See the discussion of networks beginning on page 29 of the text, including examp

Question

See the discussion of networks beginning on page 29 of the text, including example 5, to assist in the following. A section of the downtown area of a city has only one-way streets. The diagram below shows this section along with the direction of the traffic and the flow rates (measured as the average number of vehicles per hour). a. (1) Set up a system of linear equations to represent the traffic network. b. (2) Solve the system to find the unknown flow rates. c. (1) Explain the practical significance of your solution.

Explanation / Answer

(a) For Each node , we will equate the incoming traffic and the outgoing traffic.

So for Point A, the equilibrium equation

750 + x3 = 250 + x4

similarly, for other nodes

Node B : x4 + 200 = x1 + 300

For the other two nodes:

300 + x2 = 400 + x3

400 + x2 = x1 + 100

so combining linear equations

x4 - x3 = 500 ...(i)

x4  -x1 = 100...(ii)

x1 - x2 = 300..(iii)

x2 -x3 = 100...(iv)

(2) by solving all equation simultaneously,we get

The linear system can't be solved as there are infiinte soultions.

by if we say x1 = x

then x2 = x - 300 ; x3 = x - 400 and x4 = x + 100

(c) We can here see that for any x we can understand the whole traffic system. In any closed loop or network system, if we know the flow of even one traffic line, we can calculate the entire network flow design system.

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