Utility optimization framework for a distributed traffic control of urban road networks

ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Utility

optimization

framework

for

a

distributed

traffic

control

of

urban

road

networks

Tung

Le

a

_,

_Hai

_L.

_Vu

b ,∗

_,

_Neil

_Walton

c

_,

_Serge

_P.

_Hoogendoorn

d

_,

_Péter

_Kovács

e

_,

Rudesindo

N.

Queija

e

a_{IntelligentTransportSystemsLab,Melbourne,Australia} b_{InstituteofTransportStudies,MonashUniversity,Australia} c_{SchoolofMathematics,TheUniversityofManchester,UK} d_{DelftUniversityofTechnology,TheNetherlands}

e_{Korteweg-deVriesInstituteforMathematics,UniversityvanAmsterdam,TheNetherlands}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received25May2016 Revised4October2017 Accepted6October2017 Availableonline20October2017 Keywords:

Distributedurbantrafficcontrol Utilityoptimizationframework Ginicoefficient

a

b

s

t

r

a

c

t

Routechoicebehaviorhasbeenrecognizedasanimportantfactorinthetraffic manage-mentandcontrol strategydesign.State-of-the-artresearchinthisfield(e.g.anticipatory trafficcontrolapproaches)oftenformulatesthisasabi-leveloptimizationproblemwhere users’reactions tochangesinsignal controlaretakenintoaccountinthedesign ofthe control policies.Solving such acombined optimizationproblemis possible butdifficult andcomplexwhichofteninvolvesthedeterminationofthedynamicuserequilibrium(UE). Nevertheless,anticipatorycontrolinherentlyaimstooptimizethesignalcontrolthat an-ticipatesorreactstotheusers’route choiceand thusisareactivecontrol.Furthermore, theapproaches used inanticipatorycontrol arebasedonthe couplingbetweencontrol andtrafficassignmentwheretheroutechoicebehaviorofroadusersisunrealistically as-sumedtobeinequilibrium.

Thispaper proposesa novel utility based optimizationframework to design a dis-tributedcontrolschemebasedonaso-calledBackPressureprinciple thatisscalableand proactivelyinfluencesthe drivers’routechoicebehavior. Inparticular,as opposedtothe well-studiedanticipatorycontrolandbi-leveloptimization,theproposedutility optimiza-tionformulationsolvesforasystemoptimalproblemwheretheoptimalturningfraction (asanoutputdecisionvariable) isenforced asadesirableoutcome.Theoptimalturning fractionswouldthenbeachievedinapracticaldeploymenteitherbydirectlyaskingthe userstofollowthesystemsroutingadviceorbyinfluencing, i.e.manipulating,theroute choicedecisionsmadebythemsothattheresultingturningfractionis(oriscloseto)the optimalvalue.Althoughdriversmaynotfollowthecomputedoptimalfractionsasdesire, thereexistseveralways, e.g.seeGrootetal.(2015),todesign appropriateincentivesfor driversto followtheroute guidance, andthus improvethecompliance rateand overall journeyexperienceforeveryoneinsidethenetwork.Tothisend,wedemonstrateby nu-mericalresultsthatbetternetworkperformanceisachievedusingtheproposedframework evenwithalimitedcompliancefromtheusers.Furthermore,wehaveformallyprovedthat ourproposed utilityoptimizationbasedBackPressure schemestabilizesthenetworkfor anyfeasibletrafficdemandsthusmaximizingnetworkthroughput.Inaddition,weshow

∗ _{Correspondingauthor.}

E-mailaddress:[email protected](H.L.Vu). https://doi.org/10.1016/j.trb.2017.10.004

throughsimulation and numerical results thatthe new policyoutperforms the original BackPressure-baseddistributedcontrolschemebothintermsofnetworkthroughputand othercongestionmeasuressuchastraveltime.

© 2017ElsevierLtd.Allrightsreserved.

1. Introduction

Trafficcongestionisaseriousproblemaroundtheworldandparticularlyinurbanareaswhereboththepopulationand trafficdemandgrowrapidlywithlimitedornoroomforanewinfrastructure.Thesocietalcostattributabletocongestionis considerableduetothelossinproductivity,reductioninsafetyanditsnegativeimpactontheenvironment.Todate,traffic controlisoneoftheoldestandmostcost-effectivesolutionstocombattrafficcongestion.Inparticular,ithasbeenreported thatoptimallydesignedtrafficsignalscanleadtobenefit-costratiosashighas40to1(Fhwa, 2016 ).

Sinceitsinception,traffic signalsareusedto controlconflictingtrafficflows atroadintersections,andtherehavebeen manystudiesdescribedinthe transportationliterature concerningthe designandoptimizationofisolatedorcoordinated signals.Traffic signal controllers are usually describedaseither pre-timedor vehicle-actuated(Hamilton et al., 2013 ). Pre-timed control (sometimesreferred to as“fixed time plans”)provides a repetitive cycleandits division oftiming among theconflictingmovementsattheintersection.Thetimingisrepeatedagainandagainregardlessofthetrafficdemandand canbeoptimizedinisolationorinacoordinatedmanner(Gartner et al., 1975 ).Nevertheless,thefixedtimeplansdegrade rapidly as traffic changes and they require regular update (or re-optimization) which is generally an expensive process (Papageorgiou et al., 2003 ).

Vehicle-actuated controlsdifferfrom pre-timedinthat their signal timingsare not fixed. Through theuseof vehicles detectors(forexample,inductiveloops)thistypeofcontroloptimizessignaltimingsonacycle-to-cyclebasis inrealtime. PracticalsystemsofthistypeincludeSCOOT(Hunt et al., 1981 );OPAC(Gartner, 1983 );andthehierarchicalschemeRHODES (Mirchandani and Head, 2001 ).Combinations ofboththepre-timedandvehicle-actuatedcontrol alsoexistsuch asSCATS system(Lowrie, 1982 ).

1.1. Complexityofnetwork-widecontrol

Inthe literaturevarious approacheshavebeenproposed to optimizethesignal plans.Examples includemixed-integer linearprogrammingproblemsforasingleintersection(Dujardin et al., 2011 )andfornetworkofintersections(Gartner et al., 1975 );rolling horizon optimizationusing dynamic programming, see Gartner (1983) and Mirchandani and Head (2001) ; store-and-forward models based on ModelPredictive Control (MPC) optimization(Tettamanti and Varga, 2010; Le et al., 2013 ),orMPCoptimizationwithnon-linearprediction(Shu et al., 2011 ).

However,foranetworkwithmanyintersections,theabovecentralizedoptimizationmethodsresultinexponential com-putationalcomplexityandthusarenotscalable.As notedinPapageorgiou et al. (2003) ,controlstrategiessuchasOPACin Gartner (1983) andRHODESinMirchandani and Head (2001) arenotreal-time feasibleformorethanone intersection.In fact,theyactually becameadistributedschemeviaforcedimplementationatindividualintersectionsthatareheuristically coordinatedoverthenetwork(Gartner et al., 2001 ).

On anothernote, it hasbeen widely recognized that traffic signal control anddrivers’ route choice arethe two main interdependent processes influencingtraffic congestion. While the networkoperator aims to achieve systemoptimum by controlling traffic signals,roadusersadapt routesfollowingtheir own personal objectives,e.g. by choosingthe perceived fastestwaytothedestination.Theworkof Allsop (1974) wasoneofthefirsttopointoutthedependencybetweentraffic control anddrivers’route choice. Sincethen, therehavebeen manyproposedcontrol frameworksin theliterature (often referredtoasanticipatorytrafficcontrol)toanticipatedriver’sroutechoicebehaviorwhiledeterminingtheoptimalsignal control.Particularly,thedriver’sroutechoicebehaviorhasoftenbeeninvestigatedviatheexistingmathematicalframework of dynamictraffic assignment (DTA) since their reaction to changes insignal settings willcause changes in traffic flows resultinginadifferenttrafficassignment.Themodifiedtrafficassignmentwouldthenrequireanewoptimalsignalsettings andsoon.Duetothisinterdependencebetweentrafficcontrolandtrafficassignment, theusualapproachto thiscoupled problem isto form a bi-level optimization andsolve it either by iterative optimizationand assignment procedures (van Zuylen and Taale, 20 0 0; Taale, 20 08 )orbyglobaloptimizationmodels(Chen and Ben-Akiva, 1998; Karoonsoontawong and Waller, 2010; Ukkusuri et al., 2013 ).

Althoughanticipatorycontrolrepresentsthestate-of-the-artresearchinthisfield,itfacesthesamescalabilityproblem intermsofcomplexity.Recent attempthasbeenmadebyRinaldi and Tampère (2015) towardadistributedversion ofthe anticipatoryschemebysubdividingtheoriginaloptimizationproblemintosmaller,moretractablesub-problemswhile aim-ingtoachieveglobaloptimality(Rinaldi and Tampère, 2015 ).Nevertheless,theapproachesusedinanticipatorycontrolare basedonthecouplingbetweencontrolandtrafficassignmentwheretheroutechoicebehaviorofroadusersisassumedto beinequilibrium.Suchequilibriumistypicallyonlyapplicableinthecontextofalongterm(e.g.strategicplanning) applica-tion,andisnotnecessarytrueintheinstantaneous,real-timetrafficactuateddynamicsignalcontrolscenarios.Furthermore,

solvingthisuserequilibrium(UE)viatheDTAframeworkwithrealistictraffic conditions(e.g.stochastic)iscomplex.More importantly,anticipatorycontrolinherentlyaimstooptimizethesignal controlthat adaptstotheusers’routechoice.The systemthentendstoconvergetoUEwhichissubjecttotheaggregate effectofusers’routechoice.However,theUE,even ifitisoptimized byanticipatoryframeworks,maynot presentthebest possiblethroughputthe systemmayachieve.For instant,we mayhavebetterthroughputsimply ifthe userschoosetheir routesdifferently.Inreality,thenetwork andits roadusersmight not evenbe inequilibrium. In addition,gatheringthe demandinformationwhere eachtraveler has his ownroutechoicepreferencepresentsfurtherdifficulties.

Insum, the complexityandthe suboptimal causeby users’ route choice behavior arethe two problems that we will tackleinthispaper.Tothisend, wewillproposed anovel methodologyunderpinnedby amathematical utility optimiza-tionframework solving forasystemoptimalproblemto designa distributedcontrol schemethat isscalableandactively influencesthe drivers’routechoice behavior.Ourproposed control schemewillincludetheroute choiceresponses by (a) introducingthesetH,whichisthesetofallachievableexpectedturningfractions,intotheoptimizationconstraintsand(b) introducingthecompliance probability intothesimulationstudy,i.e.usersmight chosenot tofollowthesystems advice. Theformerenablesustoenforcetheoptimalturningfractionasadesirableoutcomejointlywiththesignalcontrol.While thelattermodelstheusersresponsetoadviceviaacomplianceprobabilitythatusersmightchoosenottofollowtheroute recommendation. Remarkably, aswe show later via numerical results,better network performance is achievedusing the proposedframeworkevenwithalimitedcompliancefromtheusers.Thisdistributedframeworkenablesustodecouplethe optimalcontrolschemefromthecomplexDTAUEformulation,butstilltake intoaccount theusers’routechoice behavior andthustackleboththecomplexityandtheUEissues.

1.2.State-of-the-artofdistributedcontrol

Belowwereviewtherelatedliteratureintheareaofdistributed(ordecentralized)controlbeforedetailingtheapproach andcontributionsofthispaper.Despitetheobviousadvantagesinscalabilityandeaseinimplementation,distributedcontrol strategiesarenotyetpracticalinthecontextofurbantrafficmanagement. Itisbecausea distributedschemewillhaveto rely on detailedlocal information collectedat the intersection in real-time to be effective. distributed Such information canbe either(a) theexpectednumberofvehiclestoenter theintersectionduringthe nextcycle, or(b)the differencein trafficloadontheroadleadingintotheintersectionandthoseleadingout.Theseapproachesarenowdeployableinpractice thanksto emergingtechnologies, such assmart detectors,camerasandwirelesscommunicationenablingbetter accessto real-timetrafficdata.

Earlywork ofSmith (1980) ,aso-calledP0 localcontrolpolicy andits variants(Smith, 2011 )belong totheformer

(ap-proach(a)).FollowsbyLämmer and Helbing (2008) wheretheauthorsextendedfurthertheP0policytoincludethe

switch-ingcostbetweenphases.AheuristicmechanismwasproposedinLämmer and Helbing (2010) topreventpossibleinstability butthereremainsnoformal proofofstabilityforsuchdistributedcontrolstrategy.Notethatthenotionofstabilitywillbe formallydefinedlaterinSection 5 .

The latter(approach (b)) include work by Wunderlich et al. (2008) , Varaiya (2013) , Wongpiromsarn et al. (2012) and Le et al. (2015) all ofwhichwere basedona so-calledmaxweight orbackpressure (referto asBackPressure)algorithm (Tassiulas and Ephremides, 1992 ) originally developedforpacket schedulingin wireless data networks. TheBackPressure baseddistributedcontrolstrategyisveryattractivebecauseitcanensuremaximumpossiblenetworkthroughputwithouta prioriknowledgeofthetrafficdemand.Moreimportantlyithastheaddedbenefitofprovablestability.Inaddition,J. de Gier and Rojas (2011) reportthatforatypicalsystem,thecontrolstrategythatusescongestionobservedonbothupstreamand down stream links (i.e. BackPressure type) is more efficient and reliable in termof travel timesthan that informed by the congestionon the upstream link only (i.e. approach (a)). Further the backpressure policy can incorporate androute multicommodity flows provided a separate queue is maintained for each commodity, such a systemwas first proposed byTassiulas and Ephremides (1992) . Withsome lossofstability, canbe adapted totraffic control mechanismswhich are agnostictothecommoditiesused,seeBui et al. (2009) andZaidi et al. (2016) .

TheBackPressure mechanism, however,hasa drawback asitcan leadto a largequeuebuild up atthe upstreamof a route betweenan origin-destination(OD) pair(Stolyar, 2011 ) resulting intraffic spatialheterogeneity.This isundesirable becauselargerupstreamqueueswillincreasethespatialvariationinthedensityacrossthenetworkwhichinturnsreduces aso-called“level-of-service” inthenetwork,formoredetailsseeHoogendoorn et al. (2013) .Intheotherwords,theimpact ofcongestionworsens whentheheterogeneityoflink densitiesorqueuesincreases.Thereexistmitigation techniquesfor thisproblemintheliteratureofwirelessnetworking,butforatrafficcontrolscenario,theywouldrequiretheknowledgeof individualvehicle’sdestination(suchasinBui et al. (2009) )whichisclearlynotfeasible.

GiventheadvantagesoftheBackPressurebasedtrafficcontrol,ourmethodologyinthispaperistoforma novelutility optimizationframeworkwithanappropriatelychosenutilityfunctionthatcanreducethetrafficspatialheterogeneitywhile maximizingpossiblethroughputjustasintheoriginalBackPressurealgorithm.Tothisend,weproposetheuseofaso-called Gini-index(Foster and Sen, 1973 )asanindicationofnetworkdensityinequalitywhereaGini-index valueofzeropresents aperfectequalitywhilehighervaluepresentsgreaterdensitydeviation. Thisindexwillthenserveasameasurefortraffic spatialheterogeneityandwillbejointlyminimizedintheaboveutilityoptimizationframework.Furthermoretheproposed frameworkwilldirectlyinfluencedrivers’routechoicebehaviorbygivingoutlocalrouteguidanceadvice,forinstancewhich nextroadanusermightwanttotake,withoutassumingthattheuserwillactuallycomplywiththeadvice.Ifthedriverdoes

Table1

Comparisonbetweenourproposedpolicyandotherpolicies.

Works Centralizedor distributed Provenstability (optimalthroughput) Anticipateusers’ routechoice

Influenceusers’route choice

(TettamantiandVarga,2010) Centralized Yes No No

(Leetal.,2013) Centralized Yes No Yes

(Taale,2008;Ukkusurietal.,2013) Centralized Yes Yes No

(Smith,1980;2011;Smithetal.,2015) Distributed Yes,butnotforgeneral networks

Yes No

(Wunderlichetal.,2008;Varaiya,2013; Wongpiromsarnetal.,2012;Leetal.,2015)

Distributed Yes,forgeneral networks

No No

Ourproposedframework Distributed Yes,forgeneral networks

No Yes

indeedfollowtheadviceandenterslinkthatwasnotpartoftheoriginalplannedroute,thenshewillneedtorecalculate the routeto thedestination basedon her ownpreference. From thenetwork control’s perspective,theseadvertisedlocal turningadvicesare seenasadesirableroute choiceoftheroaduserswhichhelpstoreduce theoverall congestioninthe network.Thesignalcontrolandtrafficassignment(orturningfractions)arethenjointlyoptimizedinourutilityframework whichis inastark contrastto theanticipatorycontrol schemewherebi-level optimizationcouplingwithdynamictraffic assignmentproblemsaresolvediteratively.Inotherwords,thetrafficsignalcontrolanddesirableusers’behaviorarejointly optimizedinasingleoptimizationwhichgreatlyreducesthecomplexityoftheproblem.Besidewedonotneedtoinvolve thestepoffindingtheuserequilibriumviatheDTAformulationbasedonaparticularusers’routechoicemodel.

1.3. Researchcontributionsandpaperoutline

Theideaofjointlyoptimizingthetrafficsignalcontrolanddesirableroutechoice,thatistoinfluenceratherthanfollow drivers’behavior,wasproposedinourpreviouswork(Le et al., 2013 )butforacentralizedsignalcontrolusingmodel predic-tivecontrol(MPC).RecentworkbySmith et al. (2015) andGroot et al. (2015) followasimilarapproach,andthelatteralso providesseveralwaystodesignappropriate incentivefordriverstofollowtherouteguidance,andthusimprovethe com-pliancerateandoveralljourneyexperienceforeveryoneinsidethenetwork.Inthispaper,wewillexplorethisconceptfora distributedcontrolschemebasedontheBackPressureprinciple.Thestudyofdesignandgivingincentivetoencourageand improveusers’complianceisout ofthescopeofthispaperandwillbe investigatedinaseparate work.Nevertheless,we showvianumericalexamplesthattheBackPressurederivedfromtheproposedutilityoptimizationframeworkoutperforms the original BackPressure even with a limited user compliance (Wunderlich et al., 2008; Varaiya, 2013; Wongpiromsarn et al., 2012 ) both intermsof throughputandcongestionmetrics. See Table 1 for thesummary comparisonbetweenour proposedframeworkandotherframeworksintheliterature.

Themaincontributionsinthispaperinclude

• anovelutilityoptimizationframework togetherwiththeuseoftheGini-index asan utilityfunctionthat indicatesthe levelofnetworkspatialdensityinequality,

• anovelcontrolpolicythatisscalableandactivelyinfluenceontheusers’routechoicethatpotentiallyoutperformother distributedanticipatorycontrolpolicies.

• a new distributed optimal traffic control strategy that combines signal phasing and route guidance for urban traffic networksbasedontheBackPressuremechanism’sprinciple,

• aformalmathematicalproof thattheproposedpolicyisstable, i.e.achievinga maximumpossiblenetworkthroughput wheneverthedemandisfeasible,

• acontrolstrategythatdoesnotrequireknowledgeabouttheuser’sroutechoiceortrafficdemandwhilestillbeableto maximizethenetworkthroughputandimproveoncongestionmeasure,and

• results to demonstrateunder different level of compliance that (i) the performance improves significantly even with limitedusercompliance,(ii)arelativelysmallpercentageofcompliance(e.g.10%)isenoughtogainsignificantbenefits. The remainder of this paperis organizedas follows.We describe thenotations and network dynamicsmodel before describing ourproposed utility optimization basedBackPressure policy inSection 2 . The mainresults forstabilityofour policy are then provided in Section 3 . Forreadability most ofthe mathematical details andderivations are listed inthe Appendicesofthepaper.Section 4 presentsasimplecasestudywhereacontrol policywithusers’routechoiceinfluence couldoutperformotherdistributedanticipatorypolicies.Section 5 presentsthesimulationresultsandnumericalcomparison ofourschemewiththeexistingBackPressurepolicy.Finally,Section 6 concludesthepaper.

2. Methodology:theutilityoptimizationframework

2.1. Notationandnetworkdescription

Consideraroadtrafficnetworkinvolvingacollectionofjunctions(orintersections)J.Eachjunction j∈J consistsofa numberofin-roadsIjrepresentinganincomingroadtothatjunction.NotethatIjaredisjointandletI=∪j∈JIj.

Eachjunctionmayservedifferentcombinationsofin-roadssimultaneouslybythemeanofcontrollingtrafficsignals.We callsuchcombinationaservicephase.Aservicephaseofjunctionjispresentedbyavector

=

(

ψ

i:i∈Ij

)

where

ψ

i=ci

ifin-roadibelongstoservicephase

and

ψ

i=0otherwise,andciisthecapacityofin-roadi.Notethatin-roadcapacity

ciistheexpectednumberofcarstobeservedduringatrafficsignalcyclegiventhatthewholecycleisallocatedasagreen

signaltothein-roadiandthatiisnotemptyatanytimepointduringthecycle.LetSjdenotethesetofphasesatjunction

j.

Furthermore,letL denotetheset oflinksrepresentingthetraffic movementsata junctionofthetraffic network. We denoteii_∈Lifthereexisti_∈Ijandi_∈Ij suchthatitispossibleforvehiclestobeservedatitonextjoinati.

WeassumealljunctionsshareacommontrafficsignalcyclelengthT.Thenwecanrepresenttimeasdiscretet=0,1,... andconsideraslottedtimemodelwhereeachtimeslotcorrespondingtoatrafficlightcycle.Controldecisionsinourpolicy arethenmadeatthebeginningofeachtimeslotsimilartothepoliciespreviouslyproposedinLämmer and Helbing (2010) , Varaiya (2013) andSmith (1980) .

2.2.Networkdynamicsmodel

Letqi(t)denotethenumberofvehiclesatin-roadi∈Iatthebeginningoftimeslott.Weassumethenumberofvehicle

atan in-road cangrow unbounded. We define a vector Q

(

t

)

₌

(

qi

(

t

)

:i∈I). Let pj

(

t

)

denotethe fraction ofthe traffic

cycleattime slottatjunctionj whichisdevotedtoservicephase

.Wedefine thecontrolvector asPj=

(

pj :

∈Sj

)

.

NotethatPjaredisjointandletP=∪j∈JPj.Foranypolicy,wedefineafeasiblecontrolsetas:

j=

Pj:

∈Sj

pj ≤ 1,pj >0

∀

j∈J . (1)

Foranyin-roadi∈I,theexpectednumberofvehiclesleavingduringservicephase

is

ψ

ipj

(

t

)

,providedthein-road

isnot emptied. Accordinglyif we let therandom variable s_i(t) be the potential numberof cars served fromin-roadi at junctionjintrafficcyclet,themeanofsi(t)isgivenas

E

si

(

t

)

qi

(

t

)

= ∈Sj

ψ

ipj

(

t

)

. (2)

Inotherwords,thevariablesi(t)givestheoffered loadonin-roadi,whichisthenumberofvehiclesserved ifthereis

anindefinitequeueofvehicles.Ifthenumberofvehiclesinin-roadi isnot highenoughtobeservedduringthe right-of-way(orgreenphase)servicetime,theactualnumberofvehiclesleavingiisqi(t).Hence,thenumberofvehiclesservedat

in-roadiisarandomvariabledefinedby: fi

(

t

)

=min

si

(

t

)

,qi

(

t

)

. (3)

Foranyii∈L,let

θ

ii

(

t

)

betheproportional ofcarsserved atin-roadi thatsubsequentlyjoin in-roadiattime t.We

assume thatcars within an in-roadare homogeneous inthesense that each car atthejunction hasthesame likelihood ofjoiningeach subsequent junction.We denotethe expectation of

θ

_ii

(

t

)

by

θ

¯_ii

(

t

)

, andwrite theexpectation of turning fractionsinavectorform

¯j

(

t

)

=

(

θ

¯ii

(

t

)

:i∈J,ii∈L

)

.Inotherwords

E

θ

ii

(

t

)

si

(

t

)

,qi

(

t

)

=

_θ

¯_ii

₍

_t

₎

_{. (4)}

Notethat

¯jaredisjointandlet

¯ =∪j∈J

¯j.

Thisexpectedturningfraction,whichisknownorcan bemeasured,isanaggregatedresultofindividual routechoices andcanbe influenced by systemssuch asadvanced travelinformation system(ATIS).Assuming acertain route guidance compliance,wedefineHjasthesetofallachievableexpectedturningfractionsofjunctionj.Thesetofturningfractionsis

aclosedsethasthepropertythatiftwoturningfractionsareachievableatajunction,thenacombinationoftheseturning fractionisalsoachievable.Moreformally,weassumethatif

θ

,

φ

∈Hjandaresuchthat,forsomei,

θ

ii=

φ

ii foralli=i

thenfor

α

∈(0,1),

αθ

+

(

1−

α

)

φ

∈Hj.

Wefurtherassume that thissetisconstant andwillnotbe alteredby thequeuelengthsobservedby carswithin the network.Thus fi

(

t

)

θ

ii

(

t

)

isthenumberofcarsthatleaveinroadiand,next,joininroadi.WeletA

(

t

)

=

(

ai

(

t

)

:i∈I)∈ZI+

whereai(t)whichisastochasticquantitydenotesthenumberofexternalarrivalsatin-roadiattimet.Theai(t) assumed

Givenaservicepolicy

{

P_j

(

t

)

_,

_j

(

t

)

:j_∈J

}

∞_t=0,belowwecandefinethedynamicsofournetworkasarandomqueueing processforthein-roadiofjunctionj,asfollows,

qi

(

t+1

)

=qi

(

t

)

−fi

(

t

)

+ai

(

t

)

+ i:ii∈L

fi

(

t

)

θ

ii

(

t

)

₍₅₎

forPj(t)∈

jand

¯j

(

t

)

∈Hj.

2.3. UtilityoptimizationbasedBackPressure(U-BP)controlpolicy

Wefirstintroduce anutility functiongj(Pj(t),

j(t))thatisboundedsuchas0≤gj

(

Pj

(

t

)

,

¯j

(

t

))

≤Gforallt∈R+.One

suchfunctionisaso-calledGiniindexfunction (Foster and Sen, 1973 )whichisalwaysboundedbetween[0,1],andcanbe usedasameasureoftheinequality(orspatialheterogeneity)oftrafficwithintheurbantrafficnetwork.

GiventheadvantagesoftheBackPressure algorithmfortrafficcontrol discussedearlierinSection 1 ,theideahereisto formanovelutilityoptimizationproblemthatjointlystabilizesthenetwork(i.e.maximizespossiblenetworkthroughputas inBackPressurealgorithm)andatthesametimeminimizesthechosenutilityfunction(i.e.reducetrafficspatial heterogene-ity).Notethat theutilityfunctionisnotrestrictedtotheaggregateofindividualtravelinggain,itisratherthegeneralized costfromnetworkperspective.Theproposedpolicyisdescribedasfollows.

TheU-BPcontrolpolicy:

1. Atthe beginning ofeach traffic cycleandforeach junction j∈J,we define the weightassociated witheach service phaseatthejunctionasafunctionofthequeuesizesQ(t)andsomeinitialexpectedturningprobabilities

θ

¯_ii

(

t

)

w

(

Q

(

t

))

= i∈Ij

ψ

i

qi

(

t

)

− i:ii∈L ¯

θ

ii

(

t

)

qi

(

t

)

. (6)

2. Giventheseweights,determine thetraffic signalphase vector P_j(t) andtheexpectedturning fractionvector

¯_j

₍

t

)

for eachjunction j∈J thatarethesolutionofthefollowingminimizationproblem

minimize Pj(t),¯j(t) E[gj

(

Pj

(

t

)

,

¯j

(

t

))

|

Q

(

t

)

]− ∈Sj pj

(

t

)

w

(

Q

(

t

))

subjectto Pj

(

t

)

∈

j and

¯j

(

t

)

∈H j (7)

3. SetthecontrolsignalsatthejunctionaccordingtoPj(t)whileaimtoachievetheoptimalexpectedturningfraction

¯j

(

t

)

viagivingrouteguidanceadvicestoroadusers.

Theweights definedin(6) areused intheBackPressurepolicy asgivenbyTassiulas and Ephremides (1992) .Theycan beviewedasa“pressure” a queueplaces ondownstream queues,whichis givenbytheweightedmeanofthedifferences ofthequeuesizes.Thelargertheweightassociatewithaphase,themoreimportantitistoservethein-roadswithgreen lightsduringthatphase.

Here the optimisation problem(7) need not be linear. As queue sizesget large, the problemwill be linear andthus can be solved. Provided the function gj has a sub-gradient, then gradient descent methods will find a local-optimum.A

globaloptimum is not requiredby thealgorithm, since thesenotationslocal andglobalwill coincide asymptotically and willbecloseduringbusyperiods.Specifically,duetocompactnessofregionandconvergencetolinearfunction,thevalues of the localminima will convergeto the globaloptimum. Notice that incontrast to BackPressure policies which always serve the phase associated withthe highest weight, the proposed policy balances between higher weights and a larger inequality (or spatialheterogeneity).Thelatter isdueto the“needs” to increase queuesize closedto theorigin(source) intheBackPressurepolicy.Heregivingroute guidanceadviceisapracticalmechanismthatweusetoachieve thedesired expectedturningfractionatthejunction.WewillelaborateitinmoredetailsinSection 5 whereacomplianceprobability isintroducedtoreflectthefactthatroadusersmightnotcomplytotheadvice.

Alsonote thatthe utilitymeasures traffic queuelocallyandthusthe policycan beimplemented inadistributedway, aftereachjunctioncommunicatesqueuesizeswithitsupstreamin-roads,thephasesandturningfractionscanbecalculated. Thisdecentralizationhasnumerousadvantages:itiscomputationallyinexpensive,itdoesnotrequirecentralizedaggregation ofinformationandthusiseasiertoimplement,anditincreasestheroadnetworksrobustnesstofailures.

3. Theoreticalresults-stabilityoftheU-BPcontrolpolicy

BelowwewillformallydefineandprovethestabilitypropertyoftheproposedU-BPcontrolscheme. 3.1. Assumptions

Thenumberofcarsthat canbeserved fromanyin-roadwithinatraffic cycleisboundedSmax=maxt∈Z+,i∈Isi

(

t

)

<∞.

Thearrival isboundedAmax=maxt∈Z+,i∈Iai

(

t

)

<∞.Foranyin-roadi,thenumberofin-roadi suchthatitispossiblefor

3.2.Queueingstability

Wedefinethetotalqueuesizeoftheroadnetworktobe q

(

t

)

=

i∈I

qi

(

t

)

. (8)

Soq

(

t

)

givesthetotalnumberofcarswithintheroadnetwork.Wesaythatapolicyisstableifthelongrunnumber ofcarsinthequeueingnetworkisfinite,inparticular,

lim T→∞E

1 T T t=1 q

(

t

)

<∞. (9)

WenotethatifthequeuesizeprocesswasaMarkovchainthendefinitionofstablewouldbeequivalenttothedefinition ofpositive recurrenceforthatMarkovchain.However, theprocessthat wewilldefineneednot beaMarkovchainhence weusetheabovedefinition.

3.3.Stabilityregion

WeconsiderthesetofarrivalratesA¯₌

(

a¯_i≥0:i_∈I

)

_,forwhichthereexistsapositivevectorsP¯₌

(

p¯j :

_∈Sj,j_∈J

)

_∈

where

=∪j∈J

j,F¯=

(

f¯i:i∈I)and

¯ =

(

θ

¯ii:ii∈L

)

∈HwhereH=∪j∈JHj,namelythegreentimeproportions,the

departurerates,andtheturningfractionsrespectively,thatsatisfythefollowingconstraints

¯ ai+ i:ii∈L ¯ fi

θ

¯ii≤f¯i,

∀

j∈ J ,

∀

i∈ I j (10) ¯ fi≤ ∈Sj ¯ pj

ψ

i,

∀

i∈ I, (11)

anddenote thisset by A. Thisset is clearlyconvexby the definitionof

j and Hj.We let A◦ denote the interiorof A

wherethe above inequalities in (10) and (11) are strict. We call Athe stability region of the systemas verified by the followingpropositionandourmaintheorem.Informally,theystate thatforanydemandoutsidethestabilityregion there isnoschedulethat wouldprevent thequeuesizestogrow infinitelyinthelong run,whereas ifA¯∈A◦ _{the BackPressure}

policystabilizesthesystem.

Proposition1. Given thatthearrivals ateach time,

{

A

(

t

)

}

∞_t=1, areindependentidentically distributed random variables with expectationA¯,thenitfollowsthatifA¯∈/Athen,foranypolicy,thequeuesizes

{

Q

(

t

)

}

_t∞₌₀ areunstable.

Proof.IfA¯∈/Athenthereexistsan

>0where,foranyvectorsP¯=

(

p¯j :

∈Sj,j∈J

)

∈

,F¯=

(

f¯i:i∈I)and

¯ =

(

θ

¯ii:

i_,i_∈I)_∈Hsatisfying(11) ,thereexistsanin-roadisuchthat

¯

ai−f¯i+ i:ii∈L

¯

fi

θ

¯ii>

. (12)

Foranypolicy

{

Pj

(

t

)

,

¯j

(

t

)

: j∈J

}

∞t=0takingasuitablesubsequenceifnecessarywehavetheexistenceofthefollowing

limits ¯ fi= lim T→∞ 1 T T t=1 fi

(

t

)

,

η

¯ii= lim T→∞ 1 T T t=1 fi

(

t

)

θ

ii

(

t

)

, ¯

θ

ii= lim T→∞ 1 T T t=1 fi

(

t

)

s≤T fi

(

s

)

θ

ii

(

t

)

Notethat ¯

η

ii= lim T→∞

T t=1fi

(

t

)

T T t=1 fi

(

t

)

s≤T fi

(

s

)

θ

ii

(

t

)

= _f¯_i

_θ

¯_ii

Furtherwe claimthat

¯ ₌

₍

_θ

¯

ii:ii∈L

)

∈H.Thisholds sinceforeach i thevector

Ti =

(

θ

iiT:ii∈L

)

whosecomponents

aregivenby

θ

T ii= T t=1 fi

(

t

)

s≤T fi

(

s

)

θ

ii

(

t

)

.

Itisclearthat f¯satisfiesconstraint(11) (beingtheaverageoftermssatisfyingthisconstraint); HoweversinceA¯_∈/Ait mustbethat(12) occursforsome i.Ifwe combinethiswiththefact,thatthelongrunin-roadqueuesizemustconverge totheaveragearrivalminustheaveragedepartures,weget

lim t→∞ q

(

t

)

t = i∈I

¯ ai+ i:ii∈L ¯ fi

θ

¯ii−f¯i

>

.

Intheaboveinequality,wenotethateachterminthesummationispositivesinceitisthelimitofpositivequeuesizesand oneofthosetermsingreaterthan

by(12) .Thusweseethatforanypolicythereexistsafiniteϑsuchthatforallt>ϑ,

q

(

t

)

>t

, thus, lim T→∞ 1 T T t=1 q

(

t

)

≥lim T→∞ 1 T T t=ϑ

t=∞.

So,takingexpectations,weseethatthenetworkmustbeunstableasstatedinProposition 1 ,since(9) cannotholdas

limsup T→∞ E

1 T T t=1 q

(

t

)

≥E

liminf T→∞ 1 T T t=1 q

(

t

)

=∞, whereweappliedFatou’sLemma(Ahmed and Ahmed, 2006 ). 3.4. Maintheorem

Ourmaintheoremconsidersthestabilityofthenetwork.Namely,weprovethatundertheproposedsignalcontrolpolicy thesystemisstableforanyfeasibledemand.AsProposition 1 showsthisisthemostanypolicycanprovide.

Theorem1. IfA¯∈A◦_{,i.e.thereexistsan}

_>0suchthatA¯₊

_{1∈AthenbyapplyingtheU-BPcontrolpolicydefinedinSection} 2.32.3thelongrunaveragequeuesizesofin-roadsareboundedas

lim τ→∞E

1

τ

τ−1 t=0 q

(

t

)

≤

|

I

|

K+

J

|

G

(13) where K=1 2

Amax+DmaxSmax

2

+3S2max

2 . (14)

andthusthepolicyisstable.

The proof ofTheorem 1 isgiven inAppendix A andfollowsa somewhat standard LyapunovanalysisforBackPressure policy, e.g. see Tassiulas and Ephremides (1992) . Nevertheless, there are important features that must be considered in this particularextension. Specifically the non-standard turning fractions arepart of thecontrol mechanism andmust be considered in thedefinition ofthe stabilityregion.Also thebounded errorintroduced by thegj functionin (7) must be

dealtwithintheproof.

FromtheLittle’sTheorem,theaveragetraveltime,D¯,canbederivedasfollowed

¯ D= limτ→∞E

₁ τ τt=−01q

(

t

)

i∈Ia¯i ≤

|

I

|

K

|

J

|

G

i∈Ia¯i. (15)

4. Asimplescenariowithroutechoicemodel

Thissection illustrates the benefitof activelyinfluence users’route choice through asimple scenario.The endresults will show that a control policy that actively influenceusers’ route choice could resultinbetter overall travel delaythan the control policy that doesnot influenceon users’route choice, say Proportionalpolicy (Smith, 2011 ) andBackPressure policy (Wunderlich et al., 2008; Varaiya, 2013; Wongpiromsarn et al., 2012 ). Tomake the terminologyclear when route choice modelis involved,all control policiesin somewayinfluence onusers’ routechoice through changing thenetwork states,however,apolicythatactivelyinfluenceusers’routechoiceratherchangesthedecisionmakingcriteria.Forexample, itiscommontoassumethatusersalwayschoosethefastestroute.Apolicythatinfluencesusers’routechoicewouldhave theeffectthatsometimeusersmaychoosethelongerrouteeitherunconsciouslybasedontheinformationgiventothemor consciouslyforagreatergood.Wewillhavequantitativeanalysisforthiseffectinthefollowingexample.

Fig.1. Asimplenetwork.

WeconsiderasimplenetworktopologyinFig. 1 withasingleODpair,oneintersectionandtworoutes.Upondeparture theusershaveachoiceofwhatroutetofollowwhichissimilartoturningfractioningeneralnetworks.Route1hassmaller roadandshortdistance,sothedelaytotravelonroute1ismodeledas

b1= Q1 s1G1 = Q1 G1 (16)

whereQ1 isthenumber ofcars waitingtobe served atthe intersectionfollowedroute 1,s1 isthesaturated flow going

throughtheintersectionfromroute1ands1=1,G1isthegreentimefractionallocatedforroute1.

Route2hasbiggerroadbutlongerdistance,sothedelaytotravelonroute2ismodeledas b2=d+ Q2 s2G2 = d+ Q2 2G2 (17)

whered isa constant which represents the free flow travel time, Q2 isthe number ofcars waitingto be served atthe

intersectionfollowed route 2,s2 isthe saturated flowgoing through the intersectionfrom route 2ands2=2, G2 is the

greentimefractionallocatedforroute2.

Let

λ1

,

λ2

arethearrivalateachroute.Thereisaconstantdemandaforthisscenarioso

λ

1+

λ

2=a. (18)

Thesignalizedintersectionoperatesonfixedcycletime.Dependonthecontrolpolicy,theintersectionwillallocatethe greensignaltoeitherroute1orroute2atanypointintime,hencewehavetheconstraint

G1+G2=1. (19)

Forthisnetwork,themaximumthroughputis2whenall vehiclestravelonroute 2andall greentimeisallocatedfor route2.Similarly,theminimumthroughputis1whenallvehicletravelonroute1andallgreentimeisallocatedforroute 1.Therefore,thenetworkisalways unstableifa>2andthecontrolproblembecomestrivialifa<1.So weonlyfocuson thecase1≤a≤2.

Thequeuedynamicsaremodelledasfollows dQ1

dt =

λ

1−s1G1=

λ

1−G1 dQ2

dt =

λ

2−s2G2=

λ

2−2G2

(20)

Inthe restof thissection, we willanalyze thissystematthe userequilibrium withProportionalpolicy,BackPressure policyandtheBackPressurepolicywithusers’routechoiceinfluence.

4.1. Proportionalpolicy

TheProportionalpolicyallocatesgreentimeproportionaltothequeues.Thus,wehave G1

Q1 =

G2

Q2.

(21)

TheequilibriumstateofProportionalpolicy b1=b2 dQ1 dt =0 dQ2 dt =0 G1 Q1 = G2 Q2

G1+G2=1

λ

1+

λ

2=a. (22)

Thesolutionfortheabovesystemis G1=2−a G2=a−1

λ

1=2−a

λ

2=2a−2 Q1=2d

(

2−a

)

Q2=2d

(

a−1

)

b1=2d b2=2d. (23)

TheaveragedelayatequilibriumofProportionalpolicyis

λ

1b1+

λ

2b2

λ

1+

λ

2 =

2d. (24)

4.2. BackPressurepolicy

TheBackPressurepolicywouldallocateallgreentimetotheroutethathavelongerqueue,soatequilibrium,thequeues musthavethesamelengths

Q1=Q2. (25)

TheequilibriumstateofBackPressurepolicy b1=b2 dQ1 dt =0 dQ2 dt =0 Q1=Q2 G1+G2=1

λ

1+

λ

2=a. (26)

Thesolutionfortheabovesystemis G1=2−a G2=a−1

λ

1=2−a

λ

2=2a−2 Q1= 2d

(

2−a

)(

a−1

)

3a−4 Q2= 2d

(

2−a

)(

a−1

)

3a−4 b1= 2d

(

a−1

)

3a−4 b2= 2d

(

a−1

)

3a−4 . (27)

TheaveragedelayatequilibriumofBackPressurepolicyis

λ

1b1+

λ

2b2

λ

1+

λ

2 =

2d

(

a−1

)

3a−4 . (28)

4.3. BackPressurepolicywithinfluenceonusers’routechoice

Letsassume it isfeasible toinfluence onusers insuch a waythat thereare morepeople choose route2. Itcould be eitherincreasethespeed limitofroute2oradvertisingasmallerfreeflow traveltimeofroute2.Undertheeffectofthe newcontrolpolicy,letd0,whered0<d,bethenewfreeflowdelayperceivedbyusers.Thus,userswouldchooseroute1if

theperceivedtraveltimeonroute 1islessthantheperceivedtraveltimeonroute2,whichis Q1

G₁ <d0+2QG2₂,andchoose

route2otherwise.

TheequilibriumstateofBackPressurepolicywithinfluence Q1 G1 = d0+ Q2 2G2 dQ1 dt =0 dQ2 dt =0 Q1=Q2 G1+G2=1

λ

1+

λ

2=a. (29)

Thesolutionfortheabovesystemis G1 =2−a G2 =a−1

λ

1 =2−a

λ

2 =2a−2 Q1 = 2d0

(

2−a

)(

a−1

)

3a−4 Q2 = 2d0

(

2−a

)(

a−1

)

3a−4 b1 = 2d0

(

a−1

)

3a−4 b2 =d+ d0

(

2−a

)

3a−4 . (30)

Notethatnowb1=b2duetotheeffectinfluencedonusers.Theaveragedelayatequilibriumis

λ

1b1+

λ

2b2

λ

1+

λ

2 =2 a

(

a−1

)

d−d0 a−2 3a−4 . (31)

Asanexample,letd=2,d0=1,anda=1.9.SotheaveragedelayintheProportionalpolicy,BackPressurepolicy,and

BackPressurepolicywithusers’routechoiceinfluenceare4,2.12,and1.95respectively.Clearly,forthissimplenetworkwith certainparameters,itisindeedbeneficialtoinfluenceonusers’routechoice.

5. Numericalresults-performanceevaluationandcomparison

In this section we evaluate via simulation the performance of our proposed utility optimization based BackPressure (U-BP) traffic signal control and compare its performance with the P0 policy (Smith, 2011 ) and the BackPressure policy

(Wunderlich et al., 2008; Varaiya, 2013; Wongpiromsarn et al., 2012 ).TheP0 policyallocatesgreentimeproportionallywith

thetotal upstream queuesof thecorresponding phase, thusit will berefer to asProportionalpolicy in thissection. The existingBackPressurepolicyallocatestheentiregreentimetothephasethathasthehighestqueuebacklogdifferences be-tweentheupstreamqueuesanddownstreamqueue,andwillbereferredtoasBackPressurepolicyinthissection.Asforour Utilityoptimization-basedBackPressurepolicy,it attemptsto balancebetweenthehigherweights anda largerinequality (orspatialheterogeneity)oftrafficbysolvingajointoptimizationproblemusinganappropriatechosenutilityfunction.The Proportional,BackPressure,andourproposedpolicyschemesarebrieflysummarizedbelow,respectively.

• Proportionalpolicy(Smith, 2011 )

1. Atthebeginningofeachtimeslot,foreachjunction j∈J,calculatetheweightassociatedwitheachservicephase atthejunctionas w

(

Q

(

t

))

= i∈Ij

ψ

j i

qi

(

t

)

. (32)

2. Giventheseweights,withinthenexttimeslotassignafractionoftimeslottoeachphase

_∈Sjthatisproportional to

pj

(

t

)

= w

(

Q

(

t

))

∈Sjw

(

Q

(

t

))

Fig.2. Networktopology.

• BackPressurepolicy(Wunderlich et al., 2008; Varaiya, 2013; Wongpiromsarn et al., 2012 ):

1.At thebeginning ofeach timeslot,foreachjunction j∈J,calculatetheweightassociatedwitheach servicephase atthejunctionas w

(

Q

(

t

))

= i∈Ij

ψ

j i

qi

(

t

)

− i:ii∈L ¯

θ

ii

(

t

)

qi

(

t

)

. (34)

Hereinthispolicy,

θ

¯ii isfixedandknownprioriforeachoftheintersections.

2.Giventheseweights,assignthewholeservicetimeofthenexttimeslottophase

∗∈Sjwherew∗>w

∀

∈Sj.

• Utilityoptimization-basedBackPressure(U-BP)policyproposedinthispaper:RefertoSection 2.3 fordetails wherethe localGiniindexfunctionisdefinedas

gj

(

t+1

)

=

1

2card

(

I j

)

i∈jqi

(

t+1

)

_{i∈jk=i,k∈j}

|

qi

(

t+1

)

−qk

(

t+1

)

|

, (35)

wherecard(A)isthecardinalityofasetA.Notethattheequation(35) representsalocalGiniindexforjunctionj. WeremarkedthattheGiniindexusetheabovehasafiniteorderofmagnitudeasqueuesizesgrow.Thatiswhenthere arerelatively moderatenumberofcarsatthejunction,weexpectedtheGiniindexwillimpactonthecontrol decisionof the trafficlight. However asinjunctionbecomesbusyit becomesincreasingly pertinentwithcontrol decisionsthat bring thesystembacktostablebehaviour.Thisiswhyitisimportantfortheparametersg_jtoremainbounded.Insummary,our frameworkallowsustobalancebetweentheBackPressure,forstabilitycontrol,andtheaggregatevalueofgj,forthespatial

homogeneityinthe objective.Inequation (7) , thisweightisset to1andessentially representsthetradeofbetweenthe averagequeuelength(stability)andthesecondaryobjectiveassociatingwiththeg_jfunction(i.e.Ginicoefficientasaspatial heterogeneityindicator)whereformoderateloadstheGiniindexwilldominateensuringafairdistributionofqueuelength, andforheaviertrafficBackPressurewillbegintotakeeffectandensurethatsystem-widestabilityisensured.

5.1. Simulationsettings

We compare theperformance of theabove control strategies using asection ofthe Melbourne CBDnetwork that re-assemblesa4-by-4gridnetworktopologyasshowninFig. 2 (a).Itconsistsof16intersectionsand56roads.Forsimplicity we consideralltheroads tobebi-directional(i.e.atotal of112mono-directional links)withthree separatelanesineach directionassignedforthe threepossibletrafficmovements:going straight,turning leftandright, respectively.Eachtraffic light signal atan intersectioncontains2 phases, North-South(trafficmovement number1–6)and East-West(traffic move-mentnumber7–12) (seeFig. 2 (b)).The ingressqueues,wherevehiclesenter thenetwork,areassumedtobe infiniteand representedbyasetoflonglinksofapprox.1200minlength.Fortherestofthelinksinsidethenetwork,thelengthisset tobethesameandequalsto230mallofwhichareapproximatelytheactualdistanceoftheMelbourneCBDnetwork.Since theingressqueuesarelarge,vehicles canenterthenetworkevenwhenthereisa heavycongestioninsidetheMelbourne CBD.The speedlimit issettobe 30kmhinthe innerarea oftheCBD(asquare markedwiththedashlineinFig. 2 (a)), whileitissetto50kmhelsewhere.

Table2

Overallperformance.

Averagetraveltime(second) Averagenumberofroute advicessentpervehicle

Proportional 603 NA BP 555 NA U-BP(5%) 520 2.91 U-BP(10%) 459 2.94 U-BP(20%) 383 2.82 U-BP(30%) 369 2.75 U-BP(50%) 388 2.93 U-BP(100%) 507 3.42

Thedemand ismodeled bytwo consecutivepairs ofa30 min.peakperiodfollows bya 30min.off-peak period.The maintrafficcomesfromboththeWestandNorthandmovestowardtheEastandSouthsides, respectively;whilethe sec-ondarytraffictravelson thereversedirection(i.e.East-West andSouth-North directions).Thedemand ofthemaintraffic are18 and9 carsper minutes duringthepeak andoff-peak period,respectively. The demandof thesecondary traffic is setto be one third of the main traffic (i.e.6 and 3cars per minutes forpeak andoff-peak periods)(see Fig. 2 (a)). The traffic light cycleandsystem unit time step isset to be one minute(i.e. 60 s)where we ignore theswitching timesor transitionbetween phases in all the control schemes in our study. This overhead can be incorporated into the simula-tionby extendingthe phasetimes.Nevertheless,the qualitativeinsights gainedin thissection wouldnotchange bythat extension.

Route guidance (via e.g. advanced traveler information system, ATIS) is implemented in the U-BP scheme so that the queuelengths in each lane can be approximately maintained to be proportional to the optimal turning fractions at an individual intersection. These turning fractions are resulted from distributively solving the problem in (7) using the local utility function in (35) . In the simulation, however, individual vehicle will either follow the route guidance ad-vice, or keep its original route according to an independent compliance probability p(0≤p≤1). If the vehicle indeed chooses to comply with the advice, it will join the appropriate lane in order to make the advised turn at the inter-section andthen followthe new shortestpath to its destination afterward. Regardless of the drivers’ decision, each ve-hicle only receives at most one route guidance advice per link. In certain scenarios where it is impossible to reach the destination by following the advice (e.g. at the edges of the network)or it is too close to the intersection where lane changing isprohibited for safetyreason, vehicle will ignore the advice andfollows its current shortestpath to the destination.

BelowwewilluseanopensourcemicroscopicsimulationpackageSUMO(Simulation ofUrbanMObility)(SUMO, 2013 ) tostudythe performanceof thecontrol strategiesandobtainthe results.Duringsimulation, theexactqueue lengthsare observedandusedtomakecontroldecisioninvariouspolicies.Inparticular,theoptimalcontroldecisionsaresolvedusing Matlab(MATLAB, 2013 )anditsoptimizationtoolsbasedontheactualcontrolalgorithm,andresultsarethenfedbackinto theSUMOsimulationineverytimestep(ortrafficlightcycle).

Resultsare givenintermsofthenetworkthroughput,the totalnumberofvehicles inthenetworkover time,andthe spatialheterogeneity of traffic density in terms of the Gini index after several long simulation runs using the different controlschemes.

5.2.Resultsanddiscussions

Results for the average travel time and the average number of advices per vehicle over its entire journey using the Proportionalpolicy,theoriginal BackPressurepolicy (referredto asBP),andtheproposed utilityoptimization-based Back-Pressurepolicy(referredtoasU-BP)withdifferentcompliancepercentages(inbrackets)areshowninTable 2 .

Observethat theU-BP schemeyieldsa loweraveragetravel timeevenwhen onlyasmallpercentageofvehicles com-plying tothe route guidance advice,e.g. the travel time ofU-BP (5%)is better than boththat of theproportional policy andoriginal BPpolicy.Furthermore,theaveragetravel timedecreases asthe compliancerateincreasesupto 30%ofuser compliancewherethetrendreverses. Itisbecausetheroute guidance(bywayofthecontroller) tendsto diverttraffic to avoidthecongestedinnerarea thusimprovesthenetworkoveralltraveltime.However,whentherearetoomanyvehicles followthe routeguidance advice (e.g.in thecases of50% and100%compliance probability), theoverall benefits areout weightedbythelostintraveltime overthelongerpaths,andthus theoverallnetworkperformancedecreases.Thisis in-herentlythenatureofdistributedapproacheswheretherouteguidanceadviceisonlybasedonlocalinformation.Theroute advicesinthiscasearesolelyforthepurposeoflocalperformancewhichmightdivertraffictolongerpathsresultinginthe degradationoftheoverallnetworkperformance.Ourresultssuggestthatitmightnotbewisetomaketherouteguidance informationaccessibleforeveryvehicle.Thisisinlinewithempiricalobservationsthathavebeenreportedintheliterature (Emmerink et al., 1995; Tsubota et al., 2013 ).Nevertheless,theproposedpolicyshowsuperiorperformancecomparedwith otherdistributedpolicieswithoutturningadvice.

0 2000 4000 6000 8000 0 200 400 600 800 1000 1200 1400 (a) Second

Number of cars inside the network

0 2000 4000 6000 8000 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Second Gini index 0 2000 4000 6000 8000 0 2000 4000 6000 8000 10000 12000 14000 (c) Second

Number of cars reached destination

0 1000 2000 3000 4000 5000 6000 300 400 500 600 700 800 (d) Second

Travel time (second)

Proportional BackPressure U−BP (5%) U−BP (30%) Proportional BackPressure U−BP (5%) U−BP (30%)

Fig.3. Simulationresults.

Thenumberofaverageadvicereceivedbyavehiclevariesdependonthecomplianceprobability(seeTable2 ).Generally, the controller isprone to send more adviceswithlow compliance probability becausewhen a vehicle doesnot comply, anotheradvicewillbesenttodifferentvehicleinordertoachievethedesiredturningfractions.Thetravelroutesalsohave a roleon thenumberofadvicesasvehicleonlonger routeismore likelytoreceivemoreadvices.Thisexplainswhythe lowestnumberofadvicesoccursinthecaseofoptimaltraveltime.

Furtheranalysis ofthelevel ofcongestion, traffic spatialheterogeneity (in termsof theGiniindex), network through-put,andnetworktraveltimeusingtheProportional,theBackPressureandU-BP(with5% and30%complianceprobability) schemes are shownin Fig. 3 (a–d) respectively. Observethat theU-BP (30%)control yields a significant lower numberof totalvehicles presentinthenetwork(Fig. 3 (a)) andthus resultsina highernumberofvehiclesreaching theirdestination (i.e.increasednetworkthroughput,Fig. 3 (c))duringthe simulation.Thisisexplainedby thefactthat intheU-BP control schemewhenthenetworkiscongested,sometrafficwillbedivertedfromtheinnerCBDareatoouterconnectedroadsof theconsiderednetwork.Thisisaresultofsolving theoptimizationproblemaimingtominimizethetrafficspatial hetero-geneity(Fig. 3 (b))inthenetworkthroughtheuseoftheGiniutilityfunction.Furthermore,notethattheaveragetraveltime ofvehiclesthroughthenetworkisalsoimprovedsignificantly(seeFig. 3 (d)).

Fig. 4 shows thecongested linksin thenetwork over time for each ofthe policies. Alink is said tobe congested at a certain time if its queue length is more than 75% of the maximum numberof cars the link can physically hold.The figure showsthat congestionalmost non-exists duringthe low demandperiod fortheU-BP (30%) schemewhile persists throughouttheentiresimulationfortheotherschemes.ItimpliesthattheU-BP(30%)schemeisabletoservemorevehicles duringthepeakperiod,andresolvescongestioninthenetworkquickerresultinginahighernetworkthroughputinoverall. Insummary, ournumericalresultsshow that the proposedutility optimizationbased BackPressure (U-BP)can greatly improveon the network spatialheterogeneity whileretains theimportant stabilityfeature andgood performance ofthe originalBackPressuresignalcontrolstrategy.

0

2000

4000

6000

8000

0

20

40

60

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Fig.4. Congestionofeachcontrolscheme.

6. Conclusion

Inthispaperwe proposed anoveldistributedsignal controlstrategy basedontheBackPressure strategy.Thisstrategy doesnot requireanypriorknowledgeofthetraffic demandandonlyrequiresinformation(i.e.queuesize)that islocalto theintersection.IncontrasttopreviousBackPressure-basedstrategiesinwhichlargequeuestendtobuildupforming un-necessarycongestion,ourschememaximizestheroadnetworkthroughputwhileminimizingtheriskofcongestionforming throughanovelutilityoptimizationmathematicalframework.

Giventheproposedstrategy,wehaveformallyproventhestabilityoftheutilityoptimization-basedBackPressuresignal controlandshowedthroughsimulationandnumericalresultsthatthe newpolicy outperformstheoriginal BackPressure-baseddistributedcontrolschemebothintermsofnetworkthroughputandothercongestionmeasuressuchastraveltime. ThissignificantoutcomehasbeenachievedbyapplyingtheGini-indexasanutilityfunctioninouroptimizationframework tooptimallyobtainthesignal phasingsandtoproactivelyinfluencethedrivers’ routechoicebehavior (viaroute guidance advice)andavoidthecongestiononset.Ourresultsdemonstratedthatthenetworkperformanceimprovessignificantlyeven withlimitedusercompliance,andarelativelysmallpercentageofcompliance(e.g.10%)isenoughtogainsignificantbenefits (e.g.17%reductioninaveragetraveltime).

Acknowledgment

ThisworkwassupportedbytheAustralian Research Council (ARC)FutureFellowshipsgrantFT120100723 .

AppendixA. Proofof Theorem1 Firstletusdefinetheterm

(

t

)

= i∈I qi

(

t

)

∈Sj,j:i∈Ij

ψ

ipj

(

t

)

− i:ii∈L∈Sj,j:i∈Ij

ψ

ipj

(

t

)

θ

¯ii

(

t

)

. (A.1)

Inourfirstlemma,wearguethatitcanberearrangedtogivethesecondpartoftheobjectfunctionin(7) .

Lemma1. Thefollowingequalityholds

(

t

)

= k∈J∈Sk pk

(

t

)

i∈Ik

ψ

i

qi

(

t

)

− i:ii∈L qi

(

t

)

θ

¯ii

(

t

)

. (A.2)

Proof. Theproofisstraightforwardbyrearrangingthesum:

i∈I qi

(

t

)

∈Sj,j:i∈Ij

ψ

ipj

(

t

)

− i:ii∈L∈Sj,j:i∈Ij

ψ

ipj

(

t

)

θ

¯ii

(

t

)

= i∈I∈Sj,j:i∈Ij qi

(

t

)

ψ

ipj

(

t

)

− i∈Ii:ii∈L∈Sj,j:i∈Ij qi

(

t

)

ψ

ipj

(

t

)

θ

¯ii

(

t

)

= k∈J∈Sk pk

(

t

)

i∈Ik

ψ

i

qi

(

t

)

− i:ii∈L qi

(

t

)

θ

¯ii

(

t

)

.

Secondly,weprovetheaboundonthesamplepathsofthequeuesizeprocessusingourassumptionsinSection 3.1 .

Lemma2. ThereexistsaconstantK≥0suchthatthequeuesizeprocessobeysthebound

1 2qi

(

t+1

)

2₋1 2qi

(

t

)

2_≤q i

(

t

)

ai

(

t

)

−si

(

t

)

+ i:ii∈L si

(

t

)

θ

ii

(

t

)

+K. (A.3)

Proof. Fromthequeuelengthprocessequation,wehave

1 2qi

(

t+1

)

2₋1 2qi

(

t

)

2 =1 2

ai

(

t

)

−fi

(

t

)

+ i:ii∈L fi

(

t

)

θ

ii

(

t

)

2 +qi

(

t

)

ai

(

t

)

−fi

(

t

)

+ i:ii∈L fi

(

t

)

θ

ii

(

t

)

=1₂

ai

(

t

)

+ i:ii∈L fi

(

t

)

θ

ii

(

t

)

2 +1₂

fi

(

t

)

2 −fi

(

t

)

ai

(

t

)

+ i:ii∈L fi

(

t

)

θ

ii

(

t

)

+qi

(

t

)

ai

(

t

)

−fi

(

t

)

+ i:ii∈L fi

(

t

)

θ

ii

(

t

)

≤1 2

Amax+DmaxSmax

2 +1 2S 2 max+qi

(

t

)

ai

(

t

)

−min

(

si

(

t

)

,qi

(

t

))

+ i:ii∈L si

(

t

)

θ

ii

(

t

)

(A.4)

Intheinequalityabove,we notthatthefirsttwotermsofexpression(A.4) areboundedabovebyconstantsandthethird termisnegative.

Nowifsi(t)≥qi(t)then,since

−qimin

(

qi,si

)

=−q2i ≤ −qisi+s2i, wehavethat 1 2qi

(

t+1

)

2₋1 2qi

(

t

)

2_≤1 2

Amax+DmaxSmax

2 +1 2S 2 max+qi

(

t

)

ai

(

t

)

−si

(

t

)

+ i:ii∈L si

(

t

)

θ

ii

(

t

)

+S2 max. Ifs_i(t)_<q_i(t)then 1 2qi

(

t+1

)

2₋1 2qi

(

t

)

2_≤1 2

Amax+DmaxSmax

2 +1 2S 2 max+qi

(

t

)

ai

(

t

)

−si

(

t

)

+ i:ii∈L si

(

t

)

θ

ii

(

t

)

. Weseethatinbothcases,si(t)≥qi(t)orsi(t)<qi(t),wehavetherequiredboundwith

K=1 2

Amax+DmaxSmax

2

+3S2max

2 . (A.5)