Calculation of $P_c$

According to the given assumptions, we know that write requests arrive in the duplication queue with an arrival rate of $\lambda_w$ and leave the queue with a duplication rate of $\mu_d$. For the system to be stable, it is implied that $\lambda_w < \mu_d$, 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 $M/M/1$ 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:

$$P_{c} = 1-\frac{\lambda_w}{\mu_d}$$

$$= 1 - \frac{MTTD}{MTTW}$$ (1)

Although $P_c$ 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 $MTTD$ 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 $M/M/1$ theoretically if we redefine $MTTD$ 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.