Tuesday, March 29, 2016

Traffic model on the Wicksellian roundabout

Nick Rowe had a post about the "Wicksellian roundabout" almost exactly 2 years ago where he wanted to model an economy as money flowing around from one person to another. I took it up here. He mentioned a Japanese video that he couldn't remember a link to -- here's that video:


Nick characterized it as one car slowing down ("What happens if one car slows down temporarily?"), however in the mathematical model (of "jamitons") there is no real "cause":
However, above a critical threshold density (that depends on the model parameters) the flow becomes unstable, and small perturbations amplify. This phenomenon is typically addressed as a model for phantom traffic jams, i.e. jams that arise in the absence of any obstacles. The instabilities are observed to grow into traveling waves, which are local peaks of high traffic density, although the average traffic density is still moderate (the highway is not fully congested). Vehicles are forced to brake when they run into such waves. In analogy to other traveling waves, so called solitons, we call such traveling traffic waves jamitons.
That is to say a microscopic slowdown in one of the cars is amplified into a travelling solitary wave -- a jamiton.

This is mostly just background information. In the rest of this post (and in subsequent posts as this is a work in progress), I'm going to apply the information transfer framework to traffic where there is a cause -- a slowdown in one cell of the Wicksellian roundabout. I've used the traffic model before as an analogy for the economic information transfer model (here, here and here). Here's a 20-cell roundabout with a temporary obstacle in one cell (modeled as a decrease in probability of leaving that cell). I plot the density in that cell as well as the density in all of the cells versus time:



You can see the jam creates a travelling wave, although this one appears to propagate forward before dissipating in the normal density fluctuations. If we look at a single time frame during the jam, you can see the loss in entropy (maximum entropy occurs when the "cars" are uniformly distributed across the cells):


Information equilibrium (in economics, general equilibrium) is restored as entropy returns to its maximum over time.

9 comments:

  1. I wonder if in the vehicle case, it's the drivers' expectations that end up amplifying the problem: they look ahead, see brake lights and touch their brakes, etc.

    Also, "jamitons" sounds like what a young aspiring particle physicist is told by his parents to go put on before going to bed.

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    1. It'd be interesting to repeat the experiment with driverless cars and see if the system is more robust to avoiding jams.

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    2. In the mathematical models, the cars are effectively ideal drivers (as good as driverless cars), so no.

      Actually, ideal cars would make the jamitons *more* stable. Without some noise (jitter/dither), they would make traffic *worse*.

      Brake lights only change the perception range, so no jamitons would exist with or without "expectations". They're a fundamental property of fluid flows where constituents attempt to maintain distance, regardless of mechanism.

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    3. I was wondering why your "driverless car" equivalent seems to have traveling waves which are "dissipating in the normal density fluctuations" while those in the Japanese experiment with human drivers seem to have jamitons which can be amplified rather than dissipate.

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    4. I'm not showing the same model. As I mentioned in the beginning, that was just background information.

      The entropy model has noise -- random fluctuations of the "cars'" speed.

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  2. BTW, I first heard of this traveling wave traffic jam idea from my brother years ago: he's an RF engineer and he's had to commute through part of Los Angeles on a daily basis for many years... which gave him plenty of time to contemplate the similarity between standing or traveling waves in his RF work and the traffic situation. I'll send him a link to this... he'll be amused I'm sure. (I'm sure there have been plenty of other engineers & physicists stuck in traffic [for seemingly no reason] thinking the same thing!)

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    1. I like the jamiton pun. John Scott Russel's solitary wave of something or other, that he observed in front of ships, and measured on a canal?

      I like this post. I tweeted it. I like the bit about "no real 'cause'". Or the real cause is too high density, which translates as too low (or too inflexible) buffer stocks of money.

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    2. Cheers, Nick.

      And yes that's the reference to the original soliton. The soliton term is from the 1960s, though.

      I am going to look at the dynamics of adding/subtracting density, etc.

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