# Thread: What is Critical Velocity @ high density?

1. ## What is Critical Velocity @ high density?

Imagine a 1/4-mile (1/2 km if you're outside US) section of freeway with 3 lanes, packed with cars. If they were all crawling along a 1 mph, parade-style, we could say so-many cars per minute (CPM) were passing some point, call it x1. If we increase the velocity of all the vehicles, we could get more CPMs past the same point, say at 2 mph we would have x2.

As you keep increasing the velocity of the cars, the CPM keeps going up...to a point. The reason is, that as you increase velocity, you need to increase the car-to-car spacing, to allow for human reaction-time. At some point, the increased car-to-car spacing offsets the gain achieved by increasing velocity, so CPM goes down, instead of increasing further. So if you graph CPM vs velocity, CPM increases up to a point, then goes back down.

This swithch from increasing-CPM-with velocity to decreasing-CPM-with-velocity creates a point of instability with increasing traffic density, resulting in "caterpillaring," which all commuters should be familiar with: traffic comes to a crawl...then it opens up...then it comes to a crawl again...then it opens up...etc.

At any rate, there is some maximum velocity, call it the Critical Velocity, beyond which there is no point in going any faster in heavy traffic, because you will only get to the next clump of stopped traffic sooner. If you drive faster than this critical velocity, you are actually decreasing the overall CPM capacity of the road (see above), and contributing to the next slow-down.

I'm guessing the critical velocity is about 45 mph (20 m/s outside US), but surely someone, sometime has determined this velocity in some rational manner?

2. Well, we'd need to quantify the increase in stopping time, wouldn't we? What I was always taught was to allow two seconds time, but that (of course) results in 30 cars per minute passing any given spot regardless of velocity.

So what's the formula for determining the proper safe following distance (or time) at any given velocity?

3. Originally Posted by SeanF
Well, we'd need to quantify the increase in stopping time, wouldn't we? What I was always taught was to allow two seconds time, but that (of course) results in 30 cars per minute passing any given spot regardless of velocity.
Good point... The only factor would be the length of the car. Sorry Peter...

Originally Posted by SeanF
So what's the formula for determining the proper safe following distance (or time) at any given velocity?
That's the way that I have always heard it. I have also heard of the car lengths per 10mph... but that is just another form of the same thing. It's a judged distance anyway, so accuracy is not a big factor.

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Peter, I can't answer your question directly, but can quote a real life example that validates your arguement.

The M25 (Motorway 25) encircles London. It is extremely busy, especially to the west side, where three national motorways join it over a 20 mile stretch. The usual motorway speed limit is 70 mph, but when the traffic density gets to a certain point the speed limit is reduced to 50mph. This has the effect of increasing the CPM (cars per minute), probably due to the spacing effect you mention.

John

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Originally Posted by Peter Wilson
Imagine a 1/4-mile (1/2 km if you're outside US) section of freeway with 3 lanes, packed with cars. If they were all crawling along a 1 mph, parade-style, we could say so-many cars per minute (CPM) were passing some point, call it x1. If we increase the velocity of all the vehicles, we could get more CPMs past the same point, say at 2 mph we would have x2.

As you keep increasing the velocity of the cars, the CPM keeps going up...to a point. The reason is, that as you increase velocity, you need to increase the car-to-car spacing, to allow for human reaction-time. At some point, the increased car-to-car spacing offsets the gain achieved by increasing velocity, so CPM goes down, instead of increasing further. So if you graph CPM vs velocity, CPM increases up to a point, then goes back down.

This swithch from increasing-CPM-with velocity to decreasing-CPM-with-velocity creates a point of instability with increasing traffic density, resulting in "caterpillaring," which all commuters should be familiar with: traffic comes to a crawl...then it opens up...then it comes to a crawl again...then it opens up...etc.

At any rate, there is some maximum velocity, call it the Critical Velocity, beyond which there is no point in going any faster in heavy traffic, because you will only get to the next clump of stopped traffic sooner. If you drive faster than this critical velocity, you are actually decreasing the overall CPM capacity of the road (see above), and contributing to the next slow-down.

I'm guessing the critical velocity is about 45 mph (20 m/s outside US), but surely someone, sometime has determined this velocity in some rational manner?
Yes. I dunno where you can get the info tho. You might start with city traffic agencies like Houston Transtar. You could also try googling nonlinear dynamics and traffic.

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Keep in mind that safe stopping/following distance is dependent on more than the length of the vehicle and the reaction time of the person driving it. A 60' limousine stops much sooner and faster than a loaded 60' grain truck.

I would think that if human reaction time is a factor in calculating this, then so should be length, mass, and velocity of the vehicle; road conditions; wind; number of lanes; number of interchanges per stretch of road, etc. Some of that can be generalized, say come up with an average figure for big trucks, another for cars, then integrate the two into one representative CPM value. But I don't think there will be an easy answer, and if there is one, I would be immediately suspicious of its accuracy until I learned how the numbers were produced.

There is a stretch of highway near here that flows at about 80 mph, consistently, every day. It doesn't really experience much caterpillar effect, even in densely packed traffic. There is a cloverleaf interchange that introduces and removes traffic to and from the interstate, but the volume is excessively low. However, despite the fact that you only stand about a 15% chance of encountering a car merging at that interchange, traffic speeds always fall by 10-15 mph on approach to it. This is because somewhere up ahead, that one car is indeed merging, even if you're not around it, and traffic up ahead has to move over and make room for the merging vehicle, which has perhaps 150 feet of road to get from 25 mph to highway speeds.

In this example, we see the caterpillar effect, but its main cause seems not to be the CPM (which is altered by a rather negligible amount -- adding one car to several dozen), but rather the conditions surrounding that stretch of road. This is why I think that only taking into account how many cars there are per minute will not yield useful, real-world results. Even if everyone was driving at the Critical Velocity for that highway, they would have to slow down more for that short stretch of road where the interchange is. So this would either alter the Critical Velocity for the whole highway, or necessitate separating this short stretch from the rest.

7. Originally Posted by jseefcoot
In this example, we see the caterpillar effect, but its main cause seems not to be the CPM (which is altered by a rather negligible amount -- adding one car to several dozen), but rather the conditions surrounding that stretch of road.
Or even if that stretch is perfectly straight, no ramps, etc, it would only take one person to change lanes to start the caterpiller action. The first car brakes, the second one a little harder, the third one, etc, until the braking action corresponds with reaction.

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I drive an average of 90 interstate miles every day. I see exactly what you are talking about, except it goes the opposite way: car 1 switches lanes. Car 2 slows down a little to make sure there's room (if it even slows at all -- if the person who merged into that lane is a responsible driver, no one should have to take any action). Car 3 slows down a little less (by sacrificing some of its following distance), Car 4 even more, and Car 5 doesn't even need to tap brakes.

So yes, there is a caterpillar effect, but in this case it's extremely local and short-lived. The cars that did slow down gradually open their intervals back up to their comfort zone, and in the real world what happens is a nearly seamless merge into another lane of traffic.

If one tried to calculate a Critical Velocity using a static, unchanging stopping/following distance, I don't think the result would be representative of the real world. The less representative of the real world the figures are, the less use they will be. (That is, if you're using them to design traffic flow, or perhaps to establish a posted speed limit, which is my presumption.)

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Originally Posted by Peter Wilson
Imagine a 1/4-mile (1/2 km if you're outside US) section of freeway with 3 lanes, packed with cars. If they were all crawling along a 1 mph, parade-style, we could say so-many cars per minute (CPM) were passing some point, call it x1. If we increase the velocity of all the vehicles, we could get more CPMs past the same point, say at 2 mph we would have x2.

As you keep increasing the velocity of the cars, the CPM keeps going up...to a point. The reason is, that as you increase velocity, you need to increase the car-to-car spacing, to allow for human reaction-time. At some point, the increased car-to-car spacing offsets the gain achieved by increasing velocity, so CPM goes down, instead of increasing further. So if you graph CPM vs velocity, CPM increases up to a point, then goes back down.

This swithch from increasing-CPM-with velocity to decreasing-CPM-with-velocity creates a point of instability with increasing traffic density, resulting in "caterpillaring," which all commuters should be familiar with: traffic comes to a crawl...then it opens up...then it comes to a crawl again...then it opens up...etc.

At any rate, there is some maximum velocity, call it the Critical Velocity, beyond which there is no point in going any faster in heavy traffic, because you will only get to the next clump of stopped traffic sooner. If you drive faster than this critical velocity, you are actually decreasing the overall CPM capacity of the road (see above), and contributing to the next slow-down.

I'm guessing the critical velocity is about 45 mph (20 m/s outside US), but surely someone, sometime has determined this velocity in some rational manner?
You'll have to be far more specific about "heavy traffic." My guess is you will have a maximum speed related to density; and that it will be a curve, with the peak representing the maximum capacity of the road. The maximum capacity to me is measured by the number of cars that can get from one point to another. I drive in what I consider to be heavy traffic daily, and it is usually moving between 70 and 80 MPH. That's between 2 and 4 lanes in each direction, average spacing of about 2 to 3 car lengths (50' or so).

10. Originally Posted by JohnD
Peter, I can't answer your question directly, but can quote a real life example that validates your arguement.

The M25 (Motorway 25) encircles London. It is extremely busy, especially to the west side, where three national motorways join it over a 20 mile stretch. The usual motorway speed limit is 70 mph, but when the traffic density gets to a certain point the speed limit is reduced to 50mph. This has the effect of increasing the CPM (cars per minute), probably due to the spacing effect you mention.

John
Thanks, that's the answer I was looking for...assuming the Brits have worked this all out Next time I find myself in caterpillar conditions I'll drive 50.

There seems to be a lot of confusion in some of the other replies: yes, vehicle length, intersections, etc, effect CPM, but nonethelesss, there is always some velocity at which CPM is at its maximum.

If other areas followed London's example, I'm guessing a 2 - 3% increase in traffic through-put could be achieved. Not a whole lot; not the same as adding another lane, but a lot easier to achieve

11. Originally Posted by Peter Wilson
There seems to be a lot of confusion in some of the other replies: yes, vehicle length, intersections, etc, effect CPM, but nonethelesss, there is always some velocity at which CPM is at its maximum.
But what seems like confusion is just everyone chiming in with various factors to indicate that there is not really a maximum, although for each individual condition there may be.

In the end we are talking about human reactions, perceptions, and experience. Each stretch of road at different times is going to have a different of all these factors.

I wonder about JohnD's example. I'm not sure the cause and effect are there.
but when the traffic density gets to a certain point the speed limit is reduced to 50mph.
That, in itself says there are more cars, my assumption is that they do it for safety because at a certain density the traffic normally slows down, and those hotshots that think they can do the speed limit during those times really mess things up.
You're going to increase throughput a lot more by keeping them moving, then allowing an accident to happen.

From the OP.
The reason is, that as you increase velocity, you need to increase the car-to-car spacing, to allow for human reaction-time.
But the spacing is based on speed and reaction-time. Reaction time does not change according to speed, so spacing and speed is a one-to-one relationship in the ideal world.

As SeanF said, if reaction time is 3 seconds. Then the spacing is 3 seconds apart (regardless of distance) and the formula always works out to 20 CPM per lane no matter what speed you plug in.

12. Originally Posted by NEOWatcher
As SeanF said, if reaction time is 3 seconds. Then the spacing is 3 seconds apart (regardless of distance) and the formula always works out to 20 CPM per lane no matter what speed you plug in.
with the caveat that the spacing of 3 seconds (actually 2 seconds) is judged by the driver to be from rear bumper to front bumper, whereas the #/cars per time is measured from front bumper to front bumper. So the actual cars per time will increase slightly with speed.

A friend who is a civil engineer tells me that roads are designed to handle ~ 2000 cars / hour / lane (2 sec spacing to one sig fig) but that areas with aggressive drivers (such as my area) achieve more.

13. Originally Posted by pghnative
with the caveat that the spacing of 3 seconds (actually 2 seconds) is judged by the driver to be from rear bumper to front bumper
Yes; that's true and will modify the formula dependent on speed, but to the statement in the OP, there is no "bell" to the curve, it's all one way.

14. Peter, I've been thinking more about this, and I'm not sure you're original premise holds water.

As I mentioned earlier, cars per minute (CPM) is a simple factor of following time (SFT for "safe following time") - 3 seconds SFT means 20 CPM, 5 seconds means 12 CPM, etc. So that means that our maximum CPM will be at our minimum SFT. It doesn't seem it would ever be logical for a higher velocity to coincide with a smaller SFT, which would mean that SFT would approach zero (and CPM approach infinity) as velocity approaches zero.

Since I don't think a speed limit of zero MPH (or approaching zero) would really give us our best traffic throughput, I think there must be something other than SFT that would determine maximum CPM in the real world.

15. Originally Posted by SeanF
Since I don't think a speed limit of zero MPH (or approaching zero) would really give us our best traffic throughput
To pghnative's point, approaching zero would give us our worst throughput.
Take that one minute. One slice is taken with the car traveling a car's length, the rest is the gap between cars. As speed limit approaches 0, it takes an infinite time for the car to cross (although spacing is still 2 seconds)
As speed limit approaches infinity, the car takes no time to cross, and you still split the minute into 2 second intervals.
Or the OP's critical velocity is infinity (not counting for quantum effects).

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Originally Posted by Peter Wilson
As you keep increasing the velocity of the cars, the CPM keeps going up...to a point. The reason is, that as you increase velocity, you need to increase the car-to-car spacing, to allow for human reaction-time. At some point, the increased car-to-car spacing offsets the gain achieved by increasing velocity, so CPM goes down, instead of increasing further. So if you graph CPM vs velocity, CPM increases up to a point, then goes back down.
Actually, the CPM keeps going up with increased velocity, provided drivers are using the two-second rule (assuming dry, day conditions). This is due to the fact that less "dead" space is taken up by the vehicles themselves, so that more space is allocated to the two-second spacing.

This is the reason why, as the number of vehicles per lane per mile of roadway increases, traffic doesn't just slow down - it hits the CPM and grinds to a near halt until the logjam clears.

A single lane of traffic at 20 mph can support, including car lengths, a CPM of approximately

This is based on the two-second spacing, which comes to almost 58.7 feet, plus the length of your average car, which let's say is around 15 feet.

Thus, it takes 73.7 feet per car. At 20 mph, that comes to 2.5 seconds per car, or .4 cars per second.

Increase the speed to 60 mph, however, it takes 191 feet per car, but the time per car is reduced to 2.17 seconds per car, or .46 cars per second, which is a 15% improvement for the three-fold increase in speed.

Naturally, this involves diminishing returns, as well as increased safety issues.

The kicker, though, is what happens when cars slow to a crawl, say, an average of about 5 mph.

Now we're down to just .25 cars per second, which is a 46% reduction in the throughput of the roadway. Add stop and go traffic, and it makes it nearly twice as bad, often down to less than 10% of at-speed capacity.

Thus, the best way to avoid traffic jams isn't to necessarily build higher capacity roadways, but to avoid overloading those roadways, which is why many municipalities have installed limited-entry lighting to major traffic arteries throughout their cities. This may keep a few people from getting on the roadway for a few seconds, but a 10% reduction here can avoid a 50% or greater reduction in total throughput caused by an overload condition and the ripple effect that can have for miles behind where it initially occurs.

Amazingly enough, fluid dynamics has a similar problem wherein if the pressure in some systems is increased past a certain point, turbulence can build rapidly to the point where, despite the higher pressure, less fluid is travelling through the conduit than before. Thus, it's as important not to overload storm drains as it is not to overload highways.

17. Almost everything I wanted to say has been said by someone or another, but... this is both neat and pertinent: http://www.amasci.com/amateur/traffic/traffic1.html

As an aside, we're taught here to leave 2 seconds in city driving, and 4 seconds on the highway. Not that anyone does.

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Originally Posted by snarkophilus
Almost everything I wanted to say has been said by someone or another, but... this is both neat and pertinent: http://www.amasci.com/amateur/traffic/traffic1.html

As an aside, we're taught here to leave 2 seconds in city driving, and 4 seconds on the highway. Not that anyone does.
Excellent link, snarkophilus! I especially enjoyed the section which discussed how cruise control can help eliminate waves, which can lead to choke points and traffic jams.

19. Originally Posted by snarkophilus
Almost everything I wanted to say has been said by someone or another, but... this is both neat and pertinent: http://www.amasci.com/amateur/traffic/traffic1.html
Wow; that's still around. I've always had a hard time with his merge concept.

While it's true he's getting a higher throughput with the high speed merge, the high speed merge can only be effective if the overall traffic in both lanes is equal to, or lower than the available throughput of the single lane.

When this volume is exceeded, traffic will stop no matter how you merge.

Other than that, the site has some good points.

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