In asynchronous arbiters, failures may happen, caused by metastable operations. The purpose of this study is to derive a formula to estimate such failures in a ring arbiter as mean time between failures (MTBF), under the condition that incidences of requests issued in all devices are different from each other. The operation of the arbiter is formularized by a markov chain. This chain is used to decide the probability at which each of possible failures contributes to MTBF. The sum of such probabilities gives the MTBF which can be represented as a sum of a finite number of terms. As an example, MTBF of a ring arbiter composed of 3 cells is shown.