A stochastic model of macroscopic traffic flow: theoretical foundations

Abstract

In this thesis, a new stochastic extension of Godunov scheme based traffic flow dynamics is developed using a queuing theoretic approach. In contrast to the common approach of adding noise to deterministic models of traffic flow, the present approach considers probabilistic vehicle inter-crossing times (time headways) at various positions along the road as the source of randomness. Subsequently, time headways are used to describe stochastic vehicle counting processes. These counting processes represent the boundary flows in stochastic conservation equations of traffic flow. The advantage of this approach is that (i) non-negativity of time varying traffic variables (namely, traffic densities) is implicitly ensured, and (ii) the mean dynamic of the stochastic model is the Godunov scheme itself. Neither issue has been addressed in previous stochastic modeling approaches which extend the Godunov scheme and its special case, the cell transmission model. A Gaussian approximation of the queueing model is also proposed for purposes of model tractability. 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 function is analyzed and it is shown that the covariance matrices are bounded. Consequently, estimated covariance matrices are also bounded. As a result, 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 in a Kalman filter. The estimated queue sizes were compared to observed maximum queue sizes and the results indicate very good agreement between estimated and observed queue sizes.