Most Probable Paths to Data Loss: An Efficient Method for Reliability Evaluation of Data Storage Systems
The effectiveness of the redundancy schemes that have been developed to enhance the reliability of storage systems has predominantly been evaluated based on the mean time to data loss (MTTDL) metric. This metric has been widely used to compare schemes, to assess tradeoffs, and to estimate the effect of various parameters on system reliability. Analytical expressions for MTTDL are typically derived using Markov chain models. Such derivations, however, remain a challenging task owing to the high complexity of the analysis of the Markov chains involved, and therefore the system reliability is often assessed by rough approximations. To address this issue, a general methodology based on the direct-path approximation was used to obtain the MTTDL analytically for a class of redundancy schemes and for failure time distributions that also include real-world distributions, such as Weibull and gamma. The methodology, however, was developed for the case of a single direct path to data loss. This work establishes that this methodology can be extended and used in the case where there are multiple shortest paths to data loss to approximately derive the MTTDL for a broader set of redundancy schemes. The value of this simple, yet efficient methodology is demonstrated in several contexts. It is verified that the results obtained for RAID-5 and RAID-6 systems match with those obtained in previous work. As a further demonstration, we derive the exact MTTDL of a specific RAID-51 system and confirm that it matches with the MTTDL obtained from the methodology proposed. In some cases, the shortest paths are not necessarily the most probable ones. We establish that this methodology can be extended to the most probable paths to data loss to derive closed-form approximations for the MTTDL of RAID-6 and two-dimensional RAID-5 systems in the presence of unrecoverable errors and device failures. A thorough comparison of the reliability level achieved by the redundancy schemes considered is also conducted.