A communication network can be modeled by a graph with weighted vertices and edges corresponding to the amount of traffic from sources and expected delays at links. We give a linear algorithm for computing the sum of all delays on a weighted cactus graphs. Cactus is a graph in which every edge lies on at most one cycle. The sum of delays is equivalent to the weighted Wiener number, a well known graph invariant in mathematical chemistry. Complexity of computing Wiener polynomial on cacti is discussed.