8 ms·

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

84 points by carschno 4y ago

t_mann 4y ago
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.
- toolslive 4y ago
  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.
  - tomsto 4y ago
    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.
    - toolslive 4y ago
      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/0041164774900276 https://www.sciencedirect.com/science/article/abs/pii/004116...
  - t_mann 4y ago
    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).
capnahab 4y ago
Fascinating and something to think about as I ride the Piccadilly line escalators with my mind in neutral. Thanks.
Chris2048 4y ago
Shouldn't a down-hill escalator be able to generate electricity?
- lsllc 4y ago
  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.
- dmurray 4y ago
  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.
- MrPatan 4y ago
  What would be the minimum number of passengers it'd need before it can go by itself?.
- thombat 4y ago
  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?
- jerome-jh 4y ago
  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-passengers-to-train-stations-to-office-buildings/78716 https://www.greenchipstocks.com/articles/body-heat-from-pass...
Symbiote 4y ago
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.
- everfrustrated 4y ago
  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.
  - ajsnigrutin 4y ago
    > 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.
    - Symbiote 4y ago
      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.
    - Delphiza 4y ago
      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.
NLips 4y ago
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-london-standing https://www.shortlist.com/news/holborn-escalator-trial-tube-...
- carschno 4y ago
  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.
  - NLips 4y ago
    I think the speed _for all passengers_ was increased, because of the reduction in queueing.
- gsnedders 4y ago
  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.
- yardstick 4y ago
  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.
amelius 4y ago
Could the escalators going down generate the power for the escalators that are going up?
- KineticLensman 4y ago
  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.
  - danbruc 4y ago
    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.
    - midasuni 4y ago
      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.
  - ajsnigrutin 4y ago
    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_breaks_causing_it_to_speed_up_out_of/ 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.
  - amelius 4y ago
heavenlyblue 4y ago
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.
- rhn_mk1 4y ago
  It's probably because human fuel is in the most part obtained by solar power sequestrating carbon from the atmosphere.
  - heavenlyblue 4y ago
    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)
    - rhn_mk1 4y ago
      If you eat meat, it may be more beneficial to become a vegetarian.
- lovemenot 4y ago
  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.
  - heavenlyblue 4y ago
Reason077 4y ago
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.
- lsllc 4y ago
  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-electricity-from-renewables-outpaces-gas-and-coal-power https://www.theguardian.com/environment/2021/jan/28/uk-elect...
llampx 4y ago
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.
eynsham 4y ago
Apropos, Peter Campbell on the maintenance of escalators in the LRB: https://www.lrb.co.uk/the-paper/v24/n05/peter-campbell/why-does-it-take-so-long-to-mend-an-escalator https://www.lrb.co.uk/the-paper/v24/n05/peter-campbell/why-d....
mzs 4y ago
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.
- agucova 4y ago
  Is arithmetics not part of mathematics?
virtualritz 4y ago
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.

84 points by carschno 4y ago

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.

toolslive 4y ago
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.
- tomsto 4y ago
  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.
  - toolslive 4y ago
    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/0041164774900276 https://www.sciencedirect.com/science/article/abs/pii/004116...
- t_mann 4y ago
  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).

capnahab 4y ago

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

Chris2048 4y ago

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

lsllc 4y ago
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.
dmurray 4y ago
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.
MrPatan 4y ago
What would be the minimum number of passengers it'd need before it can go by itself?.
thombat 4y ago
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?
jerome-jh 4y ago
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-passengers-to-train-stations-to-office-buildings/78716 https://www.greenchipstocks.com/articles/body-heat-from-pass...

Symbiote 4y ago

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.

everfrustrated 4y ago
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.
- ajsnigrutin 4y ago
  > 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.
  - Symbiote 4y ago
    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.
  - Delphiza 4y ago
    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.

NLips 4y ago

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-london-standing https://www.shortlist.com/news/holborn-escalator-trial-tube-...

carschno 4y ago
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.
- NLips 4y ago
  I think the speed _for all passengers_ was increased, because of the reduction in queueing.
gsnedders 4y ago
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.
yardstick 4y ago
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.

amelius 4y ago

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

KineticLensman 4y ago
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.
- danbruc 4y ago
  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.
  - midasuni 4y ago
    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.
- ajsnigrutin 4y ago
  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_breaks_causing_it_to_speed_up_out_of/ 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.
- amelius 4y ago

heavenlyblue 4y ago

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.

rhn_mk1 4y ago
It's probably because human fuel is in the most part obtained by solar power sequestrating carbon from the atmosphere.
- heavenlyblue 4y ago
  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)
  - rhn_mk1 4y ago
    If you eat meat, it may be more beneficial to become a vegetarian.
lovemenot 4y ago
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.
- heavenlyblue 4y ago

Reason077 4y ago

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.

lsllc 4y ago
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-electricity-from-renewables-outpaces-gas-and-coal-power https://www.theguardian.com/environment/2021/jan/28/uk-elect...

llampx 4y ago

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.

eynsham 4y ago

Apropos, Peter Campbell on the maintenance of escalators in the LRB: https://www.lrb.co.uk/the-paper/v24/n05/peter-campbell/why-does-it-take-so-long-to-mend-an-escalator https://www.lrb.co.uk/the-paper/v24/n05/peter-campbell/why-d....

mzs 4y ago

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.

agucova 4y ago
Is arithmetics not part of mathematics?

virtualritz 4y ago

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.