A traffic engineer wants to know whether measurements of traffic flow entering and leaving a road network are sufficient to predict the traffic flow on each street in the network. For example, consider the network of one-way streets shown in Figure 6.4-2. The numbers in the figure give the measured traffic flows in vehicles per hour. Assume that no vehicles park anywhere within the network. If ‘possible, calculate the traffic flowsƒf1,ƒf2,ƒf3, and
ƒ 4. If this is not possible, suggest how to obtain the necessary information .
The flow into intersection I must equal the flow out of the intersection, which gives us
First check the ranks of A and [A b] using the MATLAB rank command. Both have a rank of three, which is less than the number of unknowns, so we can determine three of the unknowns in terms of the fourth. Thus we cannot determine the traffic flows based on the given measurements. This example shows that it is not always possible to find a unique, exact solution even when the number of equations equals the number of unknowns.
We can easily solve this system as follows: ƒ1 = 300 ƒ4. ƒ2 = 200 + ƒ4. and ƒ3 =
800 – ƒ4
If we could measure the flow on one of the internal roads. say. ƒ4.then we could compute the other flows. So we recommend that the engineer arrange to make this additional measurement.