Neck Brace

By understanding how the effects of incidents depend on their magnitude, duration and location, we can design better detection and removal strategies. Here, we compare two basic approaches: a system that uses “on-call” tow trucks for incident removal, with a separate surveillance method (e.g., video cameras) for detection; and a system that integrates detection and removal using roving trucks. The results suggest how to best allocate freeway surveillance and assistance resources. The benefits can be considerable.

Why the delay?

Is this scenario realistic or totally outrageous? Can we encounter an incident that does not delay our arrival to our final destination? Or, more dramatically, are there incidents that do not delay anyone? The answer to this question is yes. The incident must happen inside a queue caused by a recurrent bottleneck, and be of such short duration that it does not affect the bottleneck discharge rate—its capacity—at any point in time. This is not an easy thing for drivers to verify individually. But any driver who is late in passing through the bottleneck as a result of the incident can conclude that the incident was serious enough to cause a delay—not just to him or her, but to following drivers too. Figure 1 shows a schematic representation of an incident upstream of a bottleneck.

Critical conditions

Figures 2a and 2b are time-space diagrams for two scenarios depicting an incident upstream of a bottleneck. The diagrams follow the most basic theory of traffic flow. The black dotted lines are vehicle trajectories; the continuous lines are waves (propagation of traffic disturbances where vehicles change speeds); and the red lines are trajectories of selected vehicles, with and without the incident (i.e., actual and theoretical trajectories). Figure 2a shows a short-duration (sub-critical) incident. The typical vehicle suffers no delay at all, since it passes the bottleneck at the same time with and without the incident (i.e., the actual and theoretical vehicle trajectories meet before reaching the bottleneck). This turns out to be true for all vehicles if the traffic void in between the waves downstream of the incident does not reach the bottleneck, as occurs in our figure. Therefore, the flow through the bottleneck is never disrupted by the incident of Figure 2a.

Figure 2b shows the same incident, but with a longer (super-critical) duration. Now, the waves from the incident reach the bottleneck and reduce its flow during a starvation period. This delays every vehicle that passes the bottleneck after the beginning of the starvation period, as shown by the actual and theoretical trajectories of Figure 2b, which never meet. The incident creates “generalized” delays, which linger for the duration of the rush.

Figure 2 reveals that the critical duration increases with the incident’s distance from the bottleneck. It should now be evident that all incidents are not created equal, and that some deserve more urgent attention than others. If incidents are removed quickly enough, they do not cause any extra delay. But, if their duration exceeds a critical time the effects last for the whole rush hour. This critical time depends on the characteristics of the highway (e.g., jam density, free-flow speed, bottleneck capacity and highway capacity) and the characteristics of the incident (e.g., magnitude and location).

On the other hand, the extra delay per vehicle caused by an incident depends on three things:

• Detection time: Time it takes to detect the incident;
• Approach time: Time the roadside assistance vehicle takes to reach the incident location; and
• Removal time: Time the roadside assistance vehicle takes to physically remove the incident.

For an incident not to cause generalized delays, the sum of these three times should not exceed the critical time. Since this critical time increases linearly with the incident’s distance to the bottleneck, incidents closer to the bottleneck are more likely to cause generalized delay than more distant ones.

There are, in general, two possible scenarios. When incidents are really close to the bottleneck, it is almost certain they will affect the bottleneck and reduce its discharge rate. Let us assume there is a tow truck next to the incident when it happens, so that the detection and approach times are zero. Nonetheless, we still have to physically remove the incident. If this would exceed the critical time, a fast response (detection plus approach) time is critical because every minute saved reduces the starvation period by an equal amount. On the other hand, if an incident occurs so far from the bottleneck that generalized delays can be avoided with a slow response, the agency can act with more of a cushion.

In need of a tow

We consider here a homogeneous section of a highway (a link) directly upstream of a bottleneck, as in Figure 1, and two incident-management strategies.