On the stochastic fundamental diagram for freeway traffic: Model development, analytical properties, validation, and extensive applications

- Generate speed distributions at any given speed.
- Prove the continuity, differentiability, and convexity of the model.
- Apply this model to control traffic flow stochasticity.
- Be applicable to generate stochastic models for most regression models with scattered samples.

In this research, we apply a new calibration approach to generate stochastic traffic flow fundamental diagrams. We first prove that the percentile based fundamental diagrams are obtainable based on the proposed model. We further prove the proposed model has continuity, differentiability and convexity properties so that it can be easily solved by Gauss-Newton method. By selecting different percentile values from 0 to 1, the speed distributions at any given densities can be derived. The model has been validated based on the GA400 data and the calibrated speed distributions perfectly fit the speed-density data. This proposed methodology has wide applications. First, new approaches can be proposed to evaluate the performance of calibrated fundamental diagrams by taking into account not only the residual but also ability to reflect the stochasticity of samples. Secondly, stochastic fundamental diagrams can be used to develop and evaluate traffic control strategies. In particular, the proposed stochastic fundamental diagram is applicable to model and optimize the connected and automated vehicles at the macroscopic level with an objective to reduce the stochasticity of traffic flow. Last but not the least, this proposed methodology can be applied to generate the stochastic models for most regression models with scattered samples.