Asymptotic traffic dynamics arising in diverge-merge networks with two intermediate links

Basic road network components, such as merging and diverging junctions, contribute much to formation and propagation of traffic congestion. However, in contrast to extensive studies of traffic dynamics on a road link, little is understood for those on a road network. In this study, we provide a comprehensive analysis of asymptotic traffic dynamics arising in diverge-merge networks with two intermediate links with first-order kinematic wave models. In addition, traffic demand at the origin and traffic demand at the destination are constant and maximum, and the proportion of vehicles choosing an intermediate link is constant. First, we examine all possible asymptotic stationary states and the sufficient and necessary conditions for their existence. With an example, we find that non-stationary states can include damped (DPO) and persistent periodic oscillations (PPO), and present an equivalent dynamical system that can be used to distinguish these two types of oscillations. This study can be a springboard for better understanding of traffic dynamics, especially formation and propagation of traffic congestion, on general road networks.