Three-Dimensional Redundancy Codes for Archival Storage J.-F. Pâris, U. of Houston D. D. E. Long, U. C. Santa Cruz W. Litwin, U. Paris-Dauphine

Background Archival files  Must be kept a long time  At lowest possible cost Emphasis on  Providing highest reliability at lowest cost  Update speed is less important Focus on multi-dimensional RAID arrays  Highly reliable  Very space-efficient

A two-dimensional RAID array D11 D12 P1 D21 D22 P2 Q1Q2 Four parity disks Four parity stripes Four data disks

A better array P1 D13 D14 D34 D23 D24 D12 P2 P3 P4 Four parity disks Four parity stripes Six data disks

Can we do better? Use a three-dimensional organization Replace parity stripes by parity planes  Each parity plane will contain one parity disk  Place data disks will at the intersections of three parity planes

 Parity planes α, β, γ and δ  Four data disks αβγ,αβδ, αγδ and βγδ Example

With n p parity disks, we can protect data disks against all triple failures 2-D organizations with same number of parity disks could only protect data disks and only against all double failures Advantages

More data disks per parity disk

More protection at a lower cost

Drawback of 3-D arrays More complex update procedure  Each time we modify a data block, we have to update three parity blocks Not an issue for data that are rarely updated  Archives, media

Handling quadruple failures Only a few specific quadruple failures are fatal We show that array can tolerate fraction of all quadruple failures

Selected results Compared the MTTDL of a 3-D array with 20 data disks and 6 parity disks with those of  Two RAID arrays with 10 data disks and 3 parity disks each  60 disks using three-way mirroring to store the equivalent content of 20 data disks  A 2-D array with 21 data disks and 7 parity disks under standard stochastic assumptions

System Parameters Disk mean time to fail was assumed to be 100,000 hours (11 years and 5 months)  Corresponds to a failure rate  of 8 to 9 percent per year  High end of failure rates observed by Schroeder and Gibson and Pinheiro et al. Disk repair times varied between 12 hours and one week

Comparing MTTDLs

Conclusion 3-D RAID arrays require  Fewer parity disks than comparable RAID array organizations to achieve  Higher MTTDLs Sole limitation is cost of updates

Work in Progress Can we build zero-maintenance disk arrays?  Start with a 3-D RAID array  Add enough spares to last several years  Critical factor is failure rate of unused spares Potential for one or two MS theses  Require willingness to learn Python

Extra Slides

Our Model Device failures are mutually independent and follow a Poisson law  A reasonable approximation Device repairs can be performed in parallel Device repair times follow an exponential law  Not true but fairly robust H.-W. Kao, J.-F. Paris, T. Schwarz, S. J., and D. D. E. Long, A Flexible Simulation Tool for Estimating Data Loss Risks in Storage Arrays, Proc. MSST Symposium, May 2013.

State Diagram  State 0 is initial state   is the fraction of quadruple disk failures that result in a data loss