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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

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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