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 & Heterogeneous Media, 2015, 10 (4) : 717-748. doi: 10.3934/nhm.2015.10.717
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show all references

References:
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Math. Models Methods Appl. Sci., 12 (2002), 1801-1843. doi: 10.1142/S0218202502002343.  Google Scholar

[2]

SIAM J. Math. Anal., 43 (2011), 2384-2417. doi: 10.1137/110825145.  Google Scholar

[3]

ESAIM; Control, Optim. Calc. Var., 18 (2012), 969-986. doi: 10.1051/cocv/2011198.  Google Scholar

[4]

Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627.  Google Scholar

[5]

Quarterly Appl. Math., 70 (2012), 495-515. doi: 10.1090/S0033-569X-2012-01304-9.  Google Scholar

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Networks & Heter. Media, 10 (2015), 255-293. doi: 10.3934/nhm.2015.10.255.  Google Scholar

[7]

Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171. doi: 10.3934/dcds.2015.35.4149.  Google Scholar

[8]

Math. Models Methods Appl. Sci., 17 (2007), 1587-1617. doi: 10.1142/S021820250700239X.  Google Scholar

[9]

Discrete Contin. Dyn. Syst. B, 5 (2005), 599-630. doi: 10.3934/dcdsb.2005.5.599.  Google Scholar

[10]

SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.  Google Scholar

[11]

Proc. Roy. Soc. Edinburgh, A 133 (2003), 759-772. doi: 10.1017/S0308210500002663.  Google Scholar

[12]

J. Dyn. Control Syst., 16 (2010), 407-437. doi: 10.1007/s10883-010-9099-3.  Google Scholar

[13]

American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[14]

Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3.  Google Scholar

[15]

Springer, 2013. Google Scholar

[16]

Transp. Res., B 47 (2013), 102-126. doi: 10.1016/j.trb.2012.10.001.  Google Scholar

[17]

Transp. Res., B 45 (2011), 176-207. doi: 10.1016/j.trb.2010.05.003.  Google Scholar

[18]

Discrete Contin. Dyn. Syst. 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.  Google Scholar

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AIMS Series on Applied Mathematics, Springfield, Mo., 2006.  Google Scholar

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Ann. Inst. H. Poincar\'e, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar

[21]

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.  Google Scholar

[22]

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[23]

Transp. Res., B 53 (2013), 17-30. doi: 10.1016/j.trb.2013.01.009.  Google Scholar

[24]

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[25]

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[26]

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[27]

Reprint of the 1980 original. SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar

[28]

Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[29]

Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

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