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Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability

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  • Daganzo, Carlos F.
  • Gayah, Vikash V.
  • Gonzales, Eric J.
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    Abstract

    Recent experimental work has shown that the average flow and average density within certain urban networks are related by a unique, reproducible curve known as the Macroscopic Fundamental Diagram (MFD). For networks consisting of a single route this MFD can be predicted analytically; but when the networks consist of multiple overlapping routes experience shows that the flows observed in congestion for a given density are less than those one would predict if the routes were homogeneously congested and did not overlap. These types of networks also tend to jam at densities that are only a fraction of their routes' average jam density. This paper provides an explanation for these phenomena. It shows that, even for perfectly homogeneous networks with spatially uniform travel patterns, symmetric equilibrium patterns with equal flows and densities across all links are unstable if the average network density is sufficiently high. Instead, the stable equilibrium patterns are asymmetric. For this reason the networks jam at lower densities and exhibit lower flows than one would predict if traffic was evenly distributed. Analysis of small idealized networks that can be treated as simple dynamical systems shows that these networks undergo a bifurcation at a network-specific critical density such that for lower densities the MFDs have predictably high flows and are univalued, and for higher densities the order breaks down. Microsimulations show that this bifurcation also manifests itself in large symmetric networks. In this case though, the bifurcation is more pernicious: once the network density exceeds the critical value, the stable state is one of complete gridlock with zero flow. It is therefore important to ensure in real-world applications that a network's density never be allowed to approach this critical value. Fortunately, analysis shows that the bifurcation's critical density increases considerably if some of the drivers choose their routes adaptively in response to traffic conditions. So far, for networks with adaptive drivers, bifurcations have only been observed in simulations, but not (yet) in real life. This could be because real drivers are more adaptive than simulated drivers and/or because the observed real networks were not sufficiently congested.

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    Bibliographic Info

    Article provided by Elsevier in its journal Transportation Research Part B: Methodological.

    Volume (Year): 45 (2011)
    Issue (Month): 1 (January)
    Pages: 278-288

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    Handle: RePEc:eee:transb:v:45:y:2011:i:1:p:278-288

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    Keywords: Urban mobility Traffic congestion Macroscopic fundamental diagram;

    References

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    1. Geroliminis, Nikolas & Daganzo, Carlos F., 2008. "Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 759-770, November.
    2. Daganzo, Carlos F. & Geroliminis, Nikolas, 2008. "An analytical approximation for the macroscopic fundamental diagram of urban traffic," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 771-781, November.
    3. Daganzo, Carlos F., 2007. "Urban gridlock: Macroscopic modeling and mitigation approaches," Transportation Research Part B: Methodological, Elsevier, vol. 41(1), pages 49-62, January.
    4. Daganzo, Carlos F., 2006. "In traffic flow, cellular automata = kinematic waves," Transportation Research Part B: Methodological, Elsevier, vol. 40(5), pages 396-403, June.
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    Citations

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    Cited by:
    1. Ji, Yuxuan & Geroliminis, Nikolas, 2012. "On the spatial partitioning of urban transportation networks," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1639-1656.
    2. 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.
    3. Zhang, Lele & Garoni, Timothy M & de Gier, Jan, 2013. "A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems," Transportation Research Part B: Methodological, Elsevier, vol. 49(C), pages 1-23.
    4. Geroliminis, Nikolas & Boyacı, Burak, 2012. "The effect of variability of urban systems characteristics in the network capacity," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1607-1623.
    5. Zheng, Nan & Geroliminis, Nikolas, 2013. "On the distribution of urban road space for multimodal congested networks," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 326-341.
    6. Arnott, Richard, 2013. "A bathtub model of downtown traffic congestion," Journal of Urban Economics, Elsevier, vol. 76(C), pages 110-121.
    7. 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.
    8. Jin, Wen-Long, 2012. "The traffic statics problem in a road network," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1360-1373.
    9. Daganzo, Carlos F., 2011. "On the macroscopic stability of freeway traffic," Transportation Research Part B: Methodological, Elsevier, vol. 45(5), pages 782-788, June.
    10. Du, Jie & Wong, S.C. & Shu, Chi-Wang & Xiong, Tao & Zhang, Mengping & Choi, Keechoo, 2013. "Revisiting Jiang’s dynamic continuum model for urban cities," Transportation Research Part B: Methodological, Elsevier, vol. 56(C), pages 96-119.
    11. Geroliminis, Nikolas & Sun, Jie, 2011. "Properties of a well-defined macroscopic fundamental diagram for urban traffic," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 605-617, March.
    12. Haddad, Jack & Geroliminis, Nikolas, 2012. "On the stability of traffic perimeter control in two-region urban cities," Transportation Research Part B: Methodological, Elsevier, vol. 46(9), pages 1159-1176.
    13. Haddad, Jack & Ramezani, Mohsen & Geroliminis, Nikolas, 2013. "Cooperative traffic control of a mixed network with two urban regions and a freeway," Transportation Research Part B: Methodological, Elsevier, vol. 54(C), pages 17-36.
    14. Geroliminis, Nikolas & Sun, Jie, 2011. "Hysteresis phenomena of a Macroscopic Fundamental Diagram in freeway networks," Transportation Research Part A: Policy and Practice, Elsevier, vol. 45(9), pages 966-979, November.
    15. Leclercq, Ludovic & Geroliminis, Nikolas, 2013. "Estimating MFDs in simple networks with route choice," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 468-484.
    16. Gayah, Vikash V. & Daganzo, Carlos F., 2011. "Clockwise hysteresis loops in the Macroscopic Fundamental Diagram: An effect of network instability," Transportation Research Part B: Methodological, Elsevier, vol. 45(4), pages 643-655, May.

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