Failure Correction Techniques for Large Disk Array

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Failure Correction Techniques for Large Disk Array. Garth A. Gibson, Lisa Hellerstein et al. University of California at Berkeley. What is the problem?. Disk arrays can increase I/O bandwidth and access parallelism The chance of data loss increases with the increasing number of disk arrays. - PowerPoint PPT Presentation

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  • Failure Correction Techniques for Large Disk ArrayGarth A. Gibson, Lisa Hellerstein et al.University of California at Berkeley

  • What is the problem?Disk arrays can increase I/O bandwidth and access parallelismThe chance of data loss increases with the increasing number of disk arrays

    Figure 1. The mean time to data loss (MTTDL) in a single-erasure-correcting array.

  • Types of data failureTransient or noise-related errors: Correct by repeating the offending operation or by applying per sector error-correction facilitiesMedia defects: detect and mask at the factoryCatastrophic failures -- Head crashes or failures of the read/write or controller electronics

  • The goal of this paperAvoid loss of user dataRecover the catastrophic disk failuresMake disk arrays as reliable as an individual disk

  • Concept 1 -- erasure-correcting codes and error-correcting codesErasure-correcting codes are designed to recover erased bits in a message word An unreadable bit is called an erasureThe position of the erased bits are knownFor a catastrophic disk failure, the bits on a failed disk can be designated as unreadableError-correcting codes are designed to correct messages in which some of the bits may have been flipped, but the positions of those bits are unknown.

  • Concept 2 -- Redundancy MetricsDisk as stack of bits -ith.bit in each disk forms the ith.Codeword in the redundancy encoding

    Mean time to data loss (MTTDL): measure of reliabilityCheck disk overhead: check disks/data disksUpdate penalty: number of check disks to be updatedGroup size: the information and check disk that must be accessed during the reconstruction of a failed disk form a group

  • 1d - ParitySingle-erasure-correction schemeFor G data disks, one check disk with parity of all G disks.

    G = 4 Overhead: 1/G Update penalty: 1 Group size: G+1

  • 2d - Parity Double-erasure-correction scheme G2 data disks arranged in 2-dimensional array For each row and each column, one check disk stores parity for that row or columnG = 4 Check disk Overhead: 2G/G2 =2/G Update penalty: 2 Group size = G+1

  • N-dimensional parity (Nd-parity)N-erasure-correction schemeCheck disk overhead: NG(N-1) / GN = N/GUpdate penalty: NGroup size: G+1

  • Linear Codes Contain the original information unmodified within each codeword and compute the check bits of each codeword as the parity of subsets of the information bits

    Codeword = 1 1 1 1 Parity

  • Parity Check Matrix H = [P | I]Fig. 4 How to compute the check parity bit? H*X = 0 First row of H = [100101 100] X = [111010 x1 x2 x3] P I H*X = 1+0+0+0+0+0+x1+0+0 = 0 x1=1

  • Parity Check Matrix for 1d-parity and 2d-parityFig. 5

  • Properties of the parity check matrixExpress in terms of a parameter, t, whose value is between 0 and cH will allow any t erasures to be correctedH will allow any t errors to be detectedThe minimum number of bits in which any two codewords differ, known as the distance of the code, is at least t+1Any set of t column selected from will be linearly independent

  • Implementing ReconstructionFig. 6(a). When 4 disks fail in a 16 information disk 2d-parity array, the controllers allow us to identify which disks need to be repaired and reconstructed.

    0 10000 01100 00000 00010 00000 01000 10001 0011

  • Implementing Reconstruction cont.Fig. 6(b) Apply elementary row operations (the essence of Gaussian elimination) to find a matrix M, such that the product MB has the 4*4 identity matrix in its first four rows.

  • Elementary operationIf we interchange two equation, the new system is still equivalent to the old one.If we multiply an equation with a nonzero number, the new system is still equivalent to the old one.Replacing one equation with the sum of two equation, we obtain an equivalent system

    Example: x + y + z = 0 (1) x - 2y + 2z = 4 (2) x + 2y - z = 2 (3)

    (3) - (1) to replace (3) x + y + z = 0 (1) x - 2y + 2z = 4 (2) y - 2z = 0 (4)

    (2)-(1) to replace (2) x + y + z = 0 (1) - 3y +z = 4 (5) y - 2z = 0 (4)

    (5)+(4)*3 to replace (4) x + y + z = 0 (1) - 3y +z = 4 (5) - 5z= 10 (6)

    result: x=4, y=-2, z=-2

  • Gaussian EliminationDefinition:Using elementary operation, in every step the new matrix was exactly the augmented matrix associated to the new system. Once we obtain a triangular matrix, write the associated linear system and then solve it.

    augmented matrix:

    1 1 1 0 1 -2 2 4 1 2 -1 2

    (3) - (1) to replace (3)

    1 1 1 0 1 -2 2 4 0 1 -2 2

    (2)-(1) to replace (2)

    1 1 1 0 1 -3 1 4 0 1 -2 2

    (5)+(4)*3 to replace (4)

    1 1 1 0 0 -3 1 4 0 0 -5 10

    Example:x + y + z = 0 (1)x - 2y + 2z = 4 (2)x + 2y - z = 2 (3)

    The linear equation :

    x + y + z = 0 - 3y +z = 4 - 5z= 10

  • Implementing Reconstruction cont.Fig. 6 (C) The first 4 rows of MA describe the operations that must be performed to reconstruct our 4 disks.012 34567 89 1511 10 0 0000000 1000000001 00 0 0100010 0000010001 01 1 0100010 0100010000 00 0 0000111 0001000010 01 0 1000100 00001000

  • The position for codes with t-erasure-correctionBe implemented in softwareRun in an I/O processorSoftware learns of failures directly from disk controllers

  • ConclusionImplement the redundancy codes for disk arraysMinimize the number of check disks that must be updated whenever an information disk is updatedImprove the reliability of disk arrays

  • QuestionWhat is codeword for redundancy disk?List three redundancy metricsWhat are 1d-parity and 2d-parity schemes?What mathematical operation to be used for recovering failed disk?

    Erasure is an unreadable bit with a known positionA catastrophic disk failure is one kind of erasureThe frame is a one dimensional parity. G is the number of data disks in one group. There is one check disk storing the parity of G data disks.The figure is two dimensional parity. There are G square data disks that arranged in two- dimensional array.One check disk stores parity for that row or column.In two dimensional parity, each data disk has contribution to two groups1d - parity and two-d parity can be extended to n-dimensional parity.Linear codes consist of the information disk bits and check disk bit that is computed according to the parity of information disk bits.Linear codes can be defined as a parity check matrix.C is the number of check bits, k is the number of information bits. Vector x represents a codeword.Every column of the matrix represents the corresponding disk.Each one of this column contributes to the group in its row.For example, disk 5 contributes to this group in 1d -parity,

    Shaded column represents the failed data.Matrix H can be changed into two parts, A and B.A represents good disk and check disk, B represents the failed disk. Vector X can be divided to two parts, d and y.d denotes the good one, y denotes failed one