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Rule 184 fuzzy cellular automaton as a mathematical model for traffic flow

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Abstract

The rule 184 fuzzy cellular automaton is regarded as a mathematical model of traffic flow because it contains the two fundamental traffic flow models, the rule 184 cellular automaton and the Burgers equation, as special cases. We show that the fundamental diagram (flux–density diagram) of this model consists of three parts: a free-flow part, a congestion part and a two-periodic part. The two-periodic part, which may correspond to the synchronized mode region, is a two-dimensional area in the diagram, the boundary of which consists of the free-flow and the congestion parts. We prove that any state in both the congestion and the two-periodic parts is stable, but is not asymptotically stable, while that in the free-flow part is unstable. Transient behaviour of the model and bottle-neck effects are also examined by numerical simulations. Furthermore, to investigate low or high density limit, we consider ultradiscrete limit of the model and show that any ultradiscrete state turns to a travelling wave state of velocity one in finite time steps for generic initial conditions.

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Notes

  1. They also introduce another parameter M, but it can be eliminated with simple variable transformation and we do not discuss it for simplicity.

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Acknowledgements

We would like to thank Prof. Ralph Willox and Dr. Takafumi Mase for useful comments. KH expresses his sincere thanks to ADK Marketing Solutions Inc. for their scholarship support. This work is supported in part by joint research funding from Arithmer Inc.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Kohei Higashi & Tetsuji Tokihiro

  2. Department of Mathematical Engineering, Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo, 202-8585, Japan

    Junkichi Satsuma

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  1. Kohei Higashi
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  2. Junkichi Satsuma
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  3. Tetsuji Tokihiro
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Correspondence to Tetsuji Tokihiro.

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Higashi, K., Satsuma, J. & Tokihiro, T. Rule 184 fuzzy cellular automaton as a mathematical model for traffic flow. Japan J. Indust. Appl. Math. 38, 579–609 (2021). https://doi.org/10.1007/s13160-021-00461-3

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