### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 065, 10 pages      arXiv:1503.08669      https://doi.org/10.3842/SIGMA.2015.065
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### A CA Hybrid of the Slow-to-Start and the Optimal Velocity Models and its Flow-Density Relation

Hideaki Ujino a and Tetsu Yajima b
a) National Institute of Technology, Gunma College, Maebashi, Gunma 371-8530, Japan
b) Department of Information Systems Science, Graduate School of Engineering, Utsunomiya University, Utsunomiya 321-8585, Japan

Received March 31, 2015, in final form July 27, 2015; Published online July 31, 2015

Abstract
The s2s-OVCA is a cellular automaton (CA) hybrid of the optimal velocity (OV) model and the slow-to-start (s2s) model, which is introduced in the framework of the ultradiscretization method. Inverse ultradiscretization as well as the time continuous limit, which lead the s2s-OVCA to an integral-differential equation, are presented. Several traffic phases such as a free flow as well as slow flows corresponding to multiple metastable states are observed in the flow-density relations of the s2s-OVCA. Based on the properties of the stationary flow of the s2s-OVCA, the formulas for the flow-density relations are derived.

Key words: optimal velocity (OV) model; slow-to-start (s2s) effect; cellular automaton (CA); ultradiscretization, flow-density relation.

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