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is the sum of all storage durations, which is the life span of the data file. The derivation
of equation
(4.2)
is presented as follows:
Let event
A
j
be disk
D
surviving from
t
j
1
to
t
j
, where
j N
, the probability
that disk
D
survives from
t
0
to
t
n
can be described as
−
PAAA
(
...
)
. According to
nn
1
−
1
the property of conditional probability, we have:
PAAAPA AAPA AA
PA AAPA AA
PA APA
(
...
)
=
(
|
... )(
...
)
nn
−
1
1
n
n
−
1
1
n
−−
1
n
2
1
⋅= =
⋅
... (|
... )( |
...
)...
n
n
−
1
1
n
−
1
n
−
2
1
(| )( )
2
1
1
where
j j
1 2 1
indicates the probability of disk
D
surviving (i.e., the re-
liability of disk
D
) between
PA AA A
(|
...
)
−−
t
j
1
and
t
j
, given that
D
is alive at time
t
j
1
. Because
−
−
replica
r
has the same reliability as disk
D
,
PA AA AR
(|
...
)
=
where
R
tj
is the
j
j
−−
1
j
2
1
tj
reliability of the data file stored from
t
j
1
to
t
j
. Therefore, we have
−
PAAARR R
(
...
)
=
...
nn
−
1
1 12
t
t
tn
According to equation
(4.1)
, we have
−
λ
(
tt
−
−
)
Re
tj
=
j
j
j
1
Let
=−
−
Tt
t
j
1
, hence we have:
j
j
−
λ
T
−
λ
T
−
λ
T
PAAAee e
(
...
)
=
...
11 22
nn
nn
−
1
1
(
)
(
)
n
n
n
∑∑∑
⋅=
exp
−
λ
T
T
T
jj
j
j
j
=
1
j
=
1
j
=
1
Because
PAAART
(
...
)
=
(
)
, the above equation can be denoted as:
nn
1
−
1
RT e
()
=
−
λ
T
n
n
n
where
∑∑
, and
∑
λ
=
λ
/
jj
T
T
T
=
T
j
.
j
j
=
1
j
=
1
j
=
1
From equation
(4.2)
, it can be seen that the data reliability of one replica with a
variable disk failure rate also follows the exponential distribution, while the disk fail-
ure rate becomes the weighted mean of all the disk failure rates during the storage life
span. Therefore, equation
(4.1)
can be considered as a special case of equation
(4.2)
when the disk failure rate is a constant.
4.2.3 Generic data reliability model for multi-replicas
In previous subsections we discussed the data reliability of storing one replica.
Based on the discussions above, the novel generic data reliability model with a
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