Fair resource allocation in wireless networks in the presence of a jammer

We consider jamming in wireless networks in the framework of zero-sum games with α-utility functions. The base station has to distribute the power fairly among the users in the presence of a jammer. The jammer in turn tries to distribute its power among the channels to produce as much harm as possible. The Shannon capacity and SNIR optimization are particular cases of the proposed more general α-fairness SNIR based utility functions. Specifically, we consider two α-fairness utility functions, based on SNIR and Shifted SNIR. This game can also be viewed as a minimax problem against the nature. We show that the game has the unique equilibrium and investigate its properties. In particular, in several important cases we present the equilibrium strategies and the Jain's fairness index in closed form. It turns out that there is an important difference between SNIR and Shifted SNIR α-fairness utility functions. In the case of the SNIR based utility function all users obtain nonzero powers when α > 0. On contrary, when the Shifted SNIR based utility function is used, some users with bad channel conditions might not receive any power at all. We have also detected a surprising non-monotone behaviour of the Jain's fairness index in the case of the SNIR based utility function.