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Conservation Law Models for Traffic Flow on a Network of Roads
, 2014
"... The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into ea ..."
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The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads. If the queue sizes within the buffer are known, then the initialboundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula. Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection. The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves wellposedness for general L ∞ data, and continuity w.r.t. weak convergence. 1
Continuous Riemann solvers for traffic flow at a junction
 Discr. Cont. Dyn. Syst
"... The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers ’ turning preferences. However, various examples ..."
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The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers ’ turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem on a network of roads can be ill posed, even for initial data having small total variation. 1
Research Themes on Traffic Flow on Networks
, 2013
"... Models of traffic flow can be of two main types [1, 8, 11]. 1 Microscopic particle models, describing the position and velocity of each single car. If there are N cars on a road, one thus needs to write a system of N differential equations, one for each car. These ODEs specify how each driver adjus ..."
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Models of traffic flow can be of two main types [1, 8, 11]. 1 Microscopic particle models, describing the position and velocity of each single car. If there are N cars on a road, one thus needs to write a system of N differential equations, one for each car. These ODEs specify how each driver adjusts his velocity depending on the distance and the velocity of the vehicle ahead. 2 Macroscopic models, describing the evolution of the vehicle density (i.e. the number of cars per unit length of the road). Since the total number of cars is conserved, these models consist of one or more PDEs, usually in the form of a conservation law. Particle models are easier to simulate numerically. On the other hand, macroscopic models are mathematically more interesting and yield a better qualitative understanding of traffic patterns. Having written down a set of mathematical equations, the next steps of a mathematical analysis (i) The first and most fundamental concern is making sure that the model “well posed”. This means that, given any initial configuration, the equations determine a unique solution
Globally Optimal and Nash Equilibrium Solutions for Traffic Flow on Networks
, 2013
"... We consider a conservation law model of traffic flow on a network of roads, where drivers choose their departure times in order to minimize the sum of a departure cost and an arrival cost. Drivers can have different origins and destinations, and different cost functions. Under natural assumptions, t ..."
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We consider a conservation law model of traffic flow on a network of roads, where drivers choose their departure times in order to minimize the sum of a departure cost and an arrival cost. Drivers can have different origins and destinations, and different cost functions. Under natural assumptions, two main results have been established: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the special case of one single road, the global optimum and the Nash equilibrium are uniquely determined. 1
A DESTINATION–PRESERVING MODEL FOR SIMULATING WARDROP EQUILIBRIA IN TRAFFIC FLOW ON NETWORKS
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Research Themes on Traffic Flow on Networks
"... Models of traffic flow can be of two main types [1, 8, 11]. 1 Microscopic particle models, describing the position and velocity of each single car. If there are N cars on a road, one thus needs to write a system of N differential equations, one for each car. These ODEs specify how each driver adjus ..."
Abstract
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Models of traffic flow can be of two main types [1, 8, 11]. 1 Microscopic particle models, describing the position and velocity of each single car. If there are N cars on a road, one thus needs to write a system of N differential equations, one for each car. These ODEs specify how each driver adjusts his velocity depending on the distance and the velocity of the vehicle ahead. 2 Macroscopic models, describing the evolution of the vehicle density (i.e. the number of cars per unit length of the road). Since the total number of cars is conserved, these models consist of one or more PDEs, usually in the form of a conservation law. Particle models are easier to simulate numerically. On the other hand, macroscopic models are mathematically more interesting and yield a better qualitative understanding of traffic patterns. Having written down a set of mathematical equations, the next steps of a mathematical analysis (i) The first and most fundamental concern is making sure that the model “well posed”. This means that, given any initial configuration, the equations determine a unique solution