The Mathematics of Escalators on the London Underground (2013) [pdf] | Hacker News Reader

The Mathematics of Escalators on the London Underground (2013) [pdf]

raeng.org.uk

84 pointscarschno80 comments

On the underground you’re meant to stand on the right of escalators to leave room for those walking up on the left. However, TFL ran a trial a few years ago instructing people to stand on both sides during peak times. This actually increased overall throughout and per-person speed: throughput as you can more closely pack passengers standing rather than allowing a smaller number of walkers half the space (and those walkers are necessarily spaced out more than standers), and per-passenger speed because even though walking up the escalator would be faster, the increased throughout reduces the queue time at the escalator entrance, which has a greater effect.

To my knowledge, they didn’t move forwards with keeping the instruction permanent in peak times, presumably because it’s difficult to implement / ingrain in people.

EDIT: link https://www.shortlist.com/news/holborn-escalator-trial-tube-...

I've been wondering about that story as well. My own conclusion is that the experiment failed because the metric of average speed per passenger is wrong from a passenger's perspective. People who stand probably don't care about a hardly noticeable increase in speed, whereas people who walk are annoyed by the very noticeable decrease in speed.

I think the speed _for all passengers_ was increased, because of the reduction in queueing.

This was also only at Holborn, one of the typically most overcrowded stations on the network.

A lot of the problem was that it was relatively unpopular, even with the increase in throughput. There was some talk about having at least one walking side per set of escalators, but that's also harder from a messaging point-of-view.

One thing I’m conscious of with these crowded escalators is the exit point safety, as some people can be rather slow to move away. Making both sides standing would make me concerned about seeing accidental pileups from people not moving away quickly enough.

You get it sometimes already when the station itself is overcrowded- TfL normally starts limiting access, running congestion control etc. Fortunately most of busier places (Bank, Canary Wharf, etc) are full of regular workers who know the drill.

It’s more one off special events where causal travellers show up en masse.

Just want to comment that, that's essentially public transit within a public transit. Public transit cares for throughput, private transport (cars, left-lane of escalator) cares for latency. Queuing theory in a nutshell.

The way I understand it is that impeding someone trying to pass on an escalator in London (it doesn't have to be on the underground) is socially unacceptable. You can actually stand on whatever side you want, but you'd better be ready to get out the way if you decide to stand on the left.

Making everybody stand makes people uncomfortable because it's essentially forcing them to inconvenience other people.

This is the way I remember it in Tokyo: stand on the side, unless it's super busy. Then you jam-pack as many people as humanly possible.

> This actually increased overall throughout and per-person speed:

Increasing the average per person speed sounds like the expected result. Not necessarily a net win if the people sufficiently motivated to exit quickly to walk (who tend to bypass queues) are slowed though.

They still had the option of walking though. With two parallel escalators, 3 of the 4 "lanes" were for standing and one for walking, rather than reserving two lanes for walking when few people do so.

> On the underground you’re meant to stand on the right of escalators to leave room for those walking up on the left.

Don’t they drive on the left in the UK? The “slow” lane should be on that end.

We drive on the left, which means we walk on the right (to see traffic coming), so I guess it stems from the latter

In the US, we walk at traffic when there are cars on the road but the passing lane is the same (left) for driving roads and for pedestrian multi-lane "roads" like escalators.

In Australia, we drive on the left, but on escalators we pass on the right just as the parent comment envisioned.

Turns out it’s just legacy from the original escalator design.

“It all dates back to when the first successful escalator (there was a failed attempt to build a spiral escalator beforehand) was introduced on the tube in 1911 at Earl's Court Station. Alighting an escalator was different back then to nowadays. There was a diagonal partition — beneath which the stairs disappeared — that shunted passengers disembarking to the left. So it was decided that those walking up the escalator should stand on the left, otherwise they'd have to cut through a line of those standing. And that would've been mayhem.”

https://londonist.com/london/transport/why-don-t-we-stand-on...

The study shows this was better at peak times, especially for the goal of making sure there are no queues at the bottom of the escalators which can slow the tube down, but presumably in normal times (when there is space for standing and walking) this is worse for individuals who have a preference on walking or standing.

I get the tube to and from Holborn every day and there are always people (me included) walking and I rarely arrive at peak times.

In Moscow metro they sometimes ask (but do not enforce) via speakers to stand on both sides of the escalator. I heard this practice was brought by McKinsey from London. In my experience this sometimes works on very short escalators, and on the longer ones people seem to ignore the suggestions. I wonder about the psychological reasons of this behavior.

'Stand on the right' is relative to which way you are facing. You would still be correctly following the instruction if you are facing opposite to the direction of travel and standing in the way of 'walkers' (on your right). I find Londoners generally receive this sort of humour very poorly ;)

I took part in this trial, at Holborn Station. The escalators are very long, and few people walk. However, despite the signage, markings and announcements, when I stood on the left I still got the occasional tirade of abuse from someone attempting to walk up behind me! Most people continued to only stand on the right as a result.

Their justification for this change, as I read it back then, (the linked article doesn't get into the details) seemed to me to be completely flawed.

It is true that standers are more closely spaced together than walkers. The number quoted was a stander every 2 steps versus a walker every 3 steps. However standers spend roughly twice as much time on the escalator as walkers do. This means that walkers take up about 50% more space*time on the escalator, which is the constrained resource.

This error was exacerbated by station staff observing queue length as a goal to minimize. Walking on the left leads to a longer, and also faster-moving, queue for the left side of the escalator. People are happy to stand in this queue, because they are trying to optimize for the time they get off the escalator, not the time they get on. But station staff sees reducing this queue as a success. This is wrong - firstly they don't take into account the higher throughput for the left side. Secondly, making things less desirable definitely makes queues for them shorter, but this is not a win.

In short, standing-only can reduce total travel time for congested escalators where few people walk (obviously), but it makes things quite a lot worse for busy escalators which are at capacity for both walking and standing. (Even before you account for the fact that walkers on average care more about their speed than standers.)

The trial was at a station with 2 parallel escalators going in the same direction, giving 4 lanes of people. Instead of having 2 standing lanes and 2 walking lanes, they had 3 standing lanes and 1 walking lane. The goal was to reduce queuing for the standing lanes. The walking lanes are usually a lot less busy.

When I travelled through Holborn during this period, they had changed both of the up escalators in the main bank (ticket hall down to Central/Piccadilly fork) to 'standing only'.

I lived in London for a year or two and I rarely saw an escalator "at capacity" for walking. You can do all of the napkin math you want, but ultimately, as the article says:

> The one escalator at Holborn that allowed you to walk up the left side during the trial had a total of 115 passengers per minute, but the standing-only ones knocked that up to 151. This was only the case during peak times though – when you’ve got under 100 people on the escalators, it doesn’t really make much difference. Either way, introducing standing-only escalators meant that they only had to SHUT THE GODDAMN GATES AAAHHH once during the trial, whereas it was happening almost daily before.

I have seen lots of escalators at peak capacity for walking. Perhaps you didn't travel a lot in rush hour. People (for obvious reasons) are much more likely to walk up escalators in the morning rush hour than in the evening.

The escalators at Holborn are somewhat exceptional as was noted in the linked article, and the anecdata provided by station staff is not particularly convincing. If the station was extremely congested in the morning, it is hard to imagine that very few people chose to walk straight onto the left side of an escalator rather than queue at length to stand on the right.

'Shutting the goddamn gates' really only happens at stations like Holborn in the evening when people are trying to get out of the city centre. The up escalators at Holborn are conversely at their most busy in the morning, so this is a confusing claim at best.

As for the accusation of 'napkin math', I was simply responding to the clearly wrong napkin math which was widely cited in favor of this change. This purported to show that even at capacity, a walking left side was wasteful of escalator space. Without this flawed claim the main consideration simply becomes that if the left side is being used a lot for walkers it should be kept for walkers (obvious) and if not, not (also obvious).

It doesn't really work like that, people are lazy and even at peak times will queue at the right hand side to stand, even when the left hand side is completely empty

I got kind of mad at this decision for precisely the same reason as you.

I'd like them to put up some signage to encourage people to walk up (it's a low friction way to get some exercise) and try again. I reckon they could increase throughput and make a few people slightly healthier this way.

I've always thought of it as a classic latency vs throughput tradeoff. You sacrifice some individual latency (of the people on the left walking quickly) for improving the overall throughput of the system (and therefore worst case latency).

Why doesn't London Underground use either slowed or stopped escalators when there are no passengers?

I noticed in Berlin, escalators without people often run very slowly. They speed up as soon as someone steps on.

In Copenhagen, they stop. This is presumably more efficient, but it's easier to assume the escalator is broken, and a bit disorientating.

Where new escalators have been put in they do. Mostly this is in new stations or station entrances. Eg the new Cannon St Stn entrance.

I suspect there's no reasonable way to retrofit them to existing ones - probably requires a different type of electric motor and AC speed controller plus sensors which is getting to be a pretty big job (so expensive even if the parts aren't).

It takes TFL about 9 months to replace out an old escalator for a new one. I think I'm correct to say they have 50 year operational lives.

> It takes TFL about 9 months to replace out an old escalator for a new one. I think I'm correct to say they have 50 year operational lives.

Seriously? Why? Genuinely curious

It takes them a few days to replace the ones in local supermarkets, including the flat conveyor ones to carry the trolleys.

Probably because they're bespoke, subject to heavier use, located in confined spaces, and are expected to run 20 hours a day for 50 years.

The health and safety aspect is mind-numbingly overbearing. And for good reason. It is an underground workspace that is surrounded by the general public. When escalators are replaced the entire work area gets boxed in. There must be significant controls in terms of dust, gas (like welding equipment), noise etc. Fire is a significant risk and the underground has history that makes them anxious about fire and escalators (see Kings Cross Fire). Most of the work probably has to take place after hours, and subject to other maintenance taking place. On the underground, you can't just put a new escalator in a docker container and spin it up. The Underground is a significantly complex environment.

From a few cases I have seen, it is partly due to the very long service life. New units are substantially different from older ones, particularly in all the bits of it that you do not see as a pedestrian. Often times the retrofit involves a lot of concrete work and other work to adapt the mounting and support points to accommodate the newer designs. Couple this with the fact that you are usually doing the work in sub-optimal conditions because they are trying to keep the rest of the area open for pedestrians, so there is additional safety considerations that you don't have in a new construction install.

Reminds me of this interesting behind-the-scenes look at an escalator replacement:

https://news.ycombinator.com/item?id=28207386

I'm not sure any of them would turn off for longer than 2-3 minutes at a time. The minimum frequency would be about 12 tph (5 mins interval) but that's in each direction. You only need one person alighting from a train to cause the elevator to turn on. The less used suburban stations don't have escalators at all.

Some of the escalators on the Underground are terrifyingly deep. They could do variable speed for some of the shorter ones, but implementing that for the long escalators would cause more than a few accidents.

Yeah, I'm not sure non-Londoners appreciate this. The escalators at Angel (where I grew up) go 60m deep. Someone skiied down them and hit 30mph: https://en.wikipedia.org/wiki/Angel_tube_station#Escalator_s... https://youtu.be/rlF4nRUbKmc?t=53.

This is something I noticed experiencing undergrounds in Germany after previously only experiencing them in London - London's escalators are insane, the Cologne U-Bahn is practically surface level by comparison.

If I were to hazard a guess this is because of the soft clay that London is built on - on the one hand the soft material makes it very easy to dig tunnels, on the other hand if those tunnels are dug too shallow they will very easily destabilise the foundations of the buildings above. So you have to to build them at a depth which would be prohibitively expensive somewhere built on harder rock.

Some tube lines are very deep but there are also several which are cut-and-cover, ie at the same just-below-street depth as the NY subway.

Not only that, as Crossrail demonstrated one must also avoid the numerous existing tunnels (not just trains, but also utilities and even rivers). Not all of which are precisely mapped or even know about! Even then, the gap between tunnels in some stations is on the order of centimetres

I think part of the reason is just that they aren’t frequently replaced (I think it is often a very annoying place to install new escalators).

Personally I prefer to manually climb the stairs and I’m often surprised by how much space there is between escalators (it often seems like 50% of the width of a tunnel is just metal closing up the space between the parts of the escalators you can stand on) but I think my opinion is rare and it would be inefficient to have a flight of stairs that nearly nobody uses.

There are some outdated assumptions about the carbon intensity of electricity being made here. They seem to be assuming coal-fired power emitting 0.9kg CO2/kWh.

In reality, the actual average carbon intensity of the UK grid in 2021 was only 187g CO2/kWh: nearly 5 times less! So escalators, in the UK at least, aren't nearly as bad for the climate as this article suggests.

TBH, I was astounded at just how much CO2 was being emitted according to TFA. So 5x less is way better -- I know the UK is doing a pretty good job on renewables, esp. wind [0].

[0] https://www.theguardian.com/environment/2021/jan/28/uk-elect...

Why do those mathematical arguments about CO2 emitted never include how much CO2 would be emitted by a person (and transitively emitted, too - i.e. how much CO2 would be required to produce the required food for the person) if they did the job themselves?

I.e. somehow walking 100 steps suddenly produces 0 CO2. Which is completely not true, at least in this case the person would be breathing, let alone spending calories walking up the stairs.

It's probably because human fuel is in the most part obtained by solar power sequestrating carbon from the atmosphere.

Does that mean that if I eat meat it may be more beneficial to travel by car because a vegetarian is 1000 times more efficient than a car vs a non-vegetarian who is only 200 times more efficient than a car? (numbers are made-up)

If you eat meat, it may be more beneficial to become a vegetarian.

Because humans, trees, living things in general are carbon neutral over a pretty short duration.

The overwhelming concern is carbon that was newly put into the atmosphere from fossil fuel sources comprising millions of years' worth being released all at once (less than 200 years). Carbon that was recently removed from the atmosphere by plants which were subsequently consumed and reintroduced into the atmosphere is a very minor concern.

It's not that your perspective is wrong, just that it is dwarfed by many orders of magnitude, in respect to fossil fuels.

If instead of flying somewhere I traveled there by foot, I would surely use a lot of energy as well. Not just food and breathing, but also cooking, heating of the BnB I am staying in.

Surely you could say that this would be much less than a typing continental flight on a jet (you are probably right) and at that point the human CO2 can be considered a 0. But what exactly is that 0?

If you assume human produces exactly 0 CO2 all relative comparisons become basically infinities. It's infinitely better to walk than to cycle, it's infinitely better to walk than to fly, etc. But it's not really true. It may be several orders of magnitude better, even - but how many exactly?

I literally never saw this in the research in regards to CO2 produced.

>> It may be several orders of magnitude better, even - but how many exactly?

It's not an easy question to answer. I suspect it would be fun to try. Perhaps more for you than for me. But, aside from the pleasure of working this problem, there remains the other problem. And an answer to your question is almost entirely immaterial to that one.

I think the difference between our approaches is that you are focused on relative amounts - how exactly to account? My concern is with absolutes: a metric fuck-tonne is coming from (effectively) nowhere and going into the atmosphere in almost zero time.

It'll surely be more efficient to account for the future emission at the point of fossil extraction, with special consideration for reliable CCS.

For all of the days that you are alive you need to eat, breath, cook and heat the room that you are living in. The continental flight just allows you to eat, breath, cook, and heat in different city. You cannot account for the carbon cost of the time that you spent travelling, as flying-self will still have a carbon cost sitting around a pool waiting for walking-self to arrive.

Yeah, but if I fly I spend considerably less time on the journey.

See my comment here: https://news.ycombinator.com/item?id=31930716

For every unit of food, there is a unit of oil to produce food. I'm not sure what the ratio would be, but it surely must be there.

What you say is right. And I realise that heavenlyblue is saying much the same. It seems to be about where to account for the emissions.

I suppose walking in a foreign city is not accounted because agricultural production, transport, hospitality etc. already is accounted elsewhere.

There's surely a need to join up all the accounting up though.

I meant "walking TO a foreign city".

For example, imagine that I am taking an intercontinental flight in 8 hours:

- I spend 8 hours flying from London to Mumbai. The majority of my CO2 emissions come from the plane jet engines.

- I spend a month walking between London and Mumbai. The majority of my CO2 emissions come from the food I consume.

Note if I DID fly I would not have spent that 1 month eating, BNBing, using Air Con for a month. Which means that is the CO2 cost of my month-length walk.

Honestly, I started working this out because I thought it would be negligible. But I think you're right to doubt. Leaving aside the question of how much a human produces, since two people have suggested human output is neutral and I don't know enough to question it:

It takes an average of 0.10kcal to walk up/down a step, averaged.[1]

2.2kg of CO2 are emitted per 2000kcal of consumption (I just averaged table 3 for want of a better idea)[2]

37 steps in a staircase (TFA, 46 total - 9 flat)

3.7kcal burned, 3.7kcal * 1.1g/kcal ~= 4g CO2 per person per trip

Obviously very rough, but unless I've made an order-of-magnitude error it's in the same ballpark.

[1] https://pubmed.ncbi.nlm.nih.gov/9309638/

[2] https://nutritionj.biomedcentral.com/articles/10.1186/s12937...

Could the escalators going down generate the power for the escalators that are going up?

All escalators go up AND down - it's the direction of the side that is on the top that you notice as a user.

Also, they aren't powered by gravity but by motors.

If they were friction less, they would be in balance without passengers and would start moving downwards with passengers stepping onto them at the top. And could theoretically move a lighter group of passengers on another escalator upwards. In practice it seems very unlikely that one could make this work with reasonable effort. It already starts with the fact that you are probably not going to have well balanced streams of passengers in both directions most of the time.

Over the course of a day across the system the number of passenger-kg going up will be roughly about the same as the number going down. More going up in the morning, more going down in the evening.

But the escalators carrying the people "down" have the help of the gravity pulling all those people down and the motor works mostly as a brake...if the motors fails, it speeds up: https://www.reddit.com/r/WTF/comments/uegc45/escalator_break...

So, if the motor worked as a generator too, it would only use power when turning the empty escalator (because of friction), but when enough people are standing on it, it could actually generate power.

That's irrelevant.

The sign of the power needed/received for moving passengers up/down depends on the direction of the escalator as viewed by the user.

Yes, escalators going down can generate power for the local grid when there are people on them. Modern escalators do this. E.g. https://www.kebamerica.com/blog/vfd-regen-escalators/

It doesn't particularly power the escalator going up, it just reduces the power consumption of the whole building.

There was also the Levytator where instead of the stairs looping underneath at the end, they go around and form the opposite escalator: https://www.youtube.com/watch?v=x58272OirGU

This means that people travelling down automatically help people travelling up because they are on the same mechanically linked stairs.

This is what happens when you do not use something like LaTex to typeset a paper.

The amount of typographical/layout crimes screaming in your face on the first page already made me not want to read this – however curious the topic.

Honestly this is more 'arithmetic of escalators' than mathematics. But now I'm curious, anyone run across some related math? Graph theory and combinatorics immediately come to mind.

Is arithmetics not part of mathematics?

Apropos, Peter Campbell on the maintenance of escalators in the LRB: https://www.lrb.co.uk/the-paper/v24/n05/peter-campbell/why-d....

I hope they do an analysis on the escalators in the new BER airport.. maybe we could find out the stroke of genius that designed the airport with escalators that only go up. Emphasis on stroke.

Shouldn't a down-hill escalator be able to generate electricity?

Apparently frictional losses are too high.

This seems intuitively believable: I'm imagining an escalator disconnected from its motor and wondering whether a constant flow of people (stepping on, but not walking downhill) could keep it running at 1m/sec.

What would be the minimum number of passengers it'd need before it can go by itself?.

Doing this in a non-spherical-cow environment will be challenging, because the gravitational load varies (passengers stepping on/off) but it's probably important to keep the velocity fairly constant to avoid upsetting the balance of those standing on it, let alone those walking down it. So it likely needs the generator load to be very responsive to the behavior of a long loop of rather heavy objects with quite a lot of friction, motor driven via reducing gears.

Maybe instead of thinking of generation, it's more a question of regulating the applied motor power to keep constant velocity?

It would certainly be possible to recover some of this energy. Just like it is possible to recover some of the heat lost by people: https://www.greenchipstocks.com/articles/body-heat-from-pass...

That was my thought too, from a mechanical design point of view, perhaps some sort of funicular design perhaps where the weight of the passengers going down helps left the other side. But maybe some sort of regenerative braking on the down side charging batteries to assist the up side.

This is urgently missing some discussion of queuing theory / stochastic processes. Discussing passenger flow in terms of time averages is misleading, passengers don't arrive in continuous streams. The peak load will be many times the average, which impacts power consumption, while at other times they will be running almost empty with just a handful of passengers.

It's actually not that bad to assume a crowd is a fluid. For example it allows you to deduce where the hotspots will be when there's a panic.

Disagree. Firstly there aren’t many situations in fluid dynamics where there is a steady constraint on mass flow. Velocity just increases. (The exception is choked flow from a nozzle IIRC)

Now think about what happens when 6000 people/hr leave the station but arrive on 12 trains over the course of the hour, i.e. in surges. Each peak demand will comfortably exceed the flow rate of escalator.

You get a queue; the equivalent in fluid dynamics would be an increase in density upstream of a choked nozzle at Mach 1 but how to turn that into a relevant metric for eg queue length?

Thats also the reason - I suspect - that in many Tube stations two escalators go up and only one goes down.

Plenty of researchers went the fluid route. Also intuitively, it's viable. A simple example. If you want to step onto a train, the best place to position yourself is next to the doors of the train, not in front of them. The stream of people coming out of the train makes it so that the pressure is lowest where you stand. Anyway, there are plenty of papers on the topic.

fe this paper: https://www.sciencedirect.com/science/article/abs/pii/004116...

Yes, it can make sense for some applications (when used with care).

But check the article, they're not using fluid dynamics, they actually try to estimate capacity/load using period averages and basic arithmetic. Such an approach can lead to a massive underestimation of required capacity even in contexts where arrivals aren't as batched as they are in train stations (like a call center).

Fascinating and something to think about as I ride the Piccadilly line escalators with my mind in neutral. Thanks.

Reply on news.ycombinator.com