MAT-350 Project 1: Network Flow Analysis Using Linear Systems

Author: Justin Paul Guida
Course: MAT-350 - Applied Linear Algebra
Date: September 18, 2025

Overview

This project models a computer network flow system using systems of linear equations.
Each router in the network (A, B, C, D, and E) is represented as a node, and data transfer between routers is modeled using equations that balance inflow and outflow rates.

The project demonstrates how to construct a coefficient matrix A, a constant vector b, and solve the system Ax = b using MATLAB.
It also includes validation through LU decomposition, Cramer’s Rule, and determinant analysis.

The project is submitted as a report (.pdf) generated in MATLAB, showing full calculations, code, and annotated results.

Objectives

1. Develop a linear system of equations for a five-router network.
2. Represent the system in matrix form (Ax = b).
3. Solve the system using MATLAB’s built-in tools (rref, lu, inv, det).
4. Verify the solution using multiple algebraic methods.
5. Interpret network flow results and identify potential system bottlenecks or upgrade recommendations.

Key MATLAB Concepts Demonstrated

• Matrix Construction: Defining coefficient matrices and augmented matrices.
• Row Reduction: Solving systems using rref() for reduced row echelon form.
• LU Decomposition: Breaking down A into lower (L) and upper (U) matrices.
• Forward/Backward Substitution: Solving Ly = b and Ux = y.
• Matrix Inversion: Using inv() to compute the inverse of U.
• Determinants and Cramer’s Rule: Cross-verifying results using determinant calculations.

Results Summary

• All routers yield unique flow values with no free variables, confirming a consistent system.
• The final data flow vector was determined to be:

x = [50; 25; 30; 100; 45] Mbps

• Determinant of the system matrix A is 2.00, confirming the system’s invertibility.
• LU and Cramer’s Rule methods both reproduce identical results.
• Network recommendations include upgrading links nearing capacity to prevent congestion.

File Description

File	Description
mat350_project1_network_flow_analysis.pdf	Final report with MATLAB code, output, and written explanations.

Tools and Requirements

• MATLAB R2023a or later
• No additional toolboxes required

Educational Purpose

This project reinforces the application of linear algebra in network modeling, demonstrating how fundamental matrix operations can represent and analyze real-world data flow systems.
It highlights the practical use of MATLAB for solving and verifying linear systems in engineering and applied mathematics contexts.