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Analysis of Kinematic Waves Arising in Diverging Traffic Flow Models

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  • Wen-Long Jin

    (Department of Civil and Environmental Engineering, University of California, Irvine, Irvine, California 92697)

Abstract

Diverging junctions are important network bottlenecks, and a better understanding of diverging traffic dynamics has both theoretical and practical implications. In this paper, we present a new framework for constructing kinematic wave solutions to the Riemann problem of kinematic wave models of diverging traffic with jump initial conditions. Within this framework, the function space of weak solutions is enlarged to include interior states that occur with stationary discontinuities, and various discrete flux functions in cell transmission models of diverging traffic are used as entropy conditions to pick out unique physical solutions. Then in demand-supply space, we prove that the Riemann problem can be uniquely solved for existing discrete flux functions, in the sense that stationary states and, therefore, shock or rarefaction waves on all links, can be uniquely determined under given initial conditions. We show that the two diverge models by Lebacque and Daganzo are asymptotically equivalent. We also prove that the supply proportional and priority-based diverge models are locally optimal evacuation strategies. With numerical examples, we demonstrate the validity of the analytical solutions of interior states, stationary states, and corresponding kinematic waves. This study presents a unified framework for analyzing traffic dynamics arising at diverging junctions and could be helpful for developing emergency evacuation strategies.

Suggested Citation

  • Wen-Long Jin, 2015. "Analysis of Kinematic Waves Arising in Diverging Traffic Flow Models," Transportation Science, INFORMS, vol. 49(1), pages 28-45, February.
    • Handle: RePEc:inm:ortrsc:v:49:y:2015:i:1:p:28-45
    DOI: 10.1287/trsc.2013.0499
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    References listed on IDEAS

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    1. Daganzo, Carlos F. & Lin, Wei-Hua & Del Castillo, Jose M., 1997. "A simple physical principle for the simulation of freeways with special lanes and priority vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 31(2), pages 103-125, April.
    2. Daganzo, Carlos F., 1995. "The cell transmission model, part II: Network traffic," Transportation Research Part B: Methodological, Elsevier, vol. 29(2), pages 79-93, April.
    3. Muñoz, Juan Carlos & Daganzo, Carlos F., 2002. "The bottleneck mechanism of a freeway diverge," Transportation Research Part A: Policy and Practice, Elsevier, vol. 36(6), pages 483-505, July.
    4. Daganzo, C. F. & Cassidy, M. J. & Bertini, R. L., 1999. "Possible explanations of phase transitions in highway traffic," Transportation Research Part A: Policy and Practice, Elsevier, vol. 33(5), pages 365-379, June.
    5. Castillo, J. M. Del & Benítez, F. G., 1995. "On the functional form of the speed-density relationship--II: Empirical investigation," Transportation Research Part B: Methodological, Elsevier, vol. 29(5), pages 391-406, October.
    6. Paul I. Richards, 1956. "Shock Waves on the Highway," Operations Research, INFORMS, vol. 4(1), pages 42-51, February.
    7. Jin, Wen-Long, 2010. "Continuous kinematic wave models of merging traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 44(8-9), pages 1084-1103, September.
    8. Daganzo, Carlos F., 1994. "The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory," Transportation Research Part B: Methodological, Elsevier, vol. 28(4), pages 269-287, August.
    9. Papageorgiou, Markos, 1990. "Dynamic modeling, assignment, and route guidance in traffic networks," Transportation Research Part B: Methodological, Elsevier, vol. 24(6), pages 471-495, December.
    10. Newell, G. F., 1999. "Delays caused by a queue at a freeway exit ramp," Transportation Research Part B: Methodological, Elsevier, vol. 33(5), pages 337-350, June.
    11. W.L. Jin & L. Chen & Elbridge Gerry Puckett, 2009. "Supply-demand Diagrams and a New Framework for Analyzing the Inhomogeneous Lighthill-Whitham-Richards Model," Springer Books, in: William H. K. Lam & S. C. Wong & Hong K. Lo (ed.), Transportation and Traffic Theory 2009: Golden Jubilee, chapter 0, pages 603-635, Springer.
    12. Daganzo, Carlos F., 1997. "A continuum theory of traffic dynamics for freeways with special lanes," Transportation Research Part B: Methodological, Elsevier, vol. 31(2), pages 83-102, April.
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