Alpha-Fair Resource Allocation under Incomplete Information and Presence of a Jammer

In the present work we deal with the concept of alpha-fair resource allocation in the situation where the decision maker (in our case, the base station) does not have complete information about the environment. Namely, we develop a concept of a-fairness under uncertainty to allocate power resource in the presence of a jammer under two types of uncertainty: (a) the decision maker does not have complete knowledge about the parameters of the environment, but knows only their distribution, (b) the jammer can come into the environment with some probability bringing extra background noise. The goal of the decision maker is to maximize the a-fairness utility function with respect to the SNIR (signal to noise-plus-interference ratio). Here we consider a concept of the expected a-fairness utility function (short-term fairness) as well as fairness of expectation (long-term fairness). In the scenario with the unknown parameters of the environment the most adequate approach is a zero-sum game since it can also be viewed as a minimax problem for the decision maker playing against the nature where the decision maker has to apply the best allocation under the worst circumstances. In the scenario with the uncertainty about jamming being in the system the Nash equilibrium concept is employed since the agents have non-zero sum payoffs: the decision maker would like to maximize either the expected fairness or the fairness of expectation while the jammer would like to minimize the fairness if he comes in on the scene. For all the plots the equilibrium strategies in closed form are found. We have shown that for all the scenarios the equilibrium has to be constructed into two steps. In the first step the equilibrium jamming strategy has to be constructed based on a solution of the corresponding modification of the water-filling equation. In the second step the decision maker equilibrium strategy has to be constructed equalizing the induced by jammer background noise.