# American Institute of Mathematical Sciences

December  2015, 10(4): 717-748. doi: 10.3934/nhm.2015.10.717

## Optima and equilibria for traffic flow on networks with backward propagating queues

 1 Department of Mathematics, Penn State University, University Park, Pa.16802 2 Department of Mathematics, Penn State University, University Park, PA 16802

Received  January 2015 Revised  April 2015 Published  October 2015

This paper studies an optimal decision problem for several groups of drivers on a network of roads. Drivers have different origins and destinations, and different costs, related to their departure and arrival time. On each road the flow is governed by a conservation law, while intersections are modeled using buffers of limited capacity, so that queues can spill backward along roads leading to a crowded intersection. Two main results are proved: (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.
Citation: Alberto Bressan, Khai T. Nguyen. Optima and equilibria for traffic flow on networks with backward propagating queues. Networks and Heterogeneous Media, 2015, 10 (4) : 717-748. doi: 10.3934/nhm.2015.10.717
##### References:
 [1] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci., 12 (2002), 1801-1843. doi: 10.1142/S0218202502002343. [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, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627. [5] A. Bressan, C. J. Liu, W. Shen and F. Yu, Variational analysis of Nash equilibria for a model of traffic flow, Quarterly Appl. Math., 70 (2012), 495-515. doi: 10.1090/S0033-569X-2012-01304-9. [6] A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Networks & Heter. Media, 10 (2015), 255-293. doi: 10.3934/nhm.2015.10.255. [7] A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171. doi: 10.3934/dcds.2015.35.4149. [8] 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. [9] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [10] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [11] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh, A 133 (2003), 759-772. doi: 10.1017/S0308210500002663. [12] C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for traffic flow on networks, J. Dyn. Control Syst., 16 (2010), 407-437. doi: 10.1007/s10883-010-9099-3. [13] L. C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [14] T. Friesz, Dynamic Optimization and Differential Games, Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3. [15] T. Friesz and K. Han, Dynamic Network User Equilibrium, Springer, 2013. [16] T. Friesz, K. Han, P. A. Neto, A. Meimand and T. Yao, Dynamic user equilibrium based on a hydrodynamic model, Transp. Res., B 47 (2013), 102-126. doi: 10.1016/j.trb.2012.10.001. [17] T. Friesz, T. Kim, C. Kwon and M. A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium, Transp. Res., B 45 (2011), 176-207. doi: 10.1016/j.trb.2010.05.003. [18] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [19] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [20] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincar\'e, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001. [21] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems, Complex Networks and Dynamic Systems, S V. Ukkusuri and K. Ozbay eds., Springer, New York, 2 (2013), 143-161. doi: 10.1007/978-1-4614-6243-9_6. [22] 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. [23] K. Han,T. Friesz and T. Yao, Existence of simultaneous route and departure choice dynamic user equilibrium, Transp. Res., B 53 (2013), 17-30. doi: 10.1016/j.trb.2013.01.009. [24] M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow, Math. Methods Appl. Sci., 30 (2007), 153-169. doi: 10.1002/mma.779. [25] 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. [26] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heter. Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [27] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original. SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [28] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [29] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

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##### References:
 [1] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci., 12 (2002), 1801-1843. doi: 10.1142/S0218202502002343. [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, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627. [5] A. Bressan, C. J. Liu, W. Shen and F. Yu, Variational analysis of Nash equilibria for a model of traffic flow, Quarterly Appl. Math., 70 (2012), 495-515. doi: 10.1090/S0033-569X-2012-01304-9. [6] A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Networks & Heter. Media, 10 (2015), 255-293. doi: 10.3934/nhm.2015.10.255. [7] A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171. doi: 10.3934/dcds.2015.35.4149. [8] 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. [9] Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599. [10] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [11] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh, A 133 (2003), 759-772. doi: 10.1017/S0308210500002663. [12] C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for traffic flow on networks, J. Dyn. Control Syst., 16 (2010), 407-437. doi: 10.1007/s10883-010-9099-3. [13] L. C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [14] T. Friesz, Dynamic Optimization and Differential Games, Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3. [15] T. Friesz and K. Han, Dynamic Network User Equilibrium, Springer, 2013. [16] T. Friesz, K. Han, P. A. Neto, A. Meimand and T. Yao, Dynamic user equilibrium based on a hydrodynamic model, Transp. Res., B 47 (2013), 102-126. doi: 10.1016/j.trb.2012.10.001. [17] T. Friesz, T. Kim, C. Kwon and M. A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium, Transp. Res., B 45 (2011), 176-207. doi: 10.1016/j.trb.2010.05.003. [18] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915. [19] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [20] M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincar\'e, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001. [21] M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems, Complex Networks and Dynamic Systems, S V. Ukkusuri and K. Ozbay eds., Springer, New York, 2 (2013), 143-161. doi: 10.1007/978-1-4614-6243-9_6. [22] 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. [23] K. Han,T. Friesz and T. Yao, Existence of simultaneous route and departure choice dynamic user equilibrium, Transp. Res., B 53 (2013), 17-30. doi: 10.1016/j.trb.2013.01.009. [24] M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow, Math. Methods Appl. Sci., 30 (2007), 153-169. doi: 10.1002/mma.779. [25] 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. [26] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heter. Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [27] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original. SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [28] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [29] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
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