On-Ramp Bottleneck and Shock Wave

What is modelled

A motorway can carry only so many vehicles per hour. The flow first rises with the density of cars and then falls as the road gets jammed, and the highest value is the capacity. At an on-ramp, mainline and ramp traffic share that capacity, and if their demand together is larger a queue forms and its tail moves backwards against the traffic as a shock wave. Metering the ramp keeps the demand at capacity.

Equations

• Triangular fundamental diagram q = vf k below the critical density kc = w kj / (vf + w) and q = w (kj − k) above it; capacity qmax = vf kc.
• Cell transmission model: flow between cells y = min(Si, Ri+1) with sending S = min(vf k, qmax) and receiving R = min(qmax, w(kj − k)).
• Shock wave speed (q2 − q1) / (k2 − k1) between free and congested states.

Checks run in the page

• The capacity and critical density found by scanning the diagram.
• Conservation of vehicles in a simulated hour including those waiting to enter.
• The speed of the queue tail against the jump condition.
• The flow leaving the road against the capacity.

Honest simplifications

The road is one aggregated stream without lane changes, the merge priority is a fixed weight, the diagram is triangular and does not show the capacity drop seen in real queues, there are no trucks, incidents or driver differences, and the cell length and time step fix the numerical diffusion of the queue front. It illustrates traffic-flow theory and is not a traffic study.

• Press Rush hour, no metering and watch the red queue bar creep backwards while the speed map shows a red wedge.
• Press Ramp metering on and see the queue vanish while the ramp queue grows instead.
• Change the number of lanes and watch the capacity and the shock speed change.