Traffic bunching

In heavy traffic, such as on motorways in rush-hour, there is often oscillation in speed and there can even be mysterious ’emergent’ halts. The use of variable speed limits can result in everyone getting along a given stretch of road quicker.

Soros (worth reading) has written an article that suggests that this is all to do with the humanity and ‘thinking’ of the drivers, and that something similar is the case for economic and financial booms and busts. This might seem to indicate that ‘mathematical models’ were a part of our problems, not solutions. So I suggest the following thought experiment:

Suppose a huge number of  identical driverless cars with deterministic control functions all try to go along the same road, seeking to optimise performance in terms of ‘progress’ and fuel economy. Will they necessarily succeed, or might there be some ‘tragedy of the commons’ that can only be resolved by some overall regulation? What are the critical factors? Is the nature of the ‘brains’ one of them?

Are these problems the preserve of psychologists, or does mathematics have anything useful to say?

Dave Marsay

About Dave Marsay
Mathematician with an interest in avoiding 'bad' reasoning.

8 Responses to Traffic bunching

1. Mathematics definitely has something useful to say. This kind of thing always brings to my mind a fact I learned a long time ago in undergrad chemical engineering. Suppose you have a stable but uncontrolled process — that is, a small variation in the feedstock will change the composition of the output but won’t cause it to diverge or oscillate. Even if non-linear, such a process can be well-modeled by a first-order approximation, leading to dynamics described by a first-order differential equation. The solution of such an equation is an exponential decay. (It’s a decay because we assumed that the process was stable — otherwise we have serious problems, and should not be mixing those input components together!)

To sum up: suppose such a process is running at steady state. Hit the process with a jump change in the composition of the feedstock, and the output composition will transition to a new steady state by an exponential decay.

Now put a feedback controller on the process. The controller measures the output and adjusts the composition of the feedstock to try to hit a target output composition. A necessary condition for the controller to be able to do that is that the feedback turns the system into one well-modeled by a second-order differential equation (at *least* second-order). But second-order differential equations can have unstable solutions, oscillating solutions, and combinations of oscillations and instability.

In the ideal case of a well-tuned controller, when the process is hit with a jump change in the composition of the feedstock, the controller changes the flow rates of the components such that after a transient bump, the output returns to the original steady state. But for a poorly tuned controller, the output composition can overshoot the target, leading to “ringing”, and it can even lead to positive feedback, in which each cycle overshoots the target more that the previous one.

If we understand human-driven cars and/or automated driverless cars as under the control of a feedback controller that seeks to make the difference between the current state and a target state vanish, it’s easy to see the analogy.

• Oh, that fact I learned: in order to have the potential for perfect control (in the sense of making deviations from the target vanish in finite time), the system must also be given the potential for instability due to positive feedback.

2. A Traffic Modelling Professional says:

Another vote for mathematics! The behaviour discussed in the first para can be replicated using some relatively simple mathematical models that describe how drivers respond to various stimuli (e.g. when they choose to brake and accelerate, depending on what the vehicle in front is doing – so-called ‘car following models’). The same models can then be used to predict how successful a proposed strategy to ‘smooth’ traffic flow will be.

• Dave Marsay says:

Thanks. Before the crash I tried to use what your paper calls ‘spontaneous shock-waves’ as a familiar analogy to possible booms and busts in heterogeneous economies. I still do not understand while this didn’t lead people to reconsider their preconceptions.

To improve the analogy between my thought experiment and the paper one would want to gradually increase the flow by 2% a year by adjusting the controller parameters. I am not sure why the intuitions are different in the two cases.

3. I like the description +coreyyanofsky gives, but, surely, as in any control system, the accuracy of responses to abrupt changes depends upon having either (a) a rapid sampling rate of the state of everything pertinent (including the overhead of processing it), or (b) a good model which can predict the state of everything pertinent given fewer samples. To the degree the driverless can provide either of these, especially the former, such as is provided to Google driverless cars by their installed LIDAR, it should be possible to “station keep” at short ranges in perfect safety, even if something unexpected goes on ahead. Moreover, from what I understand, driverless cars models not only the vehicles ahead of them, but all the vehicles their sensors can reliably pinpoint, permitted a composite view of evolving situations, even to the degree they can identify a car which is driving unsafely, and take appropriate evasives.

• Dave Marsay says:

Thanks, some interesting thoughts. I wonder, is it possible for a human driver to avoid getting sandwiched between two cars when the traffic suddenly bunches? Is it possible for a driverless car? I have in mind an analogy with algorithmic trading that can avoid huge losses. In this case the algorithm can bail out, but this isn’t always possible if you are in the middle lane of a motorway. So maybe Google cars will avoid busy motorways?

4. David Harold Chester says:

The nature of a roughly steady traffic flow is that it can be unstable. Suppose the vehicle ahead of a particular driven vehicle, slows (due to an external effect like the wind or wanting to better read a road sign). Then the driver of the current vehicle notices the brake lights and that the gap between this vehicle and his/her own becomes less and also this second vehicle slows. But due to the reaction time, the need for braking is more abrupt and so in series the subsequent vehicles all slow down progressively more. However, there is a secondary effect. The driver of a slower vehicle will not see the need to brake so hard as all that, due to the already reduced speeds of the vehicles nearby, so behind the most forward critical vehicle just described there will gradually be a series of slowing vehicles until their speed has dropped to a safe but non-zero amount. The same effect happens in reverse when a driver’s vehicle finds that it can increase speed. This results in bunches of traffic forming. Its nature is clearly unstable.

I have a book with a diagram showing how this occurs in a structural sense, and by a similar way how the effect of a climate change can result in a developing famine (in India), which can become worse or reduced over time. It should be possible to write some relatively simple differential equations and determine which parameters are significant in each case.

• Dave Marsay says:

When I first experienced this I thought of a central heating thermostat: these tend to have a gap between the temperature below which they switch on and the temperature above which they switch off. Otherwise a boiler might switch on and off frequently, annoyingly. With this in mind I tried to leave a gap big enough for reaction+braking, but allowed it to reduce whilst alert and braking, then easing off the brakes when the car in front stopped braking, allowing the gap to build up again. Rather than sophisticated ‘mathematical modelling’ I thought it expedient to leave a larger initial gap, and adjust the gaps through practice (depending on car in front, road conditions, …). Unfortunately, on a multi-lane highway a car behind will generally overtake and fill the gap, particularly when a large gap is required (e.g., wet road).

I quite get that if all the cars were autonomous this could be made much safer. But what about mixed traffic?

A similar effect occurs in economies: for example, one government might build up enough capability and spare capacity to cope with shocks such as epidemics, but then another might come along and see them as wasteful.

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