  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. The main performance analysis result of the paper is that
this type of distributed adaptation can be analyzed using stochastic geometry
in the proportional fair case. In this case, we show that, when the nodes form
a homogeneous Poisson point process in the Euclidean plane, the distribution of
the optimal MAP can be obtained from that of a certain shot noise process
w.r.t. the node Poisson point process and that the mean utility can also be
derived from this distribution. We discuss the difficulties to be faced for
analyzing the performance of the other cases (maximum throughput and max-min
fairness). Numerical results illustrate our findings and quantify the gains
brought by spatial adaptation in such networks.
