Thursday, March 20, 2008

Experimental Traffic Jams

You may have already come across this nice movie of a large-scale physics experiment:

From Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam by Yuki Sugiyama et al., New J. Phys. 10 (2008) 033001 (including movies).

If ever you have been driving on a crowded highway, chances are high that you have taken part in a similar "experiment", just that no one has captured it on film and put it on YouTube. This happened to me last Monday on my way to work: First, I got stuck in a traffic jam at the merging of three lanes into two - no wonder in rush-hour traffic. But then there was a second full stop, a few kilometres down the road, and for no obvious reason at all - no construction site, no junction, no accident... it was the classical phantom traffic jam.

This kind of annoying phenomenon occurs on roads with a steady traffic flow if the distances between cars become too small: As soon as a car slows down a bit for whatever reason, the following car must break also, and so on. And because drivers are humans and have a reaction time, they break ever later, and ever stronger, and at one point, they come to a full stop. This stop then moves "upstream", in the opposite direction of the traffic flow - it's a shockwave-like phenomenon that has been intensively studied by German physicists since the 1990s.

But it seems that no one so far has checked the models that describe the phantom jams in a controlled fashion, and so, the Japanese guys have set up an experiment: Take 22 cars, put them on a road, and tell the drivers to go on at a constant speed. As so often in physics, periodic boundary conditions are a useful trick to simulate a much larger system - the cars are driving on a circular track. It doesn't take long before the shockwave develops.

Here is a chart, taken from the paper, that shows the evolution of the flow of the cars:

The horizontal axis shows distance along the circular track, the vertical axis indicates time. The lines trace the paths of each of the 22 cars. The flatter the line, the higher the speed, and a vertical segment of the line means halt. One can see how a perturbation of the steady flow set in after just 40 seconds around metre 150 of the track. At closer inpection, the culprit seems to be a car that was a bit slower than the others for a while. Speeding up (the kink in the orange circle) doesn't help - the following cars have to break, and the phantom traffic jam can't be avoided anymore. The plot shows nicely how the perturbation - the zone of zero velocity (aka the jam) - travels at constant speed in the direction opposite to the traffic flow.

Too bad - phantom traffic jams just happen, it's all physics...

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Anonymous said...

As a driver, is there any thing I can do to reduce these spontaneous jams?
If a gap appears in front of me, should I try and close it, or just match velocity with the car in front of me? Is there any kind of braking strategy that smooths the kinks out?

Georg said...

Hello Stefan,
I remember faintly an article in SciAm
The authors made experiments with two
cars coupled by a steel wire to measure
distance and rel. speed.
From this experiments they derived a
differential equation regarding driving
behaviour in single lane situations.
I do not remember details, but one:
they recommended to cut "gaps" in the
traffic to one of the tunnels leading
to/from manhattan.
This was achieved by appropriate control
of traffic lights, result was, that
there were no longer those virtual jams,
because those gaps absorbed them in
an early stage.
Although some capacity was wasted to the
gaps, the effective capacity of the tunnel rose.
Regards
Georg

Bee said...

To add my own rather unqualified opinion on the matter, it seems to me these phantom jams are much more common in the USA and in Canada than in Germany, at least that is my impression. Then I am lead to speculate why so:

First, I was guessing it could be some general driving behaviour, either cultural or an artifact of plenty SUVs and automatic gear, but it doesn't seem plausible. Reason is that e.g. people in Toronto drive completely different than in LA, but the highway phenomenon is the same. Also, the road conditions aren't so dramatically different that this would have much of an effect.

So then I thought it might be the traffic rules, not people's behaviour. The most obvious difference is that on German highways passing on the right isn't allowed. That in itself doesn't seem to have much to do with the phantom jams, but a result of the no-passing-on-the-right rule is that the speed on the lanes on the highways are much better separated in that the left lane is the fast one. If I drive in the USA, the traffic on all lanes flows about the same speed. I would guess it is some kind of an optimal configuration in this case. In Germany, people keep much more to the right (unless there is very little traffic, like in the middle of the night cars seem to prefer middle lanes even if the right one is empty).

Either way, now as a result of this equal distribution over lanes, it happens fairly often that one has a front of drivers all going at the same speed that one can not pass at all. This is something that rarely happens in Germany - possibly if a beetle tries to pass a truck who tries to pass another truck, but then you can bet there will be a BMW honking and cursing at the poor beetle. Anyway, what I am trying to say is if you make a small perturbation now in that front (possibly by coincidence, or because something flies over the highway, or some collection of potholes etc), the traffic behind has no chance than to brake. If that would happen only on one or two lanes, at some point people could dilute the pressure to the other lanes.

As to what to do to reduce them, I too would like to know! One thing that seems plausible is to get away as fast as possible if you're through so the jam can resolve from the front. And then of course what you ought to do when stuck in it, you all change lanes to the right so I can pass by on the left ;-p

(Aside: Another difference that I can't quite pin down are the speed limits for truck drivers. If there are some they seem to be mostly ignored in North America - I am always stunned if I get passed by a truck when I'm already well above the speed limit (which I of course never do, just for the record) - don't they have no log to report or something?)

michaeldcassidy said...

I do think automatic transmissions help bring on phantom traffic jams. Drivers with automatics have much less control over the speed/acceleration/deceleration of their cars and tend sharply accelerate and decelerate.

Arun said...

People doing something other than driving (polishing their nails or yakking on the cell phone or turning to admonish their squalling brats in the back seats or reading the newspaper or plucking their eyebrows or using the electric shaver, etc. etc.) I think are the first hazard that needs to be cured.

Bee said...

I'd think that hazard should be taken care of by natural selection ;-) But more seriously, since I (in principle, I somehow got around it) had to take a road test to get a driver's licence in Ontario I bought the official handbook which explicitly recommends not to shave or use a laptop while driving (but hey, it's only a recommendation).

stefan said...

Hi anonymous,

As a driver, is there any thing I can do to reduce these spontaneous jams?
If a gap appears in front of me, should I try and close it, or just match velocity with the car in front of me? Is there any kind of braking strategy that smoothes the kinks out?

From my understanding of what is going on, the most important thing to avoid are differences in velocity between cars following each other.

So, I would say, if a gap appears in front of you, and it is not there because you have slowed down, than this means that the car in front of you has speeded up. Chances are high this car will have to brake again soon, because it is just faster than the average traffic flow, and then it is probably the best not to try to close the gap. Of course, then someone may pass you and try to fill in - this is quite a foolproof recipe to generate these jams ;-)

The only way to possibly smooth out the kinks once it is there is to break early and gently when approaching the jam, not always easy...

As a rule, I think one should try to drive with as smooth a velocity as possible and avoid strong accelerations and breaking - but don't blame me if you got stuck anyway ;-)

Best, Stefan

stefan said...

Hi Georg,

ah, that's interesting! So by inserting the gaps, they effectively reduced the car density? Well, and any jam would not cross the gap, and thus they would make fewer problems.

Best, Stefan

stefan said...

Dear Bee,

I am also quite sure that the driving-on-the-right command on German highways matters. From my experience, phantom jams happen very often in situations when there is a dense flow of cars on all lanes with approximately the same speed. It's of course difficult to disentangle then whether density of cars or equal traffic flow over all lanes is the more important factor.

I am not convinced at all that "to get away as fast as possible" once one is through is a good idea. Since the impatient BMWs typically speed up much faster than the Twingos, this immediately creates the most suitable conditions for the next jam to emerge. I have experienced this quite often already, the highway Frankfurt-Heidelberg is a great laboratory for these kinds of experiments ;-)

Best, Stefan

stefan said...

Hi Michael,

hm, about the automatic gear - it may be because they are more inert with respect to breaking and acceleration?

I have just checked out a bit the theory... in the most simplest form, the phantom traffic jam sets in once the gradient of the velocities of the cars goes beyond a certain threshold. The gradient of velocity is the difference between the speed of two subsequent cars, Δv, divided by their distance, Δs. The velocity gradient has the dimension of an inverse time, and the criterion for the onset of jams is

Δv/Δs > 1/(2τ)

where τ is called relaxation time. This is equation (24) in the long Helbing review.

The left-hand side is quite obvious: the higher the differences in velocity, the higher the chance of a built-up of a jam. The relaxation time τ, on the right-hand side, quantifies how fast the cars can adopt their speed to a changing traffic situation. The shorter the relaxation time, the bigger the threshold (which is the inverse of relaxation time), and the more stable the traffic flow against the formation of jams. My guess is that cars with automatic gears have longer relaxation times (drivers can adjust speed not as fast as with manual gear), thus the flow is less stable.

The nature of the relaxation time seems to be a bit tricky. In the paper Congested Traffic States in Empirical Observations and Microscopic Simulations (cond-mat/0002177) Martin Treiber, Ansgar Hennecke and Dirk Helbing quote actual numbers for τ: However, realistic velocity relaxation times are of the order of 10 s (city traffic) to 40 s (freeway traffic) and therefore much larger than reaction delay times (of the order of 1 s). This has surprised me. So, reaction time seems to set a lower limit for τ and hence, the jam threshold is always finite - but jams happen for other reasons than just reaction time (of course, the idea is that after the reaction time, the driver behaves appropriately...) And for the freeway traffic, this means that jams can set in once Δv/Δs is larger than 1/80 s - or Δv = 0.5 m/s = 1.8 Km/h for a car-to-car distance of 40 metre. That's not much!

Best, Stefan

Dr Who said...

In Europe, it is considered to be really bad to drive slowly in the leftmost lane, and people coming up behind you have no hesitation in flashing their lights, sounding their horn etc to get you to move out of the way. That kind of "impatience" [which is entirely justified in my opinion] is not accepted in anglo-saxonia, unfortunately, and in some places it could lead to serious trouble. I don't understand the anti-BMW stuff; of course if you have a fast car, you want to be able to drive quickly and not be blocked by slow people. The Americans have a saying: "Close the gap, or get out of the way!" which I think all drivers should take to heart. [Actually, the correct version is, "Close the @#\$%ing gap, or \$%^&ing get out of the !@#\$ing way, you useless sack of \$%%&ing ^&*(!!!!!!"]

Phil Warnell said...

Hi Stefan,

Nice post, I’m surprised that Tommaso and Amara haven’t chimed in on this all ready. Back on Feb. 7 Tommaso over at Quantum Survivor’s Diary put up a post called “Explaining Traffic Jambs” that holds relevance to this one. It was focused on the more general topic as to what cases traffic jambs, not just restricted to phantom the variety. It was actually Amara who filled in the blanks on this one. Well actually more like filled in my blanks. It appears she wrote a paper in this regard some 11 years ago. It seems that this can be all viewed from the Lagrangian (least action perspective) which your article alludes to. That is, the longer delay in action requiring a greater response manifests larger jambs. Therefore to avoid traffic jambs the limit would be, instant response coupled with no action. When I look at it this way it seems not only intuitive yet natural.

Best,

Phil

amaragraps said...

There are six New Horizons meetings at SwRI in Boulder this week; my own presentation of results was today (results and talk went well :-) ). Sorry I missed this, but Phil gave the link. Thanks for mentioning my old summary, Phil. Also, I should say that the references inside might be useful for anyone here here wanting to follow up more on the topic.

stefan said...

Hi Phil, Amara -

thanks for the pointer - I had not taken notice of Tommaso's post, curious coincidence, as this was just six weeks ago...

Since I have listed some references to the original papers about this stuff, I maybe should add one more: In Dynamical model of traffic congestion and numerical simulation, M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama state the stability criterion I have mentioned in my last comment (Phys. Rev. E 51 (1995) 1035-1042). By the way, it seems that three of the autors of that theory/simulation paper are now also authors of the experimental paper in the New Journal of Physics.

I have the impression that the models now used are pretty complex... anyway, searching the arxiv for traffic jam yields a suprisingly big number of hits.

Best, Stefan

PS: Amara, what did you talk about? Will you blog about it?

stefan said...

BTW, Phil,

That is, the longer delay in action requiring a greater response manifests larger jambs. Therefore to avoid traffic jambs the limit would be, instant response coupled with no action.

Right, this is exactly what is encoded in the Bando criterion, that the threshold for the onset of jams is an inverse relaxation time, τ. The shorter this time - instant reaction being the limit - the higher the threshold, and the less jams.

Best, Stefan

PS: Hm, when Amara refers to a Lagrangian method, I guess she thinks about a special method to solve hydrodynamical equations (Lagrange vs. Euler hydrodynamics) - not sure if this relates to minimal action. Although, early slight reaction to changes in velocity probably minimizes action.

Phil Warnell said...

Hi Stefan,

“thanks for the pointer - I had not taken notice of Tommaso's post, curious coincidence, as this was just six weeks ago...”

You should not take this as a failing, yet simply as another example of the consequence of the limits imposed by nature. That is, the more time taken in writing, the less time available to read. I would rather that you write more and read less, for I acknowledge that you have in the past done much of what is needed of that. Not to mention that reading still remains for you as a requirement, so you might continue to put bread on the table ;-)

Best,

Phil

P.S. lately I find myself deleting more comments. I apologize for this. It’s not that I change my mind it’s simply as of late I’ve caught my typos and can’t tolerate them as much as before. I am thus torn between being understood for my limitations and my unwillingness to expose them. It could be considered as vanity, yet I justify this as being a requirement mandated by clarity :-) Ironically, although I have long recognized the truth of the Lagrangian method, inspired Fermat, as it relates to “a stitch in time saves nine”, I still haven’t been able to apply it to myself :-)

amaragraps said...

Dear Stefan,

I won't write a blog entry at scientificblogging.com, if that's what you mean, but I can say something here.

My current SwRI position is financially supported by (Pluto) New Horizons, a new space mission for me, but many aspects of the operations and especially the people are familiar because the planetary science community is relatively small. So an intro to what took place this week in Boulder, then I'll say what were the meetings.

The New Horizons mission got off with a bang, from launch to the Jupiter (5AU) flyby was only about a year. So after launch, the NH team had to initially check-out the instruments at the same time as they were sequencing for the Jupiter flyby, which made for a 'compressed' year. Some of the in-space calibrations and tests that one would normally make early in a space mission needed to be postponed until after the Jupiter flyby. That flyby however, was also a good 'check-out' of the instruments, and the great science it acquired was a kind of bonus. There is a topical Science magazine issue devoted to NH that was published last Fall, that gave the science highlights, if you want to look.

Now because the Pluto Encounter won't occur until July 2015, the spacecraft is mostly in 'hibernation' until then (usually the Student Dust Counter will be the only instrument operating), and woken up for about one month each year for 'Annual Check-Outs'. During these unique months, each instrument has a set of tests and calibrations to perform, confirming old results, trying new observations, all to prepare and confirm the mission's readiness for Pluto. Because the expertise and funding and other aspects for sequencing the commands for the Encounter exist _now_, and because the Pluto Encounter is _difficult_ (about 12,000 km from the surface of Pluto, but remember it has at least three moons, with one of them, Charon, almost as large as Pluto), and will need extensive testing in the next years, the Pluto Encounter sequences are being written _now_. The team hopes to have it finished by the end of summer.

So the NH meetings that took place this past week were: Pluto Encounter Planning (Sunday and Wednesday afternoon, New Horizons Science Team Meeting number 17 (Monday and Tuesday), Pluto Encounter Science Requirements Review (Wednesday), Ralph Instrument team meeting (Thursday morning), Science Operations Center Meeting (Thursday), and the Final Instrument Commissioning Review (Friday, today). Some of these meetings sound like what they were, others probably sound cryptic.

One example of the latter: Pluto Encounter Science Requirements Review was about ... everything ... regarding the mission, Each of the science goals, what instruments will be used for that science goal, prioritization of the science goals, implementation methods for each science goals, what backup plan if there is a failure. Then the spacecraft: how much fuel in total, how much other power, how much data storage, how many thruster firings left, how much each of the operations 'cost' in fuel, power, data storage, thruster counts. Then all about the ground stations: which radio receiver will be operating over what bandwidth range, and over which parts of the mission. All of these were finalized before launch, so now, all of these items were re-evaluated after-in-flight-use after the Jupiter flyby, that is, to verify that all looks ok for the Pluto encounter.

On Thursday, I made my first presentation to the New Horizons group, in particular the Ralph team. The result of my burning the last 3 months, i.e. what I started with immediately when I arrived in Boulder. It was all about the calibration of the stars in the imager field of view, that is, that the number of photons from the identified stars matched what is expected for an object at that distance onto the CCD detector at various wavelengths. The instrument behavior is almost perfect ('INcredible,' Alan Stern said), we have a line slope that equals 1 (expected = actual) in some of the Pan Frames of the MVIC part of the Ralph instrument of the M6 and M7 star cluster fields. These Messier clusters are among the calibration star clusters for this instrument. Given that the spectral types in the star clusters are varied, that was a good test to see how well the instrument responds to point sources of different temperatures. So the results so far are excellent, and my talk went well. The other aspects of Ralph such as (very low) optical distortion and the other NH instruments are also performing excellently (is that a word?). So what I learned in this week of meetings is that the New Horizons space mission and the instruments and the operations part of the mission are in top form.

stefan said...

Dear Amara,

thank you a lot for your comment about New Horizons - this all sounds very exciting! Maybe I should write a post about the mission sometime. Ralph is the main telescope and camera system? Does this name have a special meaning?

Best, Stefan

amaragraps said...

Dear Stefan: The instrument: Ralph is named after Alice's husband on the 1950s American TV comedy The Honeymooners. :-)

bellamy said...

This is because 1) there is too much deviation in operational parametres of the devices and humans involved; 2) cos everybody's a fuckin individual. (Military units, in 'drill', make up for this.) In any case, all the more reason for automated transport....except...people wanna be fuckin individuals. Alas.

Bee said...

Hi bellamy,

I'd hope the fucking individuals do their fucking elsewhere than on the highway, is this really a necessary adjective? Besides this, I don't agree with your conclusion. The actual reason is there's too much traffic. People being individual is very likely a better choice than automatizing reactions. The latter might work better in most situations, but if it fails, it fails far more dramatically. If the system is homogeneous it's more prone to catastrophic errors, whereas 'individual' heterogeneity can smooth these errors out. I.e. the jams occur more often, which is annoying, but at least they aren't fatal very often. Chose for yourself what you prefer. Best,

B.

Bee said...

Dear Stefan,

Since the impatient BMWs typically speed up much faster than the Twingos, this immediately creates the most suitable conditions for the next jam to emerge.

Well, you shouldn't drive on the BMW lane with your Twingo ;-)

Best,

B.

Bee said...

Hi Michael,

I do think automatic transmissions help bring on phantom traffic jams. Drivers with automatics have much less control over the speed/acceleration/deceleration of their cars and tend sharply accelerate and decelerate.

My experience is actually the other way round, drivers with automatics try to avoid acceleration/deceleration - especially when coupled with a tempomat you put yourself into the driver's seat, set the speed to 70 mph, open a coke and watch the trees pass by.

Besides this, I've been driving an automatic for some years, and I can't say it changed much about my driving behaviour. Also, I think the newer automatic transmissions accelerate quite well. So I'm not sure whether the breaking threshold is affected all that much by the car's details or is more a result of habit.

Best,

B.

Phil Warnell said...

Hi Bee,

While I would not care that it appears that I agree with our adjective and adverb limited friend, his comment and your response does raise a question. That question is how statistically safer would a totally automated system have to be, before drivers would be willing to surrender their control to a computer?

I am sure that with further improvements in technology, emergency subroutines and so forth that this could be brought well below the current human related risk factors within the foreseeable future. Despite this I would still be certain that most would resist such a system for some time, not out of a reluctance of giving up personal freedom, yet unwillingness to give up control to automation out of fear.

My question then would be, at what point statistically, should this be realized to be irrational fear as apposed to rational. As and example many people who fear flying are not so because they are not aware of how safe it is as a form of transport, rather they fear they are helpless to effect their fate if something goes wrong. On the other side, most know that turning the key in their vehicle everyday is the riskiest thing we will ever do; and yet for most they perceive little danger. This is just another example of the hurtles science will have to overcome in effecting our future.

Best,

Phil

bellamy said...

Bee - indeed, it was necessary, as it helped exemplify all sorts of implications that Phil then went on to flesh out. It was, for those of a certain perception, a sort of poetic affectation - but functional. Of course, those who are easily offended - ...or, offended at all, alas - would disagree.

Also, ya all use tools in a particular fashion when working together to enhance your work, yes? This isn't a fundamental condition people invoke in life. {da-dump}

Bee said...

Hi bellamy,

I wasn't offended, just surprised by the unmotivated occurence of fucking individuals on the 401. Best,

B.

Michael F. Martin said...

Could this be a splay state?

http://people.maths.ox.ac.uk/~porterm/research/alex_surf_final.pdf

stefan said...

Hi Michael,

synchronization in coupled oscillators is an interesting topic on its own, but I don't see a direct link to the traffic jams. Maybe there is some mapping of the degrees of freedom to a model of coupled oscillators, but what might it be?

Cheers, Stefan

Michael F. Martin said...

Isn't there a frequency associated with each of those cars in units loops / sec? And they all start off at the same frequency and different equally/spaced phase differences?

I'm more asking than stating. After reading the linked paper, I remembered this post and simply wondered whether you couldn't model that propagating wave as a propagating phase difference with the Kuramoto model. I wish I had more time to puzzle these things out...

stefan said...

Isn't there a frequency associated with each of those cars in units loops / sec? And they all start off at the same frequency and different equally/spaced phase differences?

Ah, right, that's true! And when they get "in phase", this means they are all clumped in the same place on the circle. I am not aware of jam studies along these lines. Sounds interesting!

Michael F. Martin said...

http://www.scholarpedia.org/article/Weakly_coupled_oscillators

This propagating wave looks like a slight perturbation from the stable equilibrium for the linear array of phase coupled oscillators.

It's interesting to note the stability criterion. How could traffic be constrained to such criteria? All-to-all coupling isn't feasible for traffic, although it might be for other networks (like markets?).

But what about pair-wise odd coupling? That one seems to result more from the symmetry of the ring tho, so not as useful for real freeways.

I like the idea of synchronization stability criterion being used to study traffic patterns tho.

Anonymous said...

Very interesting stuff here...just found this cool video explaining Phantom Traffic Jams. https://www.blogger.com/comment.g?blogID=5430866&postID=115153094126872358&page=1