A formal proof of the optimal frame setting for Dynamic-Frame Aloha with known population size

TL;DR

This paper provides a formal proof that setting frame length equal to backlog size in Dynamic-Frame Aloha with known population size maximizes throughput, confirming the asymptotic efficiency as e^{-1}.

Contribution

It offers the first formal proof of the optimal frame setting strategy in Dynamic-Frame Aloha with known population size and analyzes its asymptotic efficiency.

Findings

Maximum efficiency achieved by setting frame length equal to backlog size.

Asymptotic efficiency approaches e^{-1}.

Provides bounds and asymptotic behavior of total transmission period.

Abstract

In Dynamic-Frame Aloha subsequent frame lengths must be optimally chosen to maximize throughput. When the initial population size ${\cal N}$ is known, numerical evaluations show that the maximum efficiency is achieved by setting the frame length equal to the backlog size at each subsequent frame; however, at best of our knowledge, a formal proof of this result is still missing, and is provided here. As byproduct, we also prove that the asymptotical efficiency in the optimal case is $e^{-1}$, provide upper and lower bounds for the length of the entire transmission period and show that its asymptotical behaviour is $\sim ne-\zeta \ln (n)$, with $\zeta=0.5/\ln(1-e^{-1})$.