# 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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Springfield, MO, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Y. Hollander and J. N. Prashker, The applicability of non-cooperative game theory in transport analysis, Transportation, 33 (2006), 481-496. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] F. S. Priuli, First order mean field games in crowd dynamics,, submitted. , ().   Google Scholar [35] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar

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), 587-612. doi: 10.1017/S0956792503005266.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, Springfield, MO, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Y. Hollander and J. N. Prashker, The applicability of non-cooperative game theory in transport analysis, Transportation, 33 (2006), 481-496. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] F. S. Priuli, First order mean field games in crowd dynamics,, submitted. , ().   Google Scholar [35] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar
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