urbanstreets. Anaivesolutiontotheproblem informations are easily accesible nowadays from navigating h p consistsinexpandingtheinfrastructuretomatchtheincreas- systems, visual inspection and short-range wireless commu- [ ingdemand. However,thisisnotalwayspossibleduetolim- nication with other vehicles [11]. We show how, by conve- itationsinavailablespace,negativeoutcomesorshortnessof nientlycontrollingagentdecisions,itispossibletominimize 1 resources. Abetterapproachistocarefullytunethebehavior theoverallcongestionofthesystemandachieveperformances v 8 oftheexistinginfrastructurestoefficientlyexploittheiractual asgoodasthoseobtainedwithcomplete(global)trafficinfor- 7 structureandaccommodatethenewtrafficdemands. mation. 0 1 The study of congestion in complex networks has been The model.-Thethree ingredientsof the modelare: i) the . mainly focused on information systems, such as grid- substrate graph, ii) the vehicular dynamicsalong the streets, 1 0 computing networks or the Internet [4, 6]. In such context, iii)theroutingatstreetcrossings. 9 diversesolutionshavebeenproposedinordertoincreasethe The dynamics of vehicles takes place on top of urban 0 networkloadavoidingtheonsetofcongestion[7]. Inpartic- graphs. Weconsiderthestreetpatternofacityasaweighted v: ular,congestion-awarerouting,inwhichthenodesofthenet- graphwithN nodesandK edges. Eachedgeofthenetwork i work(therouters)redirectdynamicallytheinformationpack- representsastreet,alongwhichvehiclesmove,whereasnodes X etsacrossthelesscongestedpaths,hasprovedtoimproveno- accountofintersectionsbetweenstreets. Theweightofeach ar tablythecapacityofthenetwork[8,9].Itthusseemspossible edge is proportionalto the length of the road [19]. Here we tomakeuseofsimilarkindsofroutingstrategiesintransport assumeforsimplicitythateachedgeallowsmovementofve- networks. Ontheotherhand,transportnetworkspresentim- hiclesinbothdirections. Wehaveconsideredeightnetworks portantfeaturesdifferentfrominformationsystems. First, in representing1-squaremilesamplesofurbanstreetpatternsof a transport network the links (i.e. the roads) carry the flow real cities [19] (see Table I for details). The above city set ofvehicles,whereasthenodesarejustintersectionsbetween rangesfrom self-organizedcities, such as Bolognaand Lon- links. Therefore,onecannotneglectthedynamicsthatoccurs don,grownthroughacontinuousprocessoutofthecontrolof along the links: the quality of vehicle movement along the anycentralagency,togrid-likecitiessuchasLosAngelesand roadscharacterizesthefunctioningoftransportsystems. An- Barcelona,realizedoverashortperiodoftimeastheresultof other important feature is that, since transport networks are urbanplans,andusuallyexhibitinggridstructures. embeddedin the realspace, congestionisnotlocatedatpar- Thevehiculardynamicsalongthelinksoftheurbangraph ticular nodes of the system (such as the hubs in information is simulated by means of cellular automata [12, 13, 14]. In systems)[10]butitgeographicallyspreadsacrossthenetwork particular we use the Nagel-Schreckenberg (NaSch) model frombottlenecksanditmayeventuallyaffectalargeportion [15]. To this end, every link (street) of the city graph is di- 2 vided into a sequence of cells of equal length (5 meters) so City N K W hli α∗(L=0.06) α∗(L=0.1) thatnomorethanonevehiclecanoccupyacellateverytime Barcelona 210 323 36179 112.01 0.63 2.04 step. Eachvehicleisassignedavelocityvcellspertimestep Bologna 541 773 51219 66.26 0.87 1.54 [18]intherangev ∈[0,vmax].Wesetvmax =3,correspond- Brasilia 179 230 30910 134.39 1.52 2.09 LosAngeles 240 340 38716 113.87 1.82 2.43 ingtothetypicalmaximumvelocityof50Km/hinsideurban London 488 730 52800 72.33 1.02 1.26 areas.FollowingtheNaSchmodel,vehiclesaccelerate(decel- NewDelhi 252 334 32281 96.56 1.49 2.21 erate)whenthenextcellsareempty(occupied).Additionally, NewYork 248 419 36172 86.33 0.74 1.77 theintersectionsbetweenstreets(thenodesofthegraph)also Washington 192 303 36342 119.94 1.33 2.14 allowonlyonevehicleatagiventime,sothatseveralvehicles coming from different adjacent streets may compete for the TABLEI:Citysamplesconsidered: N andK arenumberofnodes same intersection [17]. For this reason, vehicles experience andlinksinthenetwork,W andhliarerespectivelythetotallength a slow down while approachinga roadintersection, as if the ofedgesandtheaverageedgelength(bothinmeters)[19].Wereport thevalueα∗thatmaximizestheaveragenumberofcompletedroutes endofthelanepresentedahindrance,sothattheyarriveatthe pervehicle. lastcelloftheedgewithv =0. Atthispoint,avehiclewaits to enter into the node [16] where it gets stuck until the first celloftheedgeintheproperoutgoingdirectionisfree. This waitinglockstheincomingflowsfromtheedges. Therefore, routing,α,andthenetworkload,L. First,inFig. 1weshow bottlenecksare created from the nodesand spread along the thecongestionpatternacrossthestreetsofBolognaforaload edges(roads)ofthegraph. L= 0.2,andforthreedifferentvaluesα =0,1,2. Itisclear Howvehiclesdecidetheiroutgoingdirectionwhenleaving that the larger the value of α, the more homogeneouslydis- a node? Here we implement a congestion-aware routing as tributedisthetraffic.Inordertostudyandquantifytheeffects a minimizationproblemthattakesinto accountthe lengthof ofcongestioninthevehicularmotionweanalyzetheso-called thepathandalsothetrafficalongtheoutgoingedges. When fundamentaldiagram [15]. The fundamentaldiagramrepre- a vehicle is at a node i it needs to choose a new node n in sents the network mean flux f (the average value of fluxes its neighborhoodΓi as the next hop on its path towards the onstreets)asafunctionofthetrafficloadL.Thisisshownin destinationt. Foreachoftheneighbouringnodesn,apenalty Fig.2(a)fordifferentvaluesofα. Eachofthecurvesf(L)ex- functionPnisdefinedas: hibitsthetypicalλ-inverseshapewiththetwotrafficphases: free-flow(inwhichtheflowincreasesasafunctionoftheload) Pn =(din+dnt)(1+cin)α , n∈Γi (1) atlowL,andcongested-flow(theflowdecreasesasafunction where din is the distance between nodes i and n, and cin oftheload)athighervaluesofL.Interestingly,boththevalue (cin ∈ [0,1]) represents the congestion of the link i → n, ofmaximumflowandthetransitionpointfromthefree-flow measured as the fraction of occupied cells in the link. The regimetothecongested-flowregime,increasewithα. Inpar- exponent α ≥ 0 accounts for the weight given to the local ticular,whenthevehiclesfollowshortestpaths(α = 0),they congestion in the drivers decision. The vehicle chooses the tendtoconcentrateinhighbetweennessnodesandlinks(we nodenwiththeminimumpenaltyPn. Ifcin = 0thepenalty havecheckedthatthestrongestcorrelationbetweenlinkflow functionPnisnothingelsethanthelenghtoftheshortestpath andbetweennessisindeedobtainedforα = 0). Inthiscase, tot,passingbynoden. Whencin 6=0theentireshortestpath even a small density of vehicles can result in a heavy load lengthiscorrectedbythe factor(1+cin)α. Inthisway, we at highbetweennessstreets. This givesrise to the formation assumethatthevehicleprojectsthecongestioncin ofthelink of clusters of jammed vehicles that cannot easily move and, i →nontheentirepathi→n →t. Notethat,whenα =0, although large regions of the city are quite uncongested as localcongestionplaysnoroleintheroutingandvehiclesfol- shown in Fig. 1(a), the overallflux of the networkis largely lowtheshortestpathstotheirrespectivedestinations. reduced. Weinitiallyplaceanumberofvehiclesproportionaltothe Theaboveresultisconfirmedbythesharpdecreaseofthe number of cells in the network, so that the network load, averagevehiclespeedvasafunctionofLforα=0shownin L = VC (i.e. the ratio between the number of vehicles V in Fig.2(b). Ontheotherhand,astheroutingstrategybecomes thenetworkandthetotalnumberofcellsC),isthesamefor more congestion-aware, the traffic is diverted from shortest all the cities considered. Initially, the vehicles are assigned path to free (and longer) paths causing that both the mean arandomsource(theirinitiallocation)andarandomdestina- speed of the vehicles and the mean flux on the streets are tionnode.Ateachtimestep,vehiclesmove(ifpossible)inthe largelyincreasedwith respecttothe caseα = 0, since more systemaccordingtotheNaSchrulesandthecongestion-aware vehiclesarenowabletoreachtheirdestinationswithoutbeing routing,Eq. (1). Finally, whena vehiclereachesitsdestina- blockedin congestednodes. More interestingly, we observe tionitisrandomlyassignedtoanewdestinationnode,sothat inFig.2(b)thatforα = 2and3theaveragevelocityv does Lisconstantintime. Afteraninitialtransientdynamics,the notdecrease monotonicallyas a functionof L but it reaches systemreachesasteadystateinwhichdataiscollected. a maximumvmax(α) at some load L∗(α). However,having Results.- Now we report the dynamical behavior of the a larger average velocity does not imply that the network is model in the different cities considered as a function of the working in a more efficient way. In fact, for large values of 3 (a) (b) (c) FIG.1:UrbanstreetnetworkofBologna(one-square-milesample).Linksaredrawnwithathicknessproportionaltotheircongestionc:from (a)to(c)amorecongestion-awarestrategyrulesthesameamountoftraffic,whichprogressivelyflowsinalargernumberofstreets.Thetraffic loadisfixedatL=0.2,whilethevaluesofαconsideredarerespectivelyequalto0,1and2. α, the vehicles can move faster running across longer paths theeffectoftheaveragecongestionofthepath(obtainedfrom attheexpenseofdelayingthearrivaltotheirdestinations. A theglobalknowledge)onthepathdistance. measureoftheefficiencyoftheroutingistheaveragenumber The averagenumberr of completedroutesper houris re- ofroutesrcompletedbyavehicleduringonehour. ported in Fig. 2(d). The figure shows that the routing with The value of r is reported in Fig. 2(c) as a function of L. globalknowledgedoes notperformmuch better than that of The results indicate that, because of congestion, the number Eq. (1). Additionally,atlargeloads,localknowledgeoutper- ofcompletedroutesdecreaseswhenthenumberofvehiclesin formsglobalknowledgeintermsofr. Moreover,whenglobal the city increases. The precise dependenceof r on the load, informationis takeninto account,the optimalroutingα∗ in- is related to the value of α. For α = 0, the average vehi- creases. This is related to the fact that, since congestion in cle speed has a very sharp drop as the load increases. For linksclosetovehiclelocationisalwaysup-to-dateandthere- the congestion-aware strategy with α = 1 the decrease is foreaccurate,routingbasedon localcongestionneedslower smootherthanforα = 0,whileforα = 2(α = 3)thevalue valuesofα∗ todivertvehiclesonfreestreets. ofrissmallerthanthatofα = 1whenL < 0.1(L < 0.14), butlargerwhenL > 0.1(L > 0.14). Inpractice,foragiven Conclusions.- Congestion in transportationand communi- loadL,thefunctionr(α)showsamaximumatsomeα∗ [see cationnetworksisaseriousproblemforbothpublicgoodsand insetinFig.2(c)].Thevalueofα∗isseentoincreasewiththe userstime. In thispaper we have integratedthe three essen- load L, pointingout thatthe morecongestedthe networkis, tialingredientsofvehiculartrafficinurbansettings. Namely themorecongestion-awarehastobetheroutingto reachthe the graph structure of urban patterns, a cellular automata optimalfunctioning. On theotherhand,themaximumvalue modelforvehiculardynamicsalongthelinks,andtheuseof ofr,r(α∗),decreaseswithL. Similarresultsasthoseshown acongestion-awareroutinginspiredtodatatrafficintheInter- forBolognahavebeenfoundfortheothercitiesstudied. The net. Wethushaveprovidedwithasimpleandfeasiblemodel bestroutingexponentsα∗ obtainedfortworealisticvaluesof whereonlylocalinformationabouttrafficcongestionisused thevehicledensity,namelyL=0.06andL=0.1[3],arere- toroutevehicles.Ourresultsshowhoweachindividualagent portedinTableI. Itisclearthattheoptimalvalueα∗depends canbetterorganizeitsmotionbasedonitslimitedlocalknowl- stronglyontheparticulartopologyoftheurbangraph. edgeoftrafficand,atthesametime,achieveoptimalperfor- Eventhoughapreciseinformationonglobalcongestionis mancesattheglobalsystemlevel. Wehaveimplementedthis inpracticerarelyavailable,wefinallystudythecaseinwhich modelinseveralrealcitieswithdifferentstructuralproperties eachvehicleknowsexactlythecongestionineverylinkofthe showinghowtheoptimalvehicleroutingstrategydependson network. Namely, we comparethe results obtainedwith Eq. thenetworktopology.Finally,wehaveshownthat,counterin- (1)withthoseobtainedwiththepenaltyfunction tuitively,a (unfeasible)routingbasedona globalknowledge ofcongestionisnotsuitedwhentheloadofvehiclesislarge. α Pn =(din+dnt)(1+hcinti) , (2) Theproposedroutingmodelis generalenoughto be applied where hcinti accounts of the average road congestion along toseveraltypesofhumantransportnetworks. Moreover,the thepathfromitotpassingbyn. Therefore,wenowproject dynamic and distributed nature of the model allows several 4 (a) (b) (c) (d) FIG.2: Averagestreetflow(a),averagevehiclespeed(b),andaveragenumberofcompletedroutespervehicleperhour(c)asafunctionof thenetworkloadL.Theaveragenumberofcompletedroutespervehiclefortheglobal-awarestrategyisalsoreportedforcomparisoninpanel (d).Intheinsetswereportrasfunctionα.ThenetworkconsideredisthatofthecityofBologna. applicationstobebuiltwhichmayimplementthecongestion- [8] P.Echenique, J.Go´mez-Garden˜es, andY.Moreno, Phys.Rev. aware strategy to improve the available vehicle’s navigation E70,056105(2004). systems. [9] P.Echenique, J.Go´mez-Garden˜es, andY.Moreno, Eur.Phys. Acknowledgments.- We are indebted to C. Mascolo and I. Lett.71,325(2005). [10] P. Crucitti, V. Latora and M. Marchiori, Phys. Rev. E 69, Leontiadis for their valuable comments. V. Latora was sup- 045104(R)(2004). ported by the Engineering and Physical Sciences Research [11] I.Leontiadis,C.Mascolo.“OpportunisticSpatio-TemporalDis- Council[grantnumberEP/F013442/1]. seminationSystemforVehicularNetworks”,InProceedingsof theFirstInternationalWorkshoponMobileOpportunisticNet- working(ACM/SIGMOBILEMobiOpp2007).Colocatedwith Mobisys07.PuertoRico,USA,June2007. [12] S.Wolfram,Rev.Mod.Phys.55,601(1983). [1] J.G.Wardrop,Sometheoreticalaspectsofroadtrafficresearch, [13] J.Esser,M.Schreckenberg,Int.J.Mod.Phys.C8,1025(1997). InstituteofCivilEngineers(1952). [14] D.Helbing,Phys.Rev.E57,6176(1998). [2] D.Helbing,Rev.Mod.Phys.73,10067(2001). [15] K.Nagel,M.Schreckenberg,J.Phys.IFrance2,2221(1992). [3] D.Chowdhury, L.Santen, andA.Schadschneider, Phys.Rep. [16] D.ChowdhuryandA.Schadschneider,Phys.Rev.E59R1311 329,199(2000). (1999). [4] R. Pastor-Satorras, A. Vespignani, Evolution and Structureof [17] O. Biham, A. A. Middleton, D. Levine, Phys. Rev. A 46, theInternet:AStatisticalPhysicsApproach,(CambridgeUniv. R6124(1992). Press,Cambridge,2004). [18] A timestepisassumedtobebeequal to1sec, theminimum [5] D.Belomestny,V.JentschandM.Schreckenberg,J.Phys.A36, timeofreactionofadriver. 11369(2003). [19] A. Cardillo, S. Scellato, V. Latora, S. Porta, Phys. Rev. E [6] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.U. 73 066107 (2006); P. Crucitti, V. Latora, S. Porta, Chaos 16 Hwang,Phys.Rep.424,175(2006). 015113 (2006); S. Porta, P. Crucitti, V. Latora, Enviromental [7] R. Guimera´, A. D´ıaz-Guilera, F. Vega-Redondo, A. Cabrales, andPlanningB33,705(2006). andA.Arenas,Phys.Rev.Lett.89,248701(2002). Trafficoptimizationintransport networks basedon localrouting SalvatoreScellato,1 LuigiFortuna,2,1 Mattia Frasca,2,1 Jesu´sGo´mez-Garden˜es,1,3 andVito Latora4,1 1LaboratoriosuiSistemiComplessi, ScuolaSuperiorediCatania, ViaSanNullo5/i,95123Catania, Italy 2Dipartimento di Ingegneria Elettrica, Elettronica e dei Sistemi, Universita` degli Studi di Catania, viale A. Doria 6, 95125 Catania, Italy 3DepartamentodeMatema´ticaAplicada,ESCET,UniversidadReyJuanCarlos,E-28933Mo´stoles(Madrid),Spain 4DipartimentodiFisicaeAstronomia,Universita` diCatania,andINFN,ViaS.Sofia64,95123Catania,Italy (Dated:January8,2009) Congestionintransportnetworksisatopicoftheoreticalinterestandpracticalimportance. InthisPaperwe studytheflowofvehiclesinurbanstreetnetworks. Inparticular,weuseacellularautomatamodeltosimulate 9 themotionofvehiclesalongstreets,coupledwithacongestion-awareroutingatstreetcrossings. Suchrouting 0 makes use of the knowledge of agents about traffic in nearby roads and allows the vehicles to dynamically 0 2 updatetheroutestowardstheirdestinations.Byimplementingthemodelinrealurbanstreetpatternsofvarious cities,weshowthatitispossibletoachieveaglobaltrafficoptimizationbasedonlocalagentdecisions. n a PACSnumbers:89.20.-a,89.75.-k,89.40.Bb J 8 Introduction.-Traffic optimization has always been a cru- of the system. Therefore, to study congestion-awarerouting ] h cialissueinthecontextofcommunicationandtransportation ofvehiculartrafficintransportnetworksitisnecessarytoin- p systems [1, 2, 3, 4, 5]. A transport network is a network corporatetheabovetwoingredients. - of roads, streets, pipes, power lines, or nearly any structure In this Paper, we focus on the realistic scenario in which c o whichpermitseithervehicularmovementortheflowofsome onlyalocalknowledgeofcongestionisavailableandusedby s commodity.Inmostofthedevelopedcountries,transportation theagentstomodifytheirroutes. Ourmodelisimplemented . s infrastructuresoriginallydesignedto carrya definedamount intermsofvehiculartrafficandweassumethatdriversknow c oftrafficare oftencongestedbyanoverwhelmingrequestof the shortest paths to their destinations and, simultaneously, i s resources: this is the case of railroads, airplane connections they are aware about the congestion of nearby roads. Both y and,ofcourse,urbanstreets. Anaivesolutiontotheproblem informations are easily accesible nowadays from navigating h p consistsinexpandingtheinfrastructuretomatchtheincreas- systems, visual inspection and short-range wireless commu- [ ingdemand. However,thisisnotalwayspossibleduetolim- nication with other vehicles [11]. We show how, by conve- itationsinavailablespace,negativeoutcomesorshortnessof nientlycontrollingagentdecisions,itispossibletominimize 1 resources. Abetterapproachistocarefullytunethebehavior theoverallcongestionofthesystemandachieveperformances v 8 oftheexistinginfrastructurestoefficientlyexploittheiractual asgoodasthoseobtainedwithcomplete(global)trafficinfor- 7 structureandaccommodatethenewtrafficdemands. mation. 0 1 The study of congestion in complex networks has been The model.-Thethree ingredientsof the modelare: i) the . mainly focused on information systems, such as grid- substrate graph, ii) the vehicular dynamicsalong the streets, 1 0 computing networks or the Internet [4, 6]. In such context, iii)theroutingatstreetcrossings. 9 diversesolutionshavebeenproposedinordertoincreasethe The dynamics of vehicles takes place on top of urban 0 networkloadavoidingtheonsetofcongestion[7]. Inpartic- graphs. Weconsiderthestreetpatternofacityasaweighted v: ular,congestion-awarerouting,inwhichthenodesofthenet- graphwithN nodesandK edges. Eachedgeofthenetwork i work(therouters)redirectdynamicallytheinformationpack- representsastreet,alongwhichvehiclesmove,whereasnodes X etsacrossthelesscongestedpaths,hasprovedtoimproveno- accountofintersectionsbetweenstreets. Theweightofeach ar tablythecapacityofthenetwork[8,9].Itthusseemspossible edge is proportionalto the length of the road [19]. Here we tomakeuseofsimilarkindsofroutingstrategiesintransport assumeforsimplicitythateachedgeallowsmovementofve- networks. Ontheotherhand,transportnetworkspresentim- hiclesinbothdirections. Wehaveconsideredeightnetworks portantfeaturesdifferentfrominformationsystems. First, in representing1-squaremilesamplesofurbanstreetpatternsof a transport network the links (i.e. the roads) carry the flow real cities [19] (see Table I for details). The above city set ofvehicles,whereasthenodesarejustintersectionsbetween rangesfrom self-organizedcities, such as Bolognaand Lon- links. Therefore,onecannotneglectthedynamicsthatoccurs don,grownthroughacontinuousprocessoutofthecontrolof along the links: the quality of vehicle movement along the anycentralagency,togrid-likecitiessuchasLosAngelesand roadscharacterizesthefunctioningoftransportsystems. An- Barcelona,realizedoverashortperiodoftimeastheresultof other important feature is that, since transport networks are urbanplans,andusuallyexhibitinggridstructures. embeddedin the realspace, congestionisnotlocatedatpar- Thevehiculardynamicsalongthelinksoftheurbangraph ticular nodes of the system (such as the hubs in information is simulated by means of cellular automata [12, 13, 14]. In systems)[10]butitgeographicallyspreadsacrossthenetwork particular we use the Nagel-Schreckenberg (NaSch) model frombottlenecksanditmayeventuallyaffectalargeportion [15]. To this end, every link (street) of the city graph is di- 2 vided into a sequence of cells of equal length (5 meters) so City N K W hli α∗(L=0.06) α∗(L=0.1) thatnomorethanonevehiclecanoccupyacellateverytime Barcelona 210 323 36179 112.01 0.63 2.04 step. Eachvehicleisassignedavelocityvcellspertimestep Bologna 541 773 51219 66.26 0.87 1.54 [18]intherangev ∈[0,vmax].Wesetvmax =3,correspond- Brasilia 179 230 30910 134.39 1.52 2.09 LosAngeles 240 340 38716 113.87 1.82 2.43 ingtothetypicalmaximumvelocityof50Km/hinsideurban London 488 730 52800 72.33 1.02 1.26 areas.FollowingtheNaSchmodel,vehiclesaccelerate(decel- NewDelhi 252 334 32281 96.56 1.49 2.21 erate)whenthenextcellsareempty(occupied).Additionally, NewYork 248 419 36172 86.33 0.74 1.77 theintersectionsbetweenstreets(thenodesofthegraph)also Washington 192 303 36342 119.94 1.33 2.14 allowonlyonevehicleatagiventime,sothatseveralvehicles coming from different adjacent streets may compete for the TABLEI:Citysamplesconsidered: N andK arenumberofnodes same intersection [17]. For this reason, vehicles experience andlinksinthenetwork,W andhliarerespectivelythetotallength a slow down while approachinga roadintersection, as if the ofedgesandtheaverageedgelength(bothinmeters)[19].Wereport thevalueα∗thatmaximizestheaveragenumberofcompletedroutes endofthelanepresentedahindrance,sothattheyarriveatthe pervehicle. lastcelloftheedgewithv =0. Atthispoint,avehiclewaits to enter into the node [16] where it gets stuck until the first celloftheedgeintheproperoutgoingdirectionisfree. This waitinglockstheincomingflowsfromtheedges. Therefore, routing,α,andthenetworkload,L. First,inFig. 1weshow bottlenecksare created from the nodesand spread along the thecongestionpatternacrossthestreetsofBolognaforaload edges(roads)ofthegraph. L= 0.2,andforthreedifferentvaluesα =0,1,2. Itisclear Howvehiclesdecidetheiroutgoingdirectionwhenleaving that the larger the value of α, the more homogeneouslydis- a node? Here we implement a congestion-aware routing as tributedisthetraffic.Inordertostudyandquantifytheeffects a minimizationproblemthattakesinto accountthe lengthof ofcongestioninthevehicularmotionweanalyzetheso-called thepathandalsothetrafficalongtheoutgoingedges. When fundamentaldiagram [15]. The fundamentaldiagramrepre- a vehicle is at a node i it needs to choose a new node n in sents the network mean flux f (the average value of fluxes its neighborhoodΓi as the next hop on its path towards the onstreets)asafunctionofthetrafficloadL.Thisisshownin destinationt. Foreachoftheneighbouringnodesn,apenalty Fig.2(a)fordifferentvaluesofα. Eachofthecurvesf(L)ex- functionPnisdefinedas: hibitsthetypicalλ-inverseshapewiththetwotrafficphases: free-flow(inwhichtheflowincreasesasafunctionoftheload) Pn =(din+dnt)(1+cin)α , n∈Γi (1) atlowL,andcongested-flow(theflowdecreasesasafunction where din is the distance between nodes i and n, and cin oftheload)athighervaluesofL.Interestingly,boththevalue (cin ∈ [0,1]) represents the congestion of the link i → n, ofmaximumflowandthetransitionpointfromthefree-flow measured as the fraction of occupied cells in the link. The regimetothecongested-flowregime,increasewithα. Inpar- exponent α ≥ 0 accounts for the weight given to the local ticular,whenthevehiclesfollowshortestpaths(α = 0),they congestion in the drivers decision. The vehicle chooses the tendtoconcentrateinhighbetweennessnodesandlinks(we nodenwiththeminimumpenaltyPn. Ifcin = 0thepenalty havecheckedthatthestrongestcorrelationbetweenlinkflow functionPnisnothingelsethanthelenghtoftheshortestpath andbetweennessisindeedobtainedforα = 0). Inthiscase, tot,passingbynoden. Whencin 6=0theentireshortestpath even a small density of vehicles can result in a heavy load lengthiscorrectedbythe factor(1+cin)α. Inthisway, we at highbetweennessstreets. This givesrise to the formation assumethatthevehicleprojectsthecongestioncin ofthelink of clusters of jammed vehicles that cannot easily move and, i →nontheentirepathi→n →t. Notethat,whenα =0, although large regions of the city are quite uncongested as localcongestionplaysnoroleintheroutingandvehiclesfol- shown in Fig. 1(a), the overallflux of the networkis largely lowtheshortestpathstotheirrespectivedestinations. reduced. Weinitiallyplaceanumberofvehiclesproportionaltothe Theaboveresultisconfirmedbythesharpdecreaseofthe number of cells in the network, so that the network load, averagevehiclespeedvasafunctionofLforα=0shownin L = VC (i.e. the ratio between the number of vehicles V in Fig.2(b). Ontheotherhand,astheroutingstrategybecomes thenetworkandthetotalnumberofcellsC),isthesamefor more congestion-aware, the traffic is diverted from shortest all the cities considered. Initially, the vehicles are assigned path to free (and longer) paths causing that both the mean arandomsource(theirinitiallocation)andarandomdestina- speed of the vehicles and the mean flux on the streets are tionnode.Ateachtimestep,vehiclesmove(ifpossible)inthe largelyincreasedwith respecttothe caseα = 0, since more systemaccordingtotheNaSchrulesandthecongestion-aware vehiclesarenowabletoreachtheirdestinationswithoutbeing routing,Eq. (1). Finally, whena vehiclereachesitsdestina- blockedin congestednodes. More interestingly, we observe tionitisrandomlyassignedtoanewdestinationnode,sothat inFig.2(b)thatforα = 2and3theaveragevelocityv does Lisconstantintime. Afteraninitialtransientdynamics,the notdecrease monotonicallyas a functionof L but it reaches systemreachesasteadystateinwhichdataiscollected. a maximumvmax(α) at some load L∗(α). However,having Results.- Now we report the dynamical behavior of the a larger average velocity does not imply that the network is model in the different cities considered as a function of the working in a more efficient way. In fact, for large values of 3 (a) (b) (c) FIG.1:UrbanstreetnetworkofBologna(one-square-milesample).Linksaredrawnwithathicknessproportionaltotheircongestionc:from (a)to(c)amorecongestion-awarestrategyrulesthesameamountoftraffic,whichprogressivelyflowsinalargernumberofstreets.Thetraffic loadisfixedatL=0.2,whilethevaluesofαconsideredarerespectivelyequalto0,1and2. α, the vehicles can move faster running across longer paths theeffectoftheaveragecongestionofthepath(obtainedfrom attheexpenseofdelayingthearrivaltotheirdestinations. A theglobalknowledge)onthepathdistance. measureoftheefficiencyoftheroutingistheaveragenumber The averagenumberr of completedroutesper houris re- ofroutesrcompletedbyavehicleduringonehour. ported in Fig. 2(d). The figure shows that the routing with The value of r is reported in Fig. 2(c) as a function of L. globalknowledgedoes notperformmuch better than that of The results indicate that, because of congestion, the number Eq. (1). Additionally,atlargeloads,localknowledgeoutper- ofcompletedroutesdecreaseswhenthenumberofvehiclesin formsglobalknowledgeintermsofr. Moreover,whenglobal the city increases. The precise dependenceof r on the load, informationis takeninto account,the optimalroutingα∗ in- is related to the value of α. For α = 0, the average vehi- creases. This is related to the fact that, since congestion in cle speed has a very sharp drop as the load increases. For linksclosetovehiclelocationisalwaysup-to-dateandthere- the congestion-aware strategy with α = 1 the decrease is foreaccurate,routingbasedon localcongestionneedslower smootherthanforα = 0,whileforα = 2(α = 3)thevalue valuesofα∗ todivertvehiclesonfreestreets. ofrissmallerthanthatofα = 1whenL < 0.1(L < 0.14), butlargerwhenL > 0.1(L > 0.14). Inpractice,foragiven Conclusions.- Congestion in transportationand communi- loadL,thefunctionr(α)showsamaximumatsomeα∗ [see cationnetworksisaseriousproblemforbothpublicgoodsand insetinFig.2(c)].Thevalueofα∗isseentoincreasewiththe userstime. In this Paperwe have integratedthe three essen- load L, pointingout thatthe morecongestedthe networkis, tialingredientsofvehiculartrafficinurbansettings. Namely themorecongestion-awarehastobetheroutingto reachthe the graph structure of urban patterns, a cellular automata optimalfunctioning. On theotherhand,themaximumvalue modelforvehiculardynamicsalongthelinks,andtheuseof ofr,r(α∗),decreaseswithL. Similarresultsasthoseshown acongestion-awareroutinginspiredtodatatrafficintheInter- forBolognahavebeenfoundfortheothercitiesstudied. The net. Wethushaveprovidedwithasimpleandfeasiblemodel bestroutingexponentsα∗ obtainedfortworealisticvaluesof whereonlylocalinformationabouttrafficcongestionisused thevehicledensity,namelyL=0.06andL=0.1[3],arere- toroutevehicles.Ourresultsshowhoweachindividualagent portedinTableI. Itisclearthattheoptimalvalueα∗depends canbetterorganizeitsmotionbasedonitslimitedlocalknowl- stronglyontheparticulartopologyoftheurbangraph. edgeoftrafficand,atthesametime,achieveoptimalperfor- Eventhoughapreciseinformationonglobalcongestionis mancesattheglobalsystemlevel. Wehaveimplementedthis inpracticerarelyavailable,wefinallystudythecaseinwhich modelinseveralrealcitieswithdifferentstructuralproperties eachvehicleknowsexactlythecongestionineverylinkofthe showinghowtheoptimalvehicleroutingstrategydependson network. Namely, we comparethe results obtainedwith Eq. thenetworktopology.Finally,wehaveshownthat,counterin- (1)withthoseobtainedwiththepenaltyfunction tuitively,a (unfeasible)routingbasedona globalknowledge ofcongestionisnotsuitedwhentheloadofvehiclesislarge. α Pn =(din+dnt)(1+hcinti) , (2) Theproposedroutingmodelis generalenoughto be applied where hcinti accounts of the average road congestion along toseveraltypesofhumantransportnetworks. Moreover,the thepathfromitotpassingbyn. Therefore,wenowproject dynamic and distributed nature of the model allows several 4 (a) (b) (c) (d) FIG.2: Averagestreetflow(a),averagevehiclespeed(b),andaveragenumberofcompletedroutespervehicleperhour(c)asafunctionof thenetworkloadL.Theaveragenumberofcompletedroutespervehiclefortheglobal-awarestrategyisalsoreportedforcomparisoninpanel (d).Intheinsetswereportrasfunctionα.ThenetworkconsideredisthatofthecityofBologna. applicationstobebuiltwhichmayimplementthecongestion- [8] P.Echenique, J.Go´mez-Garden˜es, andY.Moreno, Phys.Rev. aware strategy to improve the available vehicle’s navigation E70,056105(2004). systems. [9] P.Echenique, J.Go´mez-Garden˜es, andY.Moreno, Eur.Phys. Acknowledgments.- We are indebted to C. Mascolo and I. Lett.71,325(2005). [10] P. Crucitti, V. Latora and M. Marchiori, Phys. Rev. E 69, Leontiadis for their valuable comments. V. 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