Optimal back-off distribution for maximum weighted throughput in CSMA

We consider a generalized version of Carrier-Sense Multiple Access (CSMA), where the contention window size is a constant and the back-off probability distribution can be varied. We address the optimization of a weighted throughput metric, identifying the optimal back-off Probability Density Function (PDF). We give a simple fixed-point algorithm to compute the optimal PDF and prove that the solution is unique. The weighted throughput definition caters for aspects other than the mere channel utilization. It reduces to plain utilization (normalized throughput) when all weights are equal to 1. We also reconnect our result to the classic analysis of saturated non-persistent CSMA, as introduced in the seminal paper by Tobagi and Kleinrock, proving formally that the modeling assumptions of that work, that lead to a Geometric PDF of back-off, actually correspond to the throughput-optimal choice, provided that the ratio of the Geometric PDF is suitably chosen.