# American Institute of Mathematical Sciences

December  2015, 10(4): 857-876. doi: 10.3934/nhm.2015.10.857

## 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

Received  September 2014 Revised  January 2015 Published  October 2015

In this paper we propose a LWR-like model for traffic flow on networks which allows to track several groups of drivers, each of them being characterized only by their destination in the network. The path actually followed to reach the destination is not assigned a priori, and can be chosen by the drivers during the journey, taking decisions at junctions.
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.
Citation: Emiliano Cristiani, Fabio S. Priuli. A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks. Networks and Heterogeneous Media, 2015, 10 (4) : 857-876. doi: 10.3934/nhm.2015.10.857
##### References:
 [1] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Euro. J. Appl. Math., 14 (2003), 587-612. 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), 2384-2417. 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), 969-986. 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), 627-648. 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), 255-293. doi: 10.3934/nhm.2015.10.255. [6] A. Bressan and F. S. Priuli, Infinite horizon noncooperative differential games, J. Differential Equations, 227 (2006), 230-257. 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), 4149-4171. 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), 379-394. 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), 519-552. 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, V. Mazalov (eds.), Game Theory and Applications, Vol. 16, Chap. 4, Nova Publishers, New York, 2013. [11] G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria, SIAM J. Control Optim., 47 (2008), 1330-1350. doi: 10.1137/060672832. [12] G. Carlier and F. Santambrogio, A continuous theory of traffic congestion and Wardrop equilibria, J. Math. Sci., 181 (2012), 792-804. 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), 1587-1617. 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), 1-15. 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), 1-19. 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), 605-629. 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), Article ID 679056, 19 pages. doi: 10.5402/2011/679056. [18] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012. [19] C. S. Fisk, Game theory and transportation systems modelling, Transportation Res. Part B, 18 (1984), 301-313. 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), 1155-1176. doi: 10.1137/040605503. [21] M. Garavello, The LWR traffic model at a junction with multibuffers, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 463-482. 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), 1915-1938. 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), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Springfield, MO, 2006. [25] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optim. Theory Appl., 126 (2005), 589-616. 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), 1066-1087. 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), 813-826. 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), 481-496. [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), 1572-1589. 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), 317-345. 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), 154-166. 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), 607-625. doi: 10.1145/79147.214078. [33] F. S. Priuli, Infinite horizon noncooperative differential games with non-smooth costs, J. Math. Anal. Appl., 336 (2007), 156-170. 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), 42-51. 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), 767-768. doi: 10.1680/ipeds.1952.11362.

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##### References:
 [1] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Euro. J. Appl. Math., 14 (2003), 587-612. 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), 2384-2417. 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), 969-986. 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), 627-648. 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), 255-293. doi: 10.3934/nhm.2015.10.255. [6] A. Bressan and F. S. Priuli, Infinite horizon noncooperative differential games, J. Differential Equations, 227 (2006), 230-257. 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), 4149-4171. 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), 379-394. 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), 519-552. 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, V. Mazalov (eds.), Game Theory and Applications, Vol. 16, Chap. 4, Nova Publishers, New York, 2013. [11] G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria, SIAM J. Control Optim., 47 (2008), 1330-1350. doi: 10.1137/060672832. [12] G. Carlier and F. Santambrogio, A continuous theory of traffic congestion and Wardrop equilibria, J. Math. Sci., 181 (2012), 792-804. 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), 1587-1617. 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), 1-15. 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), 1-19. 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), 605-629. 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), Article ID 679056, 19 pages. doi: 10.5402/2011/679056. [18] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012. [19] C. S. Fisk, Game theory and transportation systems modelling, Transportation Res. Part B, 18 (1984), 301-313. 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), 1155-1176. doi: 10.1137/040605503. [21] M. Garavello, The LWR traffic model at a junction with multibuffers, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 463-482. 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), 1915-1938. 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), 261-283. doi: 10.4310/CMS.2005.v3.n3.a1. [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Springfield, MO, 2006. [25] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optim. Theory Appl., 126 (2005), 589-616. 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), 1066-1087. 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), 813-826. 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), 481-496. [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), 1572-1589. 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), 317-345. 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), 154-166. 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), 607-625. doi: 10.1145/79147.214078. [33] F. S. Priuli, Infinite horizon noncooperative differential games with non-smooth costs, J. Math. Anal. Appl., 336 (2007), 156-170. 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), 42-51. 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), 767-768. doi: 10.1680/ipeds.1952.11362.
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