Optimal capacity of p-peristent CSMA protocols

n this paper we deal with the characterization and computation of the $p$ value, say $p_{opt}$, corresponding to the maximum protocol capacity in $p$-persistent CSMA protocols. The contribution of this paper is twofold. First, we give an analytical justification, and a numerical validation of a heuristic formula widely used in the literature to characterize the $p_{opt}$. Second, we provide closed formulas for the $p_{opt}$, and we show that the optimal capacity state, given the message length distribution, is characterized by an invariant figure: the $Mp_{opt}$ product.