Shortcut method for reliability comparisons in RAID

Given that the reliability of each disk in a disk array during its useful lifetime is given as r = 1 − ϵ with ϵ ≪ 1, we show that the reliability of a RAID disk array tolerating all possible n − 1 disk failures can be specified as R ≈ 1 − anϵn, where an is the smallest nonzero coefficient in the corresponding asymptotic expansion, e.g., for n-way replication R = 1 − ϵn. We compare the reliability of several mirrored disk organizations, which provide tradeoffs between reliability and load balancedness (after disk failure) by comparing their a2 values, which can be obtained via a partial reliability analysis taking into account a few disk failures. We next use asymptotic expansions to compare the reliability of hierarchical RAID disk arrays, which combine replication and rotated parity disk arrays (RAID5 and RAID6). Finally, we argue that the mean time to data loss in systems with repair is related to the reliability without repair. As part of this discussion we show how to estimate the mean time to data loss in RAID5 and RAID6 disk arrays without resorting to transient analysis.