Minimum replication for meeting the data reliability requirement - Reliability Assurance of Big Data in the Cloud - page 40

inequation is a monotonically decreasing function of t. Therefore , while RA (1 k is
determined, variable t cannot exceed a certain value. When the longest storage dura-
tion of the data file is reached, the right-hand side of the inequation equals to RA (1)
k

,

which is

− λ t − λ t

(1) 1 1 = −− −

e )(1 e )

1 2

RA (5.7)

k

t

Therefore, by solving equation (5.7) , the longest storage duration of the data file
can be obtained.
In general, our minimum replication calculation approach determines the minimum
replication for meeting the data reliability requirement based on inequation (5.6) and
equation (5.7) . By using inequation (5.6) , we are able to justify whether the storage
with one replica suffices to meet the data reliability requirement, and hence the mini-
mum r ep l ic a number (i.e., either one replica or two replicas) could be determined. 1
Given λλ , 12 as the average failure rates of the corresponding disks, by solving equa-

tion (5.7) , the longest storage duration of the data file while meeting RA (1)
k

can be

predicted.

5.1.2 Optimization of the minimum replication
calculation formulas
Inequation (5.6) and equation (5.7) are the keys for calculating the minimum replica-
tion for meeting the data reliability requirement. However, it could be difficult to solve
equation (5.7) in its current form due to two reasons:

• First, due to the variable nature of the average disk failure ra te , it changes along with
the storage duration and the exact age of the disk. Therefore, λ becomes a function of
variable t . Considering this factor, the solving process of equation (5.7) becomes very
complicated.
• Second, as a direct consequence of the variable average disk failure rate, the longest storage
duration of the data file changes from time to time. Therefore, every time when the longest
storage duration of the data file is needed, the process of solving equation (5.7) needs to be
conducted again. As will be mentioned later, in our data reliability assurance solution the
longest storage duration of the data file could be used many times throughout the life span
of each data file. Therefore, the overhead for solving equation (5.7) could be very big.

In general, solving complicated equation (5.7) could be a time-consuming and ex-
pensive process. In particular, the involvement of function λ () and calculation of the
longest storage duration of data files for more than once have made the problem even
worse. To address this issue, optimizations need to be conducted for the minimum
replication calculation approach.

1 We do not predict the longest storage duration of data file with a single replica. If inequation (5.6) does
not hold, it means that one replica simply cannot provide satisfactory data reliability assurance for any time;
and if inequation (5.6) holds, it means that one replica is sufficient for storing the data file for the expected
storage duration, and hence the prediction is not needed.