# Traffic Flow Theory as Applied to ITS

##### Length: Approximately eight hours.

## Description

Traffic flow theories aim to describe in a precise way the interactions between the vehicles, their operators and the roadway system. As such, these theories are an indispensable component for all models and tools that are being used in the design and operation of streets and highways. This course examines a number of theoretical approaches to traffic flow using a mathematical approach.

## Objectives

Upon completion of the course, participants will be able to:

- Identify the relationship among the basic traffic stream characteristics;
- Select and design a traffic surveillance system for measuring traffic, and calculate from raw field data the traffic stream characteristics;
- Select the appropriate traffic stream model to analyze different traffic conditions;
- Explain in simple terms the mechanics of the driving task of car following;
- Explain the interactions among traffic streams at unsignalized intersections and merge/diverge points;
- Explain how traffic stream models can be used in the ITS environment (i.e. automated highways; advanced traffic management and control, etc.)

## Outline

**Lesson 1: Introduction to Traffic Flow Theory for ITS**

What is Traffic Flow Theory?

What is Important about Traffic Flow Theory for ITS?

Common Issues

Do I Really Need to Know This Stuff?

Let’s Get Started!

**Lesson 2: The Traffic Stream and Traffic Dynamics**

The Time-Space Diagram

Tracking a Single Vehicle

Tracking Several Vehicles

Counting Cars

Counting Cars (2)

Generalized Definition of Flow

Usefulness of Generalized Definitions

Looking at Hypothetical Aerial Photographs

Looking at Hypothetical Aerial Photographs

Variables

Generalized Definition of Density

Average Speed

Measuring Speed at the Side of the Road

Space Mean Speed Exercise

Time-Mean Speed

Time-Mean Speed (cont’d)

Aerial Observations of Speed

Mean Speeds: Summary

Wait a Minute…Isn’t Space Mean Speed a Harmonic Mean?

**Lesson 3: Fundamental Equations**

Stationary Traffic

Decomposing Traffic

Space and Time Mean Speeds in Stationary Traffic

Space and Time Mean Speeds in Stationary Traffic (cont’d)

Labeling Passing Vehicles

Imaginary Race Cars

Two-Dimensional Counting Functions

Cumulative Plots

Cumulative Plots: What This Graph Means?

Counts During Time Intervals

Simulation vs. Mathematical Modeling

Smoothing Out the Steps

Smoothing Out the Steps (cont’d)

Playing with Mud

Zooming Out to Consider Many Vehicles

Instantaneous flow q (x,t)

Instantaneous flow q (x,t) (cont’d)

Slicing Along the Other Axis

Slicing Along the Other Axis (cont’d)

Instantaneous Density k (x,t)

Partial Derivatives

Conservation Laws

Conservation Laws (cont’d)

Conservation Laws (cont’d)

Fundamental Diagram

Fundamental Diagram (cont’d)

Fundamental Diagram (cont’d)

Interface (Shockwaves)

Interface Speed (Mathematical)

Interface Speed (Mathematical) (cont’d)

Interface Speed (Graphical)

**Lesson 4: Shock Analysis**

Shock Analysis Example

Problem Set-Up

Problem Set-Up (cont’d)

The Light Turns Red

The Light Turns Red (cont’d)

The Light Turns Green

The Light Turns Green (cont’d)

What Else is There to Learn From This Picture?

## Author

Dr. Chronis Stamatiadis

Dept. of Civil Engineering

University of Massachusetts Lowell