Stability analysis of the classical car-following model

This paper investigates the stability of the classical car-following model (for example, Chandler et al., Operations Research, 6, 165–184, 1958; Herman et al., Operations Research, 7, 86–106, 1959; Wilhelm and Schmidt, Transportation Engineering Journal (ASCE) 99, 923–933, 1973). Conditions for local and asymptotic stability as defined in the references cited are established for the linear model. These differ from those in the literature in two ways. First, it will be shown that, in the autonomous model when the product of the coefficient of proportionality α and the reaction time τ is less than or equal to 1/e, there exist oscillatory solutions with higher frequencies than 2π, although there are none with lower frequencies. Secondly, asymptotic stability is considered along with local stability. The derived condition for asymptotic stability is both necessary and sufficient. In addition, the condition depends on the frequency of the forcing term, with the sufficient condition for the asymptotic stability found in the literature being included as a special case. The nonlinear model is considered by linearization and numerical integrations. Some practical values of parameters are tested for the stability of the model. The analyses in this paper are extended to consider different values of α and τ for different drivers in the line.