A stochastic model of traffic flow: Gaussian approximation and estimation

A Gaussian approximation of the stochastic traffic flow model of Jabari and Liu (2012) is proposed. The Gaussian approximation is characterized by deterministic mean and covariance dynamics; the mean dynamics are those of the Godunov scheme. By deriving the Gaussian model, as opposed to assuming Gaussian noise arbitrarily, covariance matrices of traffic variables follow from the physics of traffic flow and can be computed using only few parameters, regardless of system size or how finely the system is discretized. Stationary behavior of the covariance dynamics is analyzed and it is shown that the covariance matrices are bounded. Consequently, Kalman filters that use the proposed model are stochastically observable, which is a critical issue in real time estimation of traffic dynamics. Model validation was carried out in a real-world signalized arterial setting, where cycle-by-cycle maximum queue sizes were estimated using the Gaussian model as a description of state dynamics. The estimated queue sizes were compared to observed maximum queue sizes and the results indicate very good agreement between estimated and observed queue sizes.

► An analytically tractable Gaussian model of (stochastic) first-order traffic flow.
► Analysis of Lipschitz continuity and (weak-sense) differentiability of disjunctive flux functions.
► A recipe for computing large state covariance matrices using few parameters and discussion of their properties.
► A preliminary validation of the model using Kalman filtering in a real-world setting.