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On my daily commute to work, I find a sign on all escalators, stating

Stand right, walk left

The idea behind this, obviously, is that you should leave space for people in a hurry, so they can run up the escalator and catch the train, while the majority of people just waits until they're transported to the top.

I have the gut feeling that this rule is not valid in all cases, ie. depending on the amount of people who want to use said escalator. I think that when there are a lot of people, even those in a hurry would be better off if this rule would be (temporarily) ignored by everybody, since the waiting time before the escalator can be reduced.

Now, the question is this: Given a few assumptions, where is the tipping point to use the full escalator width (which, by the way would have space for two rows of people)?

According to my observations, I think we could assume the following things:

  • The ratio of people in a hurry is around 5 percent
  • There is space for two people next to each other. So we have either two rows of people standing, or one row (half capacity), leaving space for the "runners"
  • People who are in a hurry roughly double their speed up the escalator
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This seems a strange homework question. –  Alex Becker Jan 11 '12 at 7:17
    
It's not really homework, but it seems to me that it looks and feels like something you might encounter as homework. And I had to tag it something :) –  Dave Vogt Jan 11 '12 at 7:23
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I retagged it as "soft-question", which seems most fitting to me due to its current non-rigorous formulation. –  Alex Becker Jan 11 '12 at 7:28
    
When the number of people in the queue at the foot of the staircase is greater than 11 percent more than the number of stairs. –  p.s. Jan 24 '12 at 3:47

2 Answers 2

The tipping point is whenever the number of standers exceeds the capacity of the right lane in the absence of any runners. That is, some standers will prefer to stand on the left, if there are no runners behind them, presumably because the right lane is full. At this point, when a runner appears, some standers will cooperate out of politeness and "squeeze" to make more room on the left, allowing runners to pass. As the flow (and density) increases, eventually a stander will find it increasingly uncomfortable to squeeze and will simply continue standing on the left and not allow a runner to pass; the left lane will now be blocked and all runners will be forced to stand behind other standing would-be runners, except for the front-runner who is blocked behind an obstinate or helpless stander. Perhaps a fight will break out between a runner and a blocking stander, but two blocked runners are likely to accept their mutual situation and continue standing without conflict. So keeping the density and flow rate constant, on the basis of the preceding I conjecture that the rate of conflict is proportional to the number of blocking standers, and not the number of runners, given that the passengers may have various opinions; to increase the number of runners, and therefore the flow rate, some standers need to get out of the way somehow, so we can consider the possibility that some standers are willing to become runners temporarily...

But you seem to be asking something else: when it should be permitted to stand on the left, or else run in the middle occupying both lanes? ("... where is the tipping point to use the full escalator width?") The former analysis at least shows that it depends on your tolerance for conflict, the density of the escalator, your preference between right and left, and the presence of people immediately behind trying to run past you.

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Why not stand on the left, and walk on the right?; like the slow cars are on the left, and the fast cars are on the right.

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There're very few countries in the world (Japan, UK,...) where slow cars are on the left. In most countries slow cars are closer to the right. The choice of the side in the case of escalators is purely conventional. The choice of the side on the roads might has some historical and medical reasons. –  TZakrevskiy Mar 27 at 10:55

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