A Formal Proof of the Optimal Frame Setting for Dynamic-Frame Aloha With Known Population Size

摘要

In dynamic-frame Aloha, subsequent frame lengths must be optimally chosen to maximize throughput. When the initial population size 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, to the best of our knowledge, a formal proof of this result is still missing, and is provided here. As byproduct, we also prove that the asymptotic efficiency in the optimal case is , provide tight upper and lower bounds for the length of the entire transmission period, and show that its asymptotic behavior is with .