Traffic, Networks, and Elliptic Equations
Posted by Phil Aaronson at 6:53 PM
Living in the Bay Area, I spend a lot of time in traffic, even when I ride my bike to work. Since I'm sitting in traffic anyway, I spend a lot of time thinking about traffic. As a system I mean. If you rummage around in your old math books from college there's a pretty classic equation in there governing traffic flow. The solution's there, and since I've forgotten most of my math, I'll let it stay there, boxed up with my mechanical pencils and my HP-15C calculator. Oh, that was a good calculator by the way.
Not very interesting stuff I know, but what is interesting, is that the raw equation for traffic flow is "hyperbolic". In english, that means that shocks (really radical changes in velocity) should be occurring all the time when you drive in traffic. That doesn't really happen in practice, when you drive you can look a bit ahead and slow down more gradually. Good drivers don't wait for the break lights on the car in front of them to start their deceleration. They know its coming. They look ahead. They anticipate.
Those mathematical geniuses have thought of that too, and try to account for that as well by adding in additional terms to the classic equation. This tends to alter it from a shock filled "hyperbolic" class of problem to "parabolic", which is a good thing for traffic. But not great. What that means: information propagates one way. In other words, drivers look ahead well, but rarely to the side, and even more rarely behind. Which feels right, doesn't it?
There is another class of equation called "elliptic". What if traffic were elliptic? If it were, then your speed and the decisions you make would be impacted by the cars in front of you, like now, but those behind and to the sides would have an impact as well. The only time I see anyone's speed impacted by what's going on behind them is if a BMW rides right up from behind, inches away from their bumper and sits there for a while. Even then not so much. But if it were really elliptic then everything changes. You know that spot where the highway goes from 4 lanes down to 3 and it clogs up for miles and miles and you sit in bumper to bumper traffic? That's not elliptic. If things really were elliptic, you'd accelerate through that stretch, yes actually go faster not slower. Elliptic good.
That sounds a little crazy doesn't it? Impossible? Well ok, that is extreme, but this is really where the internet in your car craze should be going. Instead of trying to let drivers, God forbid, look at web pages in traffic or carry on phone conversations, which will only make traffic worse. They anticipate less and make things, as a system more "hyperbolic", more bad. It should focus instead on propagating information that will help make traffic less.
If there were a way for you to broadcast your driving actions to not just the car behind you, which is all you're reaching now, but to say those 30 cars around you locally, both in front, and behind, traffic problems would decrease. Why? Because this is how elliptic systems behave. Ok, not very reassuring. But thinking about it a bit, I think it feels true even in the real world. Maybe not 4 lanes of highway down to 3 going faster kind of better, but better.
If you knew that a corvette, say a mile or two back in your lane was coming up on you at 150mph would you move over well before the 'vette got even close? I think I would in a second. If you were coming up on a red light and you knew that 10 cars behind you all wanted to go right on red, would you go into the right lane quite as quick? Would you move over a lane more often to let faster moving traffic through if you knew for sure that the car two lanes over wasn't also going to move into that same lane at the same time?
Its these kinds of small effects that sum to a system that runs a whole lot smoother. This is what networking traffic should REALLY be about, exactly that, networking the traffic about the traffic. I'll be the first to admit I'm talking off the back of an envelope, but the cumulative benefits of knowing more about what the cars around you has such a potentially huge benefit. Not just in terms of getting you to your destination in one piece, but faster as well.
Not very interesting stuff I know, but what is interesting, is that the raw equation for traffic flow is "hyperbolic". In english, that means that shocks (really radical changes in velocity) should be occurring all the time when you drive in traffic. That doesn't really happen in practice, when you drive you can look a bit ahead and slow down more gradually. Good drivers don't wait for the break lights on the car in front of them to start their deceleration. They know its coming. They look ahead. They anticipate.
Those mathematical geniuses have thought of that too, and try to account for that as well by adding in additional terms to the classic equation. This tends to alter it from a shock filled "hyperbolic" class of problem to "parabolic", which is a good thing for traffic. But not great. What that means: information propagates one way. In other words, drivers look ahead well, but rarely to the side, and even more rarely behind. Which feels right, doesn't it?
There is another class of equation called "elliptic". What if traffic were elliptic? If it were, then your speed and the decisions you make would be impacted by the cars in front of you, like now, but those behind and to the sides would have an impact as well. The only time I see anyone's speed impacted by what's going on behind them is if a BMW rides right up from behind, inches away from their bumper and sits there for a while. Even then not so much. But if it were really elliptic then everything changes. You know that spot where the highway goes from 4 lanes down to 3 and it clogs up for miles and miles and you sit in bumper to bumper traffic? That's not elliptic. If things really were elliptic, you'd accelerate through that stretch, yes actually go faster not slower. Elliptic good.
That sounds a little crazy doesn't it? Impossible? Well ok, that is extreme, but this is really where the internet in your car craze should be going. Instead of trying to let drivers, God forbid, look at web pages in traffic or carry on phone conversations, which will only make traffic worse. They anticipate less and make things, as a system more "hyperbolic", more bad. It should focus instead on propagating information that will help make traffic less.
If there were a way for you to broadcast your driving actions to not just the car behind you, which is all you're reaching now, but to say those 30 cars around you locally, both in front, and behind, traffic problems would decrease. Why? Because this is how elliptic systems behave. Ok, not very reassuring. But thinking about it a bit, I think it feels true even in the real world. Maybe not 4 lanes of highway down to 3 going faster kind of better, but better.
If you knew that a corvette, say a mile or two back in your lane was coming up on you at 150mph would you move over well before the 'vette got even close? I think I would in a second. If you were coming up on a red light and you knew that 10 cars behind you all wanted to go right on red, would you go into the right lane quite as quick? Would you move over a lane more often to let faster moving traffic through if you knew for sure that the car two lanes over wasn't also going to move into that same lane at the same time?
Its these kinds of small effects that sum to a system that runs a whole lot smoother. This is what networking traffic should REALLY be about, exactly that, networking the traffic about the traffic. I'll be the first to admit I'm talking off the back of an envelope, but the cumulative benefits of knowing more about what the cars around you has such a potentially huge benefit. Not just in terms of getting you to your destination in one piece, but faster as well.
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