Modeling and Simulation of Traffic on I-485 via Linear Systems and Iterative Methods

TL;DR

This paper explores iterative methods for solving large linear systems, compares their efficiencies, and applies the best method to model and analyze traffic flow on I-485, providing insights for infrastructure planning.

Contribution

The study compares Jacobi, Gauss-Seidel, and SOR methods, develops a Python tool for selecting the fastest converging method, and applies it to real-world traffic modeling on I-485.

Findings

Gauss-Seidel is generally more efficient than Jacobi.

SOR offers the fastest convergence speed.

The traffic model identifies congestion points for infrastructure planning.

Abstract

Iterative methods such as Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) are fundamental tools in solving large systems of linear equations across various scientific fields, particularly in the field of data science which has become increasingly relevant in the past decade. Iterative methods' use of matrix multiplication rather than matrix inverses makes them ideal for solving large systems quickly. Our research explores the factors of each method that define their respective strengths, limitations, and convergence behaviors to understand how these methods address drawbacks encountered when performing matrix operations by hand, as well as how they can be used in real world applications. After implementing each method by hand to understand how the algorithms work, we developed a Python program that assesses a user-given matrix based on each method's specific convergence criteria. The program compares the spectral radii of all three methods and chooses to execute whichever will yield the fastest convergence rate. Our research revealed the importance of mathematical modeling and understanding specific properties of the coefficient matrix. We observed that Gauss-Seidel is usually the most efficient method because it is faster than Jacobi and doesn't have as strict requirements as SOR, however SOR is ideal in terms of computation speed. We applied the knowledge we gained to create a traffic flow model of the I-485 highway in Charlotte. After creating a program that generates the matrix for this model, we were able to iteratively approximate the flow of cars through neighboring exits using data from the N.C. Department of Transportation. This information identifies which areas are the most congested and can be used to inform future infrastructure development.