spillback congestion in dynamic traffic assignment: a macroscopic flow model with time-varying...

Transportation Research Part B 41 (2007) 1114–1138

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Spillback congestion in dynamic traffic assignment:A macroscopic flow model with time-varying bottlenecks

Guido Gentile *, Lorenzo Meschini, Natale Papola

Dipartimento di Idraulica Trasporti e Strade, Universita degli Studi di Roma ‘‘La Sapienza’’, Via Eudossiana, 18, 00184 Roma, Italy

Received 2 June 2005; received in revised form 16 January 2006; accepted 29 June 2006

Abstract

In this paper, we propose a new model for the within-day Dynamic Traffic Assignment (DTA) on road networks wherethe simulation of queue spillovers is explicitly addressed, and a user equilibrium is expressed as a fixed-point problem interms of arc flow temporal profiles, i.e., in the infinite dimension space of time’s functions. The model integrates spillbackcongestion into an existing formulation of the DTA based on continuous-time variables and implicit path enumeration,which is capable of explicitly representing the formation and dispersion of vehicle queues on road links, but allows themto exceed the arc length. The propagation of congestion among adjacent arcs will be achieved through the introduction oftime-varying exit and entry capacities that limit the inflow on downstream arcs in such a way that their storage capacitiesare never exceeded. Determining the temporal profile of these capacity constraints requires solving a system of spatiallynon-separable macroscopic flow models on the supply side of the DTA based on the theory of kinematic waves, whichdescribe the dynamic of the spillback phenomenon and yield consistent network performances for given arc flows. We alsodevise a numerical solution algorithm of the proposed continuous-time formulation allowing for ‘‘long time intervals’’ ofseveral minutes, and give an empirical evidence of its convergence. Finally, we carry out a thorough experimentation inorder to estimate the relevance of spillback modeling in the context of the DTA, compare the proposed model in termsof effectiveness with the Cell Transmission Model, and assess the efficiency of the proposed algorithm and its applicabilityto real instances with large networks.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Queue spillovers; Theory of kinematic waves; Network loading through implicit path enumeration; Explicit capacity const-raints; Continuous-time formulation of within-day DTA

1. Introduction

During the last decade, the within-day Dynamic Traffic Assignment (DTA) has been one of the most activefields in transport modeling (for an extensive bibliography on this topic, see Carey and Ge, 2003). Indeed, theclassical framework of static traffic assignment is often recognized to be an improper approach for the analysis

0191-2615/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trb.2007.04.011

* Corresponding author. Tel.: +39 06 44585737; fax: +39 06 44585129.E-mail addresses: [email protected] (G. Gentile), [email protected] (L. Meschini), [email protected]

(N. Papola).

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G. Gentile et al. / Transportation Research Part B 41 (2007) 1114–1138 1115

of highly congested road networks, where the formation and dispersion of vehicle queues play a decisive role.This is particularly true for those applications (variable message signs, on-board driver information, rampmetering, dynamic lane usage) where the estimation of travel times is the desired output other than the trafficflow pattern.

The presence of the temporal dimension affects the DTA by introducing the problem of loading the net-work with given path flows in such a way that the resulting arc flow temporal profiles are consistent withthe corresponding travel time temporal profiles through an arc performance function. The above question,referred to as the Continuous Dynamic Network Loading (CDNL) problem, is so important that in the liter-ature much attention has been devoted to its specific analysis (see, for instance, Xu et al., 1999). Indeed, this isthe only problem to be solved in those cases, such as highway networks and urban corridors, where the pathchoice is not involved. To our knowledge, the only existing CDNL formulation capable of taking into accountspillback congestion seems to be the event-based model proposed in Adamo et al. (1999); however, their modelcan not be easily implemented without making use of micro-simulation by its nature.

An alternative to the CDNL for describing the within-day dynamic of congestion which avoids introducingan arc performance function consists in extending at the Network level the application of a macroscopic Traf-fic Flow (NTF) model, such as the Theory of Kinematic Waves (TKW), by specifying rules on how the flowstates are transferred through road junctions. We recall here METANET proposed by Messmer and Papa-georgiou (1990), which derives from a second order approximation to the TKW, and the Cell TransmissionModel (CTM) proposed by Daganzo (1994, 1995), which is consistent with the Simplified TKW (STKW),i.e., the first order approximation of the TKW. In these models, path choices are represented only in a localform by means of temporal profiles of exogenous node splitting rates that are specified for each destination.However, to determine the splitting rates consistently with some route choice model, the departure time ofeach vehicle reaching the considered node at every given instant needs to be identified for each path (Papa-georgiou, 1990). These models intrinsically take the spillback phenomenon into account, satisfy the FIFO rulebecause vehicles are represented as particles of a mono-dimensional, partly compressible fluid (Cascetta, 2001,pp. 374–379), and can be applied to networks with many origins and destinations. However, they require adense spatial discretization, in the sense that each road link is represented as a sequence of segments or ‘‘cells’’,which is usually avoided in CDNL models.

To date, all (not event-based) CDNL and NTF models describe the temporal dimension by introducing‘‘short time intervals’’ (1–10 s), in order to exploit in the formulation the fact that a given vehicle enteringa given arc or cell during a certain time interval will exit that arc or cell not earlier than the beginning ofthe next time interval (Gentile et al., 2004). Moreover, all of them require storing the arc flows distinguishedby destination in memory. For these reasons, the resulting algorithms are quite onerous in terms of compu-tational resources and therefore are applicable only for limited size networks.

The existing DTA models that address also the spillback congestion indeed aim at combining a NTF model(for instance, Lo and Szeto, 2002; Ziliaskopoulos, 2000, use the CTM) and a path choice model into a userequilibrium, usually involving a search for dynamic shortest paths (Pallottino and Scutella, 1998), so as to per-mit the analysis of more general networks than those addressed by the CDNL problem. While correct from themodeling point of view, this approach may become unpractical from the algorithmic point of view, due to thegreat amount of computer time and memory required by the high granularity of the representation both intime and space.

As an alternative, the NTF model can be replaced by micro-simulation (Barcelo and Casas, 2002), whichinvolves tracking each single trip in the network separately through car following models, or meso-simulation(time-based in Mahmassani, 2001; event-based in Mahut et al., 2004), where the trajectory of each vehicle orvehicle packet is implicitly determined using a macroscopic traffic flow model. The main weakness of thisapproach lies in the theoretical need of performing a great number of simulations, each requiring a differentpseudo-casual sequence, in order to achieve reliable expected values of the flows and convergence to equilib-rium. However, this is practically unfeasible because computing times would be way too long and few simu-lations are actually performed, so that the resulting tools implemented in some commercial software, whileuseful for many applications, are not fully satisfactory from a modeling point of view. In addition, contrarilyto macroscopic flow models, the computing time needed to perform a micro-simulation depends on the num-ber of tracked vehicles, which in turn is proportional to travel demand.

1116 G. Gentile et al. / Transportation Research Part B 41 (2007) 1114–1138

Moreover, all the existing approaches to address DTA involve explicit path enumeration, which also callsfor considerable computation resources.

In a recent paper (Bellei et al., 2005), we proposed a new continuous-time formulation of the DTA, wherea user equilibrium is expressed as a fixed-point problem in terms of arc inflow temporal profiles, i.e., in theinfinite dimension space of time’s functions. There, it is shown that, by extending to the dynamic case theconcept of Network Loading Map (NLM), stated in Cantarella (1997) for the static case, it is no more nec-essary to introduce the CDNL as a sub-problem of the DTA, as coherence through the arc performancefunction between the travel times and the flows loaded on the network consistently to given path choiceswill be jointly attained at equilibrium. The fixed-point problem is indeed formalized by combining theNLM, yielding the arc inflow temporal profiles corresponding to given arc travel time and cost temporalprofiles, and the arc performance function, yielding the arc travel time and cost temporal profiles corre-sponding to given inflow temporal profiles. On this basis, it is possible to devise efficient assignment algo-rithms, whose complexity is equal to the one resulting in the static case multiplied by the number of timeintervals introduced.

Another important feature of the above model lays in the fact that it does not exploit the acyclic graphcharacterizing the corresponding discrete-time version of the problem. Specifically, we do not introduce thehypothesis that the longest time interval must be shorter than the smallest free flow arc travel time. Thereforeit is possible to define ‘‘long time intervals’’ (5–10 min), which allow overcoming the difficulty of solving DTAinstances on large networks.

Again in Bellei et al. (2005), with specific reference to a Logit route choice model where all the efficientpaths to each destination are implicitly considered, a formulation of the NLM is devised exploiting the con-cepts of arc conditional probability and node satisfaction. On this basis, a specific network loading procedureis also obtained as an extension to the dynamic case of Dial’s algorithm. Furthermore, the fixed-point problemaddressing the DTA is solved through an accelerated averaging algorithm, called Bather’s Method (Bottomand Chabini, 2001). The same approach was adopted in Gentile et al. (2004), where two alternative all-or-nothing assignment procedures to dynamic shortest paths, one based on virtual vehicle trajectories, the otherone based on temporal layers, are proposed to evaluate the NLM in the deterministic case. These procedurescan also be used within a Monte Carlo simulation to evaluate the NLM in the Probit case.

The main goal of this paper is the extension of our previous model to the case of queue spillovers, which is acrucial step towards a satisfying simulation of highly congested networks. Indeed, the above model is capableof representing explicitly the formation and dispersion of vehicle columns, but sets no limit to the length of thequeue which, therefore, can well exceed the length of the arc. In order to eliminate this inconsistency we willtranslate any interaction among the flows on adjacent arcs in terms of time-varying arc exit and entry capac-ities. Our approach is then to reproduce the spillback phenomenon as a hypercritical flow state, either prop-agating backwards from the final section of an arc and reaching its initial section, or originating on the latter,that reduces the capacities of the arcs belonging to its backward star and eventually influences their flow states,consistently with the definition provided by Adamo et al. (1999).

The key idea is to introduce the spillback representation directly in the arc performance function, withoutaffecting the network flow propagation model internal to the NLM. This is an important innovation, since onthis basis the DTA can still be formulated as the system of a NLM based on implicit path enumeration and ofa suitable arc performance function. The latter is provided by the Network Performance Model (NPM), whichis the system of spatially non-separable macroscopic flow models proposed here to simulate the propagation ofcongestion among adjacent arcs due to queue spillovers.

We will consider the temporal profiles of the maneuver flows at nodes as current variables of the resultingfixed-point problem (see Fig. 1). Indeed, these play a role in modeling the spillback phenomenon when theavailable capacity at a node is split among its upstream arcs. However, note that the maneuver flows coincidewith the arc inflows and outflows if mergings and diversions are separated at the graph level, as it often occursto represent turn penalties and prohibitions.

The paper is organized as follows. In Section 2, we describe the NPM. In Section 3, we recall the main fea-tures of the NLM and formulate the DTA as a user equilibrium. In Section 4, we provide an algorithm for theresulting fixed-point problem, specifying the procedures solving the NPM. In Section 5, we devise somenumerical examples on toy networks to estimate the relevance of spillback modeling, validate our model

network flow propagation model

implicit path enumeration route choice model

demand flows

arc travel times

networkloading map

network performance model

arc conditional probabilities

arc costs

arc performance function

maneuver flows

Fig. 1. Scheme of the fixed-point formulation for the DTA with spillback congestion.


on real data and compare it in terms of effectiveness with the CTM, apply the method on a large network toasses its efficiency, and finally provide empirical evidence for its convergence.

2. Network performance model

The road network is modeled as a directed graph (N,A), where N is the set of the nodes, each representing anintersection or a ‘‘centroid’’ (i.e., a trip terminal, origin or destination), and A is the set of the arcs, each repre-senting a road link between two intersections or a ‘‘connector’’ between a centroid and an intersection. The gen-eric arc a = (TL(a), HD(a)) 2 A, with tail TL(a) 2 N and head HD(a) 2 N, consists of a homogeneous channel oflength La and physical capacity Qa > 0 with a final bottleneck of saturation capacity Sa 6 Qa. Each origin

o 2 ORIG � N has no entering arcs except for one dummy link, and its exiting connectors have an infinite phys-ical capacity; each destination d 2 DEST � N has no exiting arcs except for one dummy link, and its enteringconnectors have a saturation capacity equal to the physical capacity; dummy links, which have infinite capacitiesand null length, allow defining inflows and outflows of connectors in terms of the maneuver flows at centroids.

In the following, the flow states along the arc are determined on the basis of the STKW assuming any con-cave fundamental diagram as depicted in Fig. 2, where KQa > 0 is the critical density, KJa > KQa is the jam

density, Va P Qa/KQa is the free flow speed, while the functions va(q) and wa(q) express, respectively, the vehi-

cle speed and the absolute value of the kinematic wave speed in terms of the flow q corresponding to hypercrit-ical conditions, that is for k 2 [KQa,KJa] (see Gentile et al., 2005).

Although the model formulation is general, the proposed solution algorithm and the numerical applicationsrefer to the specific case of the trapezoidal shaped fundamental diagram depicted in Fig. 3, which has a con-stant kinematic wave speed wa(q) = wa = Qa/(KJa � KQa), so that va(q) = q/(KJa � q/wa).

Let uab(s) be the maneuver flow at time s from arc a 2 A to arc b 2 A at node HD(a) = TL(b). More aggre-gated and familiar flow variables can be easily derived as follows:

faðsÞ ¼X

b2BSðTLðaÞÞubaðsÞ; ð1Þ

Qa

KJa

density

q

kKQa

flow

wa(q)

va(q) Va

Fig. 2. Concave fundamental diagram.

flow

density wa

Qa / wa

Qa

Va

Qa / Va KQa KJa

va(q)

q

Fig. 3. Trapezoidal fundamental diagram.


F aðsÞ ¼Z s

�1faðrÞ � dr; ð2Þ

/aðsÞ ¼X

b2FSðHDðaÞÞuabðsÞ; ð3Þ

where fa(s) is the inflow, Fa(s) is the cumulative inflow and /a(s) is the outflow of arc a 2 A at time s, whileBS(i) = {a 2 A: HD(a) = i} and FS(i) = {a 2 A: TL(a) = i} are, respectively, the forward and the backward

star of node i 2 N. The dummy links are introduced specifically to let (1) and (3) hold also for centroids; in-deed, the demand flows enter the network from the tail of the dummy link relative to an origin and exit thenetwork from the head of a dummy link relative to a destination.

To represent the spillback phenomenon, we assume that each arc is characterized by two time-varying bot-tlenecks, one located at the initial section and the other one located at the final section, called ‘‘entry capacity’’and ‘‘exit capacity’’, respectively.

The entry capacity, bounded from above by the physical capacity, is meant to reproduce the effect of queuespropagating backwards from the arc itself, which can reach the initial section and can thus induce spillbackconditions on the upstream arcs. In this case the entry capacity is set to limit the current inflow at a valuewhich keeps the number of vehicles on the arc equal to the storage capacity currently available. The latteris a function of the exit flow temporal profile, since the queue density along the arc changes dynamically intime and space accordingly with the STKW. Specifically, the space freed by vehicles exiting the arc at the headof the queue takes some time to become actually available at the tail of the queue, so that the jam densitymultiplied with the length is just the upper bound of the storage capacity, which in turn can be reached onlyif the queue is not moving.

The exit capacity, bounded from above by the saturation capacity, is meant to reproduce the effect of queuespillovers propagating backwards from the downstream arcs, which in turn may generate hypercritical flowstates on the arc itself. For given maneuver flows and intersection priorities, which here are assumed propor-tional to the saturation capacities, the exit capacities are obtained as a function of the entry capacities basedon flow conservation at the node.

The NPM is specified here as a circular chain of three models, namely the ‘‘exit flow and travel time modelfor time-varying capacities’’, the ‘‘entry capacity model’’, and the ‘‘exit capacity model’’, whose system can beformulated and solved through a fixed-point problem to determine the temporal profiles of entry and exitcapacity, and thus the arc travel times and costs, for given maneuver flows. The three models, described sep-arately in the following sections, are synthesized in Fig. 4, which shows how the entry capacities may be takenas current variables in the fixed-point formulation of the NPM.

2.1. Exit flow and travel time model for time-varying capacities

Assuming that the FIFO rule holds, i.e., no overtaking among vehicles can occur, we now introduce a link-based performance model for an arc with time-varying entry and exit capacities, aimed at determining the‘‘exit flow’’ temporal profile as a propagation of the inflow temporal profile to the final arc section, and thenthe corresponding travel time temporal profile. However, while the former is involved in the fixed-point

maneuver flows


arc exit flows

exit flow and travel time model

entry capacity model

arc travel times

arc entry capacities

physical capacities

arc costs

arc cost model

arc exit capacities

saturation capacities

exit capacity model

Fig. 4. Scheme of the fixed-point formulation for the NPM.


formulation of the NPM, the latter is not, and may therefore be obtained after mutually consistent entry andexit capacities have been found. Moreover, it is worth pointing out that the exit flow is by definition differentfrom the outflow, which is derived directly from the maneuver flows, although clearly the two coincide atequilibrium.

In general, a bottleneck with null length and time-varying capacity can be conveniently formulated in termsof cumulative flows so as to yield the leaving flow by contrasting the arriving flow with the throughout capac-ity, under the consideration that the former is stocked in a queue if it is not served at the moment, while thelatter cannot be stocked if it is not utilized at the moment. Let f(s), e(s) and w(s), respectively, be the arriving

flow, the leaving flow and the throughout capacity of the bottleneck at time s, and denote by F(s), E(s) and W(s)the corresponding cumulative flows:

F ðsÞ ¼Z s

�1f ðrÞ � dr; EðsÞ ¼

Z s

�1eðrÞ � dr; WðsÞ ¼

Z s

�1wðrÞ � dr; ð4Þ

we have

EðsÞ ¼ minfF ðrÞ þWðsÞ �WðrÞ : r 6 sg: ð5Þ

The above expression (5) can be explained as follows. If there is no queue at a given time s, the cumulativeleaving flow is equal to the cumulative arriving flow. If a queue arises at time r < s, from that instant untilthe queue will eventually vanish, the leaving flow equals the throughout capacity, and then the cumulativeleaving flow E(s) at time s results from adding to the cumulative arriving flow F(r) at time r the integralof the throughout capacity between r and s, that is W(s) � W(r). Notice that, if there is no queue at times, the cumulative leaving flow is the same of the case when the queue arises exactly at r = s. Based on the‘‘Newell–Luke minimum principle’’ (Daganzo, 1997; Newell, 1993) – which states that, when more thanone kinematic wave reaches a point at a same time, the flow state yielding the minimum cumulative flow dom-inates the others – the actual cumulative leaving flow at time s is the minimum among each cumulative. Leav-ing flow that would occur if the queue began at any previous instant r 6 s. Fig. 5 depicts a graphicalinterpretation of equation (5), where the temporal profile E(s) of the cumulative leaving flow is the lower en-velop of the following curves: (a) the cumulative arriving flow F(s); (b) the family of functionsW(s) + F(r) � W(r) with s > r, for every time r, each one obtained as the vertical translation of the temporalprofile relative to the cumulative throughout capacity that goes through the point (r,F(r)). No queue is pres-ent when curve (a) prevails; therefore, the queue arises at time r 0 and vanishes at time r00.

Each arc a 2 A is modeled here as a sequence of: an initial bottleneck, a running link, and a final bottleneck;where clearly the leaving flow of one element corresponds to the arriving flow of the next. Thus, we will dealwith four distinct flow temporal profiles and two time-varying capacity constraints, as depicted in Fig. 6: (1)the inflow, i.e., the arriving flow to the initial bottleneck, denoted fa(s) for each time s, with cumulative Fa(s);

F( )

E( ')

σ' '

time

vehicles

F(σ

τ) +

Ψ(

')-Ψ

(σ):

στ

≤'

Ψτ

σ(

')-Ψ

(')

στ ''

F(σ

τ

')

E( )

Ψ τ τ τ ττ

ττττ

( ) + F( ') - Ψ( ') , > ' Ψ( ) + F(σ) - Ψ(σ) , > σ

Fig. 5. Bottleneck with time-varying capacity.

fa( ) , Fa( )

final bottleneck

running link

initial bottleneck

μ τ τa( ) , Μa( ) ψ τ τ

τ ττττ τττ

a( ) , Ψa( )

γa( ) , Γa( ) a( ) , Λλ a( ) ea( ) , Ea( )

Fig. 6. Arc model and flow notation.


(2) the entry capacity of the initial bottleneck, denoted la(s), with cumulative Ma(s); (3) the leaving flow fromthe initial bottleneck, which is equal to the arriving flow to the running link, denoted ca(s), with cumulativeCa(s); (4) the leaving flow from the funning link, which is equal to the arriving flow to the final bottleneck,denoted ka(s), with cumulative Ka(s); (5) the exit capacity of the final bottleneck, denoted wa(s), with cumu-lative Wa(s); (6) the exit flow, i.e., the leaving flow from the final bottleneck, denoted ea(s), with cumulativeEa(s).

Applying Eq. (5) to the initial bottleneck, we obtain

CaðsÞ ¼ minfF aðrÞ þMaðsÞ �MaðrÞ : r 6 sg: ð6Þ

Using one of the methods presented in Gentile et al. (2005) it is possible to express the hypocritical running

time ra(s) � s, for a vehicle entering the running link at the generic time s, as a function of the previous portionof the arriving flow temporal profile, that is ca(r) at each instant r 6 s:

raðsÞ ¼ raðcaðrÞ : r 6 sÞ: ð7Þ

Then, based on the FIFO rule, the cumulative leaving flow from the running link at time s is equal to thecumulative arriving flow to the running link at time r�1

a ðsÞ, that is when the vehicle that exits the running linkat s entered it:

KaðsÞ ¼ Caðr�1a ðsÞÞ: ð8Þ

Note that, in presence of intervals with null flows, ra(s) may be not invertible at some point r; nevertheless, inthis case Ca(s) is the same for each s 2 r�1

a ðrÞ, therefore Caðr�1a ðrÞÞ is anyhow well defined.

In the particular case of a trapezoidal fundamental diagram, however, the hypocritical running time isconstant:

raðsÞ � s ¼ La=V a; ð9Þ

and Eq. (8) becomes:

KaðsÞ ¼ Caðs� La=V aÞ: ð10Þ


Finally, applying Eq. (5) to the final bottleneck, we obtain:

Fig. 7.from t

EaðsÞ ¼ minfKaðrÞ þWaðsÞ �WaðrÞ : r 6 sg: ð11Þ
By construction, ea(s) 6 wa(s) at any time s and hypercritical exit flows occur whenever ea(s) = wa(s).
Combining (6) with (7), the result and (6) with (8), the result with (11), on the basis of the definitions (1) and(4), we can express the arc exit flow model in the following compact form for all the arcs at once:

E ¼ Eðu; l;wÞ; ð12Þ
where bold symbols denote temporal profiles of vector variables.
The arc exit time is obtained here by applying the FIFO rule to the sequence of the sole running link andfinal bottleneck, i.e., without the initial bottleneck, through the following implicit expression, as depicted inFig. 7:

EaðtaðsÞÞ ¼ CaðsÞ: ð13Þ
This way we avoid computing the delay generated by the initial bottleneck twice whenever Fa(s) > Fa(r) +Ma(s) �Ma(r) for some r 6 s, since the latter is already taken into account in the travel time as the delaygenerated by the final bottlenecks of the upstream arcs due to the spillback propagation. Moreover, at equi-librium the NPM ensures that the entry capacity constraint fa(s) 6 la(s) is satisfied at any time s, and thus thedelay on the initial bottleneck is null. In presence of time intervals with null flow, expression (13) does notallow to obtain a univocal value of the exit time. To take these circumstances into account, once the cumu-lative exit flow temporal profile is known, the exit time temporal profile is calculated conventionally as
taðsÞ ¼ maxfraðsÞ; minfr : EaðrÞ ¼ CaðsÞgg: ð14Þ
Combining (6) with (7), the result and (6) with (14), on the basis of the definitions (1) and (4), we can
express the travel time model in the following compact form:

t ¼ tðu; l;EÞ: ð15Þ

2.2. Entry capacity model

In this section, we propose a new approach to represent the effect on the entry capacity of queues that, gen-erated on the final arc section by the exit capacity, reach the initial arc section, thus inducing spillbackconditions.

To explain this let us assume, for the moment, that the queue is uncompressible, i.e., only one hypercriticaldensity exists. In this case, the kinematic wave speed is infinite – from Fig. 2, it is clear that wa!1 whenKJa! KQa – so that any hypercritical flow state occurring at the final section would propagate back instan-taneously and at any instant when the queue exceeds the arc length, the entry capacity would be equal tothe exit capacity. Note however that the queue does not reach the initial section instantaneously, since here,

Γ τ

ττ

τ

τ

τ

τ

τ τ

τ

a( )

' ta( ')

Γa( ') = Λa(ra( ')) = Ea(ta( '))

time

vehicles

ra( ')

Λa( ) = Γa(ra-1( ))

Ea( )

Computation of the arc exit time-based on the cumulative leaving flow from the initial bottleneck and the cumulative exit flowhe arc by applying the FIFO rule.


consistently with the Newell–Luke minimum principle, the exiting hypercritical flow state does not prevail onthe entering hypocritical flow state until the number of vehicles that have entered the arc becomes greater thanthe number of vehicles that have exited the arc plus the storage capacity, which in this case is constant in timeand given by the arc length multiplied by the hypercritical density.

Actually, in the general case, hypercritical flow states may occur at different densities and their kinematicwave speeds are not only quite lower than the vehicular free flow speed, implying that the delay affecting thebackward translation in space from the final to the initial section of the flow states produced by the exit capac-ity is not negligible, but also somewhat different from each other, which generates a distortion in their forwardtranslation in time. Notice that the trapezoidal fundamental diagram is capable of representing the dominantdelay effect but not the distortion effect, since all backward kinematic waves have the same slope.

The spillback effect on the entry capacity is investigated here by exploiting the analytical solution of theSTKW based on cumulative flows proposed by Newell (1993). Using this approach, we can avoid evaluatingthe queue length temporal profile explicitly in order to determine the presence of spillback. Indeed, this wouldbe cumbersome, since the speed and density of the queuing vehicles vary over time and space as a function ofthe exit capacity. More simply, we will just identify the time intervals when some exiting hypercritical flowstate, propagating back along the arc, reaches the initial section and prevails on the entering hypocritical flowstate.

Indeed, the flow state occurring on the generic arc section is the result of the interaction among hypocriticalflow states coming from upstream and hypercritical flow states coming from downstream. Specifically, withreference to the initial section, the one flow state coming from upstream is the inflow, while the flow statescoming from downstream are due to the exit capacity and can be determined by back-propagating the hyper-critical portion of the cumulative exit flow temporal profile, thus yielding what we refer to as the ‘‘maximumcumulative inflow’’ temporal profile.

Accordingly to the Newell–Luke minimum principle, the flow state consistent with the spillback phenom-enon occurring at the initial section is the one implying the lowest cumulative flow. Therefore, when the cumu-lative inflow equals or overcomes the maximum cumulative inflow, so that spillback actually occurs, thederivative of the latter temporal profile may be interpreted as an upper bound to the inflow. This permitsthe proper value of the entry capacity to be determined that maintains the queue length equal to the arc length.

The instant ua(s) when the backward kinematic wave generated at time s on the final section of the genericarc a 2 A by the hypercritical exit flow ea(s) = wa(s) would reach the initial section is given by

uaðsÞ ¼ sþ La=waðeaðsÞÞ: ð16Þ

By definition, the points in time and space constituting the straight line trajectory produced by a kinematicwave are characterized by a same flow state. Moreover, Fig. 8 shows that the number of vehicles encounteredby the hypercritical wave relative to the exit flow q for any infinitesimal space ds traveled in the opposite direc-tion is equal to the time interval dsÆ [1/va(q) + 1/wa(q)] multiplied by that flow. Therefore, integrating along thearc from the final to the initial section, we obtain the cumulative flow Ha(s) that would be observed at timeua(s) in the initial section:

H aðsÞ ¼ EaðsÞ þ eaðsÞ � La � ½1=vaðeaðsÞÞ þ 1=waðeaðsÞÞ�: ð17Þ

space

time wa(q)

ds / wa(q)ds / va(q)

va(q)

ds

kinematic wave

vehicles

Fig. 8. Trajectories of a hypercritical kinematic wave and of the intersecting vehicles.


In general ua(s) is not invertible, since more than one kinematic wave may reach the initial section at thesame time. Then, based on the Newell–Luke minimum principle, the maximum cumulative inflow Ga(s) thatcould have entered the arc at time s consistently with the exit flow pattern is given by

GaðsÞ ¼ minfH aðrÞ : uaðrÞ ¼ s; eaðrÞ ¼ waðrÞ;1g: ð18Þ
Note that Ha(s) is defined only in correspondence of hypercritical exit flows; therefore, where the maximumcumulative inflow would not be defined we set it conventionally equal to infinity.
If at the generic time s the cumulative inflow Fa(s) is equal or higher than the maximum cumulative inflowGa(s), so that spillback occurs at that instant, then the entry capacity la(s) is given by the derivative dGa(s)/dsof the latter; otherwise, it is equal to the physical capacity Qa:

laðsÞ ¼dGaðsÞ=ds; if GaðsÞ 6 F aðsÞ;Qa; otherwise:

�ð19Þ

Combining (16) and (17) with (18), the result with (19), on the basis of the definitions (1) and (4), we canexpress the entry capacity model in the following compact form:

l ¼ lðu;w;E; QÞ: ð20Þ
Note that the direct dependency on the exit capacities is merely due to the need of characterizing the exit flowea(s) as to be hypercritical, that is ea(s) = wa(s), or hypocritical, i.e., ea(s) < wa(s).
With reference to the trapezoidal fundamental diagram, ua(s) is invertible. Indeed, since wa(q) = wa, basedon (16) the time when ua(r) = s is r = s � La/wa. Moreover, since q/va(q) = KJa � q/wa, based on (17) wehave: Ha(s) = Ea(s) + La Æ KJa. Therefore, Eq. (18) becomes

GaðsÞ ¼Eaðs� La=waÞ þ La � KJ a; if eaðs� La=waÞ ¼ waðs� La=waÞ;1; otherwise:

�ð21Þ

Note that in this case the maximum cumulative inflow at the time when a hypercritical kinematic wave reachesthe initial section is equal to the cumulative exit flow at the time when it was generated on the final section plusthe storage capacity La Æ KJa, which is similar to the case of an uncompressible queue, except for the delayLa/wa. If Ga(s) 6 Fa(s), then Ga(s) must be finite, so that the first case of Eq. (21) holds. Differentiating thelatter yields: dGa(s)/ds = ea(s � La/wa). Then, since ea(s � La/wa) = wa(s � La/wa), Eq. (19) becomes

laðsÞ ¼waðs� La=waÞ; if GaðsÞ 6 F aðsÞ;

Qa; otherwise:

�ð22Þ

Fig. 9 shows how, based on Eq. (21), the maximum cumulative inflow temporal profile can be obtained graph-ically through a rigid translation (thick arrows) of the cumulative exit flow temporal profile for La/wa in timeand for La Æ KJa in vehicles. Moreover, it points out that, when Ga(s) is greater than Fa(s), the queue is shorterthan La and la(s) = Qa, otherwise spillback occurs and la(s) = wa(s � La/wa).

Note that, if the queue arises at time r 0, the maximum cumulative inflow is infinity before r 0 + La/wa.Because the cumulative exit flow at r 0 is equal to the cumulative inflow at time r 0 � La/Va, and the termQa Æ LaÆ (1/Va + 1/wa) 6 La Æ KJa is equal to the maximum number of vehicles that can enter the arc duringthe interval [r 0 � La/Va,r 0 + La/wa], from (21) we have

Gaðr0 þ La=waÞP F aðr0 � La=V aÞ þ Qa � La � ð1=V a þ 1=waÞP F aðr0 þ La=waÞ: ð23Þ

The above is also true with reference to time r00 when the queue vanishes, while clearly after r00 + La/wa themaximum cumulative inflow becomes again infinity. In other words, at the beginning and at the end of anyinterval where the maximum cumulative inflow is finite, Ga(s) is not lower than Fa(s), which is consistent withthe continuity of the cumulative flow min{Ga(s),Fa(s)} at the initial section.

2.3. Exit capacity model

In this section we present a model to determine the exit capacities of upstream arcs, with reference toa given node with any number of entering and exiting arcs, on the basis of the entry capacities of the

La / Va

La / waL

aK

J a

spillback

time

time

flow

vehicles

Qa

a

inflow Fa(τ), fa(τ

τ

)

exit flow Ea(τ), ea(τ)

maximum cumulative inflow Ga( )

entry capacity τμa( )

σ' σ''

La / Va

ψ

queue

Fig. 9. Graphical determination of the entry capacity temporal profile in the case of trapezoidal fundamental diagram, piecewise constantinflow, and constant exit capacity.


downstream arcs and of the local maneuver flows. To our knowledge, this generality constitutes a minorachievement, since the other models proposed in literature allow for a maximum number of three arcs con-nected to each node. Although in principle any graph can be reduced to an equivalent one that satisfies theabove property, such a procedure represents an operative nuisance, increases the size of the network consid-erably, and introduces a modeling distortion.

To simplify the exposition, we first consider only two typologies of nodes: ‘‘mergings’’ and ‘‘diversions’’; inthis case, the maneuver flows coincide with the arc outflows and inflows, respectively.

When considering a merging x 2 N, i.e., an intersection with a singleton forward star, the problem is to splitthe entry capacity lb(s) of the arc b = FS(x) available at time s among the arcs belonging to its backward star,whose outflows compete to get through the intersection. In principle, we assume that the available capacity ispartitioned proportionally to the saturation capacity Sa of each arc a 2 BS(x) – more general partition criteriarequire the introduction of ‘‘priority constants’’ or ‘‘merging capacity ratio’’ in place of the saturation capac-ities as in Daganzo (1995) and in Kuwahara and Akamatsu (2001). This way it may happen that for some arc a

the outflow /a(s) is lower than the share of entry capacity assigned to it, so that only a lesser portion of thelatter is actually exploited. Let Xb(s) � BS(x) be the set of such arcs. The rest of the entry capacitylbðsÞ �

Pa2XbðsÞ/aðsÞ shall then be partitioned among the arcs making up the set BS(x)nXb(s). Moreover,

when no spillback phenomenon is active, i.e.,P

a2BSðxÞ/aðsÞ < lbðsÞ, the exit capacity wa(s) of each arca 2 BS(x) shall be set equal to its saturation capacity Sa. On this basis, we have:

waðsÞ ¼ Sa � nbðs;XbðsÞÞ; ð24ÞXbðsÞ ¼ fa 2 BSðxÞ : /aðsÞ < waðsÞg; ð25Þ


where we denoted for any given set of arcs X � BS(x):

nbðs;XÞ ¼lbðsÞ �

Pa2X/aðsÞP

a2BSðxÞnXSa; if X � BSðxÞ; 1; otherwise: ð26Þ

Note that a set Xb(s) satisfies jointly (24) and (25) if and only if every arc a 2 BS(x) with a saturation ratio/a(s)/Sa < nb(s, Xb(s)) belongs to Xb(s) itself and every arc a with /a(s)/Sa P nb(s, Xb(s)) does not. Sincebased on (26) nb(s,X) decreases adding to X arcs for which /a(s)/Sa > nb(s,X), while it increases removingfrom X arcs for which /a(s)/Sa < nb(s,X), and vice versa, the partition set Xb(s) can be easily proved tobe unique, and it can be simply obtained by iteratively adding to an initially empty set X* each arca 2 BS(x)nX* such that /a(s)/Sa < nb(s,X*). Finally, we shall prove that Eq. (24) yields wa(s) 6 Sa for eacharc a 2 BS(x). Assume by contradiction that nb(s,Xb(s)) > 1. Based on (24), we have that wa(s) > Sa; more-over by definition it is Sa P /a(s). Based on (25) we have then Xb(s) = BS(x), which, considering (26), con-tradicts the hypothesis. The fact that (24) holds also for arcs belonging to Xb(s) enhances the overallcontinuity of the model.

When considering a diversion x 2 N, that is an intersection with a singleton backward star, the problemis to determine at the generic time s the most severe reduction to the outflow from the arc a = BS(x)among those produced by the entry capacities of the arcs belonging to its forward star. Again, whenno arc is spilling back, the exit capacity shall be set equal the saturation capacity. When only one arcb 2 FS(x) is spilling back, that is fb(s) P lb(s), the exit capacity wa(s) scaled by the share of vehicles turn-ing on arc b is set equal to the entry capacity in order to ensure capacity conservation at the node whilesatisfying the FIFO rule applied to the vehicles exiting from arc a: wa(s) Æ fb(s)//a(s) = lb(s). When morethan one arc b 2 FS(x) is spilling back, the exit capacity is the most penalizing among the above values.On this basis, we have:

waðsÞ ¼ minfSa; lbðsÞ � /aðsÞ=fbðsÞ : b 2 FSðxÞ; fbðsÞP lbðsÞg: ð27Þ
When considering a generic node x 2 N with both mergings and diversions, the maneuver flows are to be
introduced explicitly. To achieve this generalization, firstly, models (24) and (25) are applied separately toevery arc b 2 FS(x) yielding a maneuver capacity wab(s) for each a 2 BS(x); secondly, model (27) is appliedwith reference to the maneuver capacities, that here play the role of the entry capacities, to every arca 2 BS(x), thus yielding the following exit capacities:

wabðsÞ ¼ Sa � nbðs;XbðsÞÞ; ð28Þ

XbðsÞ ¼ fa 2 BSðxÞ : uabðsÞ < wabðsÞg; ð29Þ

waðsÞ ¼ minfSa; wabðsÞ � /aðsÞ=uabðsÞ : b 2 FSðxÞ;uabðsÞP wabðsÞg; ð30Þ

where

nbðs;XÞ ¼lbðsÞ �

Pa2XuabðsÞP

a2BSðxÞnXSa; if X � BSðxÞ; 1; otherwise: ð31Þ

Combining the solution of the systems (28) and (29) with (30), on the basis of the definition (3) we canexpress the exit capacity model in the following compact form:

w ¼ wðu; l; SÞ: ð32Þ

Note that, in contrast with the models presented in the previous two sub-sections, this model is spatiallynon-separable, because the exit capacities of all the arcs belonging to the backward star of a given nodeare determined jointly, and temporally separable, because all relations refer to the same instant.

For the particular case when node x 2 N works like several separate mergings, i.e., whenuab(s) > 0) uac(s) = 0, "a 2 BS(x), "b 2 FS(x), "c 2 FS(x), with b 5 c, we introduce the hypothesis thatusers do not occupy the intersection if they cannot cross it due to the presence of a queue on their successivearc, but wait until the necessary space becomes available. Indeed, our model is not capable of addressing thedeterioration of performances due to a misusage of the intersection capacity.


2.4. Arc cost model

The cost for users entering arc a at time s is given by

Fig. 10side).

caðsÞ ¼ g � ðtaðsÞ � sÞ þ maðsÞ; ð33Þ
where ma(s) is the monetary cost, while g is the value of time. In compact form we have
c ¼ cðtÞ: ð34Þ

2.5. Formulation as a fixed-point problem

The NPM allows determining (see Fig. 4 and the left hand side of Fig. 10), for given maneuver flows atnodes, arc travel times and capacities consistent with the traffic flow theory that ensure the propagation ofcongestion through the network. It can be formulated by combining (32) with (12), the result and (32) with(20), yielding the following fixed-point problem in terms of entry capacity temporal profiles:

l ¼ lðu;wðu; l; SÞ;Eðu; l;wðu; l; SÞÞ; QÞ: ð35Þ
For given maneuver flows, the solution to (35), if any, is denoted as follows:
l ¼ l�ðuÞ: ð36Þ
Combining (36) with (32), the result and (36) with (12), the result and (36) with (15), yields a performance
function, expressing the arc travel times in terms of the maneuver flows:

t ¼ tðu; l�ðuÞ;Eðu; l�ðuÞ;wðu; l�ðuÞ; SÞÞÞ ¼ t�ðuÞ: ð37Þ
Finally, substituting (37) in (34), we have:
c ¼ cðt�ðuÞÞ ¼ c�ðuÞ: ð38Þ

p(w, c, t)D

(p, tω ; D)

p

(t)

t c

c

w

w(c, t)

network loading map

arc performance function

μ*(ϕ)

ψ(ϕ, μ)

t(ϕ, μ, E)

E(ϕ, μ, ψ)

μ

E

ψ

μ

μ(ϕ, E, ψ; Q)

ψ


Q

Sϕ

E

E(ϕ, μ, ψ)

ψ(ϕ, μ; S)

ϕ

. Variables and models of the fixed-point formulations for the NPM (left hand side) and for the DTA with spillback (right hand


3. Network loading map and user equilibrium

In the following, we briefly recall the formulation based on implicit path enumeration presented in Belleiet al. (2005), which addresses both route choice and network flow propagation. Referring to users travelingtowards a single destination d 2 DEST at the time, the formulation is based on the concepts of arc conditional

probability and node satisfaction, whose notation and definitions are introduced below:pd

aðsÞ probability of using arc a 2 A, conditional on crossing node TL(a) at time s;wd

x ðsÞ expected value of the maximum perceived utility at time s, relative to the paths Kxd connecting nodex 2 N to d which are considered by the user.

In the above paper, it is proved that the following expressions of the node satisfaction and of the arc con-ditional probability are consistent with a Logit route choice model in which users consider all and only ‘‘effi-cient’’ paths (a path is efficient if each of its arcs is efficient):

wdx ðsÞ ¼ h � ln

Xa2FSðxÞ\EAðdÞ

exp�caðsÞ þ wd

HDðaÞðtaðsÞÞh

! !; if x 6¼ d; 0; otherwise; ð39Þ

pdaðsÞ ¼ exp

�caðsÞ þ wdHDðaÞðtaðsÞÞ � wd

TLðaÞðsÞh

!; if a 2 EAðdÞ; 0; otherwise; ð40Þ

where EAðdÞ ¼ fa 2 A : ZdTLðaÞ > Zd

HDðaÞg is the set of the efficient arcs, being Zdz the ‘‘distance’’ from the generic

node z 2 N to destination d – these distances on the network are to be taken with respect to a cost patternwhich is constant in time. The solution of the triangular system formed by Eq. (39) in topological order, com-bined with Eq. (40), yields the route choice model, which can be expressed in compact form as

w ¼ wðc; tÞ; ð41Þp ¼ pðw; c; tÞ: ð42Þ

The inflow f da ðsÞ on the generic arc a 2 A, except for the dummy links of the origins, is given by the arc

conditional probability pdaðsÞ multiplied by the flow exiting from node TL (a). The latter is given, in turn,

by the sum of the outflow /dbðsÞ from each arc b 2 BS(TL(a)) \ EA(d) of its efficient backward star. For

the dummy link BS(o) of the generic origin o2 ORIG the inflow f dBSðoÞðsÞ is instead equal to the demand flow

DdoðsÞ from o to d. Then we have:

f da ðsÞ ¼ Dd

HDðaÞðsÞ; if HDðaÞ 2 ORIG; f da ðsÞ ¼ pd

aðsÞ �X

b2BSðTLðaÞÞ\EAðdÞ/d

bðsÞ; otherwise: ð43Þ

Given the exit time temporal profile of arc a, the outflow is related to the inflow temporal profile as follows:

/daðtaðsÞÞ ¼ f d

a ðsÞ=½dtaðsÞ=ds�; ð44Þ

where the weight dta(s)/ds stems from the fact that users enter the arc at a certain rate and exit it at a differentrate, which is higher than the previous one, if the travel time is decreasing, and lower, otherwise (for details,see for instance Cascetta, 2001, pp. 384–386).

The maneuver flow at the generic node x 2 N from arc a 2 BS(x) to arc b 2 FS(x) directed to d is given bythe outflow /d

aðsÞ from arc a multiplied by the arc conditional probability pdbðsÞ. Then, summing up the con-

tribution of all destinations we have:

uabðsÞ ¼X

d2DEST

pdbðsÞ � /

daðsÞ: ð45Þ

The solution of the triangular system formed by Eqs. (43) and (44) in reverse topological order, combinedwith (45), yields the network flow propagation model, which can be expressed in compact form as

u ¼ xðp; t; DÞ: ð46Þ
Combining (41) with (42) and the result with (46) yields a formulation based on implicit path enumeration ofthe NLM:
u ¼ xðpðwðc; tÞ; c; tÞ; t; DÞ ¼ x�ðc; t; DÞ: ð47Þ


On this basis the DTA can be formalized (see Fig. 10) as a fixed-point problem in terms of maneuver flowtemporal profiles by substituting into the NLM (47) the arc performance function (37) and (38):

Funct

(0) k

(1) un

(2) k

(3) l

(4) w

(5) E

(6) t =(7) c

(8) w

(9) p

(10) u

(11) u

u ¼ x�ðc�ðuÞ; t�ðuÞ; DÞ: ð48Þ

4. Solution algorithm

In this section, we outline an algorithm for solving the proposed formulation of the DTA based on implicitpath enumeration. Specifically, referring to Bellei et al. (2005) for the calculation procedures that implementthe NLM, we present in detail only the calculation procedures that implement the arc performance functionincluding the NPM.

To address numerically the continuous-time mathematical model presented in the previous sections, theperiod of analysis shall be divided into n time intervals identified by the sequence of instants s ={s0, . . . ,si, . . . ,sn}, with si < sj for any 0 6i < j 6 n. The network at time s0 and at time sn is assumed to beunloaded.

In the following we approximate the generic temporal profile g(s) of the performance and of the flow/capac-ity variables introduced in the previous sections, respectively, through a piecewise linear and a piecewise con-stant function, defined by the values gi = g(si) taken at each instant si 2 s. Under this assumption, fors 2 [si,si+1), with 0 6i 6 n � 1, in both cases we have, respectively:

gðsÞ ¼ gi þ ðs� siÞ � ðgiþ1 � giÞ=ðsiþ1 � siÞ; ð49:1ÞgðsÞ ¼ gi: ð49:2Þ

In this manner, the generic temporal profile g(s) can be then represented numerically through the (1 · n + 1)row vector g = (g0, . . . ,gi, . . . ,gn).

The algorithm is basically the same proposed in Bellei et al. (2005), to which we refer for its discussion andfor the calculation procedures concerning step (7) (arc cost model), steps (8) and (9) (route choice model), andstep (10) (network flow propagation model), which require minor modifications in order to yield the maneuverflows in lieu of the inflows. For the sake of simplicity we present here a simple MSA to solve the fixed-pointproblem instead of Bather’s method proposed in our previous paper.

ion DTA= 0,u = 0,unlm =1 *initialize the iteration counter and the maneuver flow vector*

til ku � unlmk1 6 e or k > kmax do *stop criterion*

= k + 1 *increment the iteration counter*

= l * (u) *calculate the entry capacities solving the NPM*

= w(u,l) *calculate the exit capacities*

= E(u,l,w) *calculate the cumulative exit flows*

t(u,l,E) *calculate the exit times*

¼ cðtÞ *calculate the arc costs*

= w(c, t) *calculate the node satisfactions*

= p(w,c, t) *calculate the arc conditional probabilities*

nlm = x(p, t;D) *calculate the NLM flows through the NFP model*

= u + 1/k Æ (unlm � u) *update the current equilibrium flows with the MSA*

Steps (5) (exit flow model) and (6) (travel time model) are detailed in the following, while steps (3) and (4)(involving the exit and entry capacity models) are specified next, when addressing the fixed-point probleminternal to the NPM.

Function E(u,l,w)(5.0) [f,F,/] = #(u) *calculate inflows and outflows*

(5.1) for each a 2 A *for each arc*


(5.2) C0a ¼ 0

(5.3) for i = 1 to n *for each time interval in chronological order*

*calculate the cumulative leaving flow from the initial bottleneck*

*check whether the additional entry capacity available during the interval is all utilized*

(5.4) Cia ¼ minfCi�1

a þ lia � ðsi � si�1Þ; F i

ag(5.5) i = 0(5.6) until si � La/Va > s0 do Ki

a ¼ 0, i = i + 1 *find i:si � La/Va 2 (s0,s1]*

(5.7) j = 1(5.8) for i = i to n *for each time interval in chronological order*

*propagate the cumulative flow along the running link by shifting it forward in time*

(5.9) until sj P si � La/Va do j = j + 1 *find j:si � La/Va 2 (sj�1,sj]*

(5.10) Kia ¼ Cj�1

a þ ðCja � Cj�1

a Þ � ðsi � La=V a � sj�1Þ=ðsj � sj�1Þ(5.11) E0

a ¼ 0(5.12) for i = 1 to n *for each time interval in chronological order*

*calculate the cumulative exit flow from the arc*

*check whether the additional exit capacity available during the interval is all utilized*

(5.13) Eia ¼ minfEi�1

a þ wia � ðsi � si�1Þ;Ki

ag(5.14) En

a ¼ F na

*unload the network at the end of the simulation*

The temporal profile of the cumulative leaving flow from the initial bottleneck is calculated on the basis ofthe cumulative arc inflow and the entry capacity temporal profiles consistently with (6), checking in chrono-logical order if at the end of each time interval, the additional entry capacity available during the interval is allutilized by the flow that has arrived at the initial bottleneck but has not yet left it. Then, the cumulative flow ispropagated along the whole running link shifting it forward in time by La/Va in accordance with (10). Finally,the cumulative exit flow is calculated on the basis of the cumulative flow arriving to the final bottleneck andthe exit capacity temporal profiles consistently with (11), checking in chronological order if at the end of eachtime interval, the additional exit capacity available during the interval is all utilized by the flow that hasarrived at the final bottleneck but has not yet left it. Function #(u) implements Eqs. (1)–(3).

Function t(u,E)
(6.0) [f,F,/] = #(u) *calculate inflows and outflows*
(6.1) for each a 2 A *for each arc*

(6.2) C0a ¼ 0

(6.3) for i = 1 to n *for each time interval in chronological order*

*calculate the cumulative leaving flow from the initial bottleneck*

(6.4) Cia ¼ minfCi�1

a þ lia � ðsi � si�1Þ; F i

ag(6.5) i = 0(6.6) until Ci

a > 0 do tia ¼ si þ La=V a; i ¼ iþ 1

(6.7) j = 1(6.8) for i = i to n

(6.9) until Eja P Ci

a do j = j + 1 *find j: Ej�1a < Ci

a 6 Eja

*

*calculate the exit time coherently with the FIFO rule*

(6.10) tia ¼ maxfsj�1 þ ðCi

a � Ej�1a Þ � ðsj � sj�1Þ=ðEj

a � Ej�1a Þ; si þ La=V ag

First, the temporal profile of the cumulative leaving flow from the initial bottleneck is calculated as inthe previous procedure. Then, the exit time is determined by applying the FIFO rule as in (14) to com-pute the maximum between the linear interpolation graphically depicted in Fig. 11 and the hypocriticalexit time from the running. Finally, notice that the cycle (6.9) ensures that at step (6.10) it is alwaysEj

a > Ej�1a .

The fixed-point problem appearing in step (3) can be solved by means of the following contraction algo-rithm, whose convergence is not proved. Nevertheless, when applied in practice it quite rapidly reaches a solu-tion point l that satisfies the stop criteria (3.1), even for e 0 = 0. Indeed, at each iteration the spillback

time

vehicles

i tai

cumulative leaving flow from the initial bottleneck

Γ

τ τ τ

ai

j-1 j

cumulative exit flow

Eaj

Eaj-1

Eai

Fig. 11. Arc exit time for given piece-wise linear temporal profiles of the cumulative flows.


congestion may propagate only one arc backwards, and the algorithm terminates when no further propaga-tion occurred.

Function l*(u)
(3.0) h = 0, l = Q, l* = 0 *initialize the iteration counter and the entry capacity vector*
(3.1) until kl � l*k1 6 e 0 or h > hmax do
*stop criterion*
(3.2) h = h + 1
*increment the iteration counter*
(3.3) l* = l
*memorize the current entry capacities*
(3.4) w = w(u,l;S)
*calculate the exit capacities*
(3.5) E = E(u,l,w)
*calculate the cumulative exit flows*
(3.6) l = l(u,w,E;Q)
*calculate the new entry capacities*
A ‘‘gridlock state’’ occurs when the back propagation of congestion forms a loop on the network, that is,when the spillback reaches the arc where it was generated. If the flow pattern u leads to a gridlock state, thecapacities of the arcs belonging to the loop tend asymptotically to 0, and the travel times will be infinite. Whengridlocks prevail at equilibrium, the assignment problem has no solution. But a gridlock state can come out insome step of the equilibrium algorithm, despite a solution to the assignment problem actually exists. In orderto avoid this algorithmic drawback, we introduce a maximum number of iterations in step (3.1). Clearly, if thisnumber is reached, the resulting travel times will be consistent with spillback conditions, i.e., there will besome queue exceeding the respective arc length. This does not compromise the validity of the equilibrium algo-rithm, since if we find this condition at the convergence point, we know that it does not correspond to anadmissible solution of the problem. Moreover, in real situations, gridlocks are unlikely to occur, otherwisethe network performances would be so low to induce an effect at the demand generation level.

In real applications, roundabouts are ‘‘dangerous’’ situations where gridlocks may appear. To enhance thestability of the algorithm, particular attention shall be paid in the definition of merging priorities, which in theproposed framework (see Section 2.3) requires the introduction of fictitious links with null length to set themproperly. In practice, we suggest giving priority to the links of the roundabout. Another critical factor for grid-locks is the jam density. Consider that when the congestion is very high, the vehicles tend to ‘‘squeeze’’ alsolaterally and may achieve in forming more columns than the actual lanes; this behavior of the queues can bewell represented by conferring additional jam density.

In the following we specify the procedures to calculate the exit and entry capacities.Step (3.4), which is identical to step (4) of the DTA procedure, implements the exit capacity model. Firstly,

the maneuver capacities are determined using the algorithm suggested in Section 2.3 for the case of a mergingto solve the systems (28) and (29). Secondly, Eq. (30) is applied to compute the exit capacities.

Function w(u,l;S)(3.4.0) [f,F,/] = #(u) *calculate inflows and outflows*

(3.4.1) for i = 1 to n *for each time interval*

(3.4.2) for each x 2 N *for each node*


(3.4.3) for each b 2 FS(x) *for each arc of the FS*

(3.4.4) Xib ¼ ;, EL = 0 *let the set of arcs not affected by spillback from arc b be empty*

(3.4.5) TS ¼P

a2BSðxÞSa, TF = 0 *calculate the capacity of the BS*

(3.4.6) until EL = 1 or TS = 0 do *if end loop is false and the residual capacity is not null*

(3.4.7) EL = 1 *set end loop to true*

(3.4.8) nib ¼ ðli

b � TF Þ=TS *update the spillback ratio*

(3.4.9) for each a 2 BS(x) *for each arc of the BS*

(3.4.10) wiab ¼ Sa � ni

b*set the maneuver capacity with the spillback ratio*

(3.4.11) if a 62 Xib and ui

ab < wiab then *if it is higher than the maneuver flow and a is in spillback*

(3.4.12) Xib ¼ Xi

b [ a *arc a is not affected by spillback from arc b*

(3.4.13) EL = 0 *set end loop to false*

(3.4.14) TS = TS � Sa*update the residual capacity of the BS*

TF ¼ TF þ uiab

* update the utilized capacity of the BS*

(3.4.15) for each a 2 BS(x) *for each arc of the BS*

(3.4.16) wia ¼ Sa

*set the exit capacity equal to saturation capacity*

(3.4.17) for each b 2 FS(x) *for each arc of the FS*

(3.4.18) if uiab P wi

ab then wia ¼ minfwi

ab � /ia=u

iab;w

iag *update the exit capacity*

Step (3.5) is identical to step (5) of the DTA procedure.Step (3.6) implements the entry capacity model. The entry capacity is initialized to the physical capacity.

Then, in chronological order, when the corresponding exit flow is hypercritical, based on (21) the maximumcumulative inflow is determined, and if it prevails on the cumulative inflow, based on (22) the entry capacity isupdated with the corresponding exit capacity.

Function l(u,w,E;Q)(3.6.0) [f,F,/] = #(u) *calculate inflows and outflows*

(3.6.1) for each a 2 A *for each arc*

(3.6.2) j = 1(3.6.3) for i = 1 to n *for each time interval in chronological order*

(3.6.4) lia ¼ Qa

*set the entry capacity equal to the physical capacity*

(3.6.5) until sj P si � La/wa do j = j + 1 *find j: si � La/wa 2 (sj�1, sj]*

(3.6.6) if ðEja � Ej�1

a Þ=ðsj � sj�1Þ ¼ wja then *if the exit flow is hypercritical*

*if the maximum cumulative inflow is not higher than the cumulative inflow*

*set the entry capacity equal to the corresponding exit capacity*

(3.6.7) if Ej�1a þ wj

a � ðsi � La=wa � sj�1Þ þ Qa � La � ð1=V a þ 1=waÞ 6 F ia then li

a ¼ wja

5. Numerical tests

5.1. The effects of spillback modeling

In order to investigate the behavior of the proposed model compared to that of our previous model withoutspillback, we analyze three simple examples which present intuitive solutions. We consider 100 intervals of60 s, and assume that every arc a 2 A is characterized by the same length La = 500 m and a triangular funda-mental diagram with free flow speed Va = 20 m/s, kinematic wave speed wa = 0.25 Æ Va, and physical capacityQa = 2000 veh/h. The first two examples refer to the elementary intersection depicted in Fig. 12, where noroute choice occur; while in the third example we consider the Braess network depicted in Fig. 15, wherewe assumed a Logit parameter h = 250 s.

In the first example the saturation capacities are: SA = SC = 2000 veh/h, SB = SD = 900 veh/h; and weassume that there is a constant demand for the first 33 min of simulation: D13 = 1200 veh/h,D43 = 400 veh/h and D45 = 600 veh/h. Fig. 13, which depicts the comparison between the results obtainedwith the two models, shows that the outputs are quite different both in terms of flows and travel times.

1 2 3

5

4

A

D

B

C

Fig. 12. The network of the first and second example.

DNL with spillback - inflow [veh/h]

0

200

400

600

800

1000

1200

1400

1600

1800 arc A

arc B

arc C

arc D

DNL with spillback - travel time [sec]

0

200

400600

800

1000

12001400

1600

1800

0 500 1000 1500 2000 2500 3000 3500 4000

0 500 1000 1500 2000 2500 3000 3500 4000

arc A

arc B

arc D

arc C

D

DNL without spillback - inflow [veh/h]

0

200

400

600

800

1000

1200

1400

1600

1800

0

200400600

800

1000

12001400

1600

1800

0 500 1000 1500 2000 2500 3000 3500 4000

arc A

arc B

arc C

arc D

DNL without spillback - travel time [sec]

0 500 1000 1500 2000 2500 3000 3500 4000

arc C & B

arc A

arc B

arc D

arc C

Darc C & B

Fig. 13. Output of the first example.


Without spillback, the congestion is concentrated on arc B and does not propagate upstream. Indeed thetravel time of the other arcs remains at the uncongested level, while the travel time of arc B grows until thedemand ends, since its inflow is not bounded. As a result, the travel time from node 4 to node 3 (arcs Cand B) gets very high; instead, the demand from node 4 to node 5 (arcs C and D) travels at the free flow speedand is not involved in the congestion phenomenon. Contrarily, with spillback, the travel time on arc B has anupper bound, since the congestion is transferred upstream as soon as the queue reaches the initial section.From that moment, the inflow on arc B drops to the outflow level, i.e., 900 veh/h. Since the flow D43 travelingfrom arc C to arc B is in this case lower than the share SC/ (SA + SC) of the available capacity SB that it coulduse, amounting to 450 veh/h, arc C is not affected by the spillback phenomenon, and the congestion propa-gates only on arc A. As a result, the travel time from node 4 to node 3 is much lower than the one calculatedwithout spillback, since most of the queue develops now on arc A. Note that the vehicles entering arc A aftertime 1000 will exit it after time 2000; then they shall not compete with the vehicles traveling on arc C to use theentry capacity on arc B, which explains the change of slope in their travel time occurring at time 1000 (circledin the figure).

In the second example we assume that the demand D43 increases up to 600 veh/h. The outputs of the twomodels without and with spillback are presented in Fig. 14. The situation differs from the previous one, sinceD43 is now higher than SB Æ SC/(SA + SC). Then, the congestion on arc B spills back on both arcs A and C, andthus both travel time increase substantially. As a result, also the flow traveling from node 4 to node 5 isdelayed, despite it does not use arc B. Clearly, the model without spillback is unable to represent this effect.

The aim of the third example is to show the effect of spillback on path choice. The saturation capacities ofthe Braess network depicted in Fig. 15 are: SA = SE = 2000 veh/h, SB = SC = SD = 1000 veh/h; and again weassume that there is a constant demand for the first 33 minutes of simulation: D14 = 2300 veh/h.

DNL with spillback - inflow [veh/h]

0200

400600800

1000

120014001600

18002000 arc A

arc B

arc C

arc D

DNL with spillback - travel time [sec]

0

200

400600

800

1000

12001400

1600

1800

0 500 1000 1500 2000 2500 3000 3500 40000 500 1000 1500 2000 2500 3000 3500 4000

0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000

arc A

arc B

arc C

arc D

arc C & D

DNL without spillback - inflow [veh/h]

0200

400600800

1000

120014001600

18002000 arc A

arc B

arc C

arc D

DNL without spillback - travel time [sec]

0

500

1000

1500

2000

2500 arc A

arc B

arc C

arc D

arc C & D

Fig. 14. Output of the second example.

4

A

DB

C

1

2

3

E

Fig. 15. The network of the third example.

DUE with spillback - inflow [veh/h]

0

500

1000

1500

2000

0 500 1000 1500 2000 2500 3000 3500

DUE with spillback - travel time [sec]

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500

arc A

arc B

arc C

arc D

arc E

arc A

arc B

arc C

arc D

arc E

arc A

arc B

arc C

arc D

arc E

arc A

arc B

arc C

arc D

arc E

DUE without spillback - inflow [veh/h]

0

500

1000

1500

2000

0 500 1000 1500 2000 2500 3000 3500

DUE without spillback - travel time [sec]

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500

sec sec

Fig. 16. Output of the third example.


The outputs of the two models without and with spillback are presented in Fig. 16. Without spillback, thecongestion is evenly located only on arcs C and D, so that on all the paths between node 1 and node 4 thequeue is about equal, and path A–E–D has fewer users since it is clearly not convenient. With spillback, how-ever, the queue propagates from arc C to arc A, and from arc D to arcs B and E. Moreover, the spillback effectis greater on arc B than on arc E because of their different saturation capacities. Then, after an initial growth,the queue on arc D remains constant and equal to the arc length, while the queue on arc B grows faster thanthat on arc E, so that path A–E–D now becomes competitive, as it implies a longer route but a shorter queue.


5.2. Validation of the model on real data and comparison with the CTM

In order to test the effectiveness of the proposed model we address the case of a real corridor, where thepath choice is not involved, and we compare with the measured flows the results yielded by: (a) the CTM;(b) our previous equilibrium model without spillback (from now on called DUE); (c) the equilibrium modelwith spillback presented here (from now on called DUE-SP). To be noticed that, while the CTM performsonly a network loading for given splitting rates, our DTA models also simulate the path choices, even if theyare not necessary in this case.

Specifically, we analyze one direction relative to a highly congested stretch of the Venice–Mestre highwaybelt, which is 10 km long and includes five interchanges, as depicted in Fig. 17. The trip table is made up of 19origin–destination components, each one having a specific temporal profile with 288 time intervals of 300 s,thus covering a period of analysis of 24 h. We also know the corresponding temporal profiles of flows andspeeds detected over 10 sections that are located as shown in Fig. 17. The highway has two lanes, the rampsonly one, and there are no bottlenecks; the free flow speed is 80 km/h.

In the CTM we have considered cells of 50 m, which require time intervals of 2.5 s; then, the underlyinggraph is made up of 167 cells and the period of analysis is to be divided into 34560 time intervals. Instead,the graph underlying our models is made up of 29 arcs with variable length, while the period of analysis issuitably divided into 288 intervals of 300 s.

The flow charts in Fig. 18, depicting the results yielded by DUE and DUE-SP with reference to the timeinterval of maximum congestion, make the difference between the two solutions in terms of congestion prop-agation obvious. Without spillback, the congestion arising at the Miranese interchange due to a high inflowfrom the ramp, produces a queue on the next arc, which does not propagate backwards, while with spillbackthe queue backs up to section Villabona and to the ramps, which is what actually happens in reality.

Villabona Marghera Carbonifera Miranese Castellana Favorita

Origin

Destination

Measurement section

Terraglio

A B

Fig. 17. Graph of the highway stretch.

DUE-SP DUE

Villabona

Marghera

Carbonifera

Miranese

Villabona

Marghera

Carbonifera

Miranese

hypocritical flow

hypercritical flow

Fig. 18. Flow charts for the 9.40–9.45 AM interval relative to the models with and without spillback.

Section A - flow [veh/h]

0

1000

2000

3000

4000

200 300 400 500 600 700 800

Measured Model without spillback Model with spillback Cell Transmission model

Measured Model without spillback Model with spillback Cell Transmission model

Section A - speed [km/h]

0

20

40

60

80

100

200 300 400 500 600 700 800

Section B - flow [veh/h]

0

500

1000

1500

2000

200 300 400 500 600 700 800

Section B - speed [km/h]

0

20

40

60

80

100

200 300 400 500 600 700 800

min

Fig. 19. Flows and speeds measured and calculated on two different sections.


Fig. 19 depicts the temporal profiles of flows and speeds during the morning peak hour in sections A and Bidentified in Fig. 17, showing that DUE-SP provides results similar to those yielded by the CTM and satisfac-torily close to the observed values. Contrarily, the results produced by the DUE model are quite different,especially in terms of speeds and, consequently, of travel times.

5.3. The efficiency of the algorithm

In order to assess the efficiency of the proposed algorithm, we compare its performances to those obtainedby other DTA models, with and without spillback representation. Specifically, we address the numerical

Table 1Problem sizes

Nguyen–Dupius network Sioux Falls network Rome network

Nodes (no.) 17 48 3672Arcs (no.) 23 124 9365Centroids (no.) 4 24 494Od components (no.) 4 12 96,132Trips (veh) 300 7200 264,099Time intervals (no.) 100 30 18Interval length (s) 10 60 600

Table 2Performance comparisons

Nguyen–Dupius network Sioux Falls network Rome network

DUE-SP Lo and Szeto (2002) DUE-SP Han (2003) DUE-SP DUE

Eerror (veh per interval) 0.1 0.1 1.5 1.67 5 5Iterations (no.) 23 89 28 50 104 17CPU time (s) 0.23 55 0.5 720 4566 366CPU speed (GHz) 3 0.45 3 0.45 3 3Eqv. CPU time (s) 0.23 8 0.5 108 4566 366


applications presented in Lo and Szeto (2002) and in Han (2003). We note the problem sizes in Table 1 and theperformance comparisons in Table 2, where calculation times are suitably scaled in order to take the differentCPU speeds and the convergence criteria (the norm of the difference between the equilibrium and the networkloading flow vectors) into account. A relatively high error is considered, since it is well known that the con-vergence rate of the MSA decreases rapidly.

The first instance is a deterministic DTA with spillback originally performed by a cell-based variationalinequality model adopting explicit path enumeration on the so-called Nguyen–Dupius network; refer to Loand Szeto (2002) for the description of supply and demand. Results show that the DUE-SP algorithm requiresless iterations and is quite faster in finding the solution, despite the MSA algorithm.

The second instance is a Logit DTA (h = 600 s) without spillback originally preformed by a deterministicqueue model adopting explicit path enumeration on the so-called Sioux Falls network; refer to Han (2003) forthe description of supply and demand. Also in this case DUE-SP needs less iterations to converge and its rel-ative gain in calculation time is even higher.

Finally, in order to assess the applicability of the method to real-size instances, we performed DTA byDUE and DUE-SP on the network of Rome.

5.4. The convergence of the method

In the following we empirically prove the convergence of the proposed algorithm by plotting the equilib-rium error EQER versus the number of equilibrium iterations EQIT. For this purpose, we employ the SiouxFalls network by Han (2003), with the demand level suitably increased in order to reproduce high congestion.We assume as EQER, at each iteration, the maximum absolute value of the difference between the components

0

1

2

3

4

5

6

7

8

9

10

0 2000 4000 6000 8000 10000 12000 EQIT [n˚]

veh/h

EQER [veh/h]

EQER approximation [veh/h]

EQER [veh/h]

0

2

4

6

700 750 800 850 900

n˚ of NPF iterations within one DTAiteration

0

1

2

3

700 750 800 850 900

2

6.469532

0.964

EQER EQIT −1.11465==ρ

Enlargement 50 x

Fig. 20. Convergence plot.


of the DTA and the NLM maneuver flow vectors u and unlm, and as approximating function of EQER anexponential curve whose parameters, obtained through the robust least square error method (Hampelet al., 1986), yield a cross-correlation coefficient q close to 1. The convergence plot and its 50· enlargementwith respect to the iteration axis, both depicted in the same Fig. 20, highlight the following properties:

(a) the plot exhibits an alternation between a set of DTA external iterations, characterized by a smoothlydecreasing EQER, and one iteration exhibiting a jump, a kind of ‘‘local discontinuity’’ within the errortrend, followed by a few iterations (3 in Fig. 18) after which the previous trend is restored;

(b) at each discontinuity, the number of NPF internal iterations is higher than one (only one iteration is per-formed if the capacity vector does not change);

(c) the size of the above discontinuities decreases as the number of DTA iterations increases;(d) setting aside the local discontinuities, the plot appears to monotonically tend to zero, as the approximat-

ing curve actually does.

On these bases, we can draw two interesting considerations. The first one regards the approximating curve.Since the number of iterations characterized by discontinuities is relatively negligible, as shown by the enlarge-ment in Fig. 18, a cross-correlation coefficient q close to 1 is no longer surprising. The second and more impor-tant consideration concerns the local discontinuities, which, rather surprising at first sight, turn out to beentirely consistent with the spillback phenomenon as it is modeled here: as long as the spillback patternremains almost the same, the flow error varies slightly at each successive iteration; on the contrary, when aspillback condition arises or disappears at a certain node, then the capacity vector is significantly modifiedand the flow error suffers a discontinuity.

Based on all this, we are in the position to state the empirical evidence of the algorithm convergence.

6. Conclusions

In a previous paper we proposed a new continuous-time formulation of the DTA based on implicit pathenumeration where it is no longer necessary: (a) to introduce the CDNL as a sub-problem of the DTA in orderto ensure the temporal consistency of the supply model; (b) to assume the hypothesis that the longest timeinterval is shorter than the smallest free flow arc travel time. As a result it was possible to solve large instancesof the problem in reasonable computing time, but the propagation of congestion among adjacent road linkswas not modeled.

In this paper we overcome this limit proposing a new equilibrium model formulated as a fixed-point prob-lem in terms of maneuver flow temporal profiles at nodes, which has the considerable advantage of represent-ing spillback congestion within the supply model, i.e., without affecting the NLM, thus allowing us to devisean highly efficient algorithm.

Investigating the relevance of queue spillovers in the context of the DTA on test networks, we found outthat both the equilibrium flow and travel time patterns obtained with and without modeling the spillback phe-nomenon may be considerably different, especially when the congestion level is high.

Dealing with the case of a real corridor, where the path choice is not involved, we compared the CTM byDaganzo with our equilibrium model. We found out that both models are effective, since they reproduce theobserved flows and speeds correctly.

The computing times obtained by applying our model both on toy networks available in literature and on areal-life network allow us to conclude that the proposed algorithm is definitively more efficient than other pro-cedures solving the dynamic user equilibrium.

Finally, we gave a thorough empirical evidence of its convergence.

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