Table Of ContentTrafficoptimizationintransport 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. 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).
