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A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks
1. | Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini, 19 – 00185 Rome |
2. | Istituto per le Applicazioni del Calcolo “M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, I-00185 Roma |
  The model is then used to describe three possible behaviors of drivers, associated to three different ways to solve the route choice problem: 1. Drivers ignore the presence of the other vehicles; 2. Drivers react to the current distribution of traffic, but they do not forecast what will happen at later times; 3. Drivers take into account the current and future distribution of vehicles. Notice that, in the latter case, we enter the field of differential games, and, if a solution exists, it likely represents a global equilibrium among drivers.
  Numerical simulations highlight the differences between the three behaviors and offer insights into the existence of equilibria.
References:
[1] |
S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow,, Euro. J. Appl. Math., 14 (2003), 587.
doi: 10.1017/S0956792503005266. |
[2] |
A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM J. Math. Anal., 43 (2011), 2384.
doi: 10.1137/110825145. |
[3] |
A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM Control Optim. Calc. Var., 18 (2012), 969.
doi: 10.1051/cocv/2011198. |
[4] |
A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Netw. Heterog. Media, 8 (2013), 627.
doi: 10.3934/nhm.2013.8.627. |
[5] |
A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads,, Netw. Heterog. Media, 10 (2015), 255.
doi: 10.3934/nhm.2015.10.255. |
[6] |
A. Bressan and F. S. Priuli, Infinite horizon noncooperative differential games,, J. Differential Equations, 227 (2006), 230.
doi: 10.1016/j.jde.2006.01.005. |
[7] |
A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4149.
doi: 10.3934/dcds.2015.35.4149. |
[8] |
G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379.
doi: 10.3934/dcdss.2014.7.379. |
[9] |
M. Briani and E. Cristiani, An easy-to-use numerical algorithm for simulating traffic flow on networks: Theoretical study,, Netw. Heterog. Media, 9 (2014), 519.
doi: 10.3934/nhm.2014.9.519. |
[10] |
S. Cacace, E. Cristiani and M. Falcone, Numerical approximation of Nash equilibria for a class of non-cooperative differential games,, In: L. Petrosjan, (2013). |
[11] |
G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria,, SIAM J. Control Optim., 47 (2008), 1330.
doi: 10.1137/060672832. |
[12] |
G. Carlier and F. Santambrogio, A continuous theory of traffic congestion and Wardrop equilibria,, J. Math. Sci., 181 (2012), 792.
doi: 10.1007/s10958-012-0715-5. |
[13] |
A. Cascone, C. D'Apice, B. Piccoli and L. Rarità, Optimization of traffic on road networks,, Math. Models Methods Appl. Sci., 17 (2007), 1587.
doi: 10.1142/S021820250700239X. |
[14] |
R. M. Colombo and H. Holden, On the Braess paradox with nonlinear dynamics and control theory,, J. Optim. Theory Appl., (2015), 1.
doi: 10.1007/s10957-015-0729-5. |
[15] |
Z. Cong, B. De Schutter and R. Babuška, Ant colony routing algorithm for freeway networks,, Transportation Res. Part C, 37 (2013), 1.
doi: 10.1016/j.trc.2013.09.008. |
[16] |
E. Cristiani, F. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: An environmental approach,, SIAM J. Appl. Math., 75 (2015), 605.
doi: 10.1137/140962413. |
[17] |
A. Cutolo, C. D'Apice and R. Manzo, Traffic optimization at junctions to improve vehicular flows,, International Scholarly Research Network ISRN Applied Mathematics, 2011 (2011).
doi: 10.5402/2011/679056. |
[18] |
C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506.
doi: 10.1016/j.mcm.2010.06.012. |
[19] |
C. S. Fisk, Game theory and transportation systems modelling,, Transportation Res. Part B, 18 (1984), 301.
doi: 10.1016/0191-2615(84)90013-4. |
[20] |
A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM J. Optim., 16 (2006), 1155.
doi: 10.1137/040605503. |
[21] |
M. Garavello, The LWR traffic model at a junction with multibuffers,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 463.
doi: 10.3934/dcdss.2014.7.463. |
[22] |
M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915.
doi: 10.3934/dcds.2012.32.1915. |
[23] |
M. Garavello and B. Piccoli, Source-destination flow on a road network,, Commun. Math. Sci., 3 (2005), 261.
doi: 10.4310/CMS.2005.v3.n3.a1. |
[24] |
M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).
|
[25] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, J. Optim. Theory Appl., 126 (2005), 589.
doi: 10.1007/s10957-005-5499-z. |
[26] |
M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks,, SIAM J. Sci. Comput., 25 (2003), 1066.
doi: 10.1137/S106482750241459X. |
[27] |
M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813.
doi: 10.3934/nhm.2009.4.813. |
[28] |
Y. Hollander and J. N. Prashker, The applicability of non-cooperative game theory in transport analysis,, Transportation, 33 (2006), 481. |
[29] |
A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Transportation Res. Part B, 45 (2011), 1572.
doi: 10.1016/j.trb.2011.07.011. |
[30] |
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 (1955), 317.
doi: 10.1098/rspa.1955.0089. |
[31] |
K. Nachtigall, Time depending shortest-path problems with applications to railway networks,, Euro. J. Oper. Res., 83 (1995), 154.
doi: 10.1016/0377-2217(94)E0349-G. |
[32] |
A. Orda and R. Rom, Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length,, J. Assoc. Comput. Mach., 37 (1990), 607.
doi: 10.1145/79147.214078. |
[33] |
F. S. Priuli, Infinite horizon noncooperative differential games with non-smooth costs,, J. Math. Anal. Appl., 336 (2007), 156.
doi: 10.1016/j.jmaa.2007.02.030. |
[34] |
F. S. Priuli, First order mean field games in crowd dynamics,, submitted. , (). |
[35] |
P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42.
doi: 10.1287/opre.4.1.42. |
[36] |
J. G. Wardrop, Some theoretical aspects of road traffic research,, Proc. Inst. Civ. Eng. Part II, 1 (1952), 767.
doi: 10.1680/ipeds.1952.11362. |
show all references
References:
[1] |
S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow,, Euro. J. Appl. Math., 14 (2003), 587.
doi: 10.1017/S0956792503005266. |
[2] |
A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM J. Math. Anal., 43 (2011), 2384.
doi: 10.1137/110825145. |
[3] |
A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM Control Optim. Calc. Var., 18 (2012), 969.
doi: 10.1051/cocv/2011198. |
[4] |
A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Netw. Heterog. Media, 8 (2013), 627.
doi: 10.3934/nhm.2013.8.627. |
[5] |
A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a network of roads,, Netw. Heterog. Media, 10 (2015), 255.
doi: 10.3934/nhm.2015.10.255. |
[6] |
A. Bressan and F. S. Priuli, Infinite horizon noncooperative differential games,, J. Differential Equations, 227 (2006), 230.
doi: 10.1016/j.jde.2006.01.005. |
[7] |
A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4149.
doi: 10.3934/dcds.2015.35.4149. |
[8] |
G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379.
doi: 10.3934/dcdss.2014.7.379. |
[9] |
M. Briani and E. Cristiani, An easy-to-use numerical algorithm for simulating traffic flow on networks: Theoretical study,, Netw. Heterog. Media, 9 (2014), 519.
doi: 10.3934/nhm.2014.9.519. |
[10] |
S. Cacace, E. Cristiani and M. Falcone, Numerical approximation of Nash equilibria for a class of non-cooperative differential games,, In: L. Petrosjan, (2013). |
[11] |
G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria,, SIAM J. Control Optim., 47 (2008), 1330.
doi: 10.1137/060672832. |
[12] |
G. Carlier and F. Santambrogio, A continuous theory of traffic congestion and Wardrop equilibria,, J. Math. Sci., 181 (2012), 792.
doi: 10.1007/s10958-012-0715-5. |
[13] |
A. Cascone, C. D'Apice, B. Piccoli and L. Rarità, Optimization of traffic on road networks,, Math. Models Methods Appl. Sci., 17 (2007), 1587.
doi: 10.1142/S021820250700239X. |
[14] |
R. M. Colombo and H. Holden, On the Braess paradox with nonlinear dynamics and control theory,, J. Optim. Theory Appl., (2015), 1.
doi: 10.1007/s10957-015-0729-5. |
[15] |
Z. Cong, B. De Schutter and R. Babuška, Ant colony routing algorithm for freeway networks,, Transportation Res. Part C, 37 (2013), 1.
doi: 10.1016/j.trc.2013.09.008. |
[16] |
E. Cristiani, F. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: An environmental approach,, SIAM J. Appl. Math., 75 (2015), 605.
doi: 10.1137/140962413. |
[17] |
A. Cutolo, C. D'Apice and R. Manzo, Traffic optimization at junctions to improve vehicular flows,, International Scholarly Research Network ISRN Applied Mathematics, 2011 (2011).
doi: 10.5402/2011/679056. |
[18] |
C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Math. Comput. Modelling, 52 (2010), 1506.
doi: 10.1016/j.mcm.2010.06.012. |
[19] |
C. S. Fisk, Game theory and transportation systems modelling,, Transportation Res. Part B, 18 (1984), 301.
doi: 10.1016/0191-2615(84)90013-4. |
[20] |
A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM J. Optim., 16 (2006), 1155.
doi: 10.1137/040605503. |
[21] |
M. Garavello, The LWR traffic model at a junction with multibuffers,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 463.
doi: 10.3934/dcdss.2014.7.463. |
[22] |
M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915.
doi: 10.3934/dcds.2012.32.1915. |
[23] |
M. Garavello and B. Piccoli, Source-destination flow on a road network,, Commun. Math. Sci., 3 (2005), 261.
doi: 10.4310/CMS.2005.v3.n3.a1. |
[24] |
M. Garavello and B. Piccoli, Traffic Flow on Networks,, AIMS Series on Applied Mathematics, (2006).
|
[25] |
M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, J. Optim. Theory Appl., 126 (2005), 589.
doi: 10.1007/s10957-005-5499-z. |
[26] |
M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks,, SIAM J. Sci. Comput., 25 (2003), 1066.
doi: 10.1137/S106482750241459X. |
[27] |
M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813.
doi: 10.3934/nhm.2009.4.813. |
[28] |
Y. Hollander and J. N. Prashker, The applicability of non-cooperative game theory in transport analysis,, Transportation, 33 (2006), 481. |
[29] |
A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Transportation Res. Part B, 45 (2011), 1572.
doi: 10.1016/j.trb.2011.07.011. |
[30] |
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 (1955), 317.
doi: 10.1098/rspa.1955.0089. |
[31] |
K. Nachtigall, Time depending shortest-path problems with applications to railway networks,, Euro. J. Oper. Res., 83 (1995), 154.
doi: 10.1016/0377-2217(94)E0349-G. |
[32] |
A. Orda and R. Rom, Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length,, J. Assoc. Comput. Mach., 37 (1990), 607.
doi: 10.1145/79147.214078. |
[33] |
F. S. Priuli, Infinite horizon noncooperative differential games with non-smooth costs,, J. Math. Anal. Appl., 336 (2007), 156.
doi: 10.1016/j.jmaa.2007.02.030. |
[34] |
F. S. Priuli, First order mean field games in crowd dynamics,, submitted. , (). |
[35] |
P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42.
doi: 10.1287/opre.4.1.42. |
[36] |
J. G. Wardrop, Some theoretical aspects of road traffic research,, Proc. Inst. Civ. Eng. Part II, 1 (1952), 767.
doi: 10.1680/ipeds.1952.11362. |
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