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

In dynamic-frame Aloha, subsequent frame lengths must be optimally chosen to maximize throughput. When the initial population size 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, 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 e^-1, provide tight upper and lower bounds for the length of the entire transmission period, and show that its asymptotic behavior is ~ne-ζ ln(n) with ζ =-0.5/ln (1-e^-1).