Green lights and left turns

While we’re on driving-related comments, there are (at least) two interesting things that differ between Montréal and New York:

1. The signal for a protected left turn in New York is a green arrow pointing left. In Montréal, the regular green light blinks. If you don’t know what the blinking green means, you’ll find out when the people behind you blow their horns because you’re not turning.
2. In New York, they tell you what you mayn’t do (no left turn, for instance), and anything not forbidden is permitted. In Montréal, they tell you what you may do, and anything not permitted is forbidden. If the sign has green arrows pointing straight and to the right, it means that left turns are not allowed there. (If there’s no sign, everything is OK, as in New York.)

Number 1 brings me to a recent conversation (in New York). Usually, at intersections that have green arrows, the arrow comes before the normal green light... but sometimes, it’s after. A friend pulled into a left-turn lane at a red light, and waited. The light turned (normal) green, and then later gave the green arrow, and my friend, annoyed with having had to wait longer, said, “I don’t know why they don’t have the green arrow first.”

As usual, the engineering analysis kicked in, and my first thought was that it makes no difference with regard to throughput. We arrived when the light was red, so we had to wait through the remainder of the red and the normal green before getting the arrow. But if the arrow came first, we could just as well have arrived right after it turned to normal green, and had to wait for the rest of the green and the red. Assuming a uniform distribution of arrival times, the average time you’d have to wait to get a green arrow is the same, regardless of whether the arrow comes at the start of the green cycle, or at the end.

But if the light is controlled by a sensor and there’s no red arrow (left turns are permitted during the normal green part of the cycle), it actually improves overall throughput to have the green arrow at the end. Here’s why:

With the leading arrow, left-turning cars arriving when the light is red trigger the sensor and schedule a green arrow, allowing them to turn right away. Cars arriving after that get the normal green, and have the opportunity to turn if they can. If not, they wait for the next green arrow. The green arrow, making the oncoming traffic stop, is scheduled whenever a left-turning car arrives while the light is red — or while the light is green and the car isn’t able to make its turn because of traffic.

With the trailing arrow, left-turning cars arriving when the light is red do not trigger the sensor unless they’re unable to turn during the green cycle. The only time the green arrow is scheduled is when cars remain in the left turn lane at the end of the green. That means that if the intersection is not saturated — if it’s sometimes possible for cars to make their turns without the arrow — using the trailing arrow results in less need for the green arrow, and, so, less overall delay for the traffic that isn’t turning.

Of course, if the intersection is so busy that one can effectively only turn with the arrow (or if turns are only permitted with the arrow), then we go back to the original situation: it doesn’t matter either way, leading or trailing.

That probably means that we should switch to trailing green arrows always, to improve the throughput when it matters, and to get us used to that way as a rule. But this is all napkin-scrawling; I haven’t done any proper modeling to verify that I’m right.