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Three-Dimensional Representation of Traffic Flow

Author

Listed:
  • Yasuji Makigami

    (University of California, Berkeley)

  • G. F. Newell

    (University of California, Berkeley)

  • Richard Rothery

    (General Motors Research Laboratories, Warren, Michigan)

Abstract

If one numbers vehicles consecutively along a roadway and draws the space-time trajectory of each vehicle on the same x - t graph, then this family of curves can be interpreted as the contours of a three-dimensional surface for which the third dimension is vehicle number n . The intersection of the surface with planes of constant x are the cumulative arrival curves, n vs. t , at the location x . If the surface is smoothed, the orientation of the tangent plane at any point determines the flow q , density k , and car velocity v . All commonly observed properties of traffic flow have simple geometric interpretations in this three-dimensional model.

Suggested Citation

  • Yasuji Makigami & G. F. Newell & Richard Rothery, 1971. "Three-Dimensional Representation of Traffic Flow," Transportation Science, INFORMS, vol. 5(3), pages 302-313, August.
    • Handle: RePEc:inm:ortrsc:v:5:y:1971:i:3:p:302-313
    DOI: 10.1287/trsc.5.3.302
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    Citations

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    Cited by:

    1. Smilowitz, Karen & Daganzo, Carlos & Cassidy, Michael & Bertini, Robert, 1998. "Some Observations Of Highway Traffic In Long Queues," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt8rd637pq, Institute of Transportation Studies, UC Berkeley.
    2. Jin, Wen-Long, 2018. "Unifiable multi-commodity kinematic wave model," Transportation Research Part B: Methodological, Elsevier, vol. 117(PB), pages 639-659.
    3. Cassidy, Michael J., 1998. "Bivariate relations in nearly stationary highway traffic," Transportation Research Part B: Methodological, Elsevier, vol. 32(1), pages 49-59, January.
    4. Mazaré, Pierre-Emmanuel & Dehwah, Ahmad H. & Claudel, Christian G. & Bayen, Alexandre M., 2011. "Analytical and grid-free solutions to the Lighthill–Whitham–Richards traffic flow model," Transportation Research Part B: Methodological, Elsevier, vol. 45(10), pages 1727-1748.
    5. Blake Davis & Ang Ji & Bichen Liu & David Levinson, 2020. "Moving Array Traffic Probes," Working Papers 2022-01, University of Minnesota: Nexus Research Group.
    6. Shi, Tie & Zhou, Xuesong, 2015. "A mixed integer programming model for optimizing multi-level operations process in railroad yards," Transportation Research Part B: Methodological, Elsevier, vol. 80(C), pages 19-39.
    7. Mahmoudi, Monirehalsadat & Chen, Junhua & Shi, Tie & Zhang, Yongxiang & Zhou, Xuesong, 2019. "A cumulative service state representation for the pickup and delivery problem with transfers," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 351-380.
    8. Jin, Wen-Long, 2016. "On the equivalence between continuum and car-following models of traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 543-559.
    9. Hans, Etienne & Chiabaut, Nicolas & Leclercq, Ludovic, 2015. "Applying variational theory to travel time estimation on urban arterials," Transportation Research Part B: Methodological, Elsevier, vol. 78(C), pages 169-181.
    10. van Erp, Paul B.C. & Knoop, Victor L. & Hoogendoorn, Serge P., 2018. "Macroscopic traffic state estimation using relative flows from stationary and moving observers," Transportation Research Part B: Methodological, Elsevier, vol. 114(C), pages 281-299.
    11. Florin, Ryan & Olariu, Stephan, 2020. "Towards real-time density estimation using vehicle-to-vehicle communications," Transportation Research Part B: Methodological, Elsevier, vol. 138(C), pages 435-456.
    12. Klug, Florian, 2014. "Modelling and analysis of synchronised material flow with fluid dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 404-417.
    13. Laval, Jorge A. & Leclercq, Ludovic, 2013. "The Hamilton–Jacobi partial differential equation and the three representations of traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 52(C), pages 17-30.
    14. Daganzo, Carlos F. & Laval, Jorge A., 2005. "On the numerical treatment of moving bottlenecks," Transportation Research Part B: Methodological, Elsevier, vol. 39(1), pages 31-46, January.
    15. Gu, Weihua & Gayah, Vikash V. & Cassidy, Michael J. & Saade, Nathalie, 2014. "On the impacts of bus stops near signalized intersections: Models of car and bus delays," Transportation Research Part B: Methodological, Elsevier, vol. 68(C), pages 123-140.

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