Continuity of the path delay operator for dynamic network loading with spillback

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by  Ke Han

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PLEASECITETHISARTICLEAS
Han,K.,Piccoli,B.,Friesz,T.L.,2015. Continuityofthepathdelayoperatorfordynamicnetworkloadingwith
spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009.
Continuity of the path delay operator for dynamic network loading with spillback
KeHana,∗,BenedettoPiccolib,TerryL.Frieszc
aDepartmentofCivilandEnvironmentalEngineering,ImperialCollegeLondon,UnitedKingdom.
6 bDepartmentofMathematicalSciencesandCCIB,RutgersUniversity-Camden,USA
1 cDepartmentofIndustrialandManufacturingEngineering,PennsylvaniaStateUniversity,USA.
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Abstract

8 Thispaperestablishesthecontinuityofthepathdelayoperatorsfordynamicnetworkloading(DNL)problemsbased
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on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and pre-

dicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established
]
P routeanddeparturetimechoicesoftravelers. TheLWR-basedDNLmodelisfirstformulatedasasystemofpartial
A differentialalgebraicequations(PDAEs). Wetheninvestigatethecontinuousdependenceofmergeanddivergejunc-
. tionmodelswithrespecttotheirinitial/boundaryconditions,whichleadstothecontinuityofthepathdelayoperator
h
throughthewave-fronttrackingmethodologyandthegeneralizedtangentvectortechnique. Aspartofouranalysis
t
a leadinguptothemaincontinuityresult,wealsoprovideanestimationoftheminimumnetworksupplywithoutresort
m
toanynumericalcomputation. Inparticular,itisshownthatgridlockcanneveroccurinafinitetimehorizoninthe
[ DNLmodel.

2 Keywords: pathdelayoperator,continuity,dynamicnetworkloading,LWRmodel,spillback,gridlock
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4 1. Introduction
0
1. Dynamictrafficassignment(DTA)isthedescriptivemodelingoftime-varyingflowsontrafficnetworksconsistent
0 with traffic flow theory and travel choice principles. DTA models describe and predict departure rates, departure
5 times and route choices of travelers over a given time horizon. It seeks to describe the dynamic evolution of traffic
1 in networks in a fashion consistent with the fundamental notions of traffic flow and travel demand; see Peeta and
:
v Ziliaskopoulos(2001)forsomereviewonDTAmodelsandrecentdevelopments. Dynamicuserequilibrium(DUE)
i oftheopen-looptype,whichisonetypeofDTA,remainsamajormodernperspectiveontrafficmodelingthatenjoys
X
widescholarlysupport. Itcapturestwoaspectsofdrivingbehaviorquitewell: departuretimechoiceandroutechoice
r
a (Friesz et al., 1993). Within the DUE model, travel cost for the same trip purpose isidentical for all utilized route-
and-departure-time choices. The relevant notion of travel cost is effective travel delay, which is a weighted sum of
actualtraveltimeandarrivalpenalties.
In the last two decades there have been many efforts to develop a theoretically sound formulation of dynamic
networkuserequilibriumthatisalsoacanonicalformacceptabletoscholarsandpractitionersalike. Therearetwo
essentialcomponentswithintheDUEmodels: (1)themathematicalexpressionofNash-likeequilibriumconditions;
∗Correspondingauthor
Emailaddresses:[email protected](KeHan),[email protected](BenedettoPiccoli),[email protected](TerryL.
Friesz)
PreprintsubmittedtoTransportationResearchPartB March22,2016
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PLEASECITETHISARTICLEAS
Han,K.,Piccoli,B.,Friesz,T.L.,2015. Continuityofthepathdelayoperatorfordynamicnetworkloadingwith
spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009
and (2) a network performance model, which is, in effect, an embedded dynamic network loading (DNL) problem.
The DNL model captures the relationships among link entry flow, link exit flow, link delay and path delay for any
given set of path departure rates. The DNL gives rise to the notion of path delay operator, which is viewed as a
mapping from the set of feasible path departure rates to the set of path travel times or, more generally, path travel
costs.
PropertiesofthepathdelayoperatorareoffundamentalimportancetoDUEmodels. Inparticular,continuityof
the delay operators plays a key role in the existence and computation of DUE models. The existence of DUEs is
typically established by converting the problem to an equivalent mathematical form and applying some version of
Brouwer’sfixed-pointexistencetheorem; examplesincludeHanetal.(2013c);SmithandWisten(1995);Wieetal.
(2002)andZhuandMarcotte(2000). Alloftheseexistencetheoriesrelyonthecontinuityofthepathdelayoperator.
OnthecomputationalsideofanalyticalDUEmodels,everyestablishedalgorithmrequiresthecontinuityofthedelay
operatortoensureconvergence;anincompletelistofsuchalgorithmsincludethefixed-pointalgorithm(Frieszetal.,
2013),theroute-swappingalgorithm(HuangandLam,2002),thedescentmethod(HanandLo,2003),theprojection
method(HanandLo,2002;Ukkusurietal.,2012),andtheproximalpointmethod(Hanetal.,2015a)
Ithasbeendifficulthistoricallytoshowcontinuityofthedelayoperatorforgeneralnetworktopologiesandtraffic
flow models. Over the past decade, only a few continuity results were established for some specific traffic flow
models. Zhu and Marcotte (2000) use the link delay model (Friesz et al., 1993) to show the continuity of the path
delay operator. Their result relies on the a priori boundedness of the path departure rates, and is later improved by
a continuity result that is free of such an assumption (Han et al., 2012). In Bressan and Han (2013), continuity of
thedelayoperatorisshownfornetworkswhosedynamicsaredescribedbytheLWR-Laxmodel(BressanandHan,
2011;Frieszetal.,2013),whichisavariationoftheLWRmodelthatdoesnotcapturevehiclespillback. Theirresult
also relies on the a priori boundedness of path departure rates. Han et al. (2013c) consider Vickrey’s point queue
model(Vickrey,1969)andshowthecontinuityofthecorrespondingpathdelayoperatorforgeneralnetworkswithout
invokingtheboundednessonthepathdeparturerates.
All of these aforementioned results are established for network performance models that do not capture vehicle
spillback. To the best of our knowledge, there has not been any rigorous proof of the continuity result for DNL
models that allow queue spillback to be explicitly captured. On the contrary, some existing studies even show that
thepathtraveltimesmaydependdiscontinuouslyonthepathdeparturerates,whenphysicalqueuemodelsareused.
Forexample,Szeto(2003)usesthecelltransmissionmodelandsignalcontroltoshowthatthepathtraveltimemay
depend on the path departure rates in a discontinuous way. Such a finding suggests that the continuity of the delay
operatorcouldverywellfailwhenspillbackispresent. Thishasbeenthemajorhurdleinshowingthecontinuityor
identifyingrelevantconditionsunderwhichthecontinuityisguaranteed. Thispaperbridgesthisgapbyarticulating
theseconditionsandprovidingaccompanyingproofofcontinuity.
Thispaperpresents,forthefirsttime,arigorouscontinuityresultforthepathdelayoperatorbasedontheLWR
network model, which explicitly captures physical queues and vehicle spillback. In showing the desired continuity,
weproposeasystematicapproachforanalyzingthewell-posednessoftwospecificjunctionmodels1: amergeand
a diverge model, both originally presented by Daganzo (1995). The underpinning analytical framework employs
thewave-fronttrackingmethodology(Dafermos,1972;HoldenandRisebro,2002)andthetechniqueofgeneralized
tangent vectors (Bressan, 1993; Bressan et al., 2000). A major portion of our proof involves the analysis of the
interactionsbetweenkinematicwavesandthejunctions,whichisfrequentlyinvokedforthestudyofwell-posedness
of junction models; see Garavello and Piccoli (2006) for more details. Such analysis is further complicated by the
factthatvehicleturningratiosatadivergejunctionaredeterminedendogenouslybydrivers’routechoiceswithinthe
DNLprocedure. Asaresult,specialtoolsaredevelopedinthispapertohandlethisuniquesituation.
Asweshalllatersee,acrucialstepoftheprocessaboveistoestimateandboundfrombelowtheminimumnetwork
supply,whichisdefinedintermsoflocalvehicledensitiesinthesamewayasinLebacqueandKhoshyaran(1999).
Infact,ifthesupplysomewheretendstozero(thatis,whentrafficapproachesthejamdensity),thewell-posednessof
thedivergejunctionmayfail,aswedemonstrateinSection4.2.1. Thishasalsobeenconfirmedbytheearlierstudy
ofSzeto(2003), whereawaveofjamdensityistriggeredbyasignalredlightandcausesspillbackattheupstream
junction,leadingtoajumpinthepathtraveltimes.Remarkably,inthispaperweareabletoshowthat(1)ifthesupply
1Well-posednessofamodelreferstothepropertythatthebehaviorofthesolutionhardlychangeswhenthereisaslightchangeintheini-
tial/boundaryconditions.
2
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spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009
isboundedawayfromzero(thatis,trafficisboundedawayfromthejamdensity),thenthedivergejunctionmodelis
wellposed;and(2)thedesiredboundednessofthesupplyisanaturalconsequenceofthedynamicnetworkloading
procedurethatinvolvesonlythemergeanddivergejunctionmodelswestudyhere. Thisisahighlynon-trivialresult
becauseitnotonlyplaysaroleinthecontinuityproof,butalsoimpliesthatgridlockcanneveroccurinthenetwork
loadingprocedureinanyfinitetimehorizon.
ThefinalcontinuityresultispresentedinSection5.4,followinganumberofpreliminaryresultssetoutinprevious
sections.Althoughourcontinuityresultisestablishedonlyfornetworksconsistingofsimplemergeanddivergenodes,
itcanbeextendedtonetworkswithmorecomplextopologiesusingtheprocedureofdecomposinganyjunctioninto
several simple merge and diverge nodes (Daganzo, 1995). Moreover, the analytical framework employed by this
papercanbeinvokedtotreatotherandmoregeneraljunctiontopologiesand/ormerginganddivergingrules,andthe
techniquesemployedtoanalyzewaveinteractionswillremainvalid.
Themaincontributionsmadeinthispaperinclude:
• formulation of the LWR-based dynamic network loading (DNL) model with spillback as a system of partial
differentialalgebraicequations(PDAEs);
• acontinuityresultforthepathdelayoperatorbasedontheaforementionedDNLmodel;
• a novel method for estimating the network supply, which shows that gridlock can never occur within a finite
timehorizon.
Therestofthispaperisorganizedasfollows. Section2recapssomeessentialknowledgeandnotionsregarding
theLWRnetworkmodelandtheDNLprocedure. Section3articulatesthemathematicalcontentsoftheDNLmodel
byformulatingitasaPDAEsystem. Section4introducesthemergeanddivergemodelsandestablishestheirwell-
posedness. Section 5 provides a final proof of continuity and some discussions. Finally, we offer some concluding
remarksinSection6.
2. LWR-baseddynamicnetworkloading
2.1. Delayoperatoranddynamicnetworkloading
Throughout this paper, the time interval of analysis is a single commuting period expressed as [0, T] for some
T > 0. WeletPbethesetofallpathsemployedbytravelers. Foreachpath p ∈ Pwedefinethepathdeparturerate
whichisafunctionofdeparturetimet∈[0, T]:
hp(·): [0, T] → R+
whereR+denotesthesetofnon-negativerealnumbers. Eachpathdepartureratehp(t)isinterpretedasatime-varying
pathflowmeasuredattheentranceofthefirstarcoftherelevantpath,andtheunitforh (t)isvehiclesperunittime.
p
We next define h(·) = {h (·) : p ∈ P} to be a vector of departure rates. Therefore, h = h(·) can be viewed as a
p
vector-valuedfunctionoft,thedeparturetime2.
Thesinglemostcrucialingredienttisthepathdelayoperator,whichmapsagivenvectorofdeparturerateshtoa
vectorofpathtraveltimes. Morespecifically,welet
D (t, h) ∀t∈[t , t ], ∀p∈P
p 0 f
bethepathtraveltimeofadriverdepartingattimetandfollowingpath p,giventhedeparturerateshassociatedwith
allthepathsinthenetwork. WethendefinethepathdelayoperatorD(·)bylettingD(h) = {D (·, h) : p ∈ P},which
p
isavectorconsistingoftime-dependentpathtraveltimesD (t, h).
p
2Fornotationconvenienceandwithoutcausinganyconfusion,wewillsometimesusehinsteadofh(·)todenotepathflowvectors.
3
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spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009
2.2. TheLighthill-Whitham-Richardsmodelonnetworks
WerecapthenetworkextensionoftheLWRmodel(LighthillandWhitham,1955;Richards,1956),whichcaptures
theformation,propagation,anddissipationofspatialqueuesandvehiclespillback. Discussionprovidedbelowrelies
ongeneralassumptionsonthefundamentaldiagramandthejunctionmodel,andinvolvesnoadhoctreatmentofflow
propagation,flowconservation,linkdelay,orothercriticalmodelfeatures.
We consider a road link expressed as a spatial interval [a, b] ⊂ R. The partial differential equation (PDE)
representationoftheLWRmodelisthefollowingscalarconservationlaw
∂ρ(t, x)+∂ f(cid:0)ρ(t, x)(cid:1) = 0 (t, x)∈[0, T]×[a, b] (2.1)
t x
withappropriateinitialandboundaryconditions,whichwillbediscussedindetaillater. Here,ρ(t, x)denotesvehicle
density at a given point in the space-time domain. The fundamental diagram f(·) : [0, ρjam] → [0, C] expresses
vehicle flow at (t, x) as a function of ρ(t, x), where ρjam denotes the jam density, andC denotes the flow capacity.
Throughoutthispaper,weimposethefollowingmildassumptionon f(·):
(F).Thefundamentaldiagram f(·)iscontinuous,concaveandvanishesatρ=0andρ=ρjam.
AnessentialcomponentofthenetworkextensionoftheLWRmodelisthespecificationofboundaryconditions
ataroadjunction. Derivationoftheboundaryconditionsshouldnotonlyobeyphysicalrealism,suchasthatenforced
by entropy conditions (Garavello and Piccoli, 2006; Holden and Risebro, 1995), but also reflect certain behavioral
and operational considerations, such as vehicle turning preferences (Daganzo, 1995), driving priorities (Coclite et
al., 2005), and signal controls (Han et al., 2014). Articulation of a junction model is facilitated by the notion of
Riemann Problem, which is an initial value problem at the junction with constant initial condition on each incident
link. ThereexistanumberofjunctionmodelsthatyielddifferentsolutionsofthesameRiemannProblem. Inoneline
of research, an entropy condition is defined based on a minimization/maximization problem (Holden and Risebro,
1995). In another line of research, the boundary conditions are defined using link demand and supply (Lebacque
andKhoshyaran,1999),whichrepresentthelink’ssendingandreceivingcapacities. Modelsfollowingthisapproach
includeDaganzo(1995);JinandZhang(2003)andJin(2010). ThesolutionofaRiemannProblemisgivenbythe
RiemannSolver(RS),tobedefinedbelow.
2.2.1. TheRiemannSolver
WeconsiderageneralroadjunctionJwithmincomingroadsandnoutgoingroads,asshowninFigure1.
m m+n
. . . . . . J . . . . . .
2 m+2
1 m+1
Figure1:Aroadjunctionwithmincominglinksandnoutgoinglinks.
We denote by I1,...,Im the incoming links and by Im+1,...,Im+n the outgoing links. In addition, for every i ∈
{1,...,m+n},thedynamiconI isgovernedbytheLWRmodel
i
∂ρ(t, x)+∂ f(cid:0)ρ(t, x)(cid:1) = 0 (t, x)∈[0, T]×[a, b] (2.2)
t i x i i i i
wherelinkI isexpressedasthespatialinterval[a, b],andwealwaysusethesubscript‘i’toindicatedependenceon
i i i
thelinkI. Theinitialconditionforthisconservationlawis
i
ρ(0, x) = ρˆ (x) x∈[a, b] (2.3)
i i i i
4
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Noticethattheabove(m+n)initialvalueproblemsarecoupledtogetherviatheboundaryconditionstobespecified
atthejunction. SuchasystemofcouplingequationsiscommonlyanalyzedusingtheRiemannProblem.
Definition2.1. (RiemannProblem)TheRiemannProblematthejunctionJisdefinedtobeaninitialvalueproblem
onanetworkconsistingofthesinglejunction J withmincominglinksandnoutgoinglinks,allextendingtoinfinity,
suchthattheinitialdensitiesareconstantsoneachlink:

ρi(0, x) ≡ ρˆi x∈(−∞, bi], i∈{1, ..., m}
ρ (0, x) ≡ ρˆ x∈[a , +∞), j∈{m+1, ..., m+n}
j j j
whereρˆ ∈[0, ρjam]areconstants,k=1,...,m+n.
k k
A Riemann Solver for the junction J is a mapping that, given any (m+n)-tuple of Riemann initial conditions
(cid:0)ρˆ1,...,ρˆm+n(cid:1), provides a unique (m + n)-tuple of boundary conditions (cid:0)ρ1,...,ρm+n(cid:1) such that one can solve the
initial-boundary value problem for each link, and the resulting solutions constitute a weak entropy solution of the
RiemannProblematthejunction3. AprecisedefinitionoftheRiemannSolverisgivenasfollows.
Definition2.2. (RiemannSolver)ARiemannSolverforthejunctionJwithmincominglinksandnoutgoinglinksis
amapping
(cid:89)m+n(cid:104) (cid:105) (cid:89)m+n(cid:104) (cid:105)
RS : 0, ρjam −→ 0, ρjam
k k
k=1 k=1
(cid:0) (cid:1) (cid:0) (cid:1)
ρˆ1, ..., ρˆm+n (cid:55)→ ρ1, ..., ρm+n
which relates Riemann initial data ρˆ = (cid:0)ρˆ1,...,ρˆm+n(cid:1) to boundary conditions ρ = (cid:0)ρ1,...,ρm+n(cid:1), such that the
followinghold.
(i) ThesolutionoftheRiemannProblemrestrictedoneachlinkI isgivenbythesolutionoftheinitial-boundary
k
valueproblemwithinitialconditionρˆ andboundaryconditionρ ,k=1,...,m+n.
k k
(ii) TheRankine-Hugoniotcondition(flowconservation)holds:
(cid:88)m (cid:88)m+n
f(cid:0)ρ(cid:1) = f (cid:0)ρ (cid:1) (2.4)
i i j j
i=1 j=m+1
(iii) Theconsistencyconditionholds:
RS(cid:2)RS[ρˆ](cid:3) = RS[ρˆ] (2.5)
ThreeconditionsmustbesatisfiedbytheRiemannSolver(RS).Item(i)aboverequiresthattheboundarycondition
oneachlinkmustbeproperlygivensothattheinitialvalueproblemsnotonlyhavewell-definedsolutions,butthese
solutionsmustalsobecompatibleandformasensiblesolutionatthejunction.(2.4)simplystipulatestheconservation
offlowacrossthejunction. (2.5)isadesirablepropertyandissometimesreferredtoastheinvarianceproperty(Jin,
2010).
Remark2.3. ForthesameRiemannProblem,thereexistmanyRiemannSolversthatsatisfyconditions(i)-(iii)above.
Despite their varying forms, most existing Riemann Solvers rely on a flow maximization problem at the relevant
junctionsubjecttoconstraintsrelatedtoturningratio,right-of-way,orsignalcontrols;see(Cocliteetal.,2005;Han
etal.,2014;HoldenandRisebro,1995;JinandZhang,2003)andJin(2010).
3WereferthereadertoHoldenandRisebro(1995)foradefinitionofweakentropysolutionatajunction
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2.2.2. Thelinkdemandandsupply
ForeachlinkI,weletρc bethecriticaldensityatwhichtheflowismaximized. ThedemandD(t)forincoming
i i i
linksI andthesupplyS (t)foroutgoinglinksI aredefinedintermsofthedensityneartheexitandentranceofthe
i j j
link,respectively(LebacqueandKhoshyaran,1999):

Di(t) = Di(cid:16)ρi(t, bi−)(cid:17) = Cfi(cid:0)ρ(t, b−)(cid:1) iiff ρρi((tt,, bbi−−)) <≥ ρρcic (2.6)
i i i i i i

Sj(t) = Sj(cid:16)ρj(t, aj+)(cid:17) = Cf j(cid:0)ρ (t, a +)(cid:1) iiff ρρj((tt,, aaj++)) ≤> ρρccj (2.7)
j j j j j j
Inprose,thedemandrepresentsthemaximumflowatwhichcarscanbedischargedfromtheincominglink;andthe
supply represents the maximum flow at which cars can enter the outgoing link. Notice that the demand and supply
are both expressed as functions of density, and they are always greater than or equal to the fundamental diagram
f(·)or f (·); seeFigure2foranillustration. Inoursubsequentpresentation, withoutcausingconfusionwewilluse
i j
notationsD(t)andD(ρ)interchangeablywheretheformerindicatesthedemandasatime-varyingquantity,andthe
i i
latteremphasizesdemandasafunctionofdensity. Thesameappliestothesupply.
f(!)
D(!)
S(!)
!
0 !c !jam
Figure2:Demandandsupplyasfunctionsofdensity.
3. DynamicnetworkloadingproblemformulatedasaPDAEsystem
TheaimofthissectionistoformulatetheLWR-baseddynamicnetworkloading(DNL)problemasasystemof
partialdifferentialalgebraicequations(PDAEs). TheproposedPDAEsystemusesvehicledensityandqueuesasthe
primary unknown variables, and computes link dynamics, flow propagation, and path delay for any given vector of
departurerates. ThePDAEsystemcapturesvehiclespillbackexplicitly,andaccommodatesawiderangeofjunction
typesandRiemannSolvers.
WeconsideranetworkG(A, V)expressedasadirectedgraphwithAbeingthesetoflinksandVbeingtheset
of nodes. Let P be the set of paths employed by travelers, and W be the set of origin-destination pairs. Each path
p∈Pisexpressedasanorderedsetoflinksittraverses:
p = {I , I , ...,I }
1 2 m(p)
wherem(p)isthenumberoflinksinthispath. Thereareseveralcrucialcomponentsofacompletenetworkloading
model,eachofwhichiselaboratedinasubsectionbelow. Throughouttherestofthispaper,foreachnodev∈V,we
denotebyIvthesetofincominglinksandOvthesetofoutgoinglinks.
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3.1. Within-linkdynamics
ForeachI ∈A,thedensitydynamicisgovernedbythescalarconservationlaw
i
∂ρ(t, x)+∂ (cid:2)ρ(t, x)·v(cid:0)ρ(t, x)(cid:1)(cid:3) = 0 (t, x)∈[0, T]×[a, b] (3.8)
t i x i i i i i
subject to initial condition and boundary conditions to be determined in Section 3.2. The fundamental diagram
f(ρ(cid:1) = ρ ·v(cid:0)ρ(cid:1) satisfies condition (F) stated at the beginning of Section 2.2. In order to explicitly incorporate
i i i i i
drivers’routechoices, forevery p ∈ Psuchthat I ∈ pweintroducethefunctionµp(t, x), (t, x) ∈ [0, T]×[a, b],
i i i i
which represents, in every unit of flow f(ρ(t, x)), the fraction associated with path p. We call these variables path
i i
disaggregation variables (PDV). For each car moving along the link I, its surrounding traffic can be distinguished
i
by path (e.g. 20% following path p , 30% following path p , 50% following path p ). As this car moves, such a
1 2 3
compositionwillnotchangesinceitssurroundingtrafficallmoveatthesamespeedunderthefirst-in-first-out(FIFO)
principle (i.e. no overtaking is allowed). In mathematical terms, this means that the path disaggregation variables,
µp(·, ·),areconstantsalongthetrajectoriesofcars(t, x(t))inthespace-timediagram,where x(·)isthetrajectoryofa
i
movingcaronlinkI. Thatis,
i
dµp(cid:0)t, x(t)(cid:1) = 0 ∀p suchthat I ∈ p,
dt i i
which,accordingtothechainrule,becomes
∂µp(cid:0)t, x(t)(cid:1)+∂ µp(t, x)· d x(t) = 0,
t i x i dt
whichfurtherleadstoanothersetofpartialdifferentialequationsonlinkI:
i
∂µp(t, x)+v(cid:0)ρ(t, x)(cid:1)·∂ µp(t, x) = 0 ∀p suchthat I ∈ p (3.9)
t i i i x i i
Here,ρ(t, x)isthesolutionof(3.8). Thefollowingobviousidentityholds
i
(cid:88)
µp(t, x) = 1 whenever ρ(t, x) > 0 (3.10)
i i
p(cid:51)Ii
where p (cid:51) I means“path pcontains(ortraverses)linkI”,andthesummationappearingin(3.10)iswithrespectto
i i
allsuch p. Byconvention,ifρ(t, x)=0,thenµp(t, x)=0forall p(cid:51) I.
i i i
3.2. Boundaryconditionsatanordinarynode
Forreasonthatwillbecomeclearlater,weintroducetheconceptofanordinarynode. Anordinarynodeisneither
theoriginnorthedestinationofanytrip. WeusethenotationVotorepresentthesetofordinarynodesinthenetwork.
As mentioned earlier, the partial differential equations on links incident to J are all coupled together through
a given junction model, i.e., a Riemann Solver. A common prerequisite for applying the Riemann Solver is the
determinationoftheflowdistribution(turningratio)matrix(Cocliteetal.,2005), whichreliesonknowledgeofthe
PDVsµp(t, b)forallI ∈IJ. Wedefinethetime-dependentflowdistributionmatrixassociatedwithJ:
i i i
AJ(t) = (cid:8)αJ(t)(cid:9)∈[0, 1]|IJ|+|OJ| (3.11)
ij
where by convention, we use subscript i to indicate incoming links, and j to indicate outgoing links. Each element
αJ(t)representstheturningratiosofcarsdischargedfromI thatenterdownstreamlinkI .Then,forallpthattraverses
ij i j
J,thefollowingholds.
(cid:88)
αJ(t)= µp(t, b) ∀I ∈IJ, I ∈OJ (3.12)
ij i i i j
p(cid:51)Ii,Ij
(cid:80)
ItcanbeeasilyverifiedthatαJ(t)∈[0, 1]and αJ(t)≡1accordingto(3.10).
ij j ij
WearenowreadytoexpresstheboundaryconditionsfortheordinaryjunctionJ ∈Vo. Let
|I(cid:89)J|+|OJ| |I(cid:89)J|+|OJ|
RSAJ : [0, ρjam]→ [0, ρjam]
k k
k=1 k=1
7
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spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009
be a given Riemann Solver. Notice that the dependence of the Riemann Solver on AJ has been indicated with a
superscript. TheboundaryconditionsforPDEs(3.8)read
ρ (t, b ) = RSAJ(cid:104)(cid:0)ρ(t, b−)(cid:1) , (cid:0)ρ (t, a +)(cid:1) (cid:105) ∀I ∈IJ (3.13)
k k k i i Ii∈IJ j j Ij∈OJ k
ρ(t, a) = RSAJ(cid:104)(cid:0)ρ(t, b−)(cid:1) , (cid:0)ρ (t, a +)(cid:1) (cid:105) ∀I ∈OJ (3.14)
l l l i i Ii∈IJ j j Ij∈OJ l
whereRSAJ[·]denotesthek-thcomponentofthemapping,k=1, ..., |IJ|+|OJ|.
k
Remark3.1. Intuitively,(3.13)-(3.14)meanthat,giventhecurrenttrafficstates(cid:0)ρ(t, b−)(cid:1) and(cid:0)ρ (t, a +)(cid:1)
i i Ii∈IJ j j Ij∈OJ
adjacenttothejunctionJ,theRiemannSolverRSAJ specifies,foreachincidentlinkI orI,thecorrespondingbound-
k l
aryconditionsρ (t, b )orρ(t, a). Inprose,ateachtimeinstancetheRiemannSolverinspectsthetrafficconditions
k k l l
near the junction, and proposes the discharging (receiving) flows of its incoming (outgoing) links. Such a process
isbasedontheflowdistributionmatrix AJ andoftenreflectstrafficcontrolmeasuresatjunctions. Furthermore,the
Riemann Solver operates with knowledge of every link incident to the junction, thus the boundary condition of any
relevantlinkisdeterminedjointlybyallthelinksconnectedtothesamejunction. Therefore, theLWRequationson
all the links are coupled together through this mechanism. For this reason the LWR-based DNL models are highly
challenging,boththeoreticallyandcomputationally.
TheupstreamboundaryconditionsassociatedwithPDEs(3.9)are:
f(cid:0)ρ(t, b)(cid:1)·µp(t, b)
µpj(t, aj) = i if (cid:0)ρi(t, ai)(cid:1) i ∀p suchthat {Ii, Ij}⊂ p, ∀Ij ∈OJ (3.15)
j j j
where the numerator f(cid:0)ρ(t, b)(cid:1)·µp(t, b) expresses the exit flow on link I associated with path p, which, by flow
i i i i i i
conservation,isequaltotheenteringflowoflinkI associatedwiththesamepath p;thedenominatorrepresentsthe
j
totalenteringflowoflinkI .
j
Remark3.2. Unlikethedensity-basedPDE,thePDV-basedPDEdoesnothaveanydownstreamboundarycondition
due to the fact that the traveling speeds of the PDVs are the same as the car speeds (they can be interpreted as
Lagrangianlabelsthattravelwiththecars);thusinformationregardingthePDVsdoesnotpropagatebackwardsor
spillsovertoupstreamlinks.
3.3. Flowdistributionatoriginordestinationnodes
Weconsideranodev ∈ Vthatiseithertheoriginorthedestinationofsomepath p. Oneimmediateobservation
isthattheflowconservationconstraint(2.4)nolongerholdsatsuchanodesincevehicleseitherare‘generated’(ifv
isanorigin)or‘vanish’(ifvisadestination). Asimpleandeffectivewaytocircumventthisissueistointroducea
virtuallink. Avirtuallinkisanimaginaryroadwithcertainlengthandfundamentaldiagram,andservesasabuffer
betweenanordinarynodeandanorigin/destination; seeFigure3foranillustration. Byintroducingvirtuallinksto
theoriginalnetwork,weobtainanaugmentednetworkG(A˜, V˜)inwhichallroadjunctionsareordinary,andhence
fallwithinthescopeoftheprevioussection.
s
t
J J
Figure3:Illustrationofthevirtuallinks. Left:avirtuallinkconnectinganorigin(s)toanordinaryjunctionJ. Right:avirtuallinkconnectinga
destination(t)toanordinaryjunctionJ.
LetusdenotebySthesetoforiginsintheaugmentednetworkG(A˜, V˜). Foranys∈S,wedenotebyPs ⊂Pthe
setofpathsthatoriginatefroms,andbyI thevirtuallinkincidenttothisorigin. Foreach p∈Pswedenotebyh (t)
s p
8
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PLEASECITETHISARTICLEAS
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thedeparturerate(pathflow)along p. Itisexpectedthatabuffer(point)queuemayformat sincasethereceiving
capacityofthedownstreamI isinsufficienttoaccommodateallthedeparturerates(cid:80) h (t). Forthisbufferqueue,
s p∈Ps p
denotedq (t),weemployaVickrey-typedynamic(Vickrey,1969);thatis,
s

ddtqs(t) = p(cid:88)∈Pshp(t)−Smsi(nt)(cid:110)p(cid:88)∈Pshp(t), Ss(t)(cid:111) iiff qqss((tt)) >= 00 (3.16)
whereS (t)denotesthesupplyofthevirtuallink I . Theonlydifferencebetween(3.16)andVickrey’smodelisthe
s s
time-varyingdownstreamreceivingcapacityprovidedbythevirtuallink. 4
It remains to determine the dynamics for the path disaggregation variables (PDV). More precisely, we need to
determine µp(t, a ) for the virtual link I where p ∈ Ps, and x = a is the upstream boundary of I . This will be
s s s s s
achievedusingtheVickrey-typedynamic(3.16)andtheFIFOprinciple. Specifically,wedefinethequeueexittime
function λ (t) where t denotes the time at which drivers depart and join the point queue, if any; λ (t) expresses the
s s
timeatwhichthesamegroupofdriversexitthepointqueue. Clearly,FIFOdictatesthat
(cid:90) t (cid:88)h (τ)dτ = (cid:90) λs(t) f (cid:0)ρ (τ, a )(cid:1)dτ (3.17)
p s s s
0 p∈Ps 0
wherethetwointegrandsontheleftandrighthandsidesoftheequationareflowenteringthequeueandflowleaving
thequeue,respectively. Wemaydeterminethepathdisaggregationvariablesas:
µsp(cid:0)λs(t), as(cid:1) = (cid:80) hp(ht)(t) ∀p∈Ps (3.18)
q∈Ps q
Noticethat,if(cid:80) h (t)=0,thentheflowleavingthepointqueueattimeλ (t)isalsozero;thusthereisnoneedto
q∈Ps q s
determinethepathdisaggregationvariables. Therefore,theidentity(3.18)iswelldefinedandmeaningful.
3.4. Calculationofpathtraveltimes
Withallprecedingdiscussions,wemayfinallyexpressthepathtraveltimes,whicharetheoutputsofacomplete
DNLmodel.Thepathtraveltimeconsistsoflinktraveltimespluspossiblequeuingtimeattheorigin.Mathematically,
thelinkexittimefunctionλ(t)foranyI isdefined,inawaysimilarto(3.17),as
i i
(cid:90) t f(cid:0)ρ(τ, a)(cid:1)dτ = (cid:90) λi(t) f(cid:0)ρ(τ, b)(cid:1)dτ (3.19)
i i i i i i
0 0
Forapathexpressedas p={I , I , ..., I },thetimetotraverseitiscalculatedas
1 2 m(p)
λ ◦λ ◦λ ...◦λ (t) (3.20)
s 1 2 m(p)
.
where f ◦g(t) = g(f(t))meansthecompositionoftwofunctions. Thisisduetotheassumptionthatcarsleavingthe
previouslink(orqueue)immediatelyenterthenextlinkwithoutanydelay.
3.5. ThePDAEsystem
WearenowreadytopresentagenericPDAEsystemforthedynamicnetworkloadingprocedure. Letusbeginby
summarizingsomekeynotations.
4Therighthandsideoftheordinarydifferentialequation(3.16)isdiscontinuous.Ananalyticaltreatmentofthisirregularequationisprovided
byHanetal.(2013a,b)usingthevariationalformulation.
9
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PLEASECITETHISARTICLEAS
Han,K.,Piccoli,B.,Friesz,T.L.,2015. Continuityofthepathdelayoperatorfordynamicnetworkloadingwith
spillback. TransportationResearchPartB,DOI:10.1016/j.trb.2015.09.009
G(A, V) theoriginalnetworkwithlinksetAandnodesetV;
VL thesetofvirtuallinks;
G(A˜, V˜) theaugmentednetworkincludingvirtuallinks;
S thesetoforiginsinG(A˜, V˜);
Ps thesetofpathsoriginatingfroms∈S;
Vo thesetofordinaryjunctionsinG(A˜, V˜);
IJ thesetofincominglinksofajunctionJ ∈Vo;
OJ thesetofoutgoinglinksofajunctionJ ∈Vo;
AJ(t) theflowdistributionmatrixassociatedwithjunctionJ;
RSAJ theRiemannSolverforjunctionJ,whichdependsonAJ.
WealsolistsomekeyvariablesofthePDAEsystembelow.
h (t) thepathdepartureratealong p∈P;
p
ρ(t, x) thevehicledensityonlinkI ∈A˜;
i i
µp(t, x) theproportionofflowonlinkI associatedwithpath p(pathdisaggregationvariable);
i i
q (t) thepointqueueattheorigins∈S;
s
λ (t) thepointqueueexittimefunctionatorigins∈S.
s
Givenanyvectorofpathdepartureratesh = (cid:0)h (·) : p ∈ P),theproposedPDAEsystemforcalculatingpathtravel
p
timesissummarizedasfollows.

dqdst(t) = p(cid:88)∈Pshp(t)−Smsi(nt)(cid:110)p(cid:88)∈Pshp(t), Ss(t)(cid:111) qqss((tt))>=00 ∀s∈S (3.21)
(cid:90) t (cid:88)h (τ)dτ = (cid:90) λs(t) f (cid:0)ρ (τ, a )(cid:1)dτ ∀s∈S (3.22)
p s s s
0 p∈Ps 0
(cid:90) t f(cid:0)ρ(τ, a)(cid:1)dτ = (cid:90) λi(t) f(cid:0)ρ(τ, b)(cid:1)dτ ∀I ∈A˜ (3.23)
i i i i i i i
0 0
∂ρ(t, x)+∂ (cid:2)ρ(t, x)·v(cid:0)ρ(t, x)(cid:1)(cid:3)=0 (t, x)∈[0, T]×[a, b] (3.24)
t i x i i i i i
∂µp(cid:0)t, x(cid:1)+v(cid:0)ρ(t, x)(cid:1)·∂ µp(cid:0)t, x(cid:1)=0 (t, x)∈[0, T]×[a, b] (3.25)
t i i i x i i i
µsp(cid:0)λs(t), as(cid:1) = (cid:80) hp(ht)(t) ∀s∈S, p∈Ps (3.26)
q∈Ps q
f(cid:0)ρ(t, b)(cid:1)·µp(t, b)
µpj(t, aj) = i if (cid:0)ρi(t, ai)(cid:1) i ∀p⊃{Ii, Ij} (3.27)
j j j
(cid:88)
AJ(t)=(cid:8)αJ(t)(cid:9), αJ(t)= µp(t, b) ∀I ∈IJ, I ∈OJ (3.28)
ij ij i i i j
p(cid:51)Ii,Ij
ρ (t, b ) = RSAJ(cid:104)(cid:0)ρ(t, b−)(cid:1) , (cid:0)ρ (t, a +)(cid:1) (cid:105) ∀I ∈IJ (3.29)
k k k i i Ii∈IJ j j Ij∈OJ k
ρ(t, a) = RSAJ(cid:104)(cid:0)ρ(t, b−)(cid:1) , (cid:0)ρ (t, a +)(cid:1) (cid:105) ∀I ∈OJ (3.30)
l l l i i Ii∈IJ j j Ij∈OJ l
D (t, h) = λ ◦λ ◦λ ...◦λ (t) ∀p∈P, ∀t∈[0, T] (3.31)
p s 1 2 m(p)
10