Abstract
We consider a one-dimensional traffic model with a slow-to-start rule. The initial position of the cars in \(\mathbb {R}\) is a Poisson process of parameter \(\lambda \). Cars have speed 0 or 1 and travel in the same direction. At time zero the speed of all cars is 0; each car waits a mean-one exponential time to switch speed from 0 to 1 and stops when it collides with a stopped car. When the car is no longer blocked, it waits a new exponential time to assume speed one, and so on. We study the saturated regime \(\lambda >1\) and the critical regime \(\lambda =1\), showing that in both regimes all cars collide infinitely often and each car has asymptotic mean velocity \(1/\lambda \). In the saturated regime the moving cars form a point process whose intensity tends to 1. The remaining cars condensate in a set of points whose intensity tends to zero as \(1/\sqrt{t}\). We study the scaling limit of the traffic jam evolution in terms of a collection of coalescing Brownian motions.
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Notes
This metric first considers a compactification of the xu-plane as shown in Fig. 9, and uses on the closed disk a metric equivalent to Euclidean. The distance between two paths is defined as the Hausdorff distance between their graphs on the disk, and the distance \(d(\cdot ,\cdot )\) between two collections of paths is taken as again the Hausdorff distance. What is important for us is that, for the paths that we are considering (continuous, finite, non-crossing), we have the following property. For every collection \(\mathcal {W}\) of paths and every \(\delta >0\), there is \(\delta '>0\) such that, for all \(\mathcal {W}'\) satisfying \(d(\mathcal {W},\mathcal {W}')<\delta '\), for every path in \(\mathcal {W}\) starting on \([-\delta ^{-1},\delta ^{-1}]^2\) there exists a path in \(\mathcal {W}'\) starting \(\delta \) units close in the x direction, and staying \(\delta \) units close in the u direction for all \(x \in [-\delta ^{-1},\delta ^{-1}]\) and vice-versa.
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Acknowledgements
We thank L. R. Fontes for motivating and inspiring discussions. We thank both referees for their constructive comments and criticisms. We also thank one of the referees for pointing out some issues in the original version of our proof of convergence to Brownian web and for pointing out relevant references.
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Communicated by Ivan Corwin.
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Ferrari, P.A., Rolla, L.T. Slow-to-Start Traffic Model: Traffic Saturation and Scaling Limits. J Stat Phys 180, 935–953 (2020). https://doi.org/10.1007/s10955-020-02555-7
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DOI: https://doi.org/10.1007/s10955-020-02555-7