Asymptotically Efficient Backlog Estimators for Frame Aloha

The dynamic frame Aloha protocol has been shown to reach efficiency e−1 when the number N of tags to be identified is known and approaches infinity. Results available in literature do not achieve efficiency e−1 for large N when N is unknown. In this paper we analytically show that the best tag estimation procedure let the protocol reach efficiency e−1. However, the convergence to this result shows an oscillatory behavior that on average goes like 1/ ln N. The oscillation can be reduced at will at the expenses of the convergence speed. A practical algorithm is proposed for achieving the claimed efficiency and numerical simulations are shown to validate the results.