Competency
In this project, you will demonstrate your mastery of the following competency:
- Apply matrix theory to linear equations and transformations
Scenario
You are employed as a network engineer and have been asked to analyze
a communication network to determine the current data rates and ensure
that the links aren’t at risk of “reaching capacity.” In the following
figure of the network, the sender is transmitting data at a total rate
of 100+50 = 150 megabits per second (Mbps). The data is transmitted from
the sender to the receiver over a network of five different routers.
These routers are labeled A, B, C, D, and E. The connections and data
rates between the routers are labeled as
,
,
,
, and
.
Directions
In this project, you will analyze the communication network and solve
for the unknown data rates using a variety of techniques. The system
can be modeled mathematically as a system of linear equations by writing
an equation for each node/router in the network. Each of these
equations can be written by noting that the sum of inputs must equal the
sum of outputs.
To complete the project, work on the problems described below. As you
complete each part, show your work and carefully describe how you
arrive at your final answer. The methods and conclusions need to be
clear when sharing your results with management. Any MATLAB code or
MATLAB terminal outputs you generate should be provided in your
submitted document to support your answers and work.
Develop a system of linear equations
for the network by writing an equation for each router (A, B, C, D, and E). Make sure to write your final answer as
where
is the
coefficient matrix,
is the
vector of unknowns, and
is a
vector of constants.
- Use MATLAB to construct the augmented matrix
and then perform row reduction using the rref() function. Write out your
reduced matrix and identify the free and basic variables of the system.
- Use MATLAB to
compute the LU decomposition of
, i.e., find
. For this decomposition, find the transformed set of equations
. Solve the system of equations
for the unknown vector
.
- Use MATLAB to
compute the inverse
of
using the inv() function.
Compute the solution to the original system of equations
by transforming
into
, i.e., compute
.
Check your answer for
using Cramer’s Rule.
- Use MATLAB to compute the required determinants using the det()
function. The Project One Table Template, provided in the Supporting
Materials section, shows the recommended throughput capacity of each
link in the network. Put your solution for the system of equations in
the third column so it can be easily compared to the maximum capacity in
the second column. In the fourth column of the table,
provide recommendations
for how the network should be modified based on your network throughput
analysis findings. The modification options can be No Change, Remove
Link, or Upgrade Link. In the final column,
explain
how you arrived at your recommendation.
What to Submit
To complete this project, you must submit the following:
Use the provided Project One Template as the starting point for your
project solution. Complete each portion of the template, run the
project, and then export your work as a single PDF file. Upload this PDF
document, which should show your answers and supporting work for the
problems described above. Make sure to include explanations of your
work, as well as all MATLAB code and outputs of the computations.