  We present a traffic model that extends the linear car-following model as
well as the min-plus traffic model (a model based on the min-plus algebra). A
discrete-time car-dynamics describing the traffic on a 1-lane road without
passing is interpreted as a dynamic programming equation of a stochastic
optimal control problem of a Markov chain. This variational formulation permits
to characterize the stability of the car-dynamics and to calculte the
stationary regimes when they exist. The model is based on a piecewise linear
approximation of the fundamental traffic diagram.
