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A study of the emergence of kinematic waves in targeted state car-following models of traffic

Étude de lémergence dondes cinématiques au sein de modèles de poursuite comportant un état de sécurité ciblé
Antoine Tordeux, Sylvain Lassarre et Michel Roussignol

Résumés

Nous proposons d’étudier l’émergence d’ondes cinématiques au sein de modèles microscopiques de flux de trafic. Le déplacement des véhicules est dicté par différents systèmes différentiels de poursuite reconnus, admettant une fonction d’état ciblé ainsi qu’un temps de réaction strictement positif. Au travers de simulations, nous observons deux types d’état stationnaire d’une file de véhicules évoluant sur un cercle : un état homogène pour lequel les vitesses et les espacements des véhicules sont constants et égaux, et un état hétérogène dans lequel des ondes cinématiques se propagent. Le temps de réaction et la forme de la fonction d’état ciblé semblent être des facteurs microscopiques déterminants quant à la stabilité macroscopique des états stationnaires. Une analyse linéaire de la stabilité d’un état homogène est alors entreprise avec un modèle de poursuite basique. Les résultats permettent de mettre en évidence une relation explicite entre le temps de réaction, la fonction de sécurité ciblée et la nature des états stationnaires d’une file de véhicules.

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Texte intégral

The authors thank the anonymous referees of the Cybergeo journal for helpful comments on the earlier versions of this paper.

Introduction

1A traffic flow composed of a lot of vehicles can be seen as a collective phenomenon. In a microscopic approach of modelling, inspired by statistical physics theory, local regulation rules generate macroscopic properties. In the context of traffic modelling, the flow characteristics emerge from the interactions of the vehicles and can be explained by driver behaviors. The microscopic modelling of traffic flow has belonged to the field of complex collective system theory since the late 1930s. The models aim to provide a better understanding of the underlying microscopic mechanisms that govern traffic flow performances. The microscopic model parameters have a physical meaning and their estimated values can be interpreted and checked. Microscopic car-following models, describing a coupled interaction with the predecessor, are able to reproduce several traffic flow properties, even if a driver interacts in real traffic flow with several predecessors, based on the assumption that keeping spacing distance with the predecessor is the first objective of a driver.

2The simulation of the dynamics of deterministic car-following models on a ring of a finite length reveals two types of stationary states:

  • A homogeneous one in which the flow in stationary state is composed of vehicles evolving at the same constant speed and spacing.

  • A heterogeneous one where vehicles’ speed and spacing vary during the time in stationary state. The presence of kinematic waves is generally observed, inducing successions of acceleration and deceleration phases for the vehicles and the presence of collisions (i.e. the order of vehicle varies).

3The question is about the stability of this homogeneous regime. The stability of stationary states is a classic problem in statistical physics for all dynamic systems. A frequently used approach consists in the linearization of the differential equations. When a stationary state is stable, the theory states that the stationary state is reached for a set of initial values, the size of which depends on the regularity (i.e. the linearity) of the differential equations.

4This research is triggered by the frequent observations of kinematic waves in real traffic flow. At a bottleneck, when the incoming flow is high, there is an accumulation of vehicles and a shock wave propagates up-stream at a constant speed. Althought, with any bottleneck we observe the propagation of kinematic waves appearing spontaneously when the flow density is sufficiently high. From a microscopic point of view, kinematic waves induce a well known phase of acceleration and deceleration for vehicles. For this reason, the waves are usually called stop-and-go waves. On the figure 1, vehicle positions on a lane are plotted as a function of time. The slope of the trajectories is the vehicles’ speed. This representation, frequently used in traffic engineering, gives a general view of the traffic dynamic. On the figure, obtained on a highway lane, a kinematic wave is moving backward at a constant speed. Environment, traffic signs or regulation rules at an intersection can generate waves. For instance, the traffic light cycle can generate waves for some density level. Nevertheless, kinematic waves are observed on some roads, such as a highway, whose infrastructure cannot explain their presence. It seems that drivers’ behaviors are at the origin of the waves. Experiments have been done with 22 vehicles evolving on a ring of length 230 m (Sugiyama et al, 2008). Initially, vehicles are in a homogeneous state at constant speed. Quickly (order of 1 mn), shock and rarefaction waves appear ([video]) because of small variations in driver behavior and even with homogeneous initial conditions. The spontaneous emergence of kinematic waves is the phenomenon we are going to study and explain by means of car-following models of driver behavior in a simple configuration such as a ring.

5Kinematic waves in traffic flow are an uncontrolled phenomenon with negative effects. The waves can induce strong deceleration phases that are a source of collisions. It is more uncomfortable for drivers and consumes more energy than a homogeneous flow because of unnecessary acceleration phases. In this context, it can be interesting to identify microscopic parameter values able to generate waves in car-following models. A better understanding of wave mechanism emergence due to vehicle interaction is useful to develop control strategies of the traffic flow, notably through adaptive control of vehicles’ velocity and spacing (ACC Automatic Cruise Control) or control of vehicles’ maximum speed (ISA Intelligent Speed Adaptation).

Observations of vehicles trajectories evolving on a highway. Data obtained by Treiterer (1975).

6We present in section 2 of this article, a large class of well known car-following models that incorporate a targeted space function and a strictly positive reaction time. These models assume a local interaction exclusively based on a driver and its first predecessor. In section 3, we show and study the simulation results of three famous car-following models on a ring: the Newell (1961), the Optimal velocity (Bando and Hasebe, 1995) and the Intelligent Driver (Treiber et al, 2000) models. Two types of stationary states are identified: a homogeneous state in which vehicle speed and gap are constant and equal, and a heterogeneous state in which kinematic waves spread out. The impact on the nature of the stationary state of the targeted state function form and the reaction time value is assessed through experimentation. In section 4, the linear stability of a homogeneous stationary state is analytically analyzed to obtain explicit conditions on the parameters in the case of the basic Newell car-following model. Some relations exist between the form of the targeted function, the value of the reaction time and the stability of the homogeneous state. A discussion about the behavior of a line of vehicles and the factors which keep the traffic homogeneous is proposed in the conclusion.

Form of the considered car-following models

7In this part, we describe a class of microscopic car-following models that incorporate a targeted state function, a regulation function on how to reach the targeted state and a strictly positive reaction time (Tordeux, 2010). In a car-following process, a vehicle follows his predecessor at a speed that is generally less than a maximum desired speed. The situation is such that a driver is led to adapt his behavior as a function of the neighboring vehicles behavior. It is usually admitted that a driver tries to respect a spacing distance from his predecessor according to his speed. The models describe a coupled longitudinal interaction exclusively based on the first predecessor. The variables taken into account are a vehicle and its predecessor speeds, the vehicle length and the spacing distance (figure 2).

Scheme of the model environment and the variables taken into consideration.

Model class

8In continuous time, differential equations are frequently used to model the dynamics of the vehicle motion. A model of the dynamics of the position of a vehicle can be written under the differential equation form:

where vn(t) = dxn(t) / dt is the speed of the vehicle n at time t, x is the central position and n+1 is its first predecessor. When the dynamics of the speed is described, a model can be written under the differential system form (Treiber et al, 2006a):

9In a comparable approach, a model can describe the dynamics of the acceleration by using a system of three equations (Lubashevsky et al, 2003). The regulating function R, including the vehicle length ℓ, defines the interaction between vehicles and characterises the model. In Eq. (1) of the dynamics of the position, the function is a speed. While in Eq. (2), of the dynamics of the speed, the function R is an acceleration rate.

10The reaction time of a driver seems to be an essential parameter of the car-following situation (Herman et al, 1959; Leutzbach, 1988). It has been explicitly observed and estimated around 1.3 seconds for a distracted driver and around 0.5 second for a vigilant one (Lerner, 1995). It can be integrated into a model (Chandler et al, 1958; Newell, 1961, Lubashevsky et al, 2002) in which observable variables are delayed by the reaction time:

with η the position or speed variable and Tr the reaction time. Sometimes, only the predecessor variables are affected by the reaction time (Bando et al, 1998; Tordeux et al, 2010a) :

11The predecessor position is (linearly) estimated at the time t according to the predecessor speed at time t+T r.A comparable procedure is proposed in (Treiber et al, 2006a), called temporal anticipation.

12In unstable situations in which the speed of the predecessor varies, the reaction time can generate some collisions. The respect of safety distances (or time), which depends on speed, enables drivers to compensate for the reaction time and avoid collision. In a large class of car-following model (Aw et al, 2002), a microscopic targeted state is explicitly defined by a function of targeted speed, distance or time. Let D(v) be the targeted function of the distance depending on speed and V(d) the inverse targeted function of speed depending on the distance. T(d) = d/V(d) or T(v) = D(v)/v are the targeted functions of time depending respectively on distance and speed. If the targeted time is constant, the targeted functions of speed or distance are linear. In any cases, the models admit for each spacing ΔxS and each speed vS a stationary state (ΔxS,vS) (for instance(D(vS),vS), (ΔxS,V(ΔxS)), (T(vS)vS, vS) or ΔS, ΔS/T (ΔS)) ), called fixed point and defined by:

(5) R(ΔxS, vS, vS)

for a model of position dynamics and:

(6) R(ΔxS, vS, vS) = 0

for a model of speed dynamics. When a model incorporates a monotonous targeted function, the fixed point is unique for a given speed or spacing. Some well known models of speed dynamic, in which the regulation function is proportional to the difference of speed with the predecessor (Gazis et al, 1961), admit an infinity of fixed points for a given value of speed or spacing.The introduction of a reaction time does not change the form of the microscopic pursuit state, but it can affect the stability of the homogeneous state.

Related models

13These car-following models explicitly include a targeted state through a function of targeted speed or time. In an interactive case and for a given speed (respectively a distance gap), a driver will associate a distance gap (respectively a speed).

14For instance, the Newell model (1961) is exclusively defined by a non linear function of targeted speed that depends on the distance gap. The model was studied in discrete time with a reaction time introduced in the same way as the relation given by Eq. (3). For any targeted speed function V, in continuous time, the model can be written:

15For a distance gap, the stationary state of the model is unique, defined by the couple (d,V(d)). The stationary state is the same in the Optimal Velocity model, developed by Bando and Hasebe (1995), in which a relaxation time is applied to the speed and to a function of targeted (called optimal) speed:

16A version of the Optimal Velocity model includes a reaction time (Bando et al, 1998).

17The Intelligent Driver model defined by Treiber et al (2000), in which a reaction time is added in (Treiber et al, 2006a), is written :

with f(vn,vn+1) = Tvn+vn(vn-vn+1)/(2√ab)

18The quantities a, b, ϑ, d and T are parameters. T is a targeted time, assumed constant. Some variants of the Intelligent Driver model introduce a variable targeted time gap (Treiber and Helbing, 2003; Treiber et al, 2006b). In the model, if d, ϑ >> 0, for a speed v, the stationary pursuit state is defined by the couple (vT, v).

Analysis of the stability by simulation

19In this section, the states of a line of vehicles on a ring are observed over time by simulation and for each related models. Experiments are done according to the form of the targeted state function, the reaction time value and the scheme of integration. A constant targeted time is used (i.e. a linear targeted speed function). The simulations are run with NetLogo a multi-agents simulation freeware ([NetLogo]).

20Context of the simulations

21For a line of vehicles that evolve on a 1000 m long ring, the system has two types of steady states:

  • A homogeneous state in which vehicle speeds and gap are identical (figure 3, left);

  • A heterogeneous state in which one or several kinematic waves seem to spread out indefinitely (figure 3, right).

22In figure 3, vehicle trajectories of two systems evolving towards a homogeneous and a heterogeneous stationary state are traced. The number of crossings of a given vehicle is approximately equal in both systems, attesting that average flow performances are the same in the two stationary states.

23The nature of the stationary state is assessed by the empirical variability of the vehicle speeds. This indicator is nil when the flow is homogeneous and strictly positive in a heterogeneous state. Ten independent trajectories of the system, with random initial conditions, are measured after a simulation time (2000 s) long enough to consider that the system is in a stationary state. These scenarios are repeated for different levels of traffic density and values of the targeted state related parameter . The models used in the study are the following:

  • The Newell model given by Eq. (7) with V(d) = Td and = 5

  • The Optimal Velocity model given by Eq. (8) with V(d) = Td, = 5 m and τ = 0.01 s.

  • The Intelligent Driver model given by Eq. (9) with a = b = 1 m / s2 and d = 4.

24The models parameters values are calibrated in order to give a homogeneous state stable for a nil reaction time. Thus, it is assumed that the instability of the homogeneous state is only due to the introduction of the reaction time. The maximum vehicles speed is equal to ϑ = 40 m/s. The discretization scheme used in the simulations is the second order model proposed in (Treiber et al, 2006a) for the speed dynamics model and first order explicit Euler scheme for the position dynamics model. The time step is equal to dt = 0.1s. In the heterogeneous stationary case, the vehicle speed is limited by vn ≤ (xn+1xn -) / dt at each time step to avoid collisions.

Trajectories of 40 vehicles on a ring. In black, an individual trajectory. On the left side, an example of a stable homogeneous stationary state and, and on the right side, an example of an unstable state. Model Eq. (3) and (9). Random initial conditions.

Results obtained

25The two systems of equations given by Eq. (3) and Eq. (4) with the integration of the reaction time are compared by means of the empirical speed variability of the systems in stationary state (respectively figure 4 and figure 5). For both, the targeted time parameter value seems to be a determining factor of the homogeneous stationary state stability. Except for a critical value of this parameter, the system converges towards a heterogeneous stationary state in which the variability of the vehicle speed is not nil. The phenomenon appears beyond a critical density threshold approximately equal to (ϑTy + )-1. The regulation strategies differ from one model to another but the results seem to be comparable for all the models. A transition phase appears through homogeneous and heterogeneous states, at a fixed value of the reaction time and according to the value of the targeted time gap, and vice versa. The heterogeneous states differ from one model to another because the nature of the generated waves depends on the model form. Notably the presence of collision, inducing a correction to respect the order of vehicles, often observed with the Newell and Optimal Velocity models, generates a particular form of the waves. Nevertheless, even if the heterogeneous states of the models are different, the transition between a heterogeneous state to a homogeneous one is obtained at the same value of the targeted time. Thanks to simulation experimentation we can give the conditions of an heterogeneous state stability that is different according to the procedure of integration of the reaction time. We find the conditions T > 2Ty for the first reaction time integration scheme given by Eq. (3) and T > Ty for the second scheme given by Eq. (4)).

Empirical standard deviation of vehicle speed according to different values of density and targeted time. Procedure of reaction time integration given by Eq. (3).
Empirical standard deviation of vehicle speed according to different values of density and targeted time. Procedure of reaction time integration given by Eq. (4).

26On the other hand, when the targeted time is constant, equal to 1 s in these experiments, a transition phase occurs according to the value of the reaction time. The speed distribution of the vehicles varies from a uni-modal to a bi-modal form according to the value of the reaction time and to the procedure of integration used. The uni-modal speed distribution reflects an homogeneous flow with constant vehicle speed. The bi-modal speed distribution reflects an heterogeneous flow in which some kinematic waves spread out. One mode focuses on the vehicles in a wave, with low speed. The second is centered on the highest speeds of the vehicles out of the waves. Figure 6 presents an estimation of vehicle speed in the stationary state according to the value of the reaction time and the scheme of integration used for the Newell car-following model. Clearly, the stationary state is homogeneous if 2T r<T with the first model of reaction time integration given by Eq. (3) and if T r<T with the model given by Eq. (4). The dotted line is the average vehicles speed, which decreases in heterogeneous states when the reaction time increases. This phenomenon can be due to an increase in the wave length.

Vehicle speed distribution estimated by gaussian kernels for different values of the reaction time. Newell model given by Eq (7), integration of the reaction given by, respectively, on the left, Eq. (3), and on the right, Eq. (4).

27The simulation results present a transition phase from a heterogeneous to a homogeneous stationary state according to the value of the reaction time, its scheme of integration and the slope of the targeted state function. The targeted state functions tested are linear and it could be interesting to obtain explicit conditions on the targeted function form and the reaction time value. In the next section analytical investigation is proposed on the Newell model. The aim is to obtain explicit conditions on parameters for the stability of a homogeneous stationary state, in adequacy with the simulation results.

Linear stability analysis of the Newell model

Context

28Two types of stability are generally considered (Chandler et al, 1958; Herman et al, 1959):

  • A local stability analysis of a vehicle following a leader that evolves at a constant speed,

  • A global stability analysis of a line of vehicles that evolve on a ring or on an infinite lane.

29The stability analysis is frequently used to evaluate how a variable, measuring a distance to a stationary state, evolves. The stationary steady state stability is defined in terms of perturbation analysis: perturbation absorption is said to be stable, while perturbation amplification is said to be unstable. The analysis proposed in (Bando and Hasebe, 1995, Kesting and Treiber, 2008), is appropriate in order to study the stability of a model in which no reaction time takes place. In the studies (Chandler et al, 1958; Herman et al, 1959; Bando et al, 1998) a reaction time is incorporated. In this section, the evolution of a vehicle line on a ring is considered in order to study the stability of the homogeneous stationary states. In (Kesting and Treiber, 2008; Tordeux 2010) stability conditions of the general forms of car-following models given by Eq. (1) and (2) are provided. Here, the car-following model used is a linear version of the Newell model given by Eq. (7). This model is one of the simplest ones which includes the two fundamental parameters of targeted safety state function and the reaction time. Our purpose is to evaluate the conditions on the car-following models parameters that make the homogeneous stationary state attractive.

30A flow of N vehicles evolving on a ring of length L is considered. The system is composed of N non linear differential equations in interaction. In the homogeneous steady state, denoted {H}, distance and vehicle speed are constant and equal for all vehicles:

with vn(t) = dxn(t)/dt the speed of the considered vehicle n at time t, x the central position and n+1 its first predecessor. Equivalently, an homogeneous state can be characterised by the relation:

31The stability analysis consists in studying the evolution of the difference between the system variables and homogeneous system variables of position and speed:

32An homogeneous stationary state is said to be stable when:

Initial model

33In a preliminary study, no reaction time is incorporated into the Newell model:

34V is the targeted speed function of the distance gap. is the vehicle length or more generally a minimum distance of spacing. The system is composed of N non linear interacting equations. By using a linear approximation of the function V near an homogeneous steady state {H}, the linear differential system, which describes the approximation of perturbations, becomes:

with γ = V'(L/N - ).

35The solutions of the system are a linear combination of exponential terms:

with ck = exp(i2πk / N), k = 1,…N – 1, i is a square root of the unit. Bando and Hasebe (1995) obtained the solution by expanding Fourier series. The same results can be obtained by classical matricial calculus (Tordeux et al, 2010b). Notably, for a particular initial condition, the solution can be written:

With ck = exp(i2πk / N), k = 1,…N – 1. By substituting the solution Eq. (17) in Eq. (15), the complex number zk satisfies:

36(18) zk + γ(1 - ck) = 0

zk. If with

the oscillation of this mode is enhanced as the simulation progresses and the state becomes unstable. On the other hand, if

the oscillation of this mode shrinks and the state tends towards an homogeneous steady flow.

and a stable homogeneous steady state can be found for the model given by Eq. (14) if:

(19) y > 0

37Under the assumption that the targeted speed function is strictly increasing, the model (Eq. (14)) reproduces an homogeneous flow on a ring in the stationary limit time case. Nevertheless, we show in the next section that the integration of a reaction time into the model can modify the stability condition.

Integration of the reaction time

38The two procedures described by Eq. (3) and Eq. (4) for the integration of a reaction time Tr > 0 is applied to the Newell model given by Eq. (14). To study the stability of the homogeneous state with delayed models, linear approximations in time are investigated.

Procedure of integration of the reaction time given by Eq. (3)

39When all the observable variables are delayed by the reaction time, the model Eq. (14) is written:

40The evolution scheme of the perturbation is linearly estimated by the scheme:

41With γ = V'(L/N - ) .This system is a delayed differential system of equations. Mahnke et al (2008) propose to apply a Taylor expansion of first order with respect to the reaction time Tr. The perturbation evolutions are defined by the following system of two equations:

42By substituting the solution of Eq. (17) in Eq. (22), the complex number zk satisfies:

43It can be found that

44In these conditions, the homogeneous steady state is stable for the model given by Eq. (20) when:

(24) 0 < γ < 1/(2Tr)

45The condition is in adequacy with the results obtained by simulation. The speed distribution of the vehicles is uni-modal, the flow is homogeneous, when 2Tr<T.

Procedure of integration of the reaction time given by Eq. (4)

46According to the procedure of integration of the reaction time given by Eq. (4), the Newell model is written:

47The evolution scheme of the perturbation is linearly estimated by:

with γ = V'(L/N - ). If a TAYLOR expansion of first order is applied with respect to the reaction time Tr to the quantities n(t+Tr) and a second order Taylor expansion to the quantity n (t+Tr), the scheme of perturbation evolution becomes:

48If we expand the solutions given by Eq. (17) in which the complex number satisfies:

we find that

The homogeneous steady state is stable for the model given by Eq. (25) if:

(29) 0 < γ <1/Tr

49In adequacy with the results obtained by simulation, the speed distribution of the vehicles is uni-modal, when Tr < T.

50The analytical results obtained show that the Newell car-following model, exclusively based on a function of targeted speed, produces a stable homogeneous stationary state of a vehicle line on a ring, if the targeted speed is an increasing function of the distance gap. Intuitively, this characteristic is coherent. When a reaction time is stictly positive, the results show that the slope of the targeted function is constrained by the reaction time value, if a steady homogeneous flow is required. The reaction time becomes a factor of heterogeneity of the flow. In particular the homogeneous condition is broken for sufficiently high reaction time values. The targeted time gap function, which is the distance gap divided by the targeted speed function, can be used as an indicator of the level of traffic flow homogeneous state stability. In the model studied, for a targeted time constant, the stability of the flow depends directly on the values of the pursuit time and reaction time. If the targeted time gap is variable, the nature of the variation is a determining factor. A decreasing targeted time function is a factor of instability of the homogeneous state (and conversely) because it induces a minimal value to the derivative of the targeted speed function.

Conclusion

51The results obtained show that complex collective phenomena of phantom traffic jams can easily emerge from microscopic and local interaction rules including a delay. In the class of car-following models presented, a targeted pursuit state is introduced. The system is governed by a system of delayed, non-linear differential equations in interaction. With the help of simulation, we observe that the introduction of a reaction time generates instability of a homogeneous flow and the propagation of kinematic stop-and-go waves. The linear analysis of the perturbation propagation gives an explicit condition that links the targeted state form, the reaction time and the stability of the homogeneous flow, in agreement with the simulation results and with (Treiber et al, 2006a). The homogeneous state is linearly stable, i.e. a small perturbation is absorbed, when the derivative of the targeted function of speed is less than an inverse function of the reaction time.

52There is a link between the regulation behavior of vehicle spacing and speed and the introduction of a delay into the driving. Several statistical estimations seem to agree in observing average time gap values that decrease with speed (Treiber and Helbing, 2003; Treiber et al, 2006b; Banks, 2003; Nishinari et al, 2003) suggesting high value of the targeted speed function derivative. On the other hand, the rang of observed reaction times in driver behavior is rather large (Lerner, 1995). The conditions obtained and the statistical estimation of the parameters seem to attest that drivers’ behaviors tend to produce waves. If these observed values are taken into the models, the results obtained can be a way of explaining the easy emergence of kinematic waves in real traffic flow. Nevertheless antagonist mechanisms, not taken into account in this study, may keep a flow stable and homogeneous. For instance, it has been observed that driver anticipation behaviors, including several predecessors in interaction, tend to compensate the influence of the reaction time (Treiber et al, 2006a; Treiber and Helbing, 2008). This aspect explains the absence of collisions of vehicles driving with a very small gap at high speed in real traffic data. Moreover, analytical investigation reveals that anticipation aspect is a factor of homogeneous flow stability (Wagner, 1998; Tordeux et al, 2010a). However, anticipation is not always possible. For instance, a turn, a truck, or the weather conditions can reduce visibility. The lack of driving experience, a temporary inattention of the driver or tiredness can equally alter anticipation and lead to the existence of an longer reaction time. It can be interesting to develop adaptive control of vehicle speed and spacing in order to limit the emergence of waves. Autonomous models, using vehicle speed and estimating the distance gap and the predecessor speed, reducing to nil the reaction time, can be developed to produce regulation strategies limiting wave propagation and amplification. Some studies show that small proportions of controlled vehicles (around 10 to 20 percent) give significant results of the flow properties (Davis, 2004; Kesting et al, 2007). Moreover, some car-following models estimated on real data are able to reproduce safe and comfortable performances. The statistical method approach used to estimate distance gap and predecessor speed, includes potential measurement errors and random variables. Yet, stationary state stability analyses of stochastic models with noise have to be investigated to ensure its applicability.

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Pour citer cet article

Référence électronique

Antoine Tordeux, Sylvain Lassarre et Michel Roussignol, « A study of the emergence of kinematic waves in targeted state car-following models of traffic », Cybergeo : European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 531, mis en ligne le 23 mai 2011, consulté le 17 août 2018. URL : http://journals.openedition.org/cybergeo/23660 ; DOI : 10.4000/cybergeo.23660

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Auteurs

Antoine Tordeux

LVMT, 19 rue Alfred Nobel, cité Descartes, 77455 Marne-la-Vallée, France
Corresponding author. Email address: antoine.tordeux@enpc.fr— Phone : +33 164 152 139

Sylvain Lassarre

GRETTIA - IFSTTAR, Le Descartes II, 2 Rue de la Butte Verte, 93166 Noisy le Grand, France
Email address : sylvain.lassarre@ifsttar.fr — Phone : +33 145 925 764

Michel Roussignol

LAMA - Université Paris-Est, 5 boulevard Descartes, 77454 Marne-la-Vallée, France
Email address : michel.roussignol@univ-mlv .fr — Phone : +33 160 957 528

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Droits d’auteur

© CNRS-UMR Géographie-cités 8504

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