Unifying View on Min-Max Fairness, Max-Min Fairness, and Utility Optimization in Cellular Networks

We are concerned with the control of quality of service (QoS) in wireless cellular networks utilizing linear receivers. We investigate the issues of fairness and total performance, which are measured by a utility function in the form of a weighted sum of link QoS. We disprove the common conjecture on incompatibility of min-max fairness and utility optimality by characterizing network classes in which both goals can be accomplished concurrently. We characterize power and weight allocations achieving min-max fairness and utility optimality and show that they correspond to saddle points of the utility function. Next, we address the problem of the difference between min-max fairness and max-min fairness. We show that in general there is a (fairness) gap between the performance achieved under min-max fairness and under max-min fairness. We characterize the network class for which both performance values coincide. Finally, we characterize the corresponding network subclass, in which both min-max fairness and max-min fairness are achievable by the same power allocation.