Explicit back-off rates for achieving target throughputs in CSMA/CA   networks
  CSMA/CA networks have often been analyzed using a stylized model that is fully characterized by a vector of back-off rates and a conflict graph. Further, for any achievable throughput vector $\vec \theta$ the existence of a unique vector $\vec \nu(\vec \theta)$ of back-off rates that achieves this throughput vector was proven. Although this unique vector can in principle be computed iteratively, the required time complexity grows exponentially in the network size, making this only feasible for small networks.   In this paper, we present an explicit formula for the unique vector of back-off rates $\vec \nu(\vec \theta)$ needed to achieve any achievable throughput vector $\vec \theta$ provided that the network has a chordal conflict graph. This class of networks contains a number of special cases of interest such as (inhomogeneous) line networks and networks with an acyclic conflict graph. Moreover, these back-off rates are such that the back-off rate of a node only depends on its own target throughput and the target throughput of its neighbors and can be determined in a distributed manner.   We further indicate that back-off rates of this form cannot be obtained in general for networks with non-chordal conflict graphs. For general conflict graphs we nevertheless show how to adapt the back-off rates when a node is added to the network when its interfering nodes form a clique in the conflict graph. Finally, we introduce a distributed chordal approximation algorithm for general conflict graphs which is shown (using numerical examples) to be more accurate than the Bethe approximation. 
Benny Van Houdt
cs.NI cs.IT math.IT