# How to model pedestrian flow through subway systems?

I'm a New Yorker and take the subways every day. I have a growing interest in understanding the distribution of paths people take on the subways to work every day.

I.e. if there are $n$ subway stations, I want a temporal distribution over the paths $\omega = (n_i, n_j, t)$ where $n_i$ and $n_j$ are nodes representing subway stations and $t$ is a continuous variable representing time.

I am given the expected arrival and departure times for each subway station as well as the number of people who enter and exit each station at discrete moments in time.

I am hoping someone can give me ideas on how to translate my deterministic model into one I can fit statistically, as well as reason about theoretically

## Ideal Theoretical Model

Let $N$ be the total population in the subways at any given moment in time and let $N_i$ be the number of individuals in station $i$, and $N_ij$ be the number of individuals on the subway line heading from station $i$ to station $j$. The total number of individuals in transit $$T = \sum{N_{ij}}$$

and the total number of individuals in standby is $$S = \sum{N_i}$$

Then $$N = T+S$$

I have 1.5 undergrad physics classes under my belt so I've been thinking about the system with the following analogy. Imagine the subway station to be a system of $n$ point masses(each point mass is a subway station) suspended in space, where $N_i$ represents the height of each point mass.

Each adjacent subway station exerts a downwards and upwards force on its neighbors. In particular $F_{ij}$ is the force exerted by $i$ on $j$ such that $$F_{ij} = g(N_{ij})$$ where $g$ is some unknown function.

In this light, the net force on station $i$ is defined as

$$F_{\text{net}, i} = \sum{ F_{ji} } - \sum{ F_{ij} }$$

In other words, at each moment in time, every station exerts a force on its neighbors. Actually, to be more precise, this force based model implies there are actually $n^2+n$ particles between stations and $n$ stations, therefore a total of $n^2 + 2n$ particles. The velocity of a particle is the rate at which its height decreases and the mass of a particle is a coefficient $m_{ij}$.

Having interpreted the subway station as a group of point particles with pairwise interactions under an unknown force mechanic, we can also define a potential energy configuration.

$U(N)$ defines the potential energy configuration of the system and $U_{ij}(N)$ defines the potential energy of a particle $ij$, such that

$$U(T(t)) + K(S(t)) = E(t)$$

$$K_{ij} = \frac{1}{2}m_{ij}v_{ij}^2$$

$$U_{ij} = -\int^{N_{ij}}_{0}{F_{ij}} = -N_{ij}m_{ij}$$

Fluctuations in $T$ and $S$ represent changes in the internal potential and kinetic energies of the system. Fluctuations in $N$ represents net work being done on the system, i.e.

$$W = \Delta E$$

## Issues in the Model

I still haven't defined $g$, nor defined the potential and kinetic energy functions.

## Statistical Modelling and my Question

At this point, I have a cute little physical interpretation of the subway system. But I am having trouble translating my dynamical model into one I can statistically fit to find a distribution over the ultimate paths taken by individuals over the course of a day.

• In practice, I don't know what "modelling the subway dynamics" will actually mean?
• Ought I to fit coefficients to $g$ , $U$, and $K$?
• How to formulate a rigorous statistical question given this abstract physical model.

## Or

• Alternative ways of finding the distribtion of trips, including links to literature that has already solved this problem.
• Probably there is a better solution but alternatively you can try cellular automata approach. I saw many papers about modeling of similar phenomena using such an technique. – Krzysztof Bzowski Feb 14 '17 at 12:52