Traffic flow waves - 2D ring road and space-time diagram

The intelligent driver model

Each car i accelerates according to the Intelligent Driver Model: dv/dt = a [1 - (v/v₀)⁴ - (s*/s)²], with the desired gap s* = s₀ + vT + vΔv / (2√(ab)). Here v₀ is the desired speed, T the safe time gap (the "reaction time" of the 3D scene), s₀ the minimum gap, a the comfortable acceleration, b the comfortable braking, s the bumper-to-bumper gap and Δv the speed difference to the car in front. The model never causes a collision when you brake hard.

Phantom jams

In uniform traffic every car drives at the equilibrium speed v_e(s), from s_e = (s₀ + v T) / √(1 - (v/v₀)⁴). The flow q = ρ v_e has a maximum at intermediate density; near it and above it small disturbances grow, and a stop-and-go wave appears with no bottleneck at all. The wave front moves backwards against the cars (about -15 to -20 km/h in real traffic), which is the slope of the dark stripes in the space-time diagram.

Honest simplifications: a single lane on a closed ring, identical drivers and vehicles, no lane changes, no anticipation beyond the next car, model parameters are typical textbook values and not fitted to any road; the noise slider adds random accelerations to stand in for human variation.

• Use "Free flow" and then add cars: the cars slow evenly, until a jam appears from a tiny disturbance.
• Press Brake car 1 hard in light traffic: the wave dies out; in dense traffic it grows and keeps circling.
• Raise the time gap T from 1.0 to 2.5 s: drivers are calmer and the waves vanish at the cost of capacity.
• Read the fundamental diagram: the dot leaves the green equilibrium curve when jams form.