Traffic bottleneck

Consider a stretch of highway with two lanes in one direction. Suppose that the fundamental diagram is modeled as shown here. The highway has a peak capacity of Q vehicles per hour, corresponding to a density of k_c vehicles per mile. The highway normally becomes jammed at k_j vehicles per mile.

Before capacity is reached, traffic may flow at A vehicles per hour, or a higher B vehicles per hour. In either case, the speed of vehicles is v_f (or "free flow"), because the roadway is under capacity.

Now, suppose that at a certain location x₀, the highway narrows to one lane. The maximum capacity is now limited to D’, or half of Q, since only one lane of the two is available. State D shares the same flow rate as state D', but its vehicular density is higher.

Using a time-space diagram, we may model the bottleneck event. Suppose that at time t₀, traffic begins to flow at rate B and speed v_f. After time t₁, vehicles arrive at the lighter flowrate A.

Before the first vehicles reach location x₀, the traffic flow is unimpeded. However, downstream of x₀, the roadway narrows, reducing the capacity by half—and to below that of state B. Due to this, vehicles will begin queuing upstream of x₀. This is represented by high-density state D. The vehicle speed in this state is the slower v_d, as taken from the fundamental diagram. Downstream of the bottleneck, vehicles transition to state D', where they again travel at free-flow speed v_f.

Once vehicles arrive at rate A starting at time t₁, the queue will begin to clear and eventually dissipate. State A has a flowrate below the one-lane capacity of states D and D'.

On the time-space diagram, a sample vehicle trajectory is represented with a dotted arrow line. The diagram can readily represent vehicular delay and queue length. It is a simple matter of taking horizontal and vertical measurements within the region of state D.

A slow tractor creates a moving bottleneck.

For this example, consider three lanes of traffic in one direction. Assume that a truck starts traveling at speed v, more slowly than at the free-flow speed v_f. As shown on the fundamental diagram below, speed q_u represents the reduced capacity (two-thirds of Q, i.e., 2 out of 3 lanes available) around the truck.

State A represents normal approaching traffic flow, again at speed v_f. State U, with flowrate q_u, corresponds to the queuing upstream of the truck. On the fundamental diagram, vehicle speed v_u is slower than speed v_f. But once drivers have navigated around the truck, they can again speed up and transition to downstream state D. While this state travels at free flow, the vehicle density is less because fewer vehicles get around the bottleneck.

Suppose that, at time t, the truck slows from the free-flow rate to v. A queue builds behind the truck, represented by state U. Within the region of state U, vehicles more slowly, as indicated by the sample trajectory. Because state U limits to a smaller flow than state A, the queue will back up behind the truck and eventually crowd out the entire highway (slope s is negative). If state U had the higher flow, there would still be a growing queue. However, it would not back up because the slope s would be positive.[1]