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Distribution of the Reconstruction Time
(b=500 s=7 r=7 x=1.1 MTBF=1440 rho=0.9)
Distribution of the Reconstruction Time
(b=500 s=7 r=7 x=2 MTBF=1440 rho=0.5)
Model
Simulation
Exponential
Model
Simulation
Exponential
(Model) Mean = 9.4 cycles Std.Dev. = 7.4
(Sim) Mean = 9.6 cycles Std.Dev = 7.8
(Model) Mean = 3.1 cycles Std.Dev. = 1.8
(Sim) Mean = 3.2 cycles Std.Dev = 2
0
5
10
15
20
0
20
40
60
Reconstruction time (cycles)
Reconstruction time (cycles)
Reconstruction Time (cycles)
Reconstruction Time (cycles)
(a)
x
=1
.
1
(b)
x
=2
.
0
Fig. 2.
Distribution of reconstruction time for different disk capacities
x
of 1.1 and 2
times the average amount. The average reconstruction times of simulations are respec-
tively 3.2 and 9.6 hours (Note that some axis scales are different).
First, we see that the model (dark solid line) closely matches the simulations
(blue dashed line). For example, when
x
=1
.
1 (left plot), the curves are almost
merged. Their shape is explained in the next paragraph, but notice how far they
are from the exponential. The average reconstruction times are 3.1 time steps
for the model vs 3.2 for the simulation. For
x
=2
.
0 (right plot), model is still
very close to simulation. However, in this case the exponential is much closer to
the obtained shape. In fact, the bigger the value of
x
, the closer the exponential
is. Hence, as we will confirm in the next section, the exponential distribution
is only a good choice for some given sets of parameters. Note that the tails of
the distributions are close to exponential. Keep in mind that big values of
x
are
impractical due to both storage space and bandwidth ine
ciency.
Second, we confirm the strong impact of the disk capacity. We see that for
the four considered values of
x
, the shape of distributions of the reconstruction
times are very different. When the disk capacity is close to the average number of
fragments stored per disk (values of
x
close to 1), almost all disks store the same
number of fragments (83% of full disks). Hence, each time there is a disk failure
in the system, the reconstruction times span between 1 and
C/μ
, explaining
the rectangle shape. The tail is explained by multiple failures happening when
the queue is not empty. When
x
is larger, disks also are larger, explaining that
it takes a longer time to reconstruct when there is a disk failure (the average
reconstruction time raises from 3.2 to 9.6 and 21 when
x
raises from 1.1 to 2
and 3).
We ran simulations for different sets of parameters. We present in Table 1 a
small subset of these experiments.
4.2 Where the Dead Come from?
In this section, we discuss in which circumstances the system has more probabil-
ity to lose some data. First a preliminary remark: backup systems are conceived
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