An Asymptotically Efficient Backlog Estimate for Dynamic Frame Aloha

In this paper we investigate backlog estimation procedures for Dynamic Frame Aloha (DFA) in Radio Frequency Identification (RFID) environment. In particular, we address the tag identification efficiency with any tag number $N$, including $N\rightarrow\infty$. Although in the latter case efficiency $e^{-1}$ is possible, none of the solution proposed in the literature has been shown to reach such value. We analyze Schoute's backlog estimate, which is very attractive for its simplicity, and formally show that its asymptotic efficiency is 0.311. Leveraging the analysis, we propose the Asymptotic Efficient backlog Estimate (AE$^2$) an improvement of the Schoute's backlog estimate, whose efficiency reaches $e^{-1}$ asymptotically. We further show that AE$^2$ can be optimized in order to present an efficiency very close to $e^{-1}$ for practically any value of the population size. We also evaluate the loss of efficiency when the frame size is constrained to be a power of two, as required by RFID standards for DFA, and theoretically show that the asymptotic efficiency becomes 0.356.