Estimation of traffic flows

Linear Equations > Applications > Traffic flow estimation

The basic traffic flow estimation problem involves inferring the amount of cars going through links based on information on the amount of cars passing through neighboring links.

For the simple problem above, we simply use the fact that at each intersection, the incoming traffic has to match the outgoing traffic. This leads to the linear equations

$begin{array}{rrcl} mbox{Intersection A:}&x_4 + 610&=&x_1 + 450, mbox{Intersection B:}&x_1 + 400&=&x_2 + 640, mbox{Intersection C:}&x_2 + 600&=&x_3, mbox{Intersection D:}&x_3&=&x_4 + 520. end{array}$

We can write this in matrix format: Ax = y , with

A = left( begin{array}{cccc}-1&0&0&11&-1&0&00&1&-1&00&0&1&-1 end{array}right), ;; y = left( begin{array}{c}-160240-600520end{array}right).

The matrix is nothing else than the incidence matrix associated with the graph that has the intersections as nodes and links as edges.