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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 $1, $213, 5134, and 1'5. Sender A («I») x; a «'1'» Receiver .2 8 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. 1. 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 Ax = b where A is the 5 X 5 coefficient matrix, x is the 5 X 1 vector of unknowns, and b is a 5 X 1 vector of constants. 2. Use MATLAB to construct the augmented matrix [A b] and then perform row reduction using the rref() function. Write out your reduced matrix and identify the free and basic variables of the system. 3. Use MATLAB to compute the LU decomposition of A, i.e., find A = LU. For this decomposition, find the transformed set of equations Ly = b. Solve the system of equations Ly = b for the unknown vector Y. 4. Use MATLAB to compute the inverse of Uusing the inv() function. 5. Compute the solution to the original system of equations by transforming y into x, i.e., compute x = U "ly. 6. Check your answer for $1 using Cramer's Rule. 7. 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.

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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.

  1. 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 make your final answer as  where  is the  coefficient matrix,  is the  vector of unknowns, and  is a  vector of constants.
  2. Use MATLAB to construct the augmented matrix  and then perform row reduction using the rref() function. Make out your reduced matrix and identify the free and basic variables of the system.
  3.  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 .
  4. Use MATLAB to compute the inverse of using the inv() function.
  5. Compute the solution to the original system of equations by transforming  into , i.e., compute .
  6. Check your answer for  using Cramer's Rule.
  7. 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 finish the 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.

Image transcription text

Southern New Hampshire University MAT 350 Project One Table Template Complete this template by replacing the bracketed text with the relevant information. + Network Recommended Solution Recommendation Explanation Link Capacity (Mbps) X1 60 [Insert text] [Insert text] [Insert text] X2 50 [Insert text] [Insert text] [Insert text] X3 100 [Insert text] [Insert text] [Insert text] XA 100 [Insert text] [Insert text] [Insert text] X5 50 [Insert text] [Insert text] [Insert text] O

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Project One Template

MAT350: Applied Linear Algebra

Student Name

Date


Problem 1

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 do a final answer as Ax=b where A is the 5x5 coefficient matrix, x is the 5x1 vector of unknowns, and b is a 5x1 vector of constants.

Solution:

Put your math/explanation here...


Problem 2

Use MATLAB to construct the augmented matrix [A b] and then perform row reduction using the rref() function. Write out your reduced matrix and identify the free and basic variables of the system.

Solution:

%code




Problem 3

Use MATLAB to compute the LU decomposition of A, i.e., find A = LU. For this decomposition, find the transformed set of equations Ly = b, where y = Ux. Solve the system of equations Ly = b for the unknown vector y.

Solution:

%code






Problem 4

Use MATLAB to compute the inverse of U using the inv() function.

Solution:

%code




Problem 5

Compute the solution to the original system of equations by transforming y into x, i.e., compute x = inv(U)y.

Solution:

%code




Problem 6

Check your answer for  using Cramer's Rule. Use MATLAB to compute the required determinants using the det() function.

Solution:

%code






Problem 7

The Project One Table Template, provided in the Project One Supporting Materials section in Brightspace, 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.

Solution:

Fill out the table in the original project document and export your table as an image. Then, use the Insert tab in the MATLAB editor to insert your table as an image.

This template is the starting point for your solution to this project. Download the template, open in MATLAB, and complete each portion.

Answer & Explanation
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