# Traffic bunching

January 23, 2014 4 Comments

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?

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.

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.

See, e.g., http://www.paramics-online.com/insight/modelling-cooperative-driving-in-congestion-shockwaves-on-a-freeway-network/

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.