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Train Passengers’ Load Model

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Years of taking trains finally caused me to spend some time on a simple question – why people are standing in overloaded trains, while there are available seats at the same train? Apparently simple question might have a bit complex answer.

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First, let’s make few assumptions.

1. The train is long and each passenger does not know what is happening in the next car of the train without going there.

2. Most people do not communicate with each other and not pass information about other places and of cause they do not agree to spread uniformly

3. People on the station do not know the distribution of passengers in coming train a priori

4. To make it interesting, lets assume there are several stops for the train i.e. there are more than one iteration and not all people come at the same time

5. There are no big groups of passengers that travel together and there are at mostly up to 2 people that know each other and would like to seat together.

Ok. I think it should be enough and we can start thinking about a model.

Initial spread

First important question is why passengers are spread non-uniformly in across the train and on station in the first place?

If it would be, then we would not see the phenomenon we try to explain.

First reason – asymmetric entrance to the station platform, narrow platform, very long platform, short time of waiting of cause would cause heterogeneous spread of waiting people that otherwise would successfully spread uniformly due to diffusion process. Here you can easily model it by applying the diffusion flux equation from physics. Of cause it makes the train passengers model even more extreme, but even uniformly distributed waiting people (at the second+ station) would behave according to the model we would like to find if the train is not uniformly loaded.

Second reason is psychology of passengers during the load. Even equal amount of people entering all doors of the train in the first station, small amount of  people would leave people in first car they enter, so spread would be uniform. But what happens if amount is rising?

Let’s take the example of train car that has 2 rows of 10×4 seats i.e. 80 seats total. People that enter love their privacy and try to get 1 or 2 seats of each group of 4 seats – meaning the situation would not change if there are up to 40 passengers for each car. Interesting thing happens when there are more than 40. To avoid impact on privacy and inconvenience of breaking the personal space circle, people start to move to another car. Remember that situation there should be similar and people would proceed to move further. Up until when? And here there is an interesting variable of passenger transition energy. It can be physical or mental energy, but there is a driving force that is going down as we proceed our journey through the train.

If we would plot it, it possibly might look like this:

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So at the beginning people has a lot of energy and it goes down as they search for an available pair of seats. The density is low at this stage and people can go relatively far since it looks like very soon they would find a free pair and there is no physical resistance (the path is empty). How far they can go? Up until the energy will pass some threshold level of patience or up until the termination – the end/beginning of the train. That is why most probably we see often this kind of train load (green – less, red – more):

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What happens if there are already more than half of load (>40 seating passengers) within the car i.e. there are here and there groups of 3 people seating together? in this case people would be likely to abandon the search for a better place and create another triad or join to existing ones and create group of 4. The reason for that might be estimation of reduced likelihood that there are empty places somewhere else (remember the assumption of not knowing that happens in other places) and comfort decision to join/create a large group when there are already other people that did it.

So existing groups of 3-4 if they know each other or people with physical limitations that simply cannot go far can be that tipping point that would cause transition from 50% load to ~100% load of a train car.

So most probably the load would look like this:

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Standing passengers

We can see various reasons for heterogeneity of a train load, but why it comes to a such extreme condition that people are standing?

If the non-uniform distribution was Energy-distance dependency, here another factor becomes critical  – passenger transition resistance. Not only the very existence of people cause a mental resistance to move further, but also physical resistance due to such aspects as people standing on the way (even temporary) or passenger belongings.

If the density is low (few people are sitting here and there), than the energy we have to spend to move is low. At some point, the resistance becomes significant so we start to feel the pressure and the energy requirement for transition rises up until some saturation point. We can move for some time even with increased resistance, but if it becomes too high, than we give up and we are ready to sit anywhere. Something like this:

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This is a point where the phenomenon we are analyzing is happening. Too low energy to move further due to passed distance and too high high resistance together with local load saturation to 100% would cause people to give up.

What can be done?

1. Distribution. Station platform architecture has to ensure good and fast diffusion of passengers and their uniform distribution. Multiple entrances, wide platform would reduce the formation of “clot” and chance of standing passengers. Dual floor train is better in this terms as the train becomes twice shorter for the same amount of passengers.

2. Feedback. The assumption that people do not know what is happening in other cars can be broken and reduce amount of people that would stand (the feedback that would add “mental energy” to proceed despite resistance) or even would cause uniform distribution in the first place. It can be voice notification where are lots of free places on the train (which is done sometimes) or visual representation of the load across the train (similarly to charts above). The feedback can be for waiting passengers, so they could prepare for arriving train and reduce energy spent to overcome the internal resistance.

3. Direct trains would have reduced phenomenon as it would leverage a single-time load from well distributed platform (if it is so).

Well… those were my thoughts on my daily routine. Would be interesting to build the computer model and simulate tipping points, various scenarios, validate various station platform architectures and trains. Need to find a student and give it as a project… of cause if it was not already done somewhere else (did not find during quick search).

😉

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Author: Andrey Gabdulin

www.gabdulin.com Product Development

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