  Starting from a non-local version of the Prigogine-Herman traffic model, we
derive a natural hierarchy of kinetic discrete velocity models for traffic flow
consisting of systems of quasi-linear hyperbolic equations with relaxation
terms. The hyperbolic main part of these models turns out to have several
favourable 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.
