n this letter, we present an analysis tool for the convergence of a turbo decoder using the min-sum algorithm. First of all, we classify cycles on a turbo code factor graph. We define and develop a quantity which characterizes the trellis convergence and a probability of a message round on a cycle. Our analysis is then applied to an interleaver design for a turbo decoder using the min-sum algorithm. Since the probability of a message round is closely related to the convergence property, the constructed interleaver optimizes the convergence property. The performance is compared with several referential interleavers.