Deriving the Upper Bound of the Number of Sensors Required to Know All Link Flows in a Traffic Network

It is demonstrated that the minimum number of sensors required to know all link flows in a traffic network can be determined only if path information is available. However, not all paths need to be enumerated but, at most, a small subset defining the rank $r_{w}$ of the link-path incidence matrix ${\bf W}$ . If this rank for a reduced subset of paths is already $m - n$ , where $m$ and $n$ are the number of links and noncentroid nodes, respectively, we can conclude that $m - n$ sensors are sufficient. It is also shown that the formulas providing the dependent link flows in terms of the independent link flows can be obtained by the node-based or path-based approaches with the same results only when $r_{w} = m - n$ . Finally, an algorithm to obtain the small subsets of linearly independent path vectors is given. The methods are shown by a parallel network example and the Ciudad Real and Cuenca networks, for which the savings in link counts with respect to the $m - n$ bound are larger than 16%. The corresponding savings in path enumeration are larger than 80%.