Traffic Parameters
Traffic Statistics
Flow q (cars/step)—
Mean speed v̄—
Stopped cars—
Jam clusters—
Phase—
Fundamental Diagram
Flow q = ρ·v̄ vs density ρ. Red dot = current state. Peak flow at ρ_c ≈ 1/(1+1/v_max).
Nagel-Schreckenberg model: phantom jams, spontaneous congestion, and fundamental diagrams
Flow q = ρ·v̄ vs density ρ. Red dot = current state. Peak flow at ρ_c ≈ 1/(1+1/v_max).
The NaSch model is a minimal cellular automaton for traffic flow with four update rules per timestep: (1) Acceleration: v → min(v+1, v_max); (2) Slowing: v → min(v, gap) where gap is distance to next car; (3) Randomization: v → max(v−1, 0) with probability p; (4) Movement: car moves v cells forward.
Despite its simplicity, the model reproduces real traffic phenomena: the fundamental diagram (flow q = ρv̄ vs density ρ) shows a capacity drop, a free-flow phase, and a congested phase. Most remarkably, the randomization step p creates phantom jams — stop-and-go waves that propagate backward with no external cause, just like on real highways. The space-time diagram (road rows = successive timesteps) shows these jams as backward-moving stripes.