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

Abstract

In Dynamic-Frame Aloha subsequent frame lengths must be optimally chosen to maximize throughput. When the initial population size $\mathcal{N}$ 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 $\zeta =0.5/\ln (1-e^{-1})$.

22 pages, submitted to IEEE Trans. on Information Theory

References in corpus (1)

• An Asymptotically Efficient Backlog Estimate for Dynamic Frame Aloha

Cited by in corpus (1)

• An Asymptotically Efficient Backlog Estimate for Dynamic Frame Aloha