[CGMthrowaway]

Of course braking/change in velocity creates waves. But this effect is overemphasized in my opinion. Locally analyzed, traffic can be simplified incredibly by observing that a lane's maximum throughput is simply given by following spacing, measured in time.

If drivers are using a 2 second following distance, commonly taught in driving school, then max throughput is simply

  1 car / 2 sec

If you double following distance, you halve the throughput. If you halve following distance, you double your throughput. The throughput of a (full, i.e. rush-hour) road has nothing to do with speeds of people driving, and everything to do with following distance.

[mnw21cam]

This assumes that a 2 second interval is appropriate for all travelling speeds.

This assumption is untrue at very low speeds, particularly when it takes longer than 2 seconds for a car to pass a point. For instance if we assume cars are 4m long, then with an interval of 2 seconds the cars would be touching at 4.47mph

The assumption is also untrue at very high speeds. You'll want a larger gap. That's partly because at such high speeds the ability of a vehicle to decelerate differs - if a vehicle with good brakes does an emergency stop and the car behind it has a respectable 2 second gap but has worse brakes then they can end up colliding. It's also partly because a 2 second gap at very high speeds means the car in front is further away, and that can cause a greater delay before the driver realises what is happening. As a third reason a greater margin needs to be used at very high speeds simply because the consequences of a crash are that much greater and should therefore be avoided even more than at lower speeds.

Therefore there is a kind of U-shaped curve in the "safe" following interval, and consequently a speed at which safe throughput is maximised.

That's why variable speed limits have been introduced in various places. For instance, in the UK which normally has a 70mph speed limit on motorways, in very high traffic conditions this can be lowered using electronic signs to increase the safe throughput of the road. It's commonly reduced to 50mph, though it does get lowered further in sections approaching a queue of vehicles that has actually stopped.

There's also the issue of speed oscillations. With a high speed limit and vehicles following too closely, a little variation in speed in one vehicle can turn into a larger variation in the following vehicles, causing a backwards-travelling wave of braking (sometimes to an absolute halt) and speeding up again. Lowering the speed limit reduces this.

[CGMthrowaway]

By 2 sec following distance I am referring to their back bumper to your front bumper. So cars "overlapping" in your example is not possible

If you want 4 sec gap at higher speeds that's fine, the formula is speed-independent for throughput, not speed-independent for following distance. If you want 4 seconds at high speed then use 4 sec instead of 2 sec (i.e. 1 car/ 4 sec)

>There's also the issue of speed oscillations. With a high speed limit and vehicles following too closely, a little variation in speed in one vehicle can turn into a larger variation in the following vehicles, causing a backwards-travelling wave of braking (sometimes to an absolute halt) and speeding up again. Lowering the speed limit reduces this.

"Lowering the speed limit reduces oscillations." Exactly, that is my whole point, that (again, locally analyzed) you can ignore the waves, and instead look only at the following distance of the slowest car in the lane, to determine throughput of the road behind that car. Your idea of "lowering the speed limit" to eliminate waves is the same net effect on throughput as observing that the throughput cannot exceed that given by the longest-following car on the road.

[mnw21cam]

A niggle - if you are referring to a 2 second gap between the back bumper and the following front bumper, then the formula is no longer speed independent, as you need to add the small overhead to account for the time taken for the length of the vehicle to pass as well. This will be small enough to be mostly negligible except at low speeds.

[CGMthrowaway]

Totally.

[consp]

If you take this measurement as the goal it would be best at near standstill speed (around 4m/s). If you want to maximize the traveled distance of the group it is around 60kph, which is the metric most people actually care about when discussing throughput.

[CGMthrowaway]

1 car / (actual max following time) no matter the speed. Where do you see speed in the equation? It's not. You can put it in of course but my point is it's unnecessary if you know following distance, which is theoretically more invariable than speed on a road anyway

[fransje26]

> If you double following distance, you halve the throughput. If you halve following distance, you double your throughput.

That postulate breaks down as soon as you move away from a laminar traffic assumption and include distracted drivers, lane changes, and weather influences. Which is why the wave theory model is important to understand the propagation of perturbations and their effect on maximum throughput.

> The throughput of a (full, i.e. rush-hour) road has nothing to do with speeds of people driving, and everything to do with following distance.

And yet, in the limit case of a bumper-to-bumper situation (or, in fluid dynamics parlance, an incompressible flow), the variable determining the change in mass flow-rate is the velocity of the medium. Mimetically, we could also look at ants. To ease congestion in a bumper-to-bumper situation, they accelerate.

[CGMthrowaway]

YES to all! You're so close. Drivers do not accelerate in bumper-to-bumper the way ants do. They maintain a 2sec (or whatever they are trained) following time instead. Which therefore dictates velocity (car lengths per following time). Thus the limiter on flow-rate is actually following time!