At the encoding, if bo,bl,..., bm_l is a string of information bytes, according to (11) ..... 1987. M. Blaum,. J. Brady,. J. Bruck and J. Menon,. âEVENODD: An Optimal.
EVENODD:
An
Optimal
Scheme
Failures Mario
Blaurn*
in RAID Jim
Bradyt
*IBM
Alrnaden
for
Tolerating
Double
Disk
Architectures Jehoshua
Bruck*
Jai
Menon*
Research Center
San Jose, CA 95120 {blaurn,bruck,menonjm}
(%dmaden.ibm.com
tIBM ssJj San Jose, CA 95193
Abstract We present ODD,
for
a novel
tolerating
architectures. for
EVENODD
tolerating
regani
to both storage the addition
consists
of simple
advantage
parity
that
exclusive-OR which
is typically
dard RAID-5
controllers. on standani
Hence,
controllers
The only optimal
two extra disks) is based on Reed-Solomon corncting
codes, requires computation
and results example,
in a more
we show
operations
involved
disk array
with
required
that
complex
implementation.
the number
in implementing
is complete.
First,
However, factors
growth
tional
(i.e.
accelerate,
loss of no more than improving
one disk will
For
quately
disk MTBF compensate
in a
15 disks is about 50% of the number
As a result,
when using the RS scheme.
Disk Arrays
or decreasing
of interest
and in attempting
Introduction Disk arrays
5 disk arrays, signing highly
[16], in particular
number
RAID-3
the exclusive-OR
of disks is maintained
of data
from
on a redundant
[4, 5, 8, 13].
correcting
and RAID-
ity than
have become an accepted way for deavailable and reliable disk subsystems.
In such arrays, When
taneously
codes simple
terminology,
higher
disk.
diska fail simul-
(in coding
capabiltheory
whose lc~cation is
Theoretically,
in order
to retrieve
(erased)
disks, we need at least two
data loss (MTTDL)
for recovering
is proportional
245 @ 1994 IEEE
in Large
lost in two failed
redundant disks (in coding the Singleton bound [11]).
$03.00
of
it will not.
correcting
is an error
by exclusive-ORing the data on the surviving diska, and writing this into a spare disk. The mean time to
1063-6897/94
that
known).
some
a disk fails, the data on it can be reconstructed
of such a system
in the number
the use of erasure-
is suggested
an erasure
can ade-
to design systems that
For this,
[8] with parity
whether
MTTR
has arisen
will not lose data even when multiple
1
tradi-
prove to be inade-
and concludes
a lot
that
[6] explores
for the increase
disks per installation,
videa~ and fax.
from the simultaneous
quate by the year 2000 [6]. Also,
fields
for data are
and bIy the in-
it was shown
arrays which can protect
because
are Ibecoming
requirements
caused by normal
of
the average
is growing
disk form
Second, installation
Such
when the number
of disks in an installation
As these trends
known
of exclusive-OR EVENODD
are lost if a second
crease in new forms of data like audio,
(RS) error-
over finite
is small.
increasing,
can be
storage
disks in the subsystem
smaller.
without
previously
redundant
MTTDL
number
in stan-
EVENODD
RAID-5
changes.
scheme that employes
present
disk [16]. Data
have acceptable
of two reasons.
requires
failures
of diska and the mean time to reconstruct
the failed
arrays
A ma-
it only
proportional
disk fails before the reconstruction
with
disks and
computations. is that
to the square of
and inversely
(MTTR)
scheme
EVENODD
of only two redundant
implemented any hardware
known
between
(MTBF)
the number
in RAID
that is optimal
and performance.
of EVENODD
hardware,
we call EVEN-
is the first
double disk failures
employs jor
method,
up to two disk failures
to the square of the disk mean time
the
information
the information
theory, this is known as A natural scheme, then, lost
in two
disks,
is
using
the so called
Reed-Solomon
codes [11].
ever, Reed-Solomon
codes involve
nite fields.
be desirable
It would
exclusive-OR parity.
operations
This
capabllit
operations
in [17], although
drawback
error propagation. volutional
type,
Moreover, there
the end of the data.
tion
there
corresponding
this code
For higher
redundancy
correcting
error in such applications. as follows:
the encoding scheme.
decoding
after the failure
procedure
of one or two disks.
capability,
by comparing
Reed-Solomon
codes.
In Section
mance issues of our scheme.
Therefore,
present
the problem
exclusive-OR
still
operations
is finding
codes based on
and of block
type.
in [1, 4, 9, 10] and later generfllzed
in [5] for multiple
erssures.
slthough
very
tations
simple,
(which
In this paper,
we present
procedure
that
and does not
We also present EVENODD single
parity
twice
recursive
as many simple
exclusive-OR
parity
scheme;
computations.
requires
operations
of
a prime
to
array quired
with
of exclusive
OR oper-
EVENODD
in a disk
15 disks is about
50% of the number
to having
an optimal
procedure,
re
number
of operahas the
operations
are very efiicient.
a disk sector is modified, wiU need to be modified here that correcting dancy
EVENODD
algorithms.
corresponds
Hence,
efficient
arbitrary
capacity
symbols
separately.
an Abelian
EVENODD
m -1
magnetic
over, the decoding
recording algorithm
on
can handle
by assuming
that
(all the informa-
the presentation,
we will
(in
fact,
group).
in terms of the redun-
of m – 1
in some of our
each symbol
is a bit.
to assume that
the sym-
it can be shown
A practical
that
our
are elements
implementation
in
is to
as an 8-bit byte, and to assume that (half
a sector).
Notice
that
257
number.
Baaed on the assumptions tolerating
of
works for disks with each block
assume that
it is not necessary
we assume
m – 1 symbols
For simplicity,
= 256 symbols
is a prime
two disk
failures
above,
the problem
can be described
of
as fol-
lows:
and decoding
can be used in other
Problem
applications where there is a need of correcting two erased symbols with low complexity; for example, in multi-track
m is
has no effect
no information
by treating
consider a symbol
when
to a new 2-erasure
encoding
that
EVENODD
of disks simply
to simplify
bols are bhmry
only two other disk sectors at the same time. We note
code which is optimal
and has very
In particular,
requirement
scheme works even when the symbols
advantages that the encoding procedure can be implemented using existing parity hardware and that small write
of RAID-5
We assume
on it. Our procedure
examples,
EVENODD
of
but it is easy to
each of the m disks has only
However, In addition
number
In order
when using the RS scheme.
tions for the encoding
This
is, our
tion bits are O).
we
codes; for example,
the number
That
is an extension
is dedicated),
of EVENODD.
are disks with
information
in implementing
number.
an arbitrary there
parity
among
effects when re-
are performed. which
the
It is pos-
the redundancy
is distributed).
the optimaMy
about
since
operations
with
m disks while
in the last two disks.
is of a scheme
parity
that
involved
7, we
axe m + 2 disks
how it can be made an extension
(where
have two redundant disks. EVENODD is substantially more efficient when compared to Reed-Solomon ations
there
in the first
to distribute
(where
imagine
We
compared
is optimal
write
RAID-4
of implementation
this
is stored
description
oper-
procedure.
EVENODD
in Section
remarks.
that
stored
however,
peated
en-
it to that of the traditional
codes.
of traditional
6 we discuss perfor-
Finally,
all disks in order to avoid bottleneck
scheme aa well as to the scheme based
on Reed-Solomon the
sible,
operations.
the complexity
and compared
redundancy
at the encoding
decoding
assume
the information
to implement
a novel and efficient
involve
In
Encoding We will
compu-
is based on exclusivc+OR
a simple
have calculated
recursive
hardware)
small write
2
those solutions,
and hard
exclusive-OR
process and during
ations
involve
are inefficient
using existing
coding
still
operations.
A solu-
tion was obtained
However,
some concluding
the
be used
of implementation
it to that
the codes in [7, 14, 15] have the same dhmchantages.
will
In Section 4 we ad-
of small write
of EVENODD
used by our
3 we present
which
Section 5 we address the complexity
at
in the next sec-
procedure
In Section
dress the implementation
is an infinite
since the code is of con-
is an overhead
we describe
new EVENODD
when the error correcting
y of the code is broken,
one random
The paper is organized
over fi-
to have codes doing
only, as in the case of simple
was achieved
has the following
correct
How-
array,
Consider
number,
the (m – 1) x (m+
such that
symbol
aij,
2) O~
i
