Reliability of Erasure Coded Storage Systems: A Geometric Approach

We consider the probability of data loss, or equivalently, the reliability function for an erasure coded distributed data storage system. Data loss in an erasure coded system depends on probability distributions for the disk repair duration and the disk failure duration. In previous works, the data loss probability of such systems has been studied under the assumption of exponentially distributed disk failure and disk repair durations, using well-known analytic methods from the theory of Markov processes. These methods lead to an estimate of the integral of the reliability function. Here, we address the problem of directly calculating the data loss probability for general repair and failure duration distributions. An exact form and a closed limiting form is developed for the probability of data loss and it is shown that the probability of the event that a repair duration exceeds a failure duration is sufficient for characterizing the data loss probability for highly reliable systems. While it is known that mean repair duration and mean failure duration are not sufficient for this purpose, simple characterizations have not appeared in the literature. For the case of constant repair duration, we explore a geometric approach that relies on the computation of volumes of a family of polytopes that are related to the code. An exact calculation is provided and an upper bound on the data loss probability is obtained by posing the problem as a set avoidance problem. Theoretical calculations are compared to simulation results.