Question

Analysis of Traffic Flow

This question is based on the Traffic Flow Example in Section 2.2 of the text (pages 110-111). Read through that example before working the problem. Use the matrices A, B, C and M = CBA from this example in the following. Be careful when you copy the entries in these matrices from the text into Matlab
. (a) Generate the following: x = 1000*rvect(2), y = A*x, z = B*y, w = C*z by Matlab. The vectors x, y, z, w describe the random traffic flows through succesive parts of the network. (2) Calculate [1 1]*x, [1 1 1]*y, [1 1 1]*z, [1 1 1 1]*w using Matlab. (1) Explain the meaning of each answer in terms of the traffic flow through a particular part in the network. (1)
(b) Suppose that on a particular day the traffic flow gives the vector y = [270 126 704]’. Give the equation relating y and x and solve it for x using Matlab as in Question 3. Note that these vectors x and y are not the same as the random vectors you generated in part (a). Now you are starting with y and trying to solve for x. (1) Is the entering traffic flow vector x that you just found uniquely determined in this case? Explain using the general theory of solving Ax = b from Question 4. (1)
(c) Let w = [100 200 300 400]’ . Give the equation relating w and x and use Matlab as in Question 3 to determine if w can be an output traffic vector. Note that these vectors x and w are not the same as the random vectors you generated in part (a). Now you are starting with w and trying to find x. (Hint: Calculate rref([M w]) to see if the equation relating x and w is consistent.

Answer 1

The student is working with concepts of matrix multiplication and linear algebra to analyze traffic flow in a network using MATLAB. Calculating the total traffic flow using vector multiplication with matrices A, B, and C represents different stages of the traffic through the network. The problems involve solving linear equations to find traffic states and confirm the consistency of these states with network conditions.

Step-by-step explanation:

The student's question pertains to the analysis of traffic flow through networks, which can be studied using matrix algebra and vector multiplication in MATLAB. The matrices A, B, and C represent the network's transitions from one section to another, and the vector operations performed (such as A*x) represent the passage of traffic through these sections. In part (a), the product of [1 1]*x indicates the total traffic flow entering the network when vector x represents incoming traffic at two different points. Similarly, [1 1 1]*y, [1 1 1]*z, and [1 1 1 1]*w represent the total traffic flow through different network sections, with y, z, and w being the consequent traffic states after passing through matrices A, B, and C respectively.

In part (b), given a traffic state y, the student is tasked with finding the corresponding initial traffic state x using matrix inversion or other methods for solving linear equations. The problem explores the concept of unique solutions in linear algebra, specifically whether the matrix equation Ax = y always has a unique solution x. This depends on the properties of matrix A, such as whether it is invertible.

Part (c) asks the student to determine if a given traffic state w can be an output of the network for some input vector x. This involves using reduced row echelon form (rref) to assess the consistency of the system of equations described by the matrix equation Mx = w, where M is the product of matrices CBA.