Traffic flow modeling

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Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

second order model [17]. The resulting (system of) classical conservation laws usually turns out to be well-posed. Equations with non-local flux have been recently introduced in traffic flow modeling to account for the reaction of drivers or pedestrians to the surround- ing density of other individuals, see [3, 9–12, 21]. While pedestrians are likely to react to the presence of people all around them, drivers will mainly adapt their velocity to the downstream traffic, assigning a greater importance to closer ve- hicles. In this paper, we consider the following mass conservation equation for traffic flow with non-local mean velocity:
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Traffic flow modeling by conservation laws

Traffic flow modeling by conservation laws

The aim of this thesis is to investigate some mathematical models arising in traffic flow, from both analytical and numerical points of view. Traffic is a phenomenon that is hard to model and simulate due to the difficulty of reproducing the formation and the presence of traffic jams. Several approaches have been developed during the years, each one focusing on some particular traffic characteristic. In particular, researchers have started looking at traffic for different purposes as, for example, minimization of congestion, accidents, pollution, and safety issues. There are several ways of describing traffic flow and the different methods can be summarized in three big categories: microscopic models, macroscopic models and kinetic models. Microscopic models describe the trajectory of each single car in the road with an ordinary differential equation (ODE). The basic models are the car-following ones or models based on Newton’s law. The main assumption of the car-following models is that an individual car’s motion only depends on the car ahead; see [ 10 , 18 , 39 , 59 , 70 , 87 , 128 ]. Kinetic models, instead, use Boltzmann-like equations and the main quantities describing traffic are expressed with density distribution functions; see [ 1 , 115 , 121 , 123 , 124 ]. The works in thesis refer to macroscopic models where traffic is considered as a fluid. The first ones to introduce this concept were Lighthill, Whitham [ 111 ] and independently Richards [ 127 ] in the fifties. They were the first ones to describe traffic flow with equations coming from fluid dynamics, using a non linear hyperbolic partial differential equation (PDE). The Cauchy problem has successfully been extended to initial boundary value problems in [ 11 ] and then developed specifically for scalar conservation laws with genuinely nonlinear flux in [ 102 ]. More recently, several authors proposed models on networks that take into account different types of solutions at the intersections, see [ 40 , 41 , 58 , 64 , 67 , 68 , 89 , 93 , 95 , 112 ] and the references therein. In all these works, the road network is described as a graph, incoming and outgoing roads are the edges while the junctions are described by the nodes. Several models on how to distribute the traffic are proposed: in [ 40 , 41 , 58 , 67 , 68 , 95 ] the traffic is distributed according to an optimization problem, while in [ 64 , 93 ] the junction dynamics is described by a buffer and finally in [ 89 ] the traffic is distributed with a multilane model. Subsequently, different numerical methods that approximate solutions for road networks have been developed, see for example [ 3 , 32 , 89 , 106 , 109 , 110 ].
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Scalar conservation laws with moving density constraints arising in traffic flow modeling

Scalar conservation laws with moving density constraints arising in traffic flow modeling

y(0) = y 0 . (1.3) The above model was introduced in [10] to model the effect of urban transport systems, such as buses, in a road network. Other macroscopic models for moving bottlenecks in road traffic were recently proposed by [3, 12]. Compared to those approaches, the model described by (1.1) offers a more realistic definition of the slower vehicle speed and a description of its impact on traffic conditions which is simpler to handle both from the analytical and the numerical point of view.

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Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result

Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result

[15] Garavello, M., Goatin, P., 2011. The Aw-Rascle traffic model with locally constrained flow. J. Math. Anal. Appl. 378 (2), 634–648. URL http://dx.doi.org/10.1016/j.jmaa.2011.01.033 [16] Giorgi, F., 2002. Prise en compte des transports en commun de surface dans la mod´elisation macroscopique de l’´ecoulement du trafic. Ph.D. thesis, Institut National des Sciences Appliqu´ees de Lyon.

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Non-local conservation laws for traffic flow modeling

Non-local conservation laws for traffic flow modeling

locally in time. We present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition. Furthermore, we propose alternative simple schemes to numerically integrate non-local multi-class systems in one space dimension. We obtain these schemes by splitting the non-local conservation laws into two different equations, namely, the Lagrangian and the remap steps. We provide some estimates recovered by approximating the problem with the Lagrangian-Antidiffusive Remap (L-AR) schemes, and we prove the convergence to weak solutions in the scalar case. We show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first and second order numerical schemes. Moreover, we recover the numerical approximation of the non-local multi-class traffic flow model proposed, presenting the multi-class version of the Finite Volume WENO (FV-WENO) schemes, in order to obtain higher order of accuracy. Simulations using FV-WENO schemes for a multi-class model for autonomous and human-driven traffic flow are presented. Finally, we introduce a traffic model for a class of non-local conservation laws at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes. We use an upwind type numerical scheme to construct a sequence of approximate solutions and we provide uniform L ∞ and BV
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Mesoscopic multiclass traffic flow modeling on multi-lane sections

Mesoscopic multiclass traffic flow modeling on multi-lane sections

We recall that the first order LWR model [ 1 , 2 ] is given by the following scalar conservation law ∂ t k + ∂ x Q(k) = 0 (1) with k(t, x) the density and Q : R → R the flow-density Fundamental Diagram (FD). One has Q (k(t, x)) := k(t, x)V (k(t, x))

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Modeling traffic on urban road networks with hyperbolic conservation laws

Modeling traffic on urban road networks with hyperbolic conservation laws

Chapter 1 Introduction to hyperbolic conservation laws and traffic flow modeling In this thesis we investigate the mathematical modeling of traffic flow, with a specific fo- cus on urban applications. Modeling traffic flow and vehicle trajectories became a necessity in the twentieth century with the rise of car mobility in modern societies. Yet, it remains a complex physical phenomenon to understand and model accurately. The complexity of road networks, the dynamics of traffic congestion, the heterogenity of vehicle sizes and performances, drivers’ behavior and the large number of vehicles involved in the modeling process represent many challenges to overcome. Furthermore, urban areas are faced today with increasing levels of congestion and pollution due to transportation. The quality of air is declining in major cities, and vehicle emissions are significantly contributing to climate change on the long-term [60, 61]. In addition, autonomous vehicles are expected to be the next revolution of urban transportation. Contrary to classical drivers, these vehicles will be routed following mathematical algorithms, which will provide full optimal control of the trajectories with respect to given constraints [95]. This optimization problem is known in transportation as the Dynamic Traffic Assignment problem [127], and requires a controllable modeling framework. In this context, traffic could in the future be routed in order to minimize its negative externalities like emissions and congestion. For all these reasons, it is urgent to develop traffic models which represent accurately the behavior of vehicles on urban road networks, and this will be the motivation of the present work.
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The Aw-Rascle traffic model with locally constrained flow

The Aw-Rascle traffic model with locally constrained flow

[17] M. J. Lighthill and G. B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A., 229:317–345, 1955. [18] C. Merkle and C. Rohde. The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques. M2AN Math. Model. Numer. Anal., 41(6):1089–1123, 2007.

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Particle approximation of a constrained model for traffic flow

Particle approximation of a constrained model for traffic flow

for traffic flow Florent Berthelin and Paola Goatin To Prof. Alberto Bressan Abstract. We rigorously prove the convergence of the micro-macro limit for particle approximations of the Aw-Rascle-Zhang equations with a maximal density constraint. The lack of BV bounds on the density vari- able is supplied by a compensated compactness argument.

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Data driven uncertainty quantification in macroscopic traffic flow models

Data driven uncertainty quantification in macroscopic traffic flow models

In this paper, we focus on a 1.25 km long rampless road stretch on the highway A50 from Aubagne to Marseille. The road consists of three lanes and the speed limit is 90 km/hour. The data for this highway part were collected by 4 loop detectors (IDs 305, 304, 303 and 302). For the simulation, we divide the chosen road stretch into 4 segments. Each segment contains one detector. All the details about the considered traffic data and the loop detector locations are available in the git repository https://gitlab.inria.fr/acumes/dduq-traffic. The fundamental diagrams in Figure 1 illustrate the collected flow and speed data together with their corresponding densities for the inner loop 303. Most of the data points are located in the free flow phase which is visible by the clear functional relationship between density and flow or speed. The free flow phase corresponds roughly to speed values greater than 60 km/h. The more widely distributed data points in the higher density region correspond to congested regimes.
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Window Flow Control in FIFO Networks with Cross Traffic

Window Flow Control in FIFO Networks with Cross Traffic

In fact, we will show that in the case of deterministic service times of the controlled customers, the upper bound on the maximum throughput is reached when the cross flows are scaled in[r]

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Lévy flights and fractal modeling of internet traffic

Lévy flights and fractal modeling of internet traffic

The self-similarity of network traffic was observed in nu- merous papers, such as [2], [21], [24] and [6]. These and other papers showed that packet loss, buffer utilization, and response time were totally different when simulations used either real traffic data or synthetic data that included self-similarity [7], [8]. Recent papers challenge the applicability of these papers’ re- sults for today’s network traffic. For instance, the authors in [15] believe that it is time to reexamine the Poisson traffic as- sumption in the Internet backbone. They claim that traditional Poisson models can be used again to represent the characteris- tics of the aggregate traffic flow of multiplexed large numbers of independent sources [16], [31]. Their explanation is that as the amount of Internet traffic grows dramatically following the changes of network-technology, any peculiarities of the network traffic, such as burstiness, might cancel out as a result of the huge number of different multiplexed flows. The paper reports the analyses of current and historical traces of the Internet back- bone. The authors found that packet arrivals appeared Poisson at sub-second time scales, the traffic appeared nonstationary at multi-second time scales, and the traffic exhibited long-range dependence at scales of seconds and above.
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From heterogeneous microscopic traffic flow models to macroscopic models

From heterogeneous microscopic traffic flow models to macroscopic models

July 4, 2019 Abstract The goal of this paper is to derive rigorously macroscopic traffic flow models from mi- croscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model.

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Air traffic conflict resolution via light propagation modeling

Air traffic conflict resolution via light propagation modeling

2.2 Light Modeling We use light propagation analogy. Light propagates in space under Descartes law [5]: the tra- jectory of a light ray is the shortest path in time. The distance and travel time are correlated by a local metric called index. The analogy we use is to replace the index by a cost function for the aircraft trajectory: we consider the refractive index as a measure of congestion or so-called

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A macroscopic traffic flow model accounting for bounded acceleration

A macroscopic traffic flow model accounting for bounded acceleration

(1.1) ∂tρ + ∂xf (ρ) = 0, x ∈ R, t > 0, where ρ = ρ(t, x) ∈ [0, ρ max ] denotes the density of cars, f : ρ 7→ f (ρ) = ρv(ρ) is the flow-density fundamental diagram and v : ρ 7→ v(ρ) is a decreasing function representing the mean velocity of the flow. This first model has been afterwards improved, in order to capture additional characteristics of traffic dynamics. We refer the reader to [ 16 ] for a recent review of available models. Nonetheless, all these models are based on non-linear conservation laws, and may thus display discontinuous solutions in space and time, giving rise to infinite acceleration or deceleration rates for traffic. In this work, we focus on situations in which the solutions to the LWR model present an unbounded acceleration of traffic. This is the case when the traffic conditions display downward jumps in density, as it happens for example when a traffic light turns green. At these locations, the solution to the classical LWR model consists of a rarefaction wave, accounting for an instantaneous jump from a lower velocity to a higher velocity, which corresponds to an infinite acceleration of the leading vehicle and bounded but unrealistic acceleration values for the following ones. This prevents any coupling of the LWR model (and also second order models like [ 29 , 33 ] and [ 3 , 34 ]) with consumption and pollution models (see for example [ 31 ] and references therein), in which the acceleration component plays a key role.
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Conservation Laws with Unilateral Constraints in Traffic Modeling

Conservation Laws with Unilateral Constraints in Traffic Modeling

Abstract Macroscopic models for both vehicular and pedestrian traffic are based on conser- vation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to these constraints, and numerical integra- tions of the resulting models are included to show their practical usability.

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From heterogeneous microscopic traffic flow models to macroscopic models

From heterogeneous microscopic traffic flow models to macroscopic models

Abstract The goal of this paper is to derive rigorously macroscopic traffic flow models from mi- croscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model.

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A multi-lane macroscopic traffic flow model for simple networks

A multi-lane macroscopic traffic flow model for simple networks

Figure 2: Solutions to (1.1)–(1.7)–(1.9), with M ` = {1, 2, 3}, M r = {1, 2} and initial data (3.2) at time t = 1. V ` = 1.5: left V r = 1, right V r = 2. Focus on the queue forming before x = 0 and compare the two cases, V r < V ` and V r > V ` . When the maximal speed diminishes, the queue is longer and the number of vehicles in the queue is greater with respect to the case of increasing maximal speed: for x < 0, in the former case it is more difficult for vehicles in lane 3 to pass in lane 2, since here the decrease in the maximal speed diminishes the flow at x = 0.
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Speed limit and ramp meter control for traffic flow networks

Speed limit and ramp meter control for traffic flow networks

4 Numerical Results This section collects the numerical results corresponding to three different sce- narios. The first example is concerned with variable speed limits to control traffic flow on a road network similar to the Frankfurter Kreuz, which is a fa- mous German Autobahn interchange. Here we vary discretization as well as control grid parameters to study their influence on the optimal solution. In the second example we consider the combined optimization of variable speed limits (VSL) and ramp metering. The topology of the third example is quite similar to the second one, but refers to a real world case study.
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Comparative study of macroscopic traffic flow models at road junctions

Comparative study of macroscopic traffic flow models at road junctions

0), the vehicles are stuck at the junction (γ ` = γ r 1 = γ r 2 = 0), even if some of them could proceed in the other outgoing road. In order to overcome this problem, non-FIFO models have been developed, see for instance [12]. Non-FIFO model allows some flow through the junction even if one of the outgoing road is fully congested. Using the same notation as before, the non-FIFO rule reads as follows, in terms of the demand and supply functions (2.3):

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