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Let
b
r
be the number of buffers (each
SY
max
bytes) allocated for the rebuild process. Then
at time
t
1) tracks for track group
i
are completely retrieved. Due to disk
asynchrony, some of the disks may have completed retrieving track
i
earlier than
l
i
and have
started reading subsequent tracks. In particular, the earliest time for a disk to start reading
tracks
i
=
l
i
, all (
N
D
−
+
1 will simply be equal to
e
i
. Let
be the minimum time for retrieving a track:
W
−
1
=
α
+
t
hsw
+
(5.40)
Then a disk can retrieve up to track
l
i
−
e
i
i
+
(5.41)
by time
t
l
i
. In the worst case, all but the last disk have performed early retrievals and the
buffer requirement (in number of tracks) will be given by
=
l
i
−
(
N
D
−
e
i
b
r
=
(
N
D
−
1)
+
2)
+
1
(5.42)
where the first term is the buffers for reading track group
i
; the second term is the buffers for
early retrievals; and the last term is the buffer for writing to the spare disk.
Finally, the maximum buffer requirement can be obtained from
1
∀
max
(
N
D
−
l
i
−
(
N
D
−
i
SY
max
e
i
B
r
=
1)
+
2)
+
(
N
D
−
D
asyn
(
N
D
−
1
SY
max
=
1)
+
2)
+
(5.43)
of which equation (5.37) becomes a special case of equation (5.43) with
D
asyn
=
0.
5.6.2 Active Disk Synchronization
In deriving the buffer requirement in equation (5.43), we assumed that disks that completed
reading a track earlier than others will continue reading the subsequent tracks. While this
appears to be making efficient use of disk time, it is in fact counter-productive. Unlike trans-
action processing (OLTP) applications, residual disk time in a continuous-media server will
not be used for retrieving additional media blocks due to the periodicity of the disk schedule.
Therefore, even if there is residual disk time after reading all media blocks and rebuild tracks,
the disk will just sit idle until the next service round.
This observation motivates us to introduce an active disk synchronization (ADS) scheme
to further reduce the buffer requirement in equation (5.43). Specifically, track retrievals for
the surviving disks under ADS are actively synchronized according to the slowest disk. For
example, in reading track group
i
, all disks will start their retrieval for track
i
at time
t
l
i
−
1
instead of
t
i
−
1
,
j
for disk
j
as shown in Figure 5.8. Note that the added delay will not affect
the normal retrieval process or the rebuild process as they are dimensioned according to the
worst-case scenario.
=
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