Abstract
A cellular automaton (CA) describes traffic dynamics in a completely discrete way: Space is subdivided into cells, time into time steps, and derived quantities such as speed or acceleration are integer multiples of the corresponding basic units. Cellular automata are easy and fast to simulate. However, due to their discrete nature, they reproduce real-life traffic only in a schematic way. This is particularly true for the Nagel-Schreckenberg Model as the simplest and most generic representative of traffic-related CA. Besides this model, two more refined cellular automata are presented.
As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.
Albert Einstein
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Notes
- 1.
This summary is not complete. For example, traffic flow can also be described by stochastic queuing models which are discrete in the space and state variables and continuous or discrete in time (master equations or Markov chains, respectively). These models are not discussed here.
- 2.
Notice that, in the context of cellular automata, \(i\,{=}\,x_{\alpha }\), \(t\), and \(v\) are index-like integer quantities. Consequently, equations such as \(x(t+1)\,{=}\,x+v\) are formally correct although they may look terribly erroneous to any physics teacher. Since this naming is a de-facto standard in the scientific literature on microscopic traffic CA, we have adopted it although variable names such as \(i_{\alpha }\) for \(x_{\alpha }\) and \(j\) for \(t\) would make the index-like nature of the variables more explicit. In order to avoid confusion, in this chapter, dimensional physical quantities are denoted with a sub- or superscript “phys”.
- 3.
Most continuous models describe the lateral position in a discrete way with a lane index, as well.
- 4.
Simply because it is hard to obtain meaningful results, otherwise.
- 5.
To be consistent with the literature, we denote the gap by \(g\) instead of \(s\), in this chapter.
- 6.
The propagation velocity of perturbations of free traffic is always positive.
- 7.
Remember that cellular automata only make sense for aggregated phenomena. If \(p_0>p\), the average acceleration of stopped vehicles is lower than that of moving vehicles.
- 8.
See pp. 411 in the book The Physics of Traffic by B.S. Kerner (Springer, 2004).
- 9.
For details, the reader is referred to the two monographes of Kerner on this topic, “The Physics of Traffic” (Springer, 2004), and “The Long Road to Three-Phase Traffic Theory” (Springer, 2009).
- 10.
This artifact vanishes if the synchronized state is eliminated by reducing the range factor \(k\) from its published value \(2.55\) to \(k\,{=}\,1\) or by setting \(v_p\,{=}\,v_0\).
- 11.
When implementing continuous social-force pedestrian models, the main problem is the bookkeeping to obtain references to the pedestrians of the local environment. This is typically solved by a virtual grid such that, at any time, each pedestrian is associated with an element of this grid. Since CAs are based on a fixed regular grid with predetermined neighboring relations, no such problems arise for CAs.
- 12.
Graphics processors are optimized for calculations with real numbers, so the speed advantage of using integers is essentially lost.
- 13.
When presenting CA simulations, many authors even do not bother to translate the results into physical units.
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Further Reading
Further Reading
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Von Neumann, J., Burks, A.: Theory of self-reproducing automata. (1966)
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Schadschneider, A., Chowdhury, D., Nishinari, K.: Stochastic Transport in Complex Systems: From Molecules to Vehicles. Elsevier 2010
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Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Physics Reports 329 (2000) 199–329
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Maerivoet, S., DeMoor, B.: Cellular automata models of road traffic. Physics Reports 419 (2005) 1–64
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Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. Journal de Physique I France 2 (1992) 2221–2229
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Barlovic, R., Santen, L., Schadschneider, A., Schreckenberg, M.: Metastable states in cellular automata for traffic flow. Euro. Phys. Journal 5 (1998) 793
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Kerner, B.S.: The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory. Springer (2004)
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Kerner, B.S.: The Long Road to Three-Phase Traffic Theory. Springer (2009)
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Treiber, M., Kesting, A. (2013). Cellular Automata. In: Traffic Flow Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32460-4_13
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DOI: https://doi.org/10.1007/978-3-642-32460-4_13
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