1. Disk Failures
Xiaqing He
ID: 204
Dr. Lin

2. Content
Focus on : “How to recover from disk crashes” -- common term RAID = redundancy array of independent disks Mirroring—RAID level 1; Parity checks--RAID 4; Improvement--RAID 5; RAID 6;

3. 1) Mirroring The simplest scheme to recovery from Disk Crashes
Mirror: making two or more copied of the data on different disks
-- save data in case of one disk will fail;
--divide data on several disks and let access to several blocks at once

4. For mirroring, the only way data can be lost if there is a second (mirror/redundant) disk crash while the first (data) disk crash is being repaired.
Possibility:
Suppose:
One disk_mean time to failure : 10 years;
One of the two disk_mean time to failure : 5 years;
The process of replacing the failed disk: 3 hours=1/2920 year;
So:
the possibility of the mirror disk will fail=1/10 * 1/2920 =1/29,200;
The possibility of data loss by mirroring: 1/5 * 1/2920 = 1/146,000

5. 2)Parity Blocks Disadvanges of Mirroring: uses so many redundant disks
RAID level 4: uses only one redundant disk
How this one redundant disk works?
-- modulo-2 sum;
-- the j th bit of the redudant disk is the modulo-2 sum of the j th bits of all the data disks.
Example

6. Example Data disks: Disk1: 11110000 Disk2: 10101010 Disk3: 00111000
Redundant disk:
Disk4:

7. cont. RAID 4 Reading Similar with reading blocks from any disk;
Writing:
1) change the data disk;
2) change the corresponding block of the redundant disk;
Why?
--hold the parity checks for the corresponding blocks of all the data disks

8. cont. RAID 4_ writing For a total N data disks: 1) naïve way:
read N data disks and compute the modulo-2 sum of the corresponding blocks;
rewrite the redundant disk according to modulo-2 sum of the data disks;
2) better way:
Take modulo-2 sum of the old and new version of the data block which was rewriten;
Change the position of the redundant disk which was 1’s in the modulo-2 sum;
Example

9. Example Data disks: Disk1: 11110000 Disk2: 10101010  01100110
Modulo-2 sum of the old and new version of disk 2:
So, we need to change the positions1,2,5,6 of the redundant disk.
Redundant disk:
Disk4: 

10. Cont. RAID4_failure recovery
Redundant disk crash:
-- swap a new one and recompute data from all the data disks;
One of Data disks crash:
-- swap a new one;
-- recompute data from the other disks including data disks and redundant disk;
How to recompute?
-- take modulo-2 sum of all the corresponding bits of all the other disks

11. 3) An Improvement: RAID 5 Why need a improvement?
-- Shortcoming of RAID level 4: suffers from a bottelneck defect (when updating data disk need to read and write the redundant disk);
Principle of RAID level 5 (RAID 5):
--treat each disk as the redundant disk for some of the blocks;
Why it is feasible?
The rule of failure recovery for redundant disk and data disk is the same:
take modulo-2 sum of all the corresponding bits of all the other disks
So, there is no need to retreat one as redundant disk and others as data disks

12. Cont. RAID 5
How to recognize which blocks of each disk treat this disk as redundant disk?
-- if there are n+1 disks which were labeld from 0 to N, then we can treat the ith cylinder of disk J as redundant if J is the remainder when I is divided by n+1;
Example;

13. Cont. RAID 5_example N=3; The first disk, labeled as 0 : 4,8,12…;
The second disk, labeled as 1 : 1,5,9…;
The third disk, labeled as 2 : 2,6,10…;
……….
Suppose all the 4 disks are equally likely to be written, for one of the 4 disks, the possibility of being written:
1/4 + 3 /4 * 1/3 =1/2
N=m : 1/m +(m-1)/m * 1/(m-1) = 2/m

14. 4) Coping with multiple disk crashes
RAID 6
– deal with any number of disk crashes if using enough redundant disks
Focus on: a system of seven disks ( four data disks_numer 1-4 and 3 redundant disks_ number 5-7);
How to set up this 3*7 matrix ?
1)every column values three 1’s and 0’s except for all three 0’s;
2) column of the redundant disk has single 1’s;
3) column of the data disk has at least two 1’s;

15. Cont.) Coping with multiple disk crashes
Reading:
read form the data disks and ignore the redundant disk
Writing:
Change the data disk
change the corresponding bits of all the redundant disks

16. Cont.) Coping with multiple disk crashes
In those system which has 4 data disks and 3 redundant disk, how they can correct up to 2 disk crashes?
Suppose disk a and b failed:
find some row r in which the column for a and b are different;
Compute the correct b by taking modulo-2 sum of the corresponding bits from all the other disks other than b which have 1’s in row r;
After getting the correct b, Compute the correct a with all other disks available;
Example