Stability Analysis of Stochastic Linear Car-Following Models

Recent scholars have developed a number of stochastic car-following models that have successfully captured driver behavior uncertainties and reproduced stochastic traffic oscillation propagation. Whereas elegant frequency domain analytical methods are available for stability analysis of classic deterministic linear car-following models, there lacks an analytical method for quantifying the stability performance of their peer stochastic models and theoretically proving oscillation features observed in the real world. To fill this methodological gap, this study proposes a novel analytical method that measures traffic oscillation magnitudes and reveals oscillation characteristics of stochastic linear car-following models. We investigate a general class of stochastic linear car-following models that contain a linear car-following model and a stochastic noise term. Based on frequency domain analysis tools (e.g., Z-transform) and stochastic process theories, we propose analytical formulations for quantifying the expected speed variances of a stream of vehicles following one another according to one such stochastic car-following model, where the lead vehicle is subject to certain random perturbations. Our analysis on the homogeneous case (where all vehicles are identical) reveals two significant phenomena consistent with recent observations of traffic oscillation growth patterns from field experimental data: A linear stochastic car-following model with common parameter settings yields (i) concave growth of the speed oscillation magnitudes and (ii) reduction of oscillation frequency as oscillation propagates upstream. Numerical studies verify the universal soundness of the proposed analytical approach for both homogeneous and heterogeneous traffic scenarios, and both asymptotically stable and unstable underlying systems, as well as draw insights into traffic oscillation properties of a number of commonly used car-following models. Overall, the proposed method, as a stochastic peer, complements the traditional frequency-domain analysis method for deterministic car-following models and can be potentially used to investigate stability responses and mitigate traffic oscillation for various car-following behaviors with stochastic components.