For a long time, the circularity of the tail-biting trellis is ignored in conventional decoding algorithms of Tail-Biting Convolutional Codes (TBCC). This kind of algorithm starts decoding from the fixed location, and consequently exhibits relatively lower decoding efficiency. For the first time, this paper proves that the decoding result of the tail-biting convolutional codes is independent on the decoding starting location. It means that the Maximum Likelihood (ML) tail-biting path, which starts from any location of the tail-biting trellises, is the global ML tail-biting path. Based on this observation, a new ML decoding algorithm is proposed. The new algorithm ranks the belief-value of each location on the trellis at first, and then selects the location with the highest belief- value as the decoding starting location. Compared with other existing ML decoders, the new decoder exhibits higher convergence speed.