Link-traffic loop-free property in optimal routing for multi-class networks

Communications networks where nodes are interconnected by a generally configured manner and wherein there are multiple classes of users, each of which has its distinct generalized link costs (communication delays), are considered. Individually and overall optimal routing problems for multiclass networks are formulated along with the discussions on mutual equivalence between both problems, on the existence and uniqueness of solutions, and on the relation between the formulations with path and link flow patterns. We show that a link-traffic loop-free property holds within each class for both individually and overall optimal routing in a wide range of networks, and obtain the condition that characterizes the category of networks for which the link-traffic loop-free property holds.