The Problem of Traffic: A Mathematical Modeling Journey

How can we mathematically model traffic? Specifically we will study the problem of a single lane of cars and the perturbation from equilibrium that occurs when one car brakes, and that braking effect travels down the line of cars, amplifying as it goes along, due to the delayed reaction time of the drivers. The ultimate phenomena we would like to predict is that stop-and-go behaviour where cars don't just travel at a constant speed in rush hour, but alternate between braking and accelerating. However, when building this model our inputs are very microscopic considerations about how an individual car brakes or accelerates based on the following distance and relative velocity of the car ahead of it.

This video also aims to be an introduction to broad themes in mathematical modelling of real world problems, where we define a problem, choose the inputs to the system we will consider, make assumptions, build the model, and finally assess the model. Finally, the real piece of mathematics we are going to get out of this are called differential-delay equations, and I'll show you a bit about how to solve such equations at the end.

This video is part of the second iteration of the Summer of Math Exposition, hosted by ‪@3blue1brown‬ and ‪@LeiosLabs‬ . There are so many great videos in this in this exposition so definitely check them out by using the hashtag #SoME2.

0:00 The Challenge of Traffic
0:28 #SoME2
0:53 The Modelling Process
1:27 Defining the Problem
2:04 Choosing Which Variables to Consider
4:03 Making Assumptions
5:46 Building the Microscopic Model for Each Car
9:56 Macroscopic Equilibrium
10:34 The Relationship between Density and Velocity
16:23 Maximizing Flux and the Optimal Oensity
20:33 Modelling a Sequence of Cars
24:07 Modelling the First Car
26:05 Full Model: A Differential Delay System
27:06 Assessing the Model Graphically
29:33 Assessing the Model Qualitatively
31:45 Solving Differential Delay Systems

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