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The traffic statics problem in a road network

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  • Jin, Wen-Long

Abstract

In this study we define and solve the traffic statics problem in an open diverge-merge network based on a multi-commodity kinematic wave model, whose entropy conditions are given by invariant junction flux functions derived from macroscopic merging and diverging rules. In this problem, we are interested in finding stationary states on all links when origin demands, destination supplies, and route choice proportions are constant. After discussing the properties of four types of stationary states on a road link and presenting stationary entropy conditions at both the merging and diverging junctions, we derive a system of algebraic equations as necessary conditions for all 16 combinations of stationary states on the two intermediate links. Under different network conditions in road capacities, route choice proportions, and merging priorities, we analytically show that the traffic statics problem always admits stationary solutions, which, however, may not be unique. In particular, such stationary solutions exist even under network conditions when an initially empty diverge-merge network can settle in persistent periodic oscillations after a long time. In the future, we will be interested in discussing the stability property of the stationary states and studying the traffic statics problem in general networks. Analytical insights from the simpler traffic statics problems would be helpful for understanding complex traffic dynamics in a road network.

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  • Jin, Wen-Long, 2012. "The traffic statics problem in a road network," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1360-1373.
    • Handle: RePEc:eee:transb:v:46:y:2012:i:10:p:1360-1373
    DOI: 10.1016/j.trb.2012.06.003
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    Cited by:

    1. Smits, Erik-Sander & Bliemer, Michiel C.J. & Pel, Adam J. & van Arem, Bart, 2015. "A family of macroscopic node models," Transportation Research Part B: Methodological, Elsevier, vol. 74(C), pages 20-39.
    2. Jin, Wen-Long, 2015. "Point queue models: A unified approach," Transportation Research Part B: Methodological, Elsevier, vol. 77(C), pages 1-16.
    3. Jin, Wen-Long & Gan, Qi-Jian & Gayah, Vikash V., 2013. "A kinematic wave approach to traffic statics and dynamics in a double-ring network," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 114-131.
    4. Jin, Wen-Long, 2017. "On the stability of stationary states in general road networks," Transportation Research Part B: Methodological, Elsevier, vol. 98(C), pages 42-61.
    5. Bliemer, Michiel C.J. & Raadsen, Mark P.H. & Smits, Erik-Sander & Zhou, Bojian & Bell, Michael G.H., 2014. "Quasi-dynamic traffic assignment with residual point queues incorporating a first order node model," Transportation Research Part B: Methodological, Elsevier, vol. 68(C), pages 363-384.
    6. Jin, Wen-Long, 2013. "Stability and bifurcation in network traffic flow: A Poincaré map approach," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 191-208.
    7. Jin, Wen-Long, 2015. "On the existence of stationary states in general road networks," Transportation Research Part B: Methodological, Elsevier, vol. 81(P3), pages 917-929.
    8. Jin, Wen-Long & Gan, Qi-Jian & Lebacque, Jean-Patrick, 2015. "A kinematic wave theory of capacity drop," Transportation Research Part B: Methodological, Elsevier, vol. 81(P1), pages 316-329.
    9. Yan, Qinglong & Sun, Zhe & Gan, Qijian & Jin, Wen-Long, 2018. "Automatic identification of near-stationary traffic states based on the PELT changepoint detection," Transportation Research Part B: Methodological, Elsevier, vol. 108(C), pages 39-54.
    10. Jin, Wen-Long, 2015. "Continuous formulations and analytical properties of the link transmission model," Transportation Research Part B: Methodological, Elsevier, vol. 74(C), pages 88-103.
    11. Jin, Wen-Long & Laval, Jorge, 2018. "Bounded acceleration traffic flow models: A unified approach," Transportation Research Part B: Methodological, Elsevier, vol. 111(C), pages 1-18.
    12. Wen-Long Jin, 2020. "Stable Day-to-Day Dynamics for Departure Time Choice," Transportation Science, INFORMS, vol. 54(1), pages 42-61, January.

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