Adaptive CSMA under the SINR model: Fast convergence through local gibbs optimization

In this paper, we consider an adaptive CSMA based scheduling algorithm for a single-hop wireless network under a realistic SINR (signal-to-interference-plus-noise ratio) model for the interference, and propose an efficient local optimization based algorithm to estimate certain parameters of the algorithm called fugacities. It is known that adaptive CSMA based algorithms can achieve throughput optimality, by sampling feasible schedules from a Gibbs distribution with appropriate fugacities. Unfortunately, estimating the optimal fugacities for a desired service rate vector is an NP-hard problem. Further, the existing adaptive CSMA algorithms use a stochastic gradient descent based method, which usually entails an impractically slow (exponential in the size of the network) convergence to the optimal fugacities. In contrast, the convergence rate and the complexity of our algorithm is independent of the network size, and depends only on the neighborhood size of a link. In particular, in spatial networks where the neighborhood size does not scale with the network size, our algorithm is order optimal. We show that the proposed algorithm corresponds exactly to performing the well-known Bethe approximation to the underlying Gibbs distribution. We also consider two special cases of the SINR interference model and obtain the corresponding fugacities in closed form. Numerical results indicate that the proposed method achieves extremely fast convergence to near-optimal fugacities, and often outperforms the convergence rate of the stochastic gradient descent by a few orders of magnitude.