Fuzzy Approaches to Proportional Fairness

Proportional fairness has been shown to maximize the aggregate utility of rate control for elastic traffic in a resource sharing communication network, and has been applied to a broad range of resource allocation problems. For a refined analysis, however, the representation of proportional fairness as a relation between vectors with positive components will often not provide the level of detail that is needed. Therefore, we study the representation as a fuzzy relation, and propose several ways to specify a measure function to allocate a degree of (proportional) relatedness. The approaches are based on combinatorial aspects, especially size and number of related sub vectors, and geometric aspects, especially the minimum effort to change a vector to become related. A case study demonstrates that the introduced fuzzy fairness relations can be used to numerically evaluate the suitability of the maxmin fair state to represent the proportional fair state in a resource sharing network problem with maximum link capacities.