How to kill your waiting time?

On my way to school, I always drive past a crossroad. In my waiting time for the traffic lights to turn green for me, I observe the pattern of the traffic lights.The pattern is not hard to discern (see the figure below):

Scheme 1: In the day time, first, the left-turn lights for both directions turn green, so that cars in the left-turn lane (B1 and B2) can go. Then, the left-turn lights turn red, and the straight-through lights for both directions turn green, so that cars in the straight-through lane (A1 and A2) can go. The “green time” for the left-turn lights is shorter than that for the straight-through lights.

Scheme 2: In the night time, first, the lights for the east-bound lanes (A1 and B1) turn green together. Then, after they turn red, the lights for the west-bound lanes (A2 and B2) turn green together. The “green time” for lights facing both directions are equal.

The simplicity of traffic light pattern is annoying in the sense that I soon get bored in my waiting time again. So I start to think about the intuition behind such a traffic light scheme. Fortunately, I do not know anything about traffic planning, nor do I want to check their rules. Thus I would like to save this chance for myself to kill my waiting time.

At first thought, this seems to me a simple optimization problem, with the strategy depending, of course, on the target to optimize. However, today, when I stopped again at the same crossroad, I come up with a very simple but intriguing intuition to explain the traffic light scheme.

The trick of my intuition rests on how unequally cars on the straight-through lanes and cars on the left-turn lanes are distributed. It is my observation that during day time, there are more cars on the straight-through lanes (denoted by N(A1) and N(A2)) than on the left-turn lanes (denoted by N(B1) and N(B2)), while during night time, the cars on both lanes are both very few, so the difference between their numbers becomes ignorable.

Next, I shall add a well-known restriction (well-known in the sense that I suppose every driver knows it) that for the sake of safety, the left-turn cars from one direction and the straight-through cars from the opposite direction should never go at the same time.

Now let us suppose that the target is to minimize the “time wasted” in lanes. Suppose we adopt Scheme 2, in which the lights for lanes A1 and B1 turn green at the same time, and so do the lights for lanes A2 and B2. Then the “time wasted” for the east-bound cars is: TW1=t*[N(A1)-N(B1)]; for the west-bound cars is: TW2=t*[N(A2)-N(B2)], where t is the time needed for a typical car to pass the crossroad. So the total “time wasted”: TW=TW1+TW2. The “time wasted” is minimized if N(A1)=N(B1) and N(A2)=N(B2), which is usually the case for night time, but not day time.

However, if we, instead, adopt Scheme 1, in which the left-turn lanes and straight-through lanes are controlled by different traffic lights, it is thus possible to specifically tailor the length of “green time” for each light according to the average number of cars on A1/A2 and B1/B2 respectively. Ideally, if cars follow a stable and predictable flow, the “time wasted” will disappear. Moreover, the straight-through cars from both directions can go at the same time.

In sum, it is now possible to rationalize the differential traffic light scheme for day time and night time: If we target at minimizing “time wasted”, Scheme 1 is always better than Scheme 2 – that’s why we have Scheme 1 during day time! The only exception is when the number of cars in the left-turn lane and straight-though lane are equal, which implies that the two schemes work equally well. So we don’t mind having Scheme 2 during night time. Otherwise, the unequal distribution of cars during day time makes Scheme 1 look significantly better!

Yet the question remains that given Scheme 2 never beats Scheme 1, why don’t we stick to Scheme 1 all the time? I leave this question open for now, but I know R. H. Coase would say that keeping the left-turn lights and straight-through lights which face the same direction in the same color with each other (which is essentially Scheme 2) must have saved some “transaction cost”.

I could be totally wrong, since all the above are based on my waiting-time random thoughts and observations. Besides, I am setting the target of optimization naively, without much consideration for other possible targets, such as the average waiting time for a typical car. Despite all these complications, I think this is a good way to kill your waiting time in the urban traffic. I would very much appreciate any alternative explanations from the readers, who are few in number -preferably without you checking any relevant reference.

OK, I am a geek.

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