Redundancy: How Many Unreliable Spares are Needed for High Reliability and Confidence?

This paper investigates the number of redundant units needed to achieve high reliability with high confidence. The approach is developed for the case when the system failure rate is too high for a single unit to provide the required reliability over the mission duration. To achieve high reliability, N redundant units can be used, one operating unit and N - 1 spares. If the unit failure rate is f, the mission length is L, and f * L is small (not the case assumed here), the unit failure probability over the mission duration is F1 = f * L << 1. In this case, the probability that all N units will fail is Ffail = F1N, and the needed redundancy N = LN(F)/LN(F1). For the case of large f * L assumed here, F1 = f * L > 1, and F1 is the expected number of failures during the mission. (When F1 = f * L << 1, F1 is the probability that a unit will fail during the mission. When F1 = f * L > 1, F1 is the expected number of failures during the mission.) The needed redundancy, N, to achieve the required N redundant unit reliability, FN, can be computed using the cumulative Poisson distribution with mean equal to F1. The number of spares, N - 1, is increased until the probability - that the total number of failures will be less than N -1 - is equal to the required reliability. The confidence that this reliability can be achieved can be computed using the cumulative Poisson distribution or the chi-square distribution. Since the measured unit failure rate, f, has some probabilistic uncertainty, the actual failure rate will be randomly higher or lower. This means that the reliability of the N redundant systems will be overestimated about half the time. Adding more redundant units increases the confidence that the required reliability will be achieved. For a fixed number of redundant units, the expected reliability and confidence can be traded off, since lower reliability goals will be achieved with higher confidence. Both the desired reliability and confidence can be specified as initial requirements and the needed number of redundant units estimated using the measured failure rate.