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

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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

Alberto Bressan ^1, and Khai T. Nguyen ^2,

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

Abstract
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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.

Keywords: Nash equilibria., network of roads, scalar conservation laws, Traffic flows, global optima.

Mathematics Subject Classification: Primary: 49K35, 35L65; Secondary: 90B2.

Citation: Alberto Bressan, Khai T. Nguyen. Optima and equilibria for traffic flow on networks with backward propagating queues. Networks & 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. doi: 10.1142/S0218202502002343. Google Scholar
[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. Google Scholar
[3]	A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM; Control, 18 (2012), 969. doi: 10.1051/cocv/2011198. Google Scholar
[4]	A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627. doi: 10.3934/nhm.2013.8.627. Google Scholar
[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. doi: 10.1090/S0033-569X-2012-01304-9. Google Scholar
[6]	A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, 10 (2015), 255. doi: 10.3934/nhm.2015.10.255. Google Scholar
[7]	A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, Discr. Cont. Dyn. Syst., 35 (2015), 4149. doi: 10.3934/dcds.2015.35.4149. Google Scholar
[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. doi: 10.1142/S021820250700239X. Google Scholar
[9]	Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow,, Discrete Contin. Dyn. Syst. B, 5 (2005), 599. doi: 10.3934/dcdsb.2005.5.599. Google Scholar
[10]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar
[11]	R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow,, Proc. Roy. Soc. Edinburgh, A 133 (2003), 759. doi: 10.1017/S0308210500002663. Google Scholar
[12]	C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for traffic flow on networks,, J. Dyn. Control Syst., 16 (2010), 407. doi: 10.1007/s10883-010-9099-3. Google Scholar
[13]	L. C. Evans, Partial Differential Equations. Second edition,, American Mathematical Society, (2010). doi: 10.1090/gsm/019. Google Scholar
[14]	T. Friesz, Dynamic Optimization and Differential Games,, Springer, (2010). doi: 10.1007/978-0-387-72778-3. Google Scholar
[15]	T. Friesz and K. Han, Dynamic Network User Equilibrium,, Springer, (2013). Google Scholar
[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. doi: 10.1016/j.trb.2012.10.001. Google Scholar*
[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. doi: 10.1016/j.trb.2010.05.003. Google Scholar*
[18]	M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. 32 (2012), 32 (2012), 1915. doi: 10.3934/dcds.2012.32.1915. Google Scholar
[19]	M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, AIMS Series on Applied Mathematics, (2006). Google Scholar
[20]	M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincar\'e, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar
[21]	M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in, Advances in Dynamic Network Modeling in Complex Transportation Systems, 2 (2013), 143. doi: 10.1007/978-1-4614-6243-9_6. Google Scholar
[22]	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. Google Scholar*
[23]	K. Han,T. Friesz and T. Yao, Existence of simultaneous route and departure choice dynamic user equilibrium,, Transp. Res., B 53* (2013), 17. doi: 10.1016/j.trb.2013.01.009. Google Scholar*
[24]	M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow,, Math. Methods Appl. Sci., 30 (2007), 153. doi: 10.1002/mma.779. Google Scholar
[25]	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. Google Scholar
[26]	M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow,, Netw. Heter. Media, 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar
[27]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Reprint of the 1980 original. SIAM, (1980). doi: 10.1137/1.9780898719451. Google Scholar
[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. doi: 10.1098/rspa.1955.0089. Google Scholar*
[29]	P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

show all references

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. doi: 10.1142/S0218202502002343. Google Scholar
[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. Google Scholar
[3]	A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM; Control, 18 (2012), 969. doi: 10.1051/cocv/2011198. Google Scholar
[4]	A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627. doi: 10.3934/nhm.2013.8.627. Google Scholar
[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. doi: 10.1090/S0033-569X-2012-01304-9. Google Scholar
[6]	A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, 10 (2015), 255. doi: 10.3934/nhm.2015.10.255. Google Scholar
[7]	A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, Discr. Cont. Dyn. Syst., 35 (2015), 4149. doi: 10.3934/dcds.2015.35.4149. Google Scholar
[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. doi: 10.1142/S021820250700239X. Google Scholar
[9]	Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow,, Discrete Contin. Dyn. Syst. B, 5 (2005), 599. doi: 10.3934/dcdsb.2005.5.599. Google Scholar
[10]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar
[11]	R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow,, Proc. Roy. Soc. Edinburgh, A 133 (2003), 759. doi: 10.1017/S0308210500002663. Google Scholar
[12]	C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for traffic flow on networks,, J. Dyn. Control Syst., 16 (2010), 407. doi: 10.1007/s10883-010-9099-3. Google Scholar
[13]	L. C. Evans, Partial Differential Equations. Second edition,, American Mathematical Society, (2010). doi: 10.1090/gsm/019. Google Scholar
[14]	T. Friesz, Dynamic Optimization and Differential Games,, Springer, (2010). doi: 10.1007/978-0-387-72778-3. Google Scholar
[15]	T. Friesz and K. Han, Dynamic Network User Equilibrium,, Springer, (2013). Google Scholar
[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. doi: 10.1016/j.trb.2012.10.001. Google Scholar*
[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. doi: 10.1016/j.trb.2010.05.003. Google Scholar*
[18]	M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst. 32 (2012), 32 (2012), 1915. doi: 10.3934/dcds.2012.32.1915. Google Scholar
[19]	M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, AIMS Series on Applied Mathematics, (2006). Google Scholar
[20]	M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincar\'e, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar
[21]	M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in, Advances in Dynamic Network Modeling in Complex Transportation Systems, 2 (2013), 143. doi: 10.1007/978-1-4614-6243-9_6. Google Scholar
[22]	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. Google Scholar*
[23]	K. Han,T. Friesz and T. Yao, Existence of simultaneous route and departure choice dynamic user equilibrium,, Transp. Res., B 53* (2013), 17. doi: 10.1016/j.trb.2013.01.009. Google Scholar*
[24]	M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow,, Math. Methods Appl. Sci., 30 (2007), 153. doi: 10.1002/mma.779. Google Scholar
[25]	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. Google Scholar
[26]	M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow,, Netw. Heter. Media, 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar
[27]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Reprint of the 1980 original. SIAM, (1980). doi: 10.1137/1.9780898719451. Google Scholar
[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. doi: 10.1098/rspa.1955.0089. Google Scholar*
[29]	P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

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