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A kinematic wave approach to traffic statics and dynamics in a double-ring network

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  • Jin, Wen-Long
  • Gan, Qi-Jian
  • Gayah, Vikash V.

Abstract

Recently there has been much interest in understanding macroscopic fundamental diagrams of stationary road networks. However, there lacks a systematic method to define and solve stationary states in a road network with complex junctions. In this study we propose a kinematic wave approach to defining, analyzing, and simulating static and dynamic traffic characteristics in a network of two ring roads connected by a 2×2 junction, which can be either an uninterrupted interchange or a signalized intersection. This study is enabled by recently developed macroscopic junction models of general junctions. With a junction model based on fair merging and first-in-first-out diverging rules, we first define and solve stationary states and then derive the macroscopic fundamental diagram (MFD) of a stationary uninterrupted network. We conclude that the flow-density relationship of the uninterrupted double-ring network is not unique for high average network densities (i.e., when one ring becomes congested) and unveil the existence of infinitely many stationary states that can arise with a zero-speed shockwave. From simulation results with a corresponding Cell Transmission Model, we verify that all stationary states in the MFD are stable and can be reached, but show that randomness in the retaining ratio of each ring drives the network to more symmetric traffic patterns and higher flow-rates. Furthermore we model a signalized intersection as two alternate diverge junctions and demonstrate that the signalized double-ring network can reach asymptotically periodic traffic patterns, which are therefore defined as “stationary” states in signalized networks. With simulations we show that the flow-density relation is well defined in such “stationary” states, and asymptotic traffic patterns can be impacted by signal cycle lengths and retaining ratios. But compared with uninterrupted interchanges, signalized intersections lead to more asymmetric traffic patterns, lower flow-rates, and even gridlocks when the average density is higher than half of the jam density. The results are consistent between this study and existing studies, but the network kinematic wave model, with appropriate junction models, is mathematically tractable and physically meaningful. It has offered a more complete picture regarding the number and type of stationary states, their stability, and MFD in freeway and signalized networks.

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  • 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.
    • Handle: RePEc:eee:transb:v:57:y:2013:i:c:p:114-131
    DOI: 10.1016/j.trb.2013.09.004
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    3. Gupta, Namrata & Patil, Gopal R. & Vu, Hai L., 2023. "Simple abstract models to study stability of urban networks with decentralized signal control," Transportation Research Part B: Methodological, Elsevier, vol. 172(C), pages 93-116.
    4. Zhong, R.X. & Huang, Y.P. & Chen, C. & Lam, W.H.K. & Xu, D.B. & Sumalee, A., 2018. "Boundary conditions and behavior of the macroscopic fundamental diagram based network traffic dynamics: A control systems perspective," Transportation Research Part B: Methodological, Elsevier, vol. 111(C), pages 327-355.
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    6. Srivastava, Anupam & Jin, Wen-Long & Lebacque, Jean-Patrick, 2015. "A modified Cell Transmission Model with realistic queue discharge features at signalized intersections," Transportation Research Part B: Methodological, Elsevier, vol. 81(P1), pages 302-315.
    7. Xianglun Mo & Xiaohong Jin & Jinpeng Tian & Zhushuai Shao & Gangqing Han, 2022. "Research on the Division Method of Signal Control Sub-Region Based on Macroscopic Fundamental Diagram," Sustainability, MDPI, vol. 14(13), pages 1-19, July.
    8. Qi-Jian Gan & Wen-Long Jin & Vikash V. Gayah, 2017. "Analysis of Traffic Statics and Dynamics in Signalized Networks: A Poincaré Map Approach," Transportation Science, INFORMS, vol. 51(3), pages 1009-1029, August.
    9. Zhang, Lele & Finn, Caley & Garoni, Timothy M. & de Gier, Jan, 2018. "Behaviour of traffic on a link with traffic light boundaries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 116-138.
    10. 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.
    11. Rezaei, Danial & Aghayan, Iman & Hadadi, Farhad, 2021. "Studying perturbations and wave propagations by lane closures on traffic characteristics based on a dynamic approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    12. Xingliang Liu & Jian Wang & Tangzhi Liu & Jin Xu, 2021. "Forecasting Spatiotemporal Boundary of Emergency-Event-Based Traffic Congestion in Expressway Network Considering Highway Node Acceptance Capacity," Sustainability, MDPI, vol. 13(21), pages 1-17, November.
    13. Ding, Heng & Qian, Yu & Zheng, Xiaoyan & Bai, Haijian & Wang, Shiguang & Zhou, Jingwen, 2022. "Dynamic parking charge–perimeter control coupled method for a congested road network based on the aggregation degree characteristics of parking generation distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 587(C).

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