Adaptive spatial aloha, fairness and stochastic geometry

This work aims at combining adaptive protocol design, utility maximization and stochastic geometry. We focus on a spatial adaptation of Aloha within the framework of ad hoc networks. We consider quasi-static networks in which mobiles learn the local topology and incorporate this information to adapt their medium access probability (MAP) selection to their local environment. We consider the cases where nodes cooperate in a distributed way to maximize the global throughput or to achieve either proportional fair or max-min fair medium access. In the proportional fair case, we show that nodes can compute their optimal MAPs as solutions to certain fixed point equations. In the maximum throughput case, the optimal MAPs are obtained through a Gibbs Sampling based algorithm. In the max min case, these are obtained as the solution of a convex optimization problem. In the proportional fair case we show that, when the nodes form a homogeneous Poisson point process (PPP) in the Euclidean plane, the distribution of the optimal MAP can be obtained from that of a certain shot noise process with respect to this Poisson point process and that the mean utility can also be derived from this distribution. Numerical results illustrate our findings and quantify the gains brought by spatial adaptation in such networks.