Repair Time in Distributed Storage Systems - Data Management in Cloud, Grid and P2P Systems - page 104

Fig. 1. Transition around state i of the Markovian queuing model

i<μ ,the i blocks in the queue at the beginning of the time step are
reconstructed at the end. Hence, we have transitions without the term i − μ :

≤

When 0

Q i → Q 0 with prob. 1 − f

α ) v − 1 α Q i → Q β , ∀ β with prob. f (1 −

α ) T max )

Q i → Q C with prob. f (1 − (1 −

Figure 1 presents the transitions for a state i .
Analysis. Expressions to estimate the values of the bandwidth usage, the dis-
tribution of block reconstruction time and the probability of data loss can be
derived from the stationary distribution of the Markovian model. We omit here
the analysis due to lack of space, but it can be found in the research report [14].

4R su s

To validate our model, we compare its results with the ones produced by simu-
lations, and test-bed experimentation. We use a custom cycle-based simulator.
The simulator models the evolution of the states of blocks during time (number
of available fragments and where they are stored) and the reconstructions being
processed. When a disk failure occurs, the simulator updates the state of all
blocks that have lost a fragment, and starts the reconstruction if necessary. The
bandwidth is implemented as a queue for each device, respecting both BW up
and BW down constraints. The reconstructionsareprocessedinFIFOorder.
We study the distribution of the reconstruction time and compare it with the
exponential distribution, which is often used in the literature. We then discuss
the cause of the data losses. Finally, we present an important practical imple-
mentation point: when choosing the parameters of the Regenerating Code, it is
important to give to the device in charge of the repair a choice between several
devices to retrieve the data .

4.1 Distribution of Reconstruction Time

Figure 2 shows the distribution of the reconstruction time and the impact of
device asymmetry on the reconstruction time for the following scenario: N =
100, s = 7, r = 7, L r =2 MB, B = 50000, MTTF = 60 days, BW up = 128 kpbs.
All parameters are kept constant, except the disk size factor x (recall that x is
the ratio of the maximum capacity over the average amount of data per device).