Three-Dimensional Redundancy Codes for Archival Storage

1. Three-Dimensional Redundancy Codesfor Archival Storage J.-F. Pâris, U. of HoustonD. D. E. Long, U. C. Santa CruzW. Litwin, U. Paris-Dauphine

2. 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

3. A two-dimensional RAID array D12 D11 P1 P2 D22 D21 Q1 Q2 • Four parity disks • Four parity stripes • Four data disks

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

5. 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

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

7. Advantages • With npparity 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

8. More data disks per parity disk

9. More protection at a lower cost

10. 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

11. Handling quadruple failures • Only a few specific quadruple failures are fatal • We show that array can tolerate fractionof all quadruple failures

12. 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

13. System Parameters Disk mean time to fail was assumed to be 100,000 hours (11 years and 5 months) Corresponds to a failure rate l 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

14. Comparing MTTDLs

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

16. 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

17. Extra Slides

18. 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.

19. State Diagram • State 0 is initial state • a is the fraction of quadruple disk failures that result in a data loss