Macroscopic models, cumulative curves and space–time

Introduction

Fifty thousand runners, 42 km of road, a pinch point at km 3. You do not need to know who elbows whom. You need to know how many people reach the pinch each minute, and whether the road can pass them. That is a macroscopic question, and its tools are a counter and a pencil.

Cumulative curves

A cumulative curve N(x, t) is the number of people who have passed point x by time t. Its slope at a fixed point is the flow, ∂N/∂t. Draw the curve for arrivals at a gate and the curve for departures through it, and the vertical gap between them at any time is the queue; the horizontal gap at any count is how long that person waited.

Worked example. People arrive at a gate at 120 per minute for 10 minutes, then at 40 per minute. The gate passes 80 per minute.

1. For 10 minutes the queue grows at 120 − 80 = 40 per minute: 400 people at 10 minutes.
2. The 1,200th arrival (at 10 minutes) is served when 1,200 have departed: at 1,200 ÷ 80 = 15 minutes, a 5-minute wait.
3. After 10 minutes the queue shrinks at 80 − 40 = 40 per minute and is gone at 20 minutes.

Compare two points instead of two curves at one point: the vertical gap between N(x₁, t) and N(x₂, t) is the number of people between them, and dividing by the area gives the density. The bottleneck sim in lesson 1.5 draws these curves live.

Space–time diagrams

A space–time diagram plots each person’s position against time, one line per person. With distance up and time across, a steeper line is a faster runner, lines that cross are overtakes, and a horizontal line at the pinch point is a counter: count the lines crossing it in each five-minute window.

Runners start in order 1 to 12 over 5.5 minutes but reach the pinch in a different order: runner 10, starting tenth, arrives fifth. Six of the twelve arrive between 15 and 20 minutes: the busiest five minutes. Start Right–type modelling for the New York City Marathon was reported in this form, as runner counts per five-minute finish window. No peer-reviewed description of that method has been found, so treat this as a reconstruction of the idea, not the algorithm.

From counts to a continuum

Make the cells small and the crowd becomes a fluid: a density ρ(x, t) that is conserved, ∂ρ/∂t + ∂(ρv)/∂x = 0, with speed set by density through a fundamental diagram. Hughes (2002) wrote a continuum theory of pedestrian flow of this kind. A computer solves it on a chain of cells, passing between neighbours the smaller of what the upstream cell wants to send and what the downstream cell can take. The cost depends on the number of cells, not people, which is why this is the right tool for 50,000 runners over 42 km.