#### Network Flow

##### Network

A network consists of a set of points (junctions, nodes, ...) and
lines connecting some of the points. The lines (or branches) each
has a direction of flow and a value for the flow amount (either
known or unknown). For example, in this figure

there is 1 junction and three branches. One branch has a flow
amount of 25 and the other 2 are unknow but labeled as x and y.

##### Flow Rules

The rule that every flow satisfies is the conservation law. At
every junction, the flow in equals the flow out. Thus in our
example above, we know that

25 = x + y.

For particular types of networks there may be additional rules.

##### Traffic Flow

Consider the following configuration of one-way streets with
flow values:

For traffic flow the only rule is conservation, thus examining
the 4 junctions, we have 4 equations:

E + D = A + 600

A + 300 = B + 700

C + 300 = D + 200

B + 500 = C + 100

We can rewrite this as a linear system of equations:

-A + D + E = 600

A - B = 400

C - D = -100

B - C = -400

Either using MATLAB or by hand, solve this system. Either
way you should put it in augmented form and then find the
row-reduced form of the augmented matrix. You should get this

1 0 0 -1 0 -100
0 1 0 -1 0 -500
0 0 1 -1 0 -100
0 0 0 0 1 500

Thus we can conclude that E = 500 and that A = D - 100, B = D - 500
and C = D - 100, where D is free. Since we want all the flows to
be positive we see that we need D > 500.

Now we can use this solution to answer some questions.
See if you can use this solution to answer these questions:

- Suppose we need to work on road B, can we close it down
completely? If not, what is the least amount of traffic
we can have on that road?
- Same question, but for road C.
- Same question, but for road E.

Answers: (1) Yes, if B = 0, then D = 500 and all the other
flows are positive. (2) No, since D > 500, we must have
C > 400, thus the least we can have is 400. (3) No, there
is an overall conservation of flow into the city equals flow
out and thus we must have E = 500.

##### Electrical Networks

Read Section 11.2 in the book. Besides conservation (known
as Kirchoff's Current Law) we have to use Ohm's Law and
Kirchoff's Voltage Law to get our system of equations.

#### Exercises

1. For this English roundabout, find the general solution and find the smallest
possible value for each flow amount:

2. Design and solve a traffic flow problem.

3. Exercise 1 in Section 11.2

4. Exercise 4 in Section 11.2

Print out your solutions with your name and turn it in to Dr. Collins
or Eliza.

Mail: