# Mathematically modelling phantom traffic jams

You know how high traffic density always seems to lead to self-perpetuating traffic jams that have no visible cause other than the fact that everyone has slowed down? There's math to describe it:
The mathematics of such traffic jams are strikingly similar to the equations that describe detonation waves produced by explosions, said Aslan Kasimov, a lecturer in MIT's Department of Mathematics. Realizing this allowed the reseachers to solve traffic jam equations that were first theorized in the 1950s. The MIT researchers even came up with a name for this kind of gridlock - "jamiton." It's a riff on "soliton," a term used in math and physics to desribe a self-sustaining wave that maintains its shape while moving.

The equations MIT came up with are similar to those used to describe fluid mechanics, and they model traffic jams as a self-sustaining wave...

The MIT team found speed, traffic density and other factors can determine conditions that will lead to a jamiton and how quickly it will spread. Once the jam forms, the researchers say, drivers have no choice but to wait for it to clear. The new model could lead to roads designed with sufficient capacity to keep traffic density below the point at which a jamiton can form.

Kasimov found that jamitons have a "sonic point," which separates traffic flow into upstream and downstream components, much like the event horizon of a black hole. This sonic point prevents communication between these distinct components so information about free-flowing conditions just beyond the front of the jam can't reach drivers behind the sonic point. Ergo, there you sit, stuck in traffic and have no idea that the jam has no external cause, your blood pressure racing toward the stratosphere.

MIT Hopes to Exorcise 'Phantom' Traffic Jams (via Futurismic)

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1. TJ S says:

So they finally solve these equations, and the only solution to jamming that they come up with is “add more capacity to roads”?

Shouldn’t there be other ways to avoid “jamiton”, like raising/lowering speed limits, modifying the on-ramp traffic lights (those ones with only red & green lights, that only let one or two cars on at once), or limiting excessive lane changes?

2. teufelsdroch says:

Yeah, I love the idea of traffic jams as a self-organizing criticality. That is, if you’ve got people randomly hitting the gas or their brakes, you EXPECT jams to spontaneously occur for no reason at all.

Abstract from “Self-Organized Criticality and $1/f$ Noise in Traffic”:

Phantom traffic jams may emerge “out of nowhere” from small fluctuations rather than being triggered by large, exceptional events. We show how phantom jams arise in a model of single lane highway traffic, which mimics human driving behavior…We analytically derive the form of the corresponding power spectrum to be $1/f^{\alpha}$ with $\alpha =1$ exactly. This theoretical prediction agrees with our numerical simulations and with observations of $1/f$ noise in real traffic.

3. Anonymous says:

Having been stuck in traffic in the Berkeley Corridor almost daily, this theory does NOT take into account stupid drivers who leave about 8-15 car lengths between it and the next car.

The other thing this theory does not take into account is people who drive like they’re ants in a conga line. I’m used to L.A. drivers, where they swarm around slow drivers. Here in the Bay Area, people for some reason don’t like to be assertive when driving, so someone in front slows down, and the rest of the lane does too, when there’s free lanes around.

4. Takuan says:

how about licences that require people to prove they know how to drive?

5. zikman says:

waste of efforts, I say. soon cars will be extinct.

I actually remember seeing a show on Discovery or The Learning Channel about this a number of years ago. A computer model was constructed to introduce a tiny bit of variation in the speeds of vehicles travelling along a freeway, and to model the behavior of other traffic in response (car A slows from 100 km/h to 97 km/h, so car B following slows down slightly in response, and so on). The result was inevitable clogs as a minute change to speed at point A led to a virtual standstill 3 km back at point B.

None of which beats the time I got hung up in traffic in Toronto on Highway 401 because a mattress blew off a trailer of stuff someone was moving and blocked a lane for ONE HOUR before a crew came out to move it!

7. Lobster says:

Meh, it’s the math of the gridlock that bugs me. Whenever some asshole pulls out into the middle of the intersection and just SITS there, assuming they’ll be able to keep going when the other direction gets a red light, I just feel like jabbing forks in eyeballs.

Is that normal?

8. asuffield says:

There doesn’t appear to be a link to the paper here, so I can’t actually judge it, but there is nothing new in these articles. Everything described here is stuff I first heard in a presentation being given to students in the 1990s (when they tell stuff to students, it’s no longer a new development, it’s something everybody in the field knows).

What exactly is supposed to have happened here?

9. sopekmir says:

Well …

we have phantom traffic jams without solitons or jamitons – just go to “PÃ©riphÃ©rique” in Paris, or, closer to my heart, try to get to Warsaw airport from south on Monday morning :-)

In more serious mood: years ago I was shocked by the solitons and the implications of them – solitary or self-organizing wave patterns. But this math of jamitons is quite intriguing.

Cory – do you have any URL with papers or, maybe just more mathematical details?

10. prunk says:

I thought about this a while ago when a relative asked, “when there is a line of stopped cars why can’t they all start accelerating at the same rate?” Ideally all of the cars would press down on the gas at the same time (assuming the cars will all be capable of accelerating at the same rate). But the problem really is that speed of communication. Each driver will respond an instance after the driver in front of them has and thus this build up of accumulated responses hits the drivers further on much later than the first driver has already started.
Ideally I would think though that for the slowing down of a car in the front, or at least just a temporary slow and then resuming the original speed if each car did leave two car lengths between and then just reduced it to maybe one car length while waiting for the other car to accelerate again then the variation of speed would be eaten up by these buffers.

11. Anonymous says:

“Ergo, there you sit, stuck in traffic and have no idea that the jam has no external cause, your blood pressure racing toward the stratosphere.”

Why is it, exactly, that we need to know the cause of us having to slow down? Is that somehow supposed to affect the fact that we have to slow down? It’s pointless to get frustrated about traffic patterns. Knowing why we’re in a traffic jam doesn’t alter the fact that we are in a jam. Is being pissy about it going to make it better?

12. Umbriel says:

@11 — “Rubbernecking” or “Gaper Delay” exacerbates the communication problem you refer to, in that drivers are late to notice when traffic opens up ahead of them if they’re gaping at an accident, or even just pulled over cars or emergency vehicles at the side of the road or in oncoming lanes.

It’s a point of pride for me that I do _not_ slow down to look at such things unless there are exposed flames over three feet high, or body parts in my lane.

13. Anonymous says:

I love this sort of stuff. A year ago I saw some excellent .gif animations that showed traffic patterns and merging situations and was to clear jams, if only I could remember the link.

There are ways to ‘bust’ phantom traffic jams, and in response to ANON#4, those people living 15 car lengths ahead of them are the ones that are going to clear the jam. One person won’t clear the jam, but they speed up its reduction.

Much in the way Phrunk mentioned with everyone having two car lengths ahead and closing to 1 as they accelerated being a solution, having one person with lots of space does two things, lets the cars in the jam move forward, letting the wave move forward with no new cars to keep it standing still, and lets/forces more aggressive drivers move in front of them , reducing the people that will feed behind them.

It works even better in a rolling blockade situation, sometimes done by coordinated truckers or by cops. If the jam is slowing everyone down to 20mph, if you approach it from a few miles away, block all lanes with cops/truckers, slow to 50 or 40 or whatever it takes, by the time the front line blockers reach the jam it should have cleared entirely, they can then slowly speed up to normal speeds. The slow down at 40mph will dissipate much faster than the 20mph jam.

Of course that requires only moderate traffic levels and some planning. It is cheaper than building wider roads though!

14. Anonymous says:

I was very disturbed to read a study a year ago that stated that most traffic jams are caused by one bad driver, often a left lane hogger. The best idea I’ve heard for a solution is having all cars computer controlled, networked to each other. If the speed and distance was controlled by computer algorithms and not subject to human error our current infrastructure could handle much more traffic at greater transit speeds (and better fuel economy as a side bonus). I love to drive myself, so losing control doesn’t really appeal to me, but I have to admit, the idea makes sense.

15. eagleapex says:

@#4 I think it’s leaving appropriate distances between cars actually lessens these traffic waves because that distance can absorb a traffic wave and halt it’s progress.
It’s actually drivers that follow too close and over-brake (or brake at all) on the highway that starts these jamitons.

16. TJ S says:

@12:

It seems to be human nature to desire knowledge, even if we can’t react on it. Why do you think 24hr news stations are so popular?

17. Anonymous says:

Well, duh, of course the solution is to add capacityâ€¦ but only if the usage is a constant.

Problem is, it’s not.

In the Real World, whenever you add capacity, the result is that more workers buy houses further from the urban core, since the prices are cheaper and the commute is now “reasonable”; until the new capacity is exhausted and the roads are perpetually jammed, which then limits the rate of outward expansion once again.

Lather, rinse, repeat.

18. Aloisius says:

Having been stuck in traffic in the Berkeley Corridor almost daily, this theory does NOT take into account stupid drivers who leave about 8-15 car lengths between it and the next car.

Well they’re probably doing it because maintaining a constant speed is one way to unjam a phantom traffic jam.

Plus it allows people to change lanes and when they need to turn off the road instead of hitting their breaks and waiting for an opening in extreme situations.

In heavy traffic situations, I almost always leave as many car lengths in front of me as is necessary to maintain a constant speed.

19. Anonymous says:

@#12:

The frustration comes form not knowing whether the jam-up is minor, and will clear in a few minutes; or is caused by a jackknifed truck ten miles ahead blocking three lanes, which will make you two hours late to wherever you’re going.

20. Aloisius says:

erg, I mean to say I leave as many as necessary in front of me on the freeway only. I generally don’t do it in places with stop signs because of gridlock problems caused by people who think crossing an intersection when the other side is completely backed up is acceptable behavior.

21. Astin says:

I remember a post from a few years back from a fluid mechanics student in Seattle who started increasing the distance between himself and the car in front of him until he could time it so that we has moving a constant, slow speed, but never catching up to the jam. When he finally turned off the highway, his lane was moving smoothly behind him, while the other lanes were still bumper-to-bumper.

Variable speed limits BEFORE the jam might be able to lessen the jam. If you have to slow down a few miles before coming to a complete stop, it should allow the jam to break up a bit (or at least move “forward”) before you get there.

That, or build double-decker highways, where the upper deck is only avaialble as a detour when jams get to a certain size, so you’re no longer feeding the problem.

22. Anonymous says:

@#4: I, too, drive the Berkeley corridor daily.

1. If there’s room for 8-15 car lengths between two cars, it’s not a traffic jam.

2. If both cars are travelling at the same speed, it’s not a problem.

23. mdh says:

Well they’re probably doing it because maintaining a constant speed is one way to unjam a phantom traffic jam.

or to cause one,

but if you’re so much smarter than everyone else around them that just couldn’t be true.

So, is it the intelligent man’s responsibility to fix the mess by out thinking everyone around him, or should he try not to piss everyone else off by acting in a way that causes as many problems as it solves?

I say it’s your call, but check your self-estimation of your own intelligence at the curb.

and by you, i mean that jack*** I was stuck behind on I-93 yesterday, not you personally.

24. key says:

OK, there seem to be some people on this thread who know some stuff, so let me ask a question.

When you have a planned lane blockage (due to construction, for example), is it better for drivers to use both lanes for as long as possible, merging a few feet before the actual merge point? Or is it better to make a single line as soon as you know a blockage is ahead?

In certain parts of the US, you would be despised for choosing the first option, but it seems to me it’s the much more reasonable option.

25. seanboing says:

@22 Astin:

I do this regularly. If the driver ahead of me is going fast and then slowing down cyclically, I’ll give that car some more room. Basically I visualize a big rubber band between us, and I don’t accelerate as quickly as they do. The whole lane behind me eventually is running much better, much more smoothly. I am happier, and hopefully the people behind me are happier.

Though if I get too much distance in front of me then cars behind me are not so happy. I am perceived as slowing down the freeway too much. So I make sure not to leave too much space in front of me.

26. Anonymous says:

#23 Trust me, it was a jam. EVERY lane was packed, even the carpool lane, yet in this particular lane, one car left about 10 car lengths between it and the next, and the car behind it was about another 4 car lengths distance. I was behind this second car. When I finally got into the next lane and realized why the previous lane was going slower than every one else’s, I nearly blew it.

Yes, those two cars enjoyed coasting along and they enjoyed constant but slow speeds but for every other car behind them (the ones beyond the sonic point) it was a truly frustrating exercise. And the buildup behind the two cars just grew and grew.

I agree that yes, BEFORE a traffic jam happens, having tons of space between cars are better to absorb the waves. However, if a jam has already started, how can leaving more than a car-length between you and the next be anything but detrimental?

27. Takuan says:

a suggestion for the Rubbernecker Effect: emergency crews to carry eight foot portable fence posts and rolls of bright orange plastic. Upon securing the accident scene, set up visual barriers so the idiots in what should be through traffic will have nothing to look at but a blank wall and will eventually learn to KEEP MOVING!

28. TJ S says:

@26 Key:

I say, take the first opening available. From my point of view, needing to wait for cars in the blocked lane to get into the non-blocked lane is the main cause of congestion here. So, if there’s an available opening, take it. If not, it hurts nothing to wait until the last second (and can, in fact, help, since more of the available road space is being used up).

29. ThreeFJeff says:

Takuan, that will just make things worse. The rubberneckers will spend _extra_ time trying to see if there is a gap in the barrier to peek through.

30. GregLondon says:

jamiton sounds like a curse word.

31. kobrakai says:

Tailgating causes traffic jams. Don’t follow so closely and you won’t have to jam on your brakes when the person in front of you slightly touches theirs.

32. Tgg161 says:

“Thanks, Jammitin!”

33. Jerril says:

#25 posted by mdh
“or to cause one,
but if you’re so much smarter than everyone else around them that just couldn’t be true. ”

Amazingly, people here may not be talking out of their traffic-induced-furious hats. My driving school repeatedly lectured us about this, and made us watch endless videos about traffic flow.

There’s a REASON why there’s a ticket the cops can give you for travelling too close. Rearending the guy in front of you is about 3/4ths of the reason, because accidents and injuries suck ass.

The other 1/4 is because it really freeking does help traffic.

It also, incidentally, uses less gas, therefore causes less CO2 emissions. It’s also better for your car not to stop-start-stop-start in a traffic jamn like that, and hurts your leg less.

If traffic is spurting to 40km, dropping to 10km and spurting to 40km so the average travelling speed is around 20km, really, seriously, honest and for true:

Leave enough space in front of your car that you can dribble along at that average of 20km. You’re not going any faster than you were before, you’re less likely to get rear-ended by the guy behind you because you’re not making sudden movements, you’re less likely to rear-end the guy in front of you because you have space, and other drivers have room to change lanes as needed without you having to slam on your brakes when they get fed up and try to barge their way into your lane.

34. Jerril says:

It bears repeating: just because everyone around you is doing it, doesn’t mean it’s sensible or smart.

There is no wisdom in crowds, just idiot herd behavior. Stampedes and beached whales come from idiot herd behavior.

35. Aloisius says:

The other 1/4 is because it really freeking does help traffic.

It also, incidentally, uses less gas, therefore causes less CO2 emissions. It’s also better for your car not to stop-start-stop-start in a traffic jamn like that, and hurts your leg less.

It also helps your nerves and seriously reduces road rage (at least in me).

I really wish they’d simply have dynamic speed limits so that when one of these standing waves appears, they’d just all flip to 20 mph and slowly increase to help eliminate them.

36. gd23 says:

This site http://trafficwaves.org/ has some interesting ideas on unjamming traffic jams.

My anecodatal evidence seems to confirm the idea that it is better to keep moving and avoid sudden braking, as it is the ripple of braking which escalates into stop start jams. Which is why leaving extra room as a buffer works to avoid the braking.

If you want smooth traffic, act smooth in traffic?

37. SeppTB says:

@#26 Key,

In theory the best practice would be to use both lanes until the absolute end, and then cars merge alternately and smoothly. Doing this causes the least slow down for everyone. Barring that, moving over smoothly early on so you don’t hit the brakes is almost as good.

It never works though. People in the lane being merged into feel like people in the other lane are trying to cheat/cut in front of them, and will often try to block them, which slows down everyone behind them. People in the lane that is merging will sometimes stop way too early, wait forever to try and move over, causing people behind them become angry, speed ahead when they can, causing people in the merged-into-lane to feel they are “cutting in line” again and cause more problems.

In the end its almost impossible not to make everyone involved all together.

38. SeppTB says:

Thanks GD23! I was looking for that site, it is excellent.

39. gruben says:

@26: It’s preferable for both lanes to be used as long as possible. I believe this is the case for two reasons: 1) it’s a more efficient allocation of cars…if you have road, it should be used. Second, it causes people to merge at the end of the lane, which is generally easier/less stressful so mergers won’t be slowing down a lot trying to find a space as they might be if they were trying to merge early. There’s a book about this stuff…Traffic: Why We Drive the Way We Do by Tom Vanderbilt.

40. videomilitia says:

I experience this one everyday between La Brea and La Cienega on the 10 in Los Angeles. It seems to me that a major factor to reduce this would be to close down some of the existing onramps allowing more traffic to exit than to enter. Traffic always picks up after La Cienega because there aren’t as many onramps and less turns in the freeway itself. SCIENTISTS PLEASE HELP BY BUILDING A MODEL OF LA TRAFFIC!!! PLEASE!!! LOL

41. Anonymous says:

The thing that’s always amazed me about traffic is how it can be modeled by particles, as shown in the youtube video. But in the case of traffic, each particle is controlled by a powerful computer, the human brain. Even with all that power, the cars still act like particles, with no brain whatsoever.

42. Bill Beaty says:

> is it better for drivers to use both lanes for
> as long as possible

If drivers merge early, then a few “cheaters” are able to use the empty lane to zoom past everyone. But then the drivers downstream won’t let the cheaters merge back in. Yet still the cheaters can force their way in slooooowly, which brings the full lane to a near halt. A huge backup develops in the full lane, making the empty lane longer, which greatly encourages cheating.

On the other hand, if everyone merges late, and takes turns at the merge point, and both lanes flow at about the same speed …then “cheaters” don’t exist. There’s no empty lane for cheaters to use. There’s no reason for the drivers in the through-lane to block the merging cars. In fact, cars in both lanes can then open up enough space to merge easily. Things flow like meshing gears.

JAM BUSTING
http://trafficwaves.org/

VIDEO: jam in the left-hand exit lane
http://tinyurl.com/m5m9cx

43. Bill Beaty says:

> Having been stuck in traffic in the Berkeley Corridor almost daily, this theory does NOT take into account stupid drivers who leave about 8-15 car lengths between it and the next car.

But professional truckers, driving through jams with a big empty space ahead, aren’t going slow.

Their speed is the same as the car ahead of them. Regardless of the size of that space, their speed is the same as the average speed of their lane. If they were going slow, then the empty space wouldn’t be unchanging, instead it would be growing.

Truckers use this “empty space” technique to free up clogged merge zones and bust traffic jams. Recently few non-truck drivers have discovered the trick and started using it. I see these big “jam busting” spaces more and more often in traffic.

In 2004, traffic scientist L.C. Davis won awards for discovering that, if 10% – 20% of people drove with no tailgating and computer-controlled fast reaction time, then traffic jamming becomes impossible.

44. Anonymous says:

I intuitively learned the smooth driving method. I have had people zoom around me and quickly cover the 3 or 4 car length in front of me and slam on their brakes. They didn’t see that coming but I did.

I have also been in an underpowered car that could not accelerate 0-60 60-0 in 3 seconds. If all cars could and drivers safely drive them then the method would not be effective. But with trucks and slowly acceleration vehicles the net acceleration gets so slow that the jam will only dissipate late at night or something like that.

45. Bill Beaty says:

> Even with all that power, the cars still act like particles, with no brain whatsoever.

Some of us determine the average speed of the lane, then we drive at that speed. Then when the “wave” arrives, we extinguish it. Jam-waves cannot propagate past any car that’s moving at the average speed. Average-speed drivers can wipe out stop-and-go driving.

But performing this feat requires that we use the same trick the truckers are using: maintain a large empty space ahead, and let it grow and shrink as we destroy those waves.

46. klg19 says:

My brother coined a great term for traffic jams that, when one reaches the front, seem to have had no cause. He calls them “immaculate congestion.”

@47: Bill Beaty, if everyone were as smart as you, I don’t think most US cities would ever have traffic jams. What you described is the way I drive, too, based on my own mental model of traffic flow as a compressible fluid.

I was inches away from writing my master’s paper (in aerospace engineering) on applications of fluid dynamics to traffic engineering. But instead, I wrote “On the Aerodynamics of Rotating Sport Discs.” – CU Aero 90 & 92

48. teufelsdroch says:

@44

yeah the interesting conclusion is that it’s not the jerks who try to zoom ahead who create traffic, but rather the jerks who won’t let them in.

49. neurolux says:

Part of the problem is that people don’t know what the average speed is. How about every quarter mile have an LED sign that posts the average speed (call it “Recommended Speed”) for a mile before and after each sign. Some drivers will drive that speed, which will force everyone to drive that speed, and thus stop any jams before they start.

1. Antinous / Moderator says:

I read an article a few years back which described drivers who sit in their fat-ass SUVs in the fast lane going one mile under the speed limit as the Anti-Destination League.

50. NotACat says:

I’m sure I recall reading about this in SciAm years ago. I see that Traffic Waves site refers to articles in various places as far back as 1993, which sounds about right.

What’s amazing is that this is not a new phenomenon, and we still don’t have it whipped…

51. Anonymous says:

This always made my dad rage when I was young because he knew there was nothing really happening to hold up traffic. We call it the traffic slinky, and like #22 & #27 suggest, you just need to plan your speed to average out the slinky waves, then you can roll without ever stopping and fix the lane behind you.

I always thought it’s pretty simple, drivers coming up on slower, higher density traffic don’t want to slow down from their current speed, and the reaction times of each consecutive driver pools until some hit a complete stop, and then stopped traffic propagates backwards.

But you can fight the slinky! Drive at the average speed!

52. guidomax says:

An anecdote my 89-year-old grandfather often tells on this subject is that, when he was driving in convoys in North Africa during WWII, they were taught to always drive at the speed of the slowest vehicle (which, in a convoy/no passing situation, would of course be the average speed of the convoy).

This kept the convoy moving, ensured there weren’t hold-ups, and meant that they got to the destination in the fastest possible time.

He felt mighty vindicated when they started trialling variable speed limits on certain motorways, to cut down on traffic jams in exactly this way.

53. IshMEL says:

Leaving adequate space in front of you helps reduce jams because traffic is a sorting problem. Let’s say a left-lane driver realizes he has to exit soon, but the center lane is tightly packed. The left-laner will slow down, waiting for a space to free up in the center lane, causing the entire left lane to slow down, jam begins. Leaving sorting space in front of you is also safer — defensive driving!

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56. Massage Music says: