Fuzzification of Maxmin Fairness Relation Based on Subvector Dominance Degree

Maxmin fairness can be characterized as a state obtained from the application of the bottleneck flow control algorithm, allowing for a formally efficient definition of a fairness concept. But a number of problems with the related maxmin fairness vector relation, esp. non-uniqueness of the maximum, preference for improving least components, artefacts appearing in the discrete domain, and lack of numerical representation have already been noted. For this reason, we propose a fuzzy fairness, based on sub vector dominance. The fuzzy fairness, here called sub vector maxmin dominance degree, is based on the ratio of the number of all maxmin fair dominating sub vectors to the number of all possible sub vectors. It is shown how this definition extends the maxmin fairness relation, how it helps to solve the problems with maxmin fairness, and how it numerically emphasises fairness states against other states.