NagelSchreckenberg

Nagel–Schreckenberg model (NaSch) is a cellular automaton model for traffic flow on a one-dimensional lattice. Cars occupy discrete cells, and each car carries an integer velocity v in the range 0 to vmax. Time advances in synchronous steps, and the standard update consists of four sequential rules applied to all vehicles: (1) acceleration: if v < vmax then v ← v + 1; (2) deceleration for safety: let gap be the number of empty cells between a car and the one ahead; if v > gap then v ← gap; (3) stochastic braking: with probability p, if v > 0 then v ← v − 1; (4) movement: each car advances by v cells (mod L) on a ring road of length L.

The NaSch model was introduced in 1992 by Katsuhiko Nagel and Wolfhard Schreckenberg as a minimal, discrete-time, discrete-space representation of highway traffic. It is commonly studied under periodic (ring) boundary conditions, though open boundaries with injection rate alpha and removal rate beta are also used to simulate on-ramps and off-ramps. The model produces a fundamental diagram relating flow to density and exhibits a transition between freely moving and congested regimes.

A key feature of the NaSch model is the role of stochasticity through the braking probability p, which leads to spontaneous jam formation, stop-and-go waves, metastability, and hysteresis in the flow-density relationship. Despite its simplicity, the model captures essential qualitative phenomena of real traffic and serves as a benchmark for non-equilibrium statistical mechanics.

Variants and extensions include different vmax values, alternative updating schemes, multi-lane versions, overtaking rules, and more realistic acceleration schemes, making NaSch a widely used framework in traffic flow research.