A Hierarchy of Kinetic Discrete-Velocity Models for Traffic Flow Derived from a Nonlocal Prigogine-Herman Model

Starting from a nonlocal version of the Prigogine-Herman traffic model, we derive a natural hierarchy of kinetic discrete-velocity models for traffic flow consisting of systems of quasilinear hyperbolic equations with relaxation terms. The hyperbolic main part of these models turns out to have several favorable features. In particular, we determine Riemann invariants and prove richness and total linear degeneracy of the hyperbolic systems. Moreover, a physically reasonable invariant domain is obtained for all equations of the hierarchy. Additionally, we investigate the full relaxation system with respect to stability and persistence of periodic (stop-and-go-type) solutions and derive a condition for the appearance of such solutions. Finally, numerical results for various situations are presented, illustrating the analytical findings.