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Modelling and Optimisation of Flows on Networks pp 303–333Cite as

Theoretical vs. Empirical Classification and Prediction of Congested Traffic States

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

Abstract

Starting from the instability diagram of a traffic flow model, we derive conditions for the occurrence of congested traffic states, their appearance, their spreading in space and time, and the related increase in travel times. We discuss the terminology of traffic phases and give empirical evidence for the existence of a phase diagram of traffic states. In contrast to previously presented phase diagrams, it is shown that “widening synchronized patterns” are possible, if the maximum flow is located inside of a metastable density regime. Moreover, for various kinds of traffic models with different instability diagrams it is discussed, how the related phase diagrams are expected to approximately look like. Apart from this, it is pointed out that combinations of on- and off-ramps create different patterns than a single, isolated on-ramp.

First published in: The European Physical Journal B 69(4), 583–598, DOI: 10.1140/epjb/e2009-00140-5 (2009), © EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009, reproduction with kind permission of The European Physical Journal (EPJ).

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Notes

  1. 1.

    Note that traffic patterns which appear to be localized, but continue to grow in size, belong to the spatially extended category of traffic states. Therefore, “widening moving clusters” (WMC) are classified as extended congested traffic, while the similarly looking “moving localized clusters” (MLC) are not. According to Fig. 6, however, the phases of both states are located next to each other, so one could summarize both phases by one area representing “moving clusters” (MC).

  2. 2.

    A typical threshold for German freeways would be \({V }_{\mathrm{crit}} \approx 80\) km/h.

  3. 3.

    Which includes “widening moving clusters” (see Fig. 1a and footnote 1).

  4. 4.

    When averaging over spatial and temporal intervals that sufficiently eliminate effects of heterogeneity and pedal control in real vehicle traffic.

  5. 5.

    When the freeway is busy, it may happen that these two flows are different and that a queue of vehicles forms on the on-ramp. Of course, it is an interesting question to determine how the entering ramp flow depends on the freeway flow, but this is not the focus of attention here, as this formula is not required for the following considerations.

  6. 6.

    That is, “moving localized clusters” (MLC) and “widening moving clusters” (WMC), see footnote 1 and Sect. 4.2.

  7. 7.

    Since the model parameters characterize the prevailing driving style as well as external conditions such as weather conditions and speed limits, the separation lines (“phase boundaries”) and even the existence of certain traffic patterns are subject to these factors, see Sect. 7.

  8. 8.

    For example, in most models, the outflow Q out downstream of an on-ramp bottleneck decreases with the bottleneck strength and increases with the length of the on-ramp [53, 55].

  9. 9.

    The IDM parameters for plots (a) and (b) are given by v 0 = 128 km/h, T = 1 s, s 0 = 2 m, s 1 = 10 m, a = 0. 8 m/s2, and b = 1. 3 m/s2. To generate plots (c) and (d), the acceleration parameter was increased to a = 1. 3 m/s2, while the other parameters were left unchanged.

  10. 10.

    And the condition \({Q}_{\mathrm{up}} + \Delta Q < {Q}_{\mathrm{out}}\) for the gradual dissolution of the resulting congestion pattern is harder to fulfil than the condition \({Q}_{\mathrm{up}} + \Delta Q < {Q}_{\mathrm{max}}\) implied by (4).

  11. 11.

    In this connection, it is interesting to remember Kerner’s “dissolving general pattern” (DGP), which is predicted under similar flow conditions.

  12. 12.

    One may also analyze the situation with the shock wave equation: Spatially expanding congested traffic results, if the speed of the downstream shock front of the congested area (which is usually zero) minus the speed of the upstream shock front (which is usually negative) gives a positive value.

  13. 13.

    Since pinned localized clusters rarely constitute a maximum perturbation, they can also occur at higher densities and flows, as long as \({Q}_{\mathrm{up}} < {Q}_{\mathrm{c2}}\). Therefore, MLC and PLC states can coexist in the range \({Q}_{\mathrm{c1}} < {Q}_{\mathrm{up}} < {Q}_{\mathrm{c2}}\). For most traffic models and bottleneck types, congestion patterns with \({Q}_{\mathrm{tot}} \approx {Q}_{\mathrm{c1}}\) do not exist, since localized congestion patterns do not correspond to maximum perturbations. The actual lower boundary \(\tilde{{Q}}_{\mathrm{c1}}\) for the overall traffic volume \({Q}_{\mathrm{tot}}\) that generates congestion is somewhat higher than Q c1, but usually lower than \({Q}_{\mathrm{c2}}\). Considering the metastability of traffic flow in this range and the decay of the critical perturbation amplitude from ρc1 to ρc2 [10], this behavior is expected. However, for some models and parameters, one may even have \(\tilde{{Q}}_{\mathrm{c1}} > {Q}_{\mathrm{out}}\). In such cases, PLC states would not be possible under any circumstances.

  14. 14.

    Oscillatory congestion patterns upstream of off-ramps are further promoted by a behavioral feedback, since drivers may decide to leave the freeway in response to downstream traffic congestion.

  15. 15.

    The existence of “widening moving clusters”, see Sect. 4.2 and Fig. 1a, supports this view as well.

  16. 16.

    An example would be the IDM with the parameter choice s 1 = 0.

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Acknowledgements

DH and MT are grateful for the inspiring discussions with the participants of the Workshop on “Multiscale Problems and Models in Traffic Flow” organized by Michel Rascle and Christian Schmeiser at the Wolfgang Pauli Institute in Vienna from May 5–9, 2008, with partial support by the CNRS. Furthermore, the authors would like to thank for financial support by the Volkswagen AG within the BMBF research initiative INVENT and the Hessisches Landesamt für Straßen und Verkehrswesen for providing the freeway data.

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Authors and Affiliations

  1. ETH Zurich, UNO D11, Universitätstr. 41, 8092, Zurich, Switzerland

    Dirk Helbing

  2. Institute for Transport & Economics, Dresden University of Technology, Andreas-Schubert-Str. 23, 01062, Dresden, Germany

    Martin Treiber, Arne Kesting & Martin Schönhof

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  1. Dirk Helbing
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Correspondence to Dirk Helbing .

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Helbing, D., Treiber, M., Kesting, A., Schönhof, M. (2013). Theoretical vs. Empirical Classification and Prediction of Congested Traffic States. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_5

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