Non-unique flows in macroscopic first-order intersection models

Transportation Research Part B 46 (2012) 343–359

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Transportation Research Part B

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Non-unique flows in macroscopic first-order intersection models

Ruben Corthout a,⇑, Gunnar Flötteröd b, Francesco Viti a, Chris M.J. Tampère a

a Department of Mechanical Engineering, CIB/Traffic & Infrastructure, Katholieke Universiteit Leuven, Celestijnenlaan 300A, PO Box 2422, 3001 Leuven, Belgiumb Division of Traffic and Logistics, KTH – Royal Institute of Technology, Teknikringen 72, 11428 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 19 May 2011Received in revised form 27 October 2011Accepted 27 October 2011

Keywords:Dynamic Network LoadingFirst-order macroscopic intersection modelInternal supply constraintsPriority parametersSolution non-uniqueness

0191-2615/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.trb.2011.10.011

⇑ Corresponding author. Tel.: +32 (0)16 32 96 14;E-mail address: [email protected]

a b s t r a c t

Currently, most intersection models embedded in macroscopic Dynamic Network Loading(DNL) models are not well suited for urban and regional applications. This is so because so-called internal supply constraints, bounding flows due to crossing and merging conflictsinherent to the intersection itself, are missing. This paper discusses the problems that ariseupon introducing such constraints. A general framework for the distribution of (internal)supply is adopted, which is based on the definition of priority parameters that describethe strength of each flow in the competition for a particular supply. Using this representa-tion, it is shown that intersection models – with realistic behavioral assumptions, and insimple configurations – can produce non-unique flow patterns under identical boundaryconditions. This solution non-uniqueness is thoroughly discussed and approaches onhow it can be dealt with are provided. Also, it is revealed that the undesirable model prop-erties are not solved – but rather enhanced – when diverting from a point-like to a spatialmodeling approach.

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1. Introduction

In first-order macroscopic Dynamic Network Loading (DNL) models, the link model provides the demand of incominglinks and the supply of outgoing links as constraints to the intersection (or node) model. These intersection models are typ-ically point-like, i.e. without physical dimensions, combining all constraints into a strongly coupled set of equations.

The function of the intersection model is two- or three-fold. Firstly, it seeks for a consistent solution in terms of flowstransferred over the intersection, accounting for all demand and supply constraints of the adjacent links. Secondly, it imposesadditional constraints due to limited supply of conflict points on the intersection itself; these are called internal supply con-straints. The latter typically do not apply to highway junctions but can be decisive at regional and urban intersections. Thesetwo functions determine the flows over the intersection and affect the congestion dynamics in the adjacent links. Imposingadditional delay at the intersection itself, e.g. based on delay formulas such as those of Akcelik and Troutbeck (1991) andWebster (1958), constitutes a third function in models that do not explicitly capture stochastic queue formations in un-der-saturated conditions (e.g. Durlin and Henn, 2005; Yperman et al., 2007). In this paper, the focus is on the first two func-tions, which determine the flows.

This paper presents a general framework to include internal supply constraints into the intersection model analogous tohow external supply constraints of outgoing links are typically treated in the state-of-the-art. The implications of this modelextension are thoroughly discussed, most importantly the possible occurrence of solution non-uniqueness. Hence, this paperis conceptually similar to Carey (1992), who shows that the First-In–First-Out (FIFO) behavior of traffic can lead to

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344 R. Corthout et al. / Transportation Research Part B 46 (2012) 343–359

non-convexity in dynamic traffic assignment (DTA) and also proposes how to practically deal with this problem. As such, thispaper contributes to the analysis of solution non-uniqueness in the DTA problem (e.g. Carey, 1992; Daganzo, 1998).

The remainder of this article is organized as follows. Section 1.1 revisits the state-of-the-art in macroscopic first-orderintersection modeling, deploying a more general formulation of the model framework of Tampère et al. (2011) and Flötterödand Rohde (2011). Section 2.1 inquires into the various behavioral assumptions that can be made regarding intersection con-flicts. Section 2.2 then discusses the translation of these behavioral considerations into (the distribution of) internal supplyconstraint functions and how these can be embedded into the model framework presented in Section 1.1. The additionalmodeling formulations in Section 2 are intentionally kept general. While allowing various ways of detailing, it enables a gen-eral elaboration on the observation of non-unique solutions of the intersection model in Sections 3 and 4. In Section 3.1, it isshown by means of a simple example that realistic behavioral assumptions can lead to multiple flow solutions under iden-tical boundary conditions. In Section 3.2, conditions for solution (non-)uniqueness are formulated. In Section 3.3, the impli-cations of these findings are discussed; the main conclusion being that the uniqueness condition for the priority parametersin the model does not (always) intuitively correspond to realistic driver behavior. Therefore, careful consideration is neededon how to remedy the non-uniqueness and obtain one unambiguous result from the model. Section 3.4 suggests some ap-proaches. Finally, Section 4 discusses spatial intersection modeling. It is shown that also spatial models can produce differentsolutions under instantaneously identical boundary conditions, and that the result is inherently determined based on thehistory of flows. Moreover, some unrealistic and undesirable model behavior is identified. Finally, Section 5 summarizesthe article.

1.1. State-of-the-art on first-order DNL intersection models

Following an extensive review of the state-of-the-art, Tampère et al. (2011) list a set of requirements with which mac-roscopic intersection models in DNL should comply in order to achieve general consistency with the fundamental modelingprinciples of first-order traffic flow theory:

1. General applicability to any number of incoming links and outgoing links.2. Non-negativity of flows.3. Conservation of vehicles.4. Compliance with demand constraints and (internal) supply constraints.5. Flow maximization from the users’ perspective.6. Conservation of turning fractions (CTF).7. Compliance with the invariance principle of Lebacque and Khoshyaran (2005).

The first four requirements are straightforward and well-known in the literature. The latter three deserve more explana-tion. For this, we introduce the following notation: incoming (upstream) links of the considered intersection are indexed by iand outgoing (downstream) links are indexed by j. The demand (supply) of link i (j) is denoted by Si (Rj). The total demand Si

consists of partial demands Sij in the various outgoing directions, defined by the turning fractions fij so that Sij = fijSi. The flowsent by link i to link j is written as qij, and the total outflow of i is qi ¼

Pjqij.

– Flow maximization from the users’ perspective:Each flow should be actively constrained by either demand or (internal) supply. Behaviorally, this reflects the assumptionof drivers trying to advance whenever possible. This corresponds to individual flow maximization. Global flow maximiza-tion would imply a behaviorally unrealistic cooperation of drivers.

– Conservation of turning fractions (CTF):
The outflow composition of a link i (in terms of partial flows qij) must be identical to its demand composition (in terms ofpartial demands Sij). Consequently, all outflows of i are coupled through the turning fractions that are obtained from thedemand composition:
fij ¼Sij

Si¼

qij

qið1Þ

CTF implies First-In–First-Out (FIFO) at the intersection level (Daganzo, 1995). Due to CTF, the intersection model’s solu-tion is unambiguously defined by the incoming flows qi; the partial flows are then derivable as qij = fijqi.

– Compliance with the invariance principle of Lebacque and Khoshyaran (2005):The invariance principle ensures the compatibility of the intersection model with link traffic flow dynamics. If the inter-section model produces qi < Si, then i enters a congested regime. As a consequence of traffic flow dynamics, Si increasesafter some infinitesimally small time increment to the link’s capacity Ci. Any intersection model that predicts a differentoutcome for qi because of this change from Si to Ci contradicts its own initial solution and thus violates the invarianceprinciple. The solution of the intersection model should therefore be invariant to replacing Si by Ci if qi is supply con-strained (qi < Si). Analogously, if qj < Rj the solution should be invariant to an increase of Rj to Cj.

R. Corthout et al. / Transportation Research Part B 46 (2012) 343–359 345

As discussed in Tampère et al. (2011), supply constraint interaction rules (SCIRs) must be added that respect the abovelisted requirements. The SCIR represent the aggregate driver behavior at a (congested) intersection in that they combine thedemands and the (distribution of) supplies into a mutually consistent solution; therewith completing the first function of theintersection model. A general formulation of the SCIR presented earlier in Tampère et al. (2011) and Flötteröd and Rohde(2011) – so far without considering internal supplies - is given below. This formulation thus constitutes more a summarydescription of previous contributions than a new modeling proposal. For a clear understanding, the demand and supply con-straints are formulated first:

– The demand constraints imply:

1 Weaij or ai

any two2 Ind

i e Uj in

qi 6 Si 8i ð2Þ

– The supply constraint functions bRj express how the given supply Rj (in veh/h) of an outgoing link j limits the partial flowsfijqi that compete for it:

X
bRjðqÞ ¼i
fijqi � Rj 6 0 8j ð3Þ

where the vector q is composed of all qi.Based on this, the SCIR can be formulated as follows:

1. Write ‘‘i is constrained by bRj’’ as i#bRj and define the sets Uj that collect all incoming links i being constrained by outgoinglink j through

Uj ¼ fiji#bRjg ð4Þ

Analogously, sets Ui could be introduced for demand constrained links. However, this is more notationally cumbersomethan convenient. Rather, it is simply stated that for a demand constrained link qi = Si. Otherwise, qi < Si and i is supply con-strained. In this case, i must belong to a set Uj. From (3) follows that only i that wish to send flow to j (i.e. fij > 0) can belongto Uj. Also, the requirement of flow maximization implies that a constraint can only bind some flow(s) if it is completelyused up:

Uj ¼ ; () bRj < 0

Uj – ; () bRj ¼ 0 8jð5Þ

2. Finite, strictly positive1 priority parameters aij determine the rightful share of each competing i of the supplies Rj. Due toCTF, an bRj affects the total flows qi and not just the partial flows qij. Each supply constrained flow qi takes at least its rightfulshare of its determinative constraint,2 so that

qi

qi0P

aij

ai0 j8i 2 Uj; 8i0jfi0 j > 0 ð6Þ

Finally, compliance with the invariance principle requires that these aij do not depend on the demands Si.

The SCIR is fully defined by (4)–(6). Some implications and interpretations are given below.If two incoming links are constrained by the same bRj, (6) holds for both, such that the priority ratio directly determines

the flows:

qi

qi0¼ aij

ai0j8i; i0 2 Uj ð7Þ

Inserting (7) in (3) yields

8i0 2 Uj : f i0 jqi0 ¼fi0jai0 jPi2Uj

fijaij

eRj

with : eRj ¼ Rj �XiRUj

fijqi

ð8Þ

define the priority parameters aij (and aik defined in Section 2.2 for internal conflicts k) strictly positive for mathematical convenience. Algorithmically,k equal to zero should be allowable. In any case, however, at most one aij (aik) can be zero per j (k). Indeed, a well-defined ranking should exist between

movements that are in conflict.eed, if other links i0 are more restricted by some other constraint, these i0 are unable to use up their rightful share of Rj. Consequently, the share of Rj ofcreases (individual flow maximization).


That is, the competitive strength of movement ij for supply Rj is given by fijaij. As such, aij is the maximal strength (iffij = 1). It is indeed logical that the turning fractions fij play an explicit role because the competitive strength of ij for Rj obvi-ously decreases with fij. The priority parameters aij generalize the merge parameters of Daganzo (1995). For more details, seeTampère et al. (2011), Flötteröd and Rohde (2011) and Sections 2.2 and 3.1 where examples are given.

The generally formulated SCIR above are consistent with the intersection models of Tampère et al. (2011), Flötteröd andNagel (2005), Flötteröd (2008), Gentile (2010), and Flötteröd and Rohde (2011). Also, its core – the distribution of supplythrough the proportionality of aij (representing, e.g. priorities, number of (turning) lanes or link capacities) – is shared bymost other existing models. Compared to the above SCIR, they exhibit some small incompleteness or flaw, however, whichmakes them violate some requirement – as discussed in Tampère et al. (2011). To the best of our knowledge, the only modelcompliant with all requirements that has realistic SCIR substantially different from the above is that of Gibb (2011).3

The second function of intersection models, the infliction of additional supply constraints due to conflicts within theintersection itself, has not yet been considered. Although these internal supply constraints are largely responsible for thetraffic problems in many regional and urban networks, they are rarely considered in state-of-the-art models. Intersectionmodels that do include internal supply constraints are presented by Ngoduy et al. (2005), Yperman et al. (2007), Chevallierand Leclercq (2007) and Raadsen et al. (2010) (in the traditional point-like form) and by Buisson et al. (1995) and Chen et al.(2008) (for spatial models). However, none of these models comply with all of the above listed requirements. Only the modelof Flötteröd and Rohde (2011) does comply, but a uniquely converging solution algorithm in the presence of internal supplyconstraints is presented only for the special case where the incoming links can be ranked such that the flows of higher-ranked links are independent from those of lower-ranked links. Indeed, the impossibility to design a solution algorithm thatis proven to converge to a unique solution leads to the (possibly seminal) investigation of solution uniqueness presented inthat article. Motivated by the identification of a simple (three-legged) configuration that already yields non-unique flows,the authors provide a heuristic algorithm with guaranteed convergence towards a compromise solution.

2. From driver behavior to internal supply constraints

Internal supply constraints arise from:

– Traffic controls (traffic lights, ramp metering).– Crossing conflicts:� At (un)signalized intersections, between movements originating from different incoming links, heading towards dif-

ferent outgoing links.� With non-motorized traffic (pedestrians, cyclists).

– Merging conflicts:� Between flows merging into an outgoing link.� Between flows entering a roundabout, merging with flows already on the roundabout.

Conflicts between flows merging into the same outgoing link – included here for completeness – are typically consideredas external constraints in the form of the outgoing link’s supply. In fact, they can be considered both internal and external: onthe one hand, they can be dominated by congestion spilling back from the outgoing link; on the other hand, drivers (usually)evaluate crossing and merging conflicts simultaneously before traversing the intersection. In this paper, they are consideredexternal. Furthermore, only motorized traffic is considered. Also conflicts due to traffic controls are not discussed since theseare either – in case of non-adaptive control – quite straightforward to deal with or – in case of adaptive control – to the bestof our knowledge, not included in state-of-the-art DNL models.

2.1. Driver behavior in crossing and merging conflicts

In the following, three types of driver behavior are identified in solving intersection conflicts: absolute compliance to pri-ority rules, limited compliance to priority rules and turn-taking. Different driver behavior can be observed depending on traf-fic load, intersection type and geometry, and personal and cultural differences.

Absolute compliance to priority rules covers cases where an imposed ordering of the movements, rendering priority tosome major movements over other minor movements, is strictly obeyed by drivers. According to the traffic rules, this behav-ior should apply to merging and crossing conflicts at priority-controlled (and the remaining conflicts at signal-controlled)intersections under all circumstances. In reality, it can be observed only at low traffic volumes (under-saturated).

As traffic volumes increase, priority rules are less strictly obeyed due to politeness from prioritized and forcing from non-prioritized drivers. This limited compliance commonly arises as the minor movements are in nearly- to over-saturated con-ditions. Empirical documentation is given for roundabout merging in Troutbeck and Kako (1999) and for crossing and merg-ing flows in Brilon and Miltner (2005).

3 However, extending this model to include internal conflicts seems less straightforward, at least for driver behavior other than turn-taking.


In over-saturated conditions, typically some sort of turn-taking behavior occurs – in particular for merging conflicts –which can be regarded as the alternating use of the available supply by all competing movements. Hereby, the ‘turns’ oropportunities that are available for each movement are determined by the movements’ outflow capacities, possibly reducedby other conflicts (crossing, control). Empirical studies of merging behavior in over-saturated conditions at general intersec-tions are lacking. Cassidy and Ahn (2005) find that congested freeway merging occurs in a fixed, site-dependent ratio, inde-pendently of the available downstream supply. This is further confirmed in Bar-Gera and Ahn (2010), who show that thisfixed ratio is well approximated by the ratio of the number of lanes of the incoming links of the merge. From a smaller dataset, Ni and Leonard (2005) conclude that merging follows the ratio of the capacities. Finally, All-Way-Stop-Controlled(AWSC) intersections actually prescribe turn-taking for both merging and crossing conflicts. Although for AWSC intersec-tions this type of behavior is thus in compliance with the priority rules, the behavior at AWSC conflicts is categorized asturn-taking and the concepts of limited and absolute compliance are preserved for situations as described above.

Naturally, the aggregate driver behavior to be captured by macroscopic intersection models may be a mixture of differenttypes of behavior. In Fig. 1, a suggestion is provided on how aggregate driver behavior at various intersection conflicts couldbe classified. This classification is partially derived from available empirical studies, as discussed above. Since not all types ofconflicts have been (sufficiently) documented, additional assumptions are necessary. As such, until validated empirically,this classification is susceptible to discussion.

2.2. Modeling internal supply constraints in DNL intersection models

This section discusses how to model the observed behavior in internal supply constraints in DNL intersection models.External constraints (from the outgoing link supplies Rj) are generally modeled via SCIR that prescribe a proportional distri-bution (as discussed in Section 1.1). More complex theories exist to determine the restriction on minor movements fromyielding to higher prioritized movements in crossing and merging conflicts. The best-known is gap acceptance theory (seeChapters 8 and 9 in Gartner et al. (2000)); another is conflict theory (Brilon and Wu, 2001). Most often these theories assumeabsolute compliance with priority rules, but modifications exist for limited compliance (Troutbeck and Kako, 1999; Brilonand Miltner, 2005).

When embedding these theories into DNL intersection models, two particular issues need to be considered. Firstly, boththeories are designed for free flowing conditions on the major links. Gap acceptance theory in particular loses some validityin nearly-saturated and definitely in over-saturated conditions. Conflict theory is, at least theoretically, more easily extend-able to over-saturated conditions; but this has not been validated empirically. Secondly, the major flows are consideredexternally given. In a DNL intersection model, however, the major flows may also be subject to (internal) supply constraintsand therefore not known beforehand. In that respect, the DNL intersection models of Yperman et al. (2007) and Ngoduy et al.(2005) translate gap acceptance theory too directly into internal supply constraint functions, as they restrict any minor flowby the demand from the major movements. As explained in Section 1.1, this can lead to a violation of the invariance principle.Moreover, it may cause an overestimation of the constraint imposed on the minor flow (see Flötteröd and Rohde, 2011) if themajor flows themselves are supply constrained. The model of Flötteröd and Rohde (2011) treats this issue by making theminor flow dependent on the actual flows of the major movements instead of the demand. Although in Chevallier and Lecl-ercq (2007) it is shown how the less severe restriction of the minor flow due to limited (rather than absolute) compliance canbe included in the definition of the internal supply constraint function, this approach more naturally relates to absolute com-pliance. Indeed, the minor flow is limited by the major flow via their shared internal conflict, but not the other way around.We propose in general terms the following, more flexible formulation of the internal supply constraint function for an inter-nal conflict k:

dNkðqÞ 6 0 ð9Þ

where q is the vector of all qi.The general formulation (9) represents an internal supply constraint function bNk that constrains all competing flows qi

that participate in the internal conflict k through fikSi > 0; in contrast to formulations of restricted minor flows as functionsof unrestricted major flows as discussed before. The benefit of formulation (9) is its analogy with the external supply con-

Fig. 1. Driver behavior in intersection conflicts.


straint functions bRj (see (3)). The bNk are more difficult to define than the bRj though. For example, unlike the bRj, it is conceiv-able that they are non-linear. They are to be derived from gap acceptance or conflict (or any other) theory. Exactly definingthese functions bNk is beyond the scope of this paper. However, we can make the general proposition that also the distribu-tion of internal supply is governed by priority parameters aik so that, analogous to (4)–(7):

Uk ¼ fiji#bNkg ð10Þ
Uk ¼ ; () bNk < 0
ð11Þ
Uk–; () bNk ¼ 0
qi

qi0P

aik

ai0k8i 2 Uk; 8i0jfi0k > 0 ð12Þ

qi

qi0¼ aik

ai0k8i; i0 2 Uk ð13Þ

Now, it is shown how the priority parameters can be used to model different driver behavior. This discussion draws fromthe graphical representation of an intersection model’s solution space as used by Daganzo (1995).

A simple intersection with one crossing conflict bN3

� �is considered in Fig. 2. Firstly, the demand constraints S1 and S2 limit

the flow from their respective incoming link. Secondly, bN3ðq1; q2Þ 6 0 limits q1 and q2, which compete for the shared internalsupply. Say that for this intersection, the priority rules prescribe that q2 has priority over q1. Absolute compliance (case A)can then be modeled by setting aA

13 arbitrarily small and aA23 ¼ 1. Consequently, q2 consumes as much as possible (q2 = S2)

and leaves the remaining internal supply for q1. Hence, point A is the solution in this case. Limited compliance (case L)can be modeled by less extreme priorities, i.e. aL

13 > 0 and aL23 < 1. If turn-taking behavior applies to the crossing conflict

(case T), the priority parameters are based on the number of lanes or capacities of the incoming links, e.g. aT13 ¼ C1 and

aT23 ¼ C2.

In the above, the priority parameters are presented as constant values. However, an important advantage of this modelformulation is that the aij and aik can also be defined as functions of the turning fractions, capacities, but also the flows (thelatter case being proved by Flötteröd and Rohde (2011)). By doing so, transitions in driver behavior, e.g. depending on thesaturation level (see Fig. 1), may be accounted for by these priority functions. However, the remainder of this paper focuseson constant priority parameters, which already create a great deal of complexity.

In summary, this section adds general internal supply constraint functions to the model framework of Section 1.1 anddemonstrates the use of priority parameters to model different types of driver behavior in the distribution of (internal)supply.

3. Solution non-uniqueness in point-like intersection models

Based on the formal intersection modeling framework of the previous sections, it is now shown that that solution unique-ness in such models is anything but trivially guaranteed. As this model framework generally extends the proportional dis-tribution of external supply as implemented in most state-of-the-art intersection models to internal supply constraints, theanalysis in this section is general as well.

Fig. 2. Priority parameters controlling driver behavior in supply distribution.


3.1. Explanatory example

Fig. 3 shows a 2 � 4 intersection, which can be interpreted as a standard 4-leg intersection where only two inflows areconsidered. Two crossing conflicts can be identified. The priority rules for such conflicts typically state that the left-turningmovement has to yield to the straight movement coming from the opposite link, i.e. a23 = a14 = 1 and a13 and a24 arbitrarilysmall (see Section 2.2). In addition to the demand constraints, this results in the solution space as shown in Fig. 3. Due to CTF,both inflows qi are mutually dependent in both supply constraints. As such, the solution is fully determined by the total in-flows qi. The partial flows qij = fijqi are directly derivable.

In this example, the constraints and SCIR define three solutions A, B, and C. In A, the resulting flow pattern is obtainedfrom the distribution of internal supply N3, where q2 has absolute priority. Hence, q2 = S2, leaving the remaining supplyfor q1. Likewise, B results from distributing N4. Point C also meets the model definitions, with q1 and q2 being constrainedby bN3 and bN4 respectively, each leaving the remaining supply for the other flow. These solutions can be phrased in termsof the SCIR definitions (10)–(12) as follows:

q1 < S1

q2 ¼ S2

A : U3 ¼ f1g () bN3 ¼ 0 &q1

q2P

a13

a23

U4 ¼ ; () bN4 < 0

ð14Þ

q1 ¼ S1

q2 < S2

B : U3 ¼ ; () bN3 < 0

U4 ¼ f2g () bN4 ¼ 0 &q2

q1P

a24

a14

ð15Þ

q1 < S1

q2 < S2

C : U3 ¼ f1g () dN3 ¼ 0 &q1

q2P

a13

a23

U4 ¼ f2g () dN4 ¼ 0; &q2

q1P

a24

a14

ð16Þ

In summary, realistic behavioral assumptions (corresponding to the priority rules) can lead to multiple solutions.

3.2. Analysis of solution non-uniqueness

We identify the source of the solution non-uniqueness in the intersection model as the fact that inflows qi are faced withmultiple, ambiguous priority ratios in the distribution of different supplies in outgoing links j or internal conflicts k. Revis-iting the example of Section 3.1 in Fig. 3, one observes that if the priority ratios in the competition for both internal supplies

Fig. 3. Multiple solutions of the intersection model.


were identical, only one intersection point with a constraint persists and a unique solution would result. This leads to thefollowing findings.

Given arbitrary boundary conditions (demands, supplies and turning fractions) the following is a sufficient condition forsolution uniqueness:

4 Thicompethowevenon-un

5 It sviolatiouniquen

9ai; bj; bk : aij ¼ aibj 8i; j

aik ¼ aibk 8i; k

with : ai > 0 8i

bj > 0 8j

bk > 0 8k

ð17Þ

Condition (17) implies that the same priority ratio is used in the distribution of any supply between any two incom-ing links, thus ruling out ambiguity and non-uniqueness. Hereby, the factors b are scalable, as is the set of ai ’s in itsentirety. As such, all b could be set to one, thus defining single-valued priority parameters ai for each i to be appliedin the distribution of all (internal) supplies. For a proof, the reader is referred to Flötteröd and Rohde (2011), who de-scribe a uniquely converging solution algorithm for models of type (17). Specification (17) is also a necessary conditionfor the vast majority of real intersection topologies where the flows of at least two incoming links are mutually depen-dent4 in at least two common (internal or external) supply constraints (as in the example in Fig. 3). A proof of this state-ment is given in Appendix A.

Of course, depending on the boundary conditions (demands, supplies and turning fractions) ambiguous priority ratiosmay or may not induce multiple solutions in a specific case. For instance, one supply constraint may dominate and definea unique solution. More specifically, non-unique solutions for two flows qi and qi0 are possible if the boundary conditionsare such that a crossing or tangent point �qi; qi0ð Þ exists between (at least) two (internal) supply constraint functions in whichqi and qi0 are mutually dependent and this point qi; qi0ð Þ lies within the feasible domain bounded by the demand constraintsand the other supply constraints. This condition (18) is written below for an intersection point between an external supply inj and an internal supply in k. The cases of intersection points between two j or two k, or more than two supply constraints areequivalent.

9�qi; �qi0 ; j; k : bRj �qi; �qi0ð Þ ¼ bNk �qi; �qi0ð Þ ¼ 0with : 0 < �qi < Si

0 < �qi0 < Si0bRj0 �qi; �qi0ð Þ < 0 8j0 – j

bNk0 �qi; �qi0ð Þ < 0 8k0 – k

ð18Þ

Condition (18) is sufficient but not necessary. It covers the most realistic cases, but more far-fetched boundary conditionsin which non-uniqueness may exist are conceivable (see Appendix A). Graphical clarification can be obtained from revisitingagain Fig. 3. Therein, point C is the intersection point qi; q0i

� �between the two internal supply constraint functions. With

shifted or changing constraints, multiple solutions remain possible as long as C lies between A and B.Condition (18) specifies circumstances (i.e. model inputs) that may lead to non-uniqueness under a certain combination

of a’s. Multiple solutions will arise if an intersection point of the supply constraint functions exists as in (18) and the corre-sponding priority ratios point both ‘‘above’’ and ‘‘below’’ this intersection point as in Fig. 3.

3.3. Solution non-uniqueness: retrospect and corollary

As argued before, the model framework presented in Sections 1.1 and 2.2 encapsulates and extends (the main princi-ples of) most existing DNL intersection models. While solution uniqueness usually implicitly assumed in the state-of-the-art, it has been shown in the previous section that this is not trivially guaranteed;5 not even for models that only considerexternal supply constraints. However, most currently existing models implement single-valued priority parameters ai andhence have unique solutions. The models of Flötteröd and Nagel (2005), Gentile (2010), Adamo et al. (1999), Ni et al.(2006) and Taale (2008) do consider multiple-valued parameters aij. The latter three models do have a unique solution, be-cause they ignore CTF and are limited to external supply constraints. Also, Flötteröd and Nagel (2005) ultimately resort to thecomputation of one ‘‘representative’’ priority parameter. Non-uniqueness (related to internal supply constraints) has onlybeen observed by Flötteröd and Rohde (2011). It does not appear in the models of Yperman et al. (2007) and Ngoduy

s implicitly assumes CTF, since otherwise the qij are detached and the inflows qi are no longer tied in the various supplies that their movements ije for. If each qij is then subject to only one supply constraint, it can be determined via its individual aij and hence the solution is unique. In Appendix A,r, an example shows that the qij may compete for several internal supplies such that a ‘‘circular mutual dependency of flows’’ exists, which can causeiqueness regardless of CTF.hould be noted that the non-uniqueness is not solved by releasing any of the seven requirements that we adhere in our model definition. For mostns, such as to the invariance principle or flow maximization, this is straight-forward to see. The CTF requirement is a factor in most cases of non-ess, but not always (see Appendix A).


et al. (2005), since these model internal supply constraints that depend on the demands and thus there is no mutual flowdependency. Finally, pure merge models like the one of Chevallier and Leclercq (2007) naturally cannot have mutually depen-dent flows.

However, solution non-uniqueness becomes a much more prominent problem when existing models are extended withinternal supply constraints. To ensure solution uniqueness, uniform priority ratios are needed by adding condition (17) tothe model definition. However, while other modeling assumptions are derived on a behavioral basis, (17) is not – its originis to enforce uniqueness. Moreover, this technical condition (17) appears behaviorally unrealistic when introducing internalsupply constraints. Indeed, it is (often) in contradiction with how one would naturally define the priority parameters to gov-ern the distribution of internal supplies (see the example in Section 3.1). Blindly imposing single-valued priority ratios with-out any consideration of the ambiguity that seems inherent to reality is thus unadvisable. Only if the expected driverbehavior is limited to turn-taking behavior, as in AWSC intersections, reverting to this approach seems natural, e.g. withai = Ci.

Preferably, the decision of how to treat the non-uniqueness in the model should be supported by empirical research. Theempirical studies on intersection flows conducted in the past do not provide sufficient support since their focus is typicallyon various aspects of gap acceptance behavior (e.g. headway distributions) and not on validating intersection models in DNL(except for simple merges; see e.g. Bar-Gera and Ahn, 2010 and Ni and Leonard, 2005). Flow non-uniqueness has never beenreported in empirical data, but it has also never been looked for. Condition (18) identifies the rather specific circumstances inwhich the solution in the model may be non-unique. Thus, for empirical validation, these circumstances are to be sought inthe field. The need for a large amount of detailed data – not only traffic counts, but also video observation – renders thisempirical research labor- and cost-intensive.

Thus, although we find that the solution non-uniqueness in the model emanates from realistic behavioral assumptions, itremains to be seen whether or not flow ambiguity can indeed be identified in reality. If not, then hopefully empirical re-search will also allow developing stronger modeling guidelines. Regardless, however, the solution non-uniqueness that cur-rently exists in the model is undesirable for many (deterministic) applications. Therefore, we propose in the followingsection some pragmatic approaches to remedy the non-uniqueness.

3.4. Pragmatic approaches to deal with the solution non-uniqueness

The purpose of this subsection is to enable the further use of deterministic macroscopic DNL models by solving the solu-tion non-uniqueness of their intersection models. Several approaches are proposed. First, we note that in stochastic DNLmodels (e.g. Sumalee et al., 2011; Osorio et al., 2011), non-unique intersection flows can be resolved probabilistically. Justas a stochastic approach to the DTA problem yields a unique solution in distributional terms (e.g. Flötteröd et al., in press),a stochastic DNL could replace non-unique intersection flows by a unique distribution.

In deterministic DNL modeling, a transformation of the non-unique solutions into one prevailing flow pattern isneeded, which is less straightforward and prone to more subjectivity of the modeler. It may be desirable to at first allowambiguous priority parameters in the model definition, and then to alleviate the non-uniqueness by some kind of pre- orpost-processing (which must be unambiguous given the model inputs). In the following, we distinguish two types ofapproaches: (a) pre-processing the priority parameters so that the model produces a unique solution and (b) computingnon-unique solutions that result from ambiguous priority parameters and then post-processing these into one solution.In either case, the pre/post-processing is mathematically formulated as a convex combination of priority parameters/flows.

3.4.1. Pre-processing of the priority parametersFor every i with multiple-valued aij and aik, a representative ai is computed through

ai ¼X

j

wijaij þX

k

wikaik 8i

with : wij P 0&wik P 0 8i; j; kXj

wij þX

k

wik ¼ 1 8i

ð19Þ

Different choices of the weights wij and wik define different pre-processing strategies:

1. If no further information is available, uniform weights can be chosen. Other averaging schemes are thinkable but requirejustification based on supplementary modeling assumptions. The model of Flötteröd and Nagel (2005) – without internalsupply constraints – resorts to this approach.

2. One representative aij (or aik) could be selected by setting the respective wij (wik) to one and all other weights to zero. Thisapproach could be justified by the following considerations:a. The (internal) supply constraint that follows most naturally from the history of flows could be selected. However, the

analysis of Section 4 indicates that such a type of additional system memory can lead to undesirable results.


b. A ranking could be defined among all (internal) supply constraints so that the highest ranked active constraint deter-mines the solution. This ranking could be made based on the amount of competition that is present for each supply.That way, the supply that is most ‘overloaded’ would determine the flows. However, this competition or overloadingcannot be directly determined from the demands Si as this would not be compliant with the invariance principle.Rather, this ranking should be based on capacities.

This pre-processing approach comes down to indirectly imposing condition (17) in the model, while accounting for driverbehavior as realistically as possible. An advantage of this approach is that it is straightforward to implement and computa-tionally efficient.

3.4.2. Post-processing of the flowsEvery possible solution, resulting from the multiple, ambiguous priority ratios, is computed, leading to separate flow pat-

terns qr. These are then averaged into a unique resulting flow pattern q according to

q ¼X

r

wrqr8r

with : wr P 08rXr

wr ¼ 1

ð20Þ

Again, different choices of the weights define different solution strategies:

1. One could assume a uniform average solution to hold. This is in line with the assumption that a deterministic DNL rep-resents average network conditions. A naïve averaging, however, raises considerable difficulties. The average flow patternmay not comply with the requirement of individual flow maximization, or it may violate the invariance principle or somenon-linear internal supply constraint.

2. One could also select a single solution as the most plausible or representative one for the given situation (this again cor-responds to setting the respective weight wr to one and all others to zero). Again, this could be based on previous flows(assuming, e.g. that the smallest change in flow over time is the most plausible). Technically, this corresponds to selectingthe priority parameters that have led to this solution, but the selection criterion is now based on flows, which are notknown a priori.

A drawback of the flow post-processing approach is that multiple candidate solutions need to be evaluated, rendering itcomputationally intensive. Also, as stated above, a true averaging of solutions can lead to flows that are inconsistent with thebasic modeling assumptions.

Answering the question which of the above approaches or any other alternative is most realistic or most appropriate isdifficult or even impossible without further theoretical and empirical research. Moreover, the answer might differ undervarying circumstances. A trade-off between realism on the one hand and computational efficiency and limiting model com-plexity on the other seems inevitable.

4. Spatial intersection modeling

While the traditional point-like modeling approach is not disputed if internal supply constraints are disregarded, complexintersection models implement constraints that clearly have spatial characteristics. Therefore, one might consider modeling

Fig. 4. Spatial model of the example in Section 3.1.

Fig. 5. Scenario 1: S13 increases gradually.


intersections in DNL spatially, i.e. as mini-networks in which the conflict zones of crossing and merging flows are repre-sented by dummy nodes, connected by dummy links. In this section, however, it is shown that spatial models do not resolvethe problem of non-uniqueness encountered in point-like intersections.

Fig. 4 shows a spatial version of the example in Section 3.1. Two dummy links connect two dummy nodes that contain theinternal conflict zones (which are the same as in Fig. 3).

Again, the straight movements have absolute priority over the left-turning movement of the other incoming link, and theleft-turning movements do not hinder each other. For simplicity’s sake, the internal supply constraint functions are modeledidentically to external supply constraint functions, i.e. as a maximum number of vehicles that can cross the conflict zone perhour (21).

bNkðq1; q2Þ ¼ f1kq1 þ f2kq2 � Nk 6 0 k ¼ 3;4 ð21Þ

Fig. 6. Scenario 2: S24 increases gradually.


Consequently, it is possible to simulate this spatial intersection model with any state-of-the-art DNL model. Here, the LinkTransmission Model (Yperman et al., 2007) is chosen, with a minor modification6 to model the absolute compliance behaviorin the internal conflicts. For the simulation of this example, the conflict zone supplies are set to Nk = 1000 veh/h.

Firstly, two scenarios are considered in which the boundary conditions external to the intersection (i.e. the turning frac-tions and demands) are identical at the end of the simulation, but their histories are not (see Figs. 5 and 6).

The straight demands are S14 (=f14S1) and S23 (=f23S2) from incoming links 1 and 2 respectively. The left-turning demandsare S13 and S24 respectively.

In scenario 1 and 2, the left-turning demands are initially different. During the simulation, the low left-turning demand isgradually increased until it reaches the same level as the high left-turning demand from the opposing link. In the initialphase of scenario 1, S13 is low, allowing link 1 to send its full demand, since there is enough remaining supply(N3 � S23 = 200 veh/h) for the minor left-turning movement 13. Meanwhile, the left-turning movement 24 is obstructed

6 For this simple example, this requires nothing more than granting the prioritized movement its maximal share (=Sik) and passing the remaining internalsupply Nk � Sik to the minor movement. Apart from that, this modification does not have any implication for the usual working of the DNL model.


by bN4, so that a queue forms on dummy link 24 that spills back onto link 2 very quickly. This renders link 2 congested; alsothe straight movement 23 is thus held back. While S13 is gradually increased, this does not change the flows, as link 2 is nowunable to claim its maximal share of N3 (leaving N3 � q23 = 800 veh/h for link 1) due to the activated constraint bN4. In result,q1 stays dominant throughout the simulation, while q2 remains constrained. Scenario 2 is the exact opposite.

Thus, although the boundary conditions at the end of the simulation are identical in both scenarios, the resulting flowpatterns are very different. This demonstrates that also in a spatial model, solution uniqueness is not guaranteed given onlythe instantaneous boundary conditions. The spatial model inherently determines the solution on the basis of history. Whiledependency of the solution on history is not unrealistic as such, Fig. 5 shows that only a few seconds during which S13 < S24

suffice for q1 to take the upper hand and hold it for the entire simulation. (Fig. 6 is the opposite.) Clearly, this cannot be con-sidered a realistic dependence on history. Indeed, a time period equal to the spillback time over the left-turning dummy linkis enough to block the straight flow. This means that during the initial phase of a DNL, starting from an empty network, aspatial intersection model could steer the simulation results solely based on which flows happen to reach an intersectionfirst.

Finally, a nearly symmetric demand pattern is chosen for scenario 3 (Fig. 7). This results in an oscillating flow pattern,caused by back and forth propagating waves on the very short dummy links, which stabilizes very slowly. In the converged

Fig. 7. Scenario 3: Nearly symmetric demands engender unstable flows.

Fig. 8. Construction of non-unique solutions.

Fig. 9. Circular mutual dependency causing non-uniqueness (regardless of CTF).


solution, q2 dominates q1 thanks to a slightly smaller left-turning fraction. The frequency of the oscillations and the speed ofconvergence depend not only on the boundary conditions, but also on the exact spatial representation of the intersection (forinstance the length of the dummy links), which determines the spillback dynamics. This type of oscillations clearly is an arti-fact and not an interpretable model output.

In summary, the current problems of point-like intersection models cannot be bypassed in a spatial way. While at firstsight a spatial model physically detaches the mutual dependencies that lie at the origin of the ambiguity in point-like mod-els, the wave propagation over the dummy links rejoins these dependencies. This rejoined ambiguity may result in an oscil-lating flow pattern, but also in an irreversible dependency on a history of only a few seconds. Both effects are caused byspillbacks internal to the spatial intersection model and therefore the result of the (unjustified) application of macroscopicpropagation theory designed for long links on very short dummy links that fit at most a few vehicles. This model behavior isthus inherent to the spatial representation (e.g. the dummy links’ length and characteristics), which means it cannot be con-trolled by the modeler, and is clearly unrealistic and undesirable.7 Finally, two additional disadvantages of spatial models arementioned:

– How to spatially represent an intersection is a delicate matter in itself, and the result is extremely prone to gridlocks (asoccurs in the model of Chen et al. (2008)).

– From a computational perspective, it should be noted that numerical solution procedures that need to comply with theCourant–Friedrichs–Lewy condition (Courant et al., 1928) – stating in the given context that the simulation time step can-not be larger than the link travel time – such as the Godunov scheme (see Lebacque, 1996) are forced to operate at verysmall time steps because of the short dummy links in the spatial intersection model. However, this criticism does notcarry over to numerical schemes following the variational formulation of kinematic waves, which avoids this constraint(Daganzo, 2005a,b).

7 Therefore, extreme caution is advised when inheriting networks for DNL or DTA purposes directly from static models, in which complex intersections areoften modeled spatially.


5. Conclusion and outlook

This paper discusses complex intersection modeling in macroscopic, first-order DNL. Building on previous research inTampère et al. (2011) and Flötteröd and Rohde (2011), its main contributions are:

– The general definition of (internal) supply distribution through priority parameters, which can be varied to model differ-ent driver behavior.

– The analysis of solution non-uniqueness within this framework, which is found to result from realistic behavioralassumptions.

– The formulation of a sufficient (and necessary) condition for the priority parameters that ensures solution uniqueness,which, however, conflicts with a realistic representation of driver behavior.

– The description of the boundary conditions (input to the intersection model) that may induce multiple solutions.– The suggestion of pragmatic approaches to remedy the non-uniqueness in deterministic point-like models.– The elaboration on the spatial modeling approach, revealing unrealistic behavior of such models.

In this paper, many problems and concepts are discussed, but not always conclusively solved. As such, this paper opens arich field of yet undone research. To build realistic intersection models from these foundations does not only require to dealwith their inherent non-uniqueness, but also to get further to the bottom of driver behavior in intersection conflicts.

Regarding the modeling of driver behavior in DNL, several open issues remain. A classification of different types of driverbehavior is suggested, but further empirical validation on general (non-freeway) intersections is needed. In particular, spec-ifications are to be developed for the internal supply constraint functions that result from crossing and merging conflicts.Some theories exist that may serve as a basis, namely gap acceptance theory (Chapters 8 and 9 in Gartner et al. (2000))and conflict theory (Brilon and Wu, 2001; Brilon and Miltner, 2005). Hereby, we give preference to conflict theory since itseems more compatible with over-saturated conditions and is less complicated. Also, the interrelationship of (internal) sup-ply constraints due to simultaneous decision making of drivers having to traverse several conflict points constitutes a chal-lenge in future research.

Furthermore, priority functions rather than constant priority ratios may be necessary to realistically capture the transi-tions between different driver behavior, e.g. based on the saturation level. When doing so, caution is needed to preserve themodel’s compliance to the fundamental requirements listed in Section 1.1.

This should lead to a realistic, but also highly complex intersection model. Depending on the application, different trade-offs between realism, model complexity and data requirements will be desirable. Hopefully, this discussion will evolve com-parably to that of link models, where various theories – e.g. travel time functions, vertical and spatial queuing – providedifferent levels of complexity and realism.

However, the analysis of the general model framework of this paper shows that, for the majority of intersection models inthe state-of-the-art, an extension with such internal supply constraints is likely to be subject to severe modeling problemsregarding solution non-uniqueness. This paper provides technical conditions to identify this problem to the research com-munity and practical guidelines on how to treat it. To support the treatment of the solution non-uniqueness in the model,foremost empirical studies would be valuable. This paper aids by helping to understand the phenomenon (and when it oc-curs) in the model, so that these specific circumstances can be sought in the field. If the existence of multiple flow patternsunder identical boundary conditions is indeed observed, their characteristics – e.g. probability, frequency of switches, dura-tion of stable periods – and the (external) factors that lead to these characteristics – e.g. history, neighboring (signaled) inter-sections, intersection geometry – should be identified. If this empirical work disproves non-uniqueness in reality, it shouldsupport the development of stronger modeling guidelines.

In conclusion, we see good potential in further developing traditional point-like models rather than abandoning them infavor of spatial models. For this, the pragmatic approaches suggested in Section 3.4 as well as the behavioral modeling (asdescribed above) are to be further developed into unambiguous models for different types of intersections (AWSC, priority-controlled, signalized and roundabouts). Finally, also the development of efficient algorithms constitutes a considerablechallenge.

Appendix A. Proof of uniqueness condition (17)

Sufficiency of condition (17) is shown in Flötteröd and Rohde (2011). In the following, we will show that for the case ofmutually dependent flows, specific circumstances exist that render the solution non-unique for any set of a’s that does notmeet (17). Note hereby that the supply constraint functions bRj and bNk only depend on boundary conditions such as the given(internal) supply and turning fractions fij. Thus, they are independent of the model parameters aij and aik.

Assume any arbitrary, strictly positive aik; ai0k; aik0 and ai0k0 for two incoming links i and i0 that are mutually dependent intwo internal supply constraints bNk and bNk0 . Cases including external supply constraints are entirely analogous. The demandconstraints are assumed arbitrarily high, so that they never bind. Also other supply constraints are disregarded. As supplyconstraint functions are always continuous and decreasing, a setting is always conceivable so that (18) holds for these


functions. Furthermore, the boundary conditions can be chosen such that the intersection point lies between the two priorityratios as in Fig. 8.

Now, two solutions A and B exist that are both described by (13). Enforcing condition (17) leads to aikai0k¼ aik0

ai0k0) qA

iqA

i0¼ qB

iqB

i0and

hence, given that an internal supply constraint must bind, a unique solution. If (17) does not hold, an ambiguous setting asabove can always be constructed and thus it is a necessary condition for solution uniqueness for any two mutually depen-dent flows.

Moreover, it appears that the model can become insolvable if bNk and bNk0 were to coincide. In this case, (13) should holdfor both k and k0 in both A and B, which is impossible unless the priority ratios coincide. So again, this issue is only resolved ifcondition (17) holds.

Finally, we have limited the statement of (17) as a necessary condition to the simple case of two directly mutually depen-dent flows. We note that there exist cases of indirectly or circularly mutually dependent flows that may also cause solutionnon-uniqueness. This circular mutual dependency can be explained as follows: multiple flows qi1

; . . . ; qim are tied in (internal)supply constraints that each binds two different flows in a circular sequence, returning from the last i to the first, i.e.:Fi1 \ Fi2 ¼ fk1g; Fi2 \ Fi3 ¼ fk2g; . . . ; Fim \ Fi1 ¼ fkmg. (Any k could of course be replaced by a j.) In Fig. 9, an example withnon-unique solutions through such circular dependency is presented. Four internal conflicts are present, governed byright-of-way law. Hence, each flow has priority in one of its conflicts, while having to yield in the other. The solution spaceis not depicted, but one can intuitively understand that two different solutions exist. In the first solution, q1 and q3 take theirmaximal share of N1 and N3 so that q2 and q4 are restricted. In the second solution, the distribution of N2 and N4 is followed,so that q2 and q4 flow maximally and q1 and q3 are restricted.

It is clear that in this case the CTF requirement is not part of the problem of solution non-uniqueness, as all qi are unidi-rectional. Also in this example, solution uniqueness would be obtained by imposing (17).

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Non-unique flows in macroscopic first-order intersection models

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