What is a phantom traffic jam?

A phantom (or ghost) traffic jam arises without any physical obstacle � no accident, no roadwork. In the Nagel-Schreckenberg model it emerges from the random-deceleration rule: one car brakes slightly, the car behind must brake harder, creating a backward-propagating density wave. In moderate-density traffic (around 20�35% occupancy) these stop-and-go waves become self-sustaining and travel upstream at roughly 15 km/h regardless of the average speed of the cars themselves.

What is the fundamental diagram of traffic flow?

The fundamental diagram plots traffic flow (vehicles per unit time, q) against density (vehicles per unit length, ?). In free-flow conditions, q � ?�v_max � flow rises linearly with density. Beyond the critical density ?_c the road becomes congested, drivers can no longer maintain maximum speed, and flow drops sharply. The diagram has an inverted-U shape with a capacity maximum at ?_c. The Nagel-Schreckenberg model reproduces this characteristic shape quantitatively.

What is the BML model and why does it show a phase transition?

The Biham-Middleton-Levine (BML) model places cars on a 2D lattice: right-moving cars advance on even time steps, upward-moving cars on odd steps � mimicking a city grid with perfectly synchronised traffic lights. At low density all cars flow freely. Above a sharp critical density ?_c � 0.32�0.37 (depending on grid size) the system undergoes a discontinuous phase transition: the entire grid suddenly freezes into a checkerboard deadlock from which no car can ever move. The transition is abrupt � the hallmark of a first-order phase transition � and the system never self-heals.

How does a lane-drop bottleneck create a traffic queue?

When the road narrows from two lanes to one (a lane drop), the capacity halves at that point. If traffic arrives at a rate exceeding this reduced capacity, a growing queue forms upstream of the bottleneck. In the simulation, cars beyond the drop position must merge into a single lane; their speed reduces to maintain safe gaps, creating a persistent stop-and-go region even when the arrival density is well below the single-lane capacity. Increasing the on-ramp spawn rate or arrival density lengthens the jam.

Home▸Fluid Dynamics & Aerodynamics▸Traffic Flow Simulation – Nagel-Schreckenberg & BML

🧪 Traffic Flow Simulation – Nagel-Schreckenberg & BML

Simulate highway and city traffic using the Nagel-Schreckenberg cellular automaton and Biham-Middleton-Levine models. Watch phantom traffic jams emerge, observe gridlock phase transitions, and explore the fundamental diagram of traffic flow. Free interactive simulation.

Fluid Dynamics & Aerodynamics2DModerate60 FPS

⚙ Under the hood

This simulation models traffic flow using the Nagel-Schreckenberg and Bureau of Motor Carrier Statistics (BML) models. It demonstrates how vehicles interact within a roadway network, illustrating concepts like congestion and queue formation.

Nagel-SchreckenbergBML ModelsTraffic Flow

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install