Traffic Light Pattern Efficiency

Factual Questions
1

I’ve noticed a predominance of one type of traffic light pattern that seems to be horribly inefficient when it comes to managing intersections. I’ve seen one that seems (at least on paper) to be better, but I cannot for the life of me figure out why the dominant one is even used, with a few exceptions where most traffic is going to be turned, or significantly more traffic goes in one direction, etc.

Assume traffic is constant (though both roads don’t have to have equally busy) to remove the degenerate cases from the analysis. Right turns will not be considered since they can be made on Red (in most cases) and will always at least coicide with going Straight.

Also, there are times where the part of or the entire intersection must be clear for a few seconds to allow for people who may be just clearing the intersection, heavy loaded trucks, and people running yellow and/or red lights. Let us assume that this is determined by speed and traffic density such that, at least on the same road this is constant for each direction for an ALL STOP or partial stop and since the two roads are independant in this regard, we can treat this time as constant.

Traffic Pattern A:

Northbound/Southbound: Left turns
ALL STOP
Northbound/Southbound: Straight
ALL STOP
Westbound/Eastbound: Left turns
ALL STOP
Westbound/Eastbound: Straight
ALL STOP
Repeat

Degenerate case: No opposing traffic turning left, so your direction gets to go straight AND left during the Left turn phase. Your direction does not stop during the ALL STOP phase and continues going Straight in the Striaght phase.

Note that there are for four complete stops for ALL directions. And that each time a phase starts each direction must start from a full stop (if any traffic has stopped and is waiting at the light).

Traffic Pattern B:

Northbound: Left turns and Straight
Northbound Left STOP
Northbound/Southbound: Straight
Northbound Straight STOP
Southbound: Left turns and Straight
ALL STOP
Westbound: Left turns and Straight
Westbound Left STOP
Westbound/Eastbound: Straight
Wesbound Straight STOP
Eastbound: Left turns and Straight
ALL STOP
Repeat

Degenerate case: No opposing traffic turning left, so the straight phase starts sooner or lasts longer (depending on your direction) OR no opposing traffing going straight, so Left Turns and Straight phase for your direction is made longer or started sooner (depending on your direction).

Note that there is an ALL STOP only twice. There are four other partial stops, but these are not as bad as a full because traffic in at least one direction gets to continue flowing.

Analysis

Now, we’ll assume that they’ve done a study on this particular intersection and found the optimum time that each type of traffic necessitates. Let each of these times be symbolized by a function F(B, D) such that B is the bound and D is the direction (ie, Northbound left = F(N, L), Southbound Straight = F(S, St)

TA(N, S, W, E) = MAX(F(N, L); F(S, L)) + STOP + MAX(F(N, St); F(S, St)) + STOP + MAX(F(W, L); F(E, L)) + STOP + MAX(F(W, St); F(E, S)) + STOP

TB(N, S, W, E) = MAX([F(N, L) + STOP + F(S, St)]; [F(S, L) + STOP + F(N, B)]) + STOP + MAX[(F(W, L) + STOP + F(E, St)]; [F(E, L) + STOP + F(W, St)]) + STOP

Now, if we assume that traffic is approximately the same in each opposing direction (ie, F(N, L) = F(S, L), F(W, St) = F(E, St), etc.), then these two end up being about equal:

TA(N=S, W=E) = TB(N=S, W=E) = F(N, L) + STOP + F(N, St) + STOP + F(W, L) + STOP + F(W, St) + STOP

That much is probably obvious; however, what if they’re not approximately equal. Lets assume its evening rushhour such that when all F(N, X) > F(S, X) and F(W, X) > F(E, X) (obviously, in morning rushhour, these would be reversed). Then we end up with this equation for Pattern A.

TA(N>S; W>E) = F(N, L) + STOP + F(N, St) + STOP + F(W, L) + STOP + F(W, St) + STOP

This gives us an interesting comparison because this breaks into two sums both of which are greater than their potential counterparts in the first pattern:

(F(N, L) + STOP + F(N, St) + STOP) > (MAX([F(N, L) + STOP + F(S, St)]; [F(S, L) + STOP + F(N, B)]) + STOP) because F(N, St) > F(S, St) and F(N, L) > F(S,L)
(F(W, L) + STOP + F(W, St) + STOP) < (MAX[(F(W, L) + STOP + F(E, St)]; [F(E, L) + STOP + F(W, St)]) + STOP) because F(W, St) > F(E, St) and F(W, L) > F(E, L)

Therefore, if **N=S ** and W=E, then TA(N, S, W, E) = TB(N, S, W, E); however, if N!=S and W!=E, then TA(N, S, W, E) > TB(N, S, W, E). So we can say that even though O(TA) = O(TB), the average, and more likely case, is that TA > TB; or in other words, TB is not necessary less than TA, but very well may be, but TA is NEVER less than TB.

Of course, the obvious conclusion is, if TA > TB, then we can get more cycles in less time with Pattern B than with Pattern A, which would increase traffic flow. So this leads to my questions:

Is my analysis correct? I mostly didn’t feel like doing the degenerate cases, as I feel like they’ll most work out to have about the same amount of benefit to either. Even if they’re not, would they be enough to through the time in favor of Pattern A? Or, am I completely wrong?

Assuming my analysis is correct, I can’t imagine I’m the first to have done this. Why is Pattern A used at all? Are there any special cases where this one is preferable? Is it easier to program the lights to operate this way or somehow much cheaper? I have lots of anecdotal evidence from myself and others that Pattern B is faster, does anyone have any to the contrary?

Maybe if I can get a few people to help me clean up and verify the math, I can put together a letter and/or petition to my local DOT to get these kinds of changes put in. Does anyone have a better idea of who I should send this kind of simple request to? I think this could save the state tons of money over some of these silly overpasses they’re putting in that may not be necessary if they were using this alternate traffic pattern.

Many thanks to any of you brilliant mathematicians/engineers, etc. that are willing to help me out.

OT: Wasn’t sure which forum was correct. I would appreciate some helpful advice if this isn’t correct.

2

This is easier said than done. Our local news did a spot on how bad the traffic was down a particular major street due to traffic lights out of sync. They showed how if synced properly (as with an example from Phoenix) you could drive the 3 miles through a dozen or so lights without stopping. They asked the local DOT why ours weren’t in sync. They said they originally were many years ago but got out of sync during power outages, etc. They’re excuse as to why they haven’t been re-synced in the last 10-15 years, “We don’t have the time or money to do that.”
So I wouldn’t hold my breath waiting for them to act on a good idea.

3

Not sure where you are, but neither of these patterns seems to be common around here.

The typical sequencing on sensor-less intersections is
North and South Left
Lefts stop
North and South Straight
North and South Stop
East and West Left
Lefts stop
East and West Straight
East and West Stop
The left after green in your pattern B is really uncommon. I can think of only one intersection near my home that’s done that way, and it really confuses people and is less efficient overall for moving left-turns as invariably, the car at the head of the line doesn’t know it’s a left after green, so they’re daydreaming and everyone behind them starts honking.

Full stops in all directions to clear an instersection are also very uncommon. Once a direction goes yellow to red, the cross direction immediately gets green.

Where I live, most signals are on sensors, so the traffic itself determines the sequencing. There aren’t any left turn signals if there aren’t any cars queued up to turn left, for example.

4

No. Traffic does not arrive at a constant rate, it arrives in a Poisson distribution. This is important because assuming uniform traffic arrivals is not close to reality, where traffic arrives in bunches and at different times.
One possible explanation:

Suppose N-S traffic is done in style A, then we begin w/ through traffic and unprotected lefts. Next, the through traffic is stopped and the left turn movements get green arrows. When the through traffic gets the green, certain time is wasted for the lines of cars to expand (we sit closer together when we’re not moving). With the A pattern, this waste of time is doubled up; north & south through movements waste this time together rather than consecutively. With pattern B, the wasted time for beginning movement is not doubled up on; instead the through & the left share this at the same time in each direction. But since there’s generally more cars going straight, that means the wasted time is longer in the B pattern than in the A pattern.

So, wasted through = 2 seconds, let’s say, and wasted left = 1 second. With pattern A, for four directions, we have two instances of wasted through time and two of wasted left time: 2 seconds + 2 seconds + 1 second +1 second = 6 seconds wasted for cars to go from standing still to moving. With Pattern B, the wasted time will be 2 seconds + 2 seconds + 2 seconds + 2 seconds = 8 seconds wasted. So all else equal, pattern B has a lower clearing rate than pattern A.

Now think, if it takes ½ second for each car to start moving after the one before it did, and there’s ten cars in the through lane but three in the left-turn lane, then that’s 5 seconds wasted for through and 1½ seconds wasted for the left turn lane. That’s an extra 7 seconds of total intersection time wasting for pattern B.

Ultimately, call your county road commission and ask. You can also look for traffic engineering info on the Internet or in the library.

Sorry I didn’t have more help to offer.

5

I truly believe most highway departments haven’t given this as much thought as you have. Come to Massachusetts some time. We have MIT PhDs studying traffic theory in Cambridge and yet the light “patterns” if they exist at all are bewildering (left turn lanes with no left turn signals but impossible to know light delays on the other side anyone).

I have given this thought as well and I believe road efficiency could be greatly stretched with “smart lights” that use sensors and AI to direct traffic expertly at all times based on current conditions.

I have done some reading on this. Deliberate light timing studies are sporadic and may be at the whim of an obsolete signal somewhere down the line.

6

Doubtful, at least for larger units of government.

FYI, being in traffic is no more a guide to understanding traffic than turning a wrench on an assembly line is a guide to understanding factory organization. There will always be pathological cases; however, my experience in traffic (I did traffic studies for a small city for three years) suggests that there probably isn’t a lot to be done above what’s out there. Take that with a big grain of salt; maybe I’ve been lucky.

Possibly, but that might eliminate the ability to time lights along a stretch of roadway. Each light would have to be able to accomodate changing conditions at all the other lights in the system. That may be possible, but I doubt it will be practical in the near future, just in terms of solving the system for optimum movement. Every time some kid hits the pedestrian crossing button at a light, the whole system may need to be recalibrated.

It’s also the case that timing systems simply aren’t robust.

7

In Ottawa, magnetic sensors are buried at some intersections to detect cars. They’re mostly used to detect when cars are waiting in the left turn lane(to decide whether to give an advance green or not) and to detect cars on a low traffic street waiting on an intersection with a high traffic street.

8

Maybe I’m misunderstanding, but this is the scenario I was describing in Pattern A (unless I’m missing a subtlety somewhere).

They may be more uncommon in other areas. Around here, at least, there’s a full 2 or 3 seconds (sometimes longer) after all North/South stops before East/West gets to go. They seem to be longer on streets that have more traffic and/or higher speed limits. I think its to allow for the people who decide they can squeeze the yellow light or just want to run a red light; there’s also cases where a fully loaded truck, especially those carrying liquids, will not be able to finish going through the intersection before the light turns green in the other direction even if they were going the speed limit and started breaking soon after the yellow light. Of course, its a slippery slope as the longer the break gets, the “safer” it is to run the yellow, or the very first split second of the red light

9

Slight hijack, but what’s the big deal with this? I can think of plenty of junctions near me in small-town England with just this setup (replacing left with right), and nobody has a problem with it. You wait, and turn when it’s clear.

10

I’m a civil engineer but I have relatively little experience in traffic. I can tell you that there are thousands of grad students out there and DOT funded projects that study these exact questions. You’re reinventing the wheel.

I would also seriously doubt you could shave enough time off a signal to alleviate the need for an overpass. DOTs put overpasses in where there is either current significant traffic or the projection of future traffic based on census data and building plans.

I can point out one problem with your analysis: you ignored right turns. Right turn traffic can be significant and can clog interesections. If you use pattern A you can have the west/eastbound lanes making right turns while the north/southbound lanes make left turns.

If you’re interested, here’s a pdf of a short intro tutorial on factors to consider for traffic signal timing from the Texas Transportation Institute at Texas A&M. There are a lot of factors (traffic density, lane width, etc.) that go into traffic signal timing.

11

I realize this, but this isn’t the problem I’m trying to solve; I think I was a little bit unclear in how I defined it. This was made clear as I discussed with co-workers.

I see two different problems with traffic lights that need to be solved, and I think they’re mostly (if not entirely) independant. The first is minimizing the number of lights that I hit over a certain stretch of highway. This is solved (easily or not) by syncing the lights in such a way that on that stretch I stop at the fewest number of lights. I assume this is probably solved by taking into consideration this distribution, along with the speed of the traffic and how far away the lights are as in: Light one turns green, at the speed limit it takes 10 seconds to get to Light two, so Light two turns green 10 seconds later, etc. I don’t think how the cycles are handled matters much here, only that the part in the direction you’re synchronizing are appropriately aligned. I’m sure its more complicated than that, but I’m not particularly interested in that problem.

I assumed my motivation was clear when I wrote the OP (though, on rereading, it wasn’t… my fault). I’m more interested in a situation where I have to wait at a light through more than one cycle (often two or three) to get through. Thus, I can ignore the distribution, as there are almost certainly vehicles waiting to go in every direction (one of the other reasons I felt I could ignore the degenerate cases). I figure if traffic is light enough that distribution becomes an issue, that is something that is actually handled by the synchronizing of multiple lights in a string, versus the maximization of the throughput of a single light. Of course, if lights were better synchronized, it would help minimize the number of cars waiting which might help alleviate this one as well.

For simplicity’s sake, I’m assuming that sometime between when the light turned red, and the next time that light turns green, at least one vehicle had arrived, otherwise its one of the degenerate cases. I’d be happy to analyze those, but I want to make sure the other cases are correct first.

I think I’m missing something in your argument here, but I don’t understand the difference in terms of doubling up, except that they are happening at the same time instead of asynchronous. I don’t see how this “wasted through” time gets penalized on and somehow gets doubled. With Pattern B, I only and prevented from going straight when either the other side is going left, just like with Pattern A. With pattern A, here’s how it goes:

Darnit, I missed the light… wait for West/East traffic… wait for left turns… Yay, let’s get moving.

For Pattern B, that’s how exactly how Southbound goes, but its the same amount of waiting, just in a different order for Northbound:

Darnit, I missed the light… wait for left turns… wait for West/East traffic… Yay, let’s get moving.

Its similar for the left turns for Pattern A for left turns:

Darnit, I missed the light… wait for straight traffic… wait for West/East traffic… Yay, let’s get moving.

And for pattern B its, again, the same amount of waiting, just in a different order:

Darnit, I missed the light… wait for West/East traffic… wait for straight traffic… Yay, let’s get moving.

The main point I’m trying to make is that, with pattern A, for each North/South or West/East combination you have to wait for the sum of two maximums rather than the maximum of two sums. Assuming they’re the same sets of values, the first will ALWAYS be greater than or equal to the latter… some examples:

**let F(N, St) = F(S, St) = 40
let F(N, L) = F(S, L) = 20
let STOP = 3
let F(W,X) = F(E,X) = 0 ** (since we’re ignoring those cases, as they’re independant)

**TA = MAX (20, 20) + 2 + MAX(40, 40) + 2 = 64

TB = MAX ([20 + 2 + 40], [40 + 2 + 20]) + 2 = 64**

These two sums are clearly exactly the same… now let’s try if the flow isn’t the same in both directions (South has 60% the traffic that North does, but still the same sum of times as above)

**let F(N, St) = 50
let F(S, St) = 30
let F(N, L) = 12
let F(S, L) = 8
let STOP = 5
let F(W,X) = F(E,X) = 0 ** (since we’re ignoring those cases, as they’re independant)
NOTE: These are the same total times as above, just skewed to favor Northbound because its rushour in that direction

**TA = MAX (12, 8) + 2 + MAX(50, 30) + 2 = 66

TB = MAX ([8 + 2 + 50], [30 + 2 + 12]) + 2 = 62**

For TA, IF you start the Northbound traffic 4 seconds earlier to account for the four seconds less that the opposite direction didn’t need for left turns, THEN that gives you 62. You can’t do any accounting for the second max because you HAVE to wait for both directions to stop to have the cross traffic go. That’s the advantage of the asynchronous waiting. But, in MANY of the cases I’ve seen, they just give both left turns the same time, even if one empties out a lot faster than the other

In fact, we can simplify even more, as the STOP can be canceled out as a constant through both, and ignore the cross traffic. To simplify the variables, I’ll use the West/East case (no conflict between two different Ss). Plus, we can adjust TA to account for the case where the left turns in one direction end before the others do as below.

TA = MAX (F(W, L); F(E, L)) + MAX (F(W, S); F(E, S)) - CD
If both Maximums are of the same bound and difference is sufficiently large (that one is subjective, of course), then
CD = MIN (|F(W, L) - F(E, L)|; |F(W, S) - F(E, S)|)
else
**CD = 0

TB = MAX ([F(W, L) + F(E, S)]; [F(W, S) + F(E, L)])**

ONLY through that compensation difference does it make them equal. While, that CD is essentially implied in Pattern B. However, in my observations, unless there’s a large difference in the number of cars turning left from one bound versus the other, they do NOT make the time for the both turns any different. In my proposed pattern, they couldn’t make it any worse; if they were sloppy or lazy, it would be equally as bad as the current pattern at worst, but would likely be better.

12

I figured as much, but that’s why we have this forum, no?

Perhaps, but the math doesn’t seem to add up. That’s why I thought either I was doing it wrong, or I was missing some other factor that I didn’t consider.

This is not true. While Northbound is making both left turns and going straight, Eastbound can make right turns; while Southbound is making both left turns and going straight, Westbound can make right turns. The only difference is they’re not turning at the same time now.

Thanks much for the link. I’ll give it a good look. Hopefully, it will answer my questions.

13

Oops, I meant to keep the same sum of times, but I goofed, not that it changes the argument at all. I meant F(N, L) = 24 and F(S, L) = 16. such that:

**TA = MAX (24, 16) + 2 + MAX(50, 30) + 2 = 78

TB = MAX ([16 + 2 + 50], [30 + 2 + 24]) + 2 = 70**

and CD = 8.

14

If it makes you feel better, I’m pretty sure there are at least a few intersections in my hometown (Winnipeg, Manitoba) that work on the Pattern B as you’re describing it.

15

I dropped out of my civil engineering course at uni’, the rest of the course bored me before I got to grips with the subject matter concerning the equations of traffic flow.

16

Your Case B is used quite frequently in my neck of the woods.

Something new that’s starting to be deployed with greater frequency hereabouts is the “roundabout,” which is not the same as your grandfather’s traffic circle. In the few occassions that I’ve used them successfully (not gotten lost), they’re supurb! I’ve never really had problems with old-fashioned traffics circles either, though.

Finally, on the streets that are big enough to contain boulevards but for which traffic circles have not yet been invented, there is the “Michigan left turn.”. Any time I’m at a busy place outside of the area, it irritates me that they didn’t do something so simple.

I can’t speak with the authority of a traffic engineer, but as a user, me knows what me likes.

17

Well, I’m no traffic engineer, but I’ve thought about traffic patterns a bit in regards to bicycle positioning. And without going into the math, what I think Blaster is ignoring is the great value of consistency in traffic flow and traffic signals --both for safety’s sake and for efficiency in drivers understanding when and where they’re supposed to go.
In pattern A, you know on any street, you get left turns first, then the straight ahead green. But in the second, sometimes lefts come before and sometimes after the green. Now perfectly functioning automatons wouldn’t have a problem with that, but drivers are not perfect.

I agree the Massachusetts ‘hidden’ left lights (where there’s no left turn arrow; but one direction gets a green while the other side has a red) are imbecelic and insanely dangerous.

18

Sorry, I was a little testy in that reply. I get tired of reading about some lone physicist who come up with a “brilliant” solution to traffic control who’s never even discussed the problem with any civil engineers.

Actually, your solution B is the commonly seen around here. But we do see A sometimes. Usually it’s at intersections where you have a higher volume of left turns than you do thru traffic.

I’m not saying all civil engineers are perfect. There are definitely budget and manpower limitations that lead to signals being neglected for years. Or only examined after traffic has reached a point to cause complaints. There are minimum standard for wait times defined by the Highway Control Manual. This table doesn’t cut and paste well but the letters are grades and the numbers at the end are delay times in seconds.

Overpasses are always a last resort for DOTs. Structure is waaaaaay more expensive than changing signals or adding a lane. If they put one in, it’s usually for projected traffic growth or because they’re trying to maintain a minimum speed to move traffic efficiently. Signals, no matter how they’re timed, will always slow traffic. (besides, I’m a bridge engineer, I need those overpasses. :slight_smile: )

If you’re interested in reading studies done on various intersections around the country, there are lots of PDFs available on the web. I pulled up a bunch looking for “level of service signal delay” while I was looking for the above table. If youalso look for “signal traffic analysis” there’s lot of info available out there.

19

Then I think it would just be better to assume nothing about traffic arrival, if you’re also assuming that the queue is inexhaustable (for lack of a better word). Though, it remains true that the distribution of arrivals may still be relevant because the basic problem with an intersection is getting the clearing rate to be at least as large as the arrival rate. This problem exists even on highways; ever end up in a traffic jam that exists where there are no accidents, lane closures, exits or entries, &c.?

You should, actually. It’s an instructive exercise. Get a giant sheet of graph paper. On the horizontal axis put distance, and put time on the vertical. Then space out your traffic lights and draw a vertical line that begins green, turns yellow, then red in proportion to the cycle timing. Repeat so that at each light you’ve got a vertical line indicating when the reds, greens, and yellows are in both time and space. Now draw a line that goes through only the greens; that’s the speed of travel that will fit the light cycles.

If you take a real stretch of road, you know where to space the lights. If you want, you can pick the speed limit that you prefer and then from there figure out how the lights need to be timed. Now, for the fun part, once you’ve done this in one direction—so that, say, north-bound traffic gets all green at speed X—see whether the south-bound traffic also experiences sychronized lights.

If it works both ways, figure out why and what would need to change to make it not work both ways.

It won’t answer your question, but it is informative to think about.

Think about moving cars rather than time; i.e., you want to optimize your clearance rates. Since there are (generally in my experience) more through cars than left-turning cars. If you give left turning cars a full cycle, i.e., equivalent to the time the through cars get, then you are giving more time to left-hand turns than they warrant.

Assume a road with a traffic light. The light will allow people to turn left from both directions. Pattern A has the through traffic in each direction moving at the same time; then the left-turn movements happen together. With fewer left-turning cars, the amount of cycle time given to left-turning movement will be proportionally less in pattern A than it would be in pattern B.

If the cycle length is 1, then under pattern B, north-bound traffic (let’s say) gets time of ½ and then south-bound traffic gets ½. Each ½ of the cycle moves the through and left-turn movements from one direction. So in total, the through movements get time of ½ to move cars.

Under pattern A, we could have the bi-directional through traffic get time of ½ and the left-turn traffic get time of ½. But if there are fewer left-turn movements, then less time needs to be allocated to left turning movemnt. Because of this, the through traffic can be allocated, say, ¾ of the cycle to the through movement and ¼ of the cycle to the left-turn movements. With this setup, the through traffic, which is generally more volumnous, gets a higher proportion of the cycle time. Then, for example, the amount of time given to left-turn movements could be shortened until the waiting time for the average movement is equalized.

Pattern B wastes time by allocating more than needed to left-turn movements.

There are times where pattern B may be preferred; e.g., if a light has insufficient left-turn stacking, then pattern B will allow one direction to clear more quickly, since the left-turn queue is no longer blocking the through flow.

20

Do any major cities have their traffic signals controlled remotely from a central location? Or do they all have to be programmed at the light itself?