Abstract
We introduce a stochastic traffic flow model to describe random traffic accidents on a single road. The model is a piecewise deterministic process incorporating traffic accidents and is based on a scalar conservation law with space-dependent flux function. Using a Lax-Friedrichs discretization, we show that the total variation is bounded in finite time and provide a theoretical framework to embed the stochastic process. Additionally, a solution algorithm is introduced to also investigate the model numerically.
Highlights
We introduce a stochastic traffic flow model to describe random traffic accidents on a single road
Using a Lax-Friedrichs discretization, we show that the total variation is bounded in finite time and provide a theoretical framework to embed the stochastic process
Macroscopic traffic flow models based on hyperbolic conservation laws have been intensively investigated during the last decades, see [1, 2] for an overview
Summary
Macroscopic traffic flow models based on hyperbolic conservation laws have been intensively investigated during the last decades, see [1, 2] for an overview. Compared to [19], we face different challenges here: since accidents happen at various spatial positions, we use a space-dependent flux function capturing traffic accidents as spatial capacity drops in the LWR model. This is the deterministic part only and we need an appropriate model representing the stochasticity of traffic accidents. There are different works about hyperbolic equation based dynamics connected to randomness as for example random velocity fields [20, 21] and propagation of uncertainty [22] In these works, there is no influence of the conserved quantity on the stochastic nature, i.e., no bi-directional relation between the deterministic and stochastic ideas.
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