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According to the given assumptions, we know that write requests
arrive in the duplication queue with an arrival rate of
and leave the queue with a duplication rate of
. For the system to be stable, it is implied that
, otherwise the length of the duplication queue
will grow to infinity, causing the system to saturate. If the
number of requests in the queue is zero, we say that the data in
the primary node is consistent with the backup node. This
duplication queue can be modeled by an
queuing
model [31][32].
In the model, the probability of the consistent
state, i.e., the probability of an empty queue, can be calculated as follows:
Although
is derived based on the duplication process of
Protocol 1, this term can also be used in other protocols. In
Protocol 2 and 4, all data has already been duplicated to the
mirror nodes at the time when the client nodes complete the
writing access. Thus
can be thought to be 0. In Protocol 3,
at the time the client finishes the writing process, there is
still a chance that a primary node is not consistent with the its
backup node. Similarly, it can also be modeled as
theoretically if we redefine
as the difference between the
time instants when data is stored in the faster server and when
data is stored in the slower server node.
Next: Markov-Chain Model for Reliability
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Yifeng Zhu
2003-10-16