Dynamics of traffic oscillations in stochastic linear car-following models considering reaction delay: Analytical investigation and numerical analysis

In real traffic, reaction delay and stochasticity are intrinsic characteristics of human drivers. A simple approximation of the stochastic term based on experimental data reveals that the stochastic term can be decomposed into an additive term and a multiplicative term. The combined impact of the reaction delay, the additive term, and the multiplicative term on the speed variance (the bound of traffic oscillations) has been investigated analytically through closed form expressions and numerically. It is found that there is an interplay between stochasticity and delay, and traffic oscillations become more pronounced with increasing levels of stochasticity and delay. The amplitude of these oscillations, though bounded, can be significant and accompanied by an extended spatial autocorrelation even in the stable regime. Moreover, low frequency modes contribute most to the speed variance, suggesting that traffic oscillation frequencies resemble those observed in real traffic. The string stability of stochastic linear car-following models with reaction delay has also been investigated. Sufficient but not necessary string stability conditions were derived using the Lyapunov functional method. It is found that for relatively low sensitivity to spacing and high sensitivity to velocity difference, and when the instability is of a long wavelength nature, stochasticity significantly destabilizes traffic flow. The destabilizing effect of stochasticity becomes negligible when the instability is of a short wavelength nature, regardless of sensitivities to spacing and velocity difference. To complement the qualitative analysis, numerical simulations were conducted to provide quantitative insights.