Reliability of Erasure Coded Storage Systems: A Geometric Approach

TL;DR

This paper introduces a geometric method to directly compute data loss probabilities in erasure coded storage systems with general repair and failure durations, extending beyond exponential assumptions.

Contribution

It develops a geometric approach to calculate data loss probabilities for arbitrary duration distributions, including exact formulas and bounds, improving reliability analysis methods.

Findings

Closed-form limiting expression for data loss probability.

Geometric approach using polytope volume calculations.

Comparison of theoretical results with simulations.

Abstract

We consider the probability of data loss, or equivalently, the reliability function for an erasure coded distributed data storage system under worst case conditions. 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. 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 the case of constant repair duration, we develop an expression for the conditional data loss probability given the number of failures experienced by a each node in a given time window. We do so by developing 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.