Why is the queue next to me always faster when I’m queuing?

When you are waiting in line, do you have the feeling that the line next to you is moving faster?

One of the most famous examples of Murphy's Law is "the team next to you is always faster."

For example, when we shop in a supermarket and choose a line to check out, the cashier in our line will definitely not be as flexible as those in other lines; if we observe before queuing and avoid the slower line, then someone in our line will definitely complain about the price, which will slow down the entire line, etc. At this time, there is no other way except to rely on luck - when we are queuing, we can observe who is at the end of each line and see who will check out first.

If the same situation occurs on a highway, the answer is completely different, as Paul Krugman and Steve Struggle discussed in the New York Times many years ago. Let's see how to use mathematics to explain why the lane we are in is moving slowly, and this also applies to the other lane.

Is it impossible? No, it is completely possible in mathematics.

Suppose we are on a 4km long two-lane road, and there is no real slowdown queue here - the first half of the journey is driven at 10 km/h, and the second half is driven at 30 km/h. For the sake of calculation, we assume that no one is overtaking. (I know this assumption is ridiculous. After all, in Italy, car drifting is the second most popular sport after football, but let's just assume it.) What will happen then?

Obviously, the cars on both lanes will pass through this 4 km distance at the same time, and everyone knows that it is not 12 minutes, which can be regarded as driving at a speed of 20 km/h. At a speed of 10 km/h, it takes 6 minutes per km, and at a speed of 30 km/h, it takes 2 minutes per km, so it takes a total of 16 minutes.

Let's take a look at what happens. In these 16 minutes, after you run faster for 4 minutes, you will start to wonder in the remaining 12 minutes: Why are other people driving faster than me? I drew a picture, and you can see that the same situation will happen to the driver in the next lane.

But there is an even more surprising result. If the speed of the lane next to you is 5 kilometers per hour when it is crowded and 20 kilometers per hour at other times, it is obvious that they spend more time than you to pass this 4-kilometer distance. But according to the above analysis, you will still complain why others drive faster than you.

In mathematics, this behavior is known as Redelmeier's paradox.

Why doesn’t this paradox occur in the supermarket queue example? Is it because Murphy prefers the supermarket shopping cart?

Of course not. In the driving example, we assumed that the distance covered by the car was the same, so we could control the space. In the supermarket queue, this becomes insignificant because we are not interested in how long the distance is. If we change the controllable factor to time, speeding up for 20% of the total time and slowing down for the remaining 80%, then the paradox does not exist, and Murphy's Law starts to prevail again. We can figure out what happens when the lane changes suddenly and the speed of deceleration suddenly changes.

END

Text: [Italian] Maurizio Codogno

Figure: Illustrations from "Talking about Mathematics during Coffee Time", Internet pictures (deleted if infringed)