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
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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

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

• Transient or noise-related errors
• Correct by repeating the offending operation
• or by applying per sector error-correction
  facilities
• Media defects
• detect and mask at the factory
• Catastrophic failures -- Head crashes or failures
  of the read/write or controller electronics

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The goal of this paper

• Avoid loss of user data
• Recover the catastrophic disk failures
• Make disk arrays as reliable as an individual disk

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Concept 1 -- erasure-correcting codes and
error-correcting codes

• Erasure-correcting codes are designed to recover
  erased bits in a message word
• An unreadable bit is called an erasure
• The position of the erased bits are known
• For a catastrophic disk failure, the bits on a
  failed disk can be designated as unreadable
• Error-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.

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Concept 2 -- Redundancy Metrics

• Disk 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
  reliability
• Check disk overhead check disks/data disks
• Update penalty number of check disks to be
  updated
• Group size the information and check disk that
  must be accessed during the reconstruction of a
  failed disk form a group

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1d - Parity

• Single-erasure-correction scheme
• For G data disks, one check disk with parity of
  all G disks.

• Overhead 1/G
• Update penalty 1
• Group size G1

G 4
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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 column

• Check disk Overhead
• 2G/G2 2/G
• Update penalty 2
• Group size G1

G 4
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N-dimensional parity (Nd-parity)

• N-erasure-correction scheme
• Check disk overhead NG(N-1) / GN N/G
• Update penalty N
• Group size G1

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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
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Parity Check Matrix H P I
Fig. 4
How to compute the check parity bit? HX 0
First row of H 100101 100 X
111010 x1 x2 x3
P I HX 100000x100 0
x11
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Parity Check Matrix for 1d-parity and 2d-parity
Fig. 5
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Properties of the parity check matrix

• Express in terms of a parameter, t,
• whose value is between 0 and c
• H will allow any t erasures to be corrected
• H will allow any t errors to be detected
• The minimum number of bits in which any two
  codewords differ, known as the distance of the
  code, is at least t1
• Any set of t column selected from will be
  linearly independent

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Implementing Reconstruction
0 1000 0 0110 0 0000 0 0001 0 0000 0 0100 0
1000 1 0011
Fig. 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.
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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 44 identity
matrix in its first four rows.
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Elementary operation
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 x4, y-2, z-2

• If 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

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Gaussian Elimination
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
Definition 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.
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
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Implementing Reconstruction cont.
012 34567 89 15 11 10 0 0000000
10000000 01 00 0 0100010 00000100 01 01
1 0100010 01000100 00 00 0 0000111
00010000 10 01 0 1000100 00001000
Fig. 6 (C) The first 4 rows of MA describe the
operations that must be performed to reconstruct
our 4 disks.
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The position for codes with t-erasure-correction

• Be implemented in software
• Run in an I/O processor
• Software learns of failures directly from disk
  controllers

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Conclusion

• Implement the redundancy codes for disk arrays
• Minimize the number of check disks that must be
  updated whenever an information disk is updated
• Improve the reliability of disk arrays

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Question

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