On the asymptotics of the average CRI length of the slotted ALOHA collision resolution algorithm

摘要

We provide uniform asymptotic expansions of a finite sum Ln,p of essentially geometric type, which arises in the investigation of the average CRI length of the well-known slotted ALOHA collision resolution algorithm with retransmission probability p. In particular, our investigations establish large regions of uniform validity w.r.t. p as n→∞. By means of a direct asymptotic method based on the analysis of a certain sum involving binomial coefficients, we obtain an asymptotic expansion of Ln, p, which is uniformly valid for p ⩾ n-0.51/2e, n→∞. The application of some well-established methods relying on complex analysis yields another result, this time uniformly valid for p ⩾ n-0.99 as n→∞.