## 5 Performance Evaluation
### Checkpointing Interval  
Figure 9 shows the steady-state availability of our scheme via the number of backup nodes where \(\rho\) (\(\lambda\)/\(\mu\)) is Equation (6). If the rate of the watchdog timer is higher than the failure rate (\(\rho<1\)), the reliability of our system is greater than 80% when using three backup nodes. If the repair rate is identical to the failure rate (\(\rho=1\)), our system maintains a reasonable level of availability (more than 73%) using just three backup nodes. Therefore, we determined the minimum number of backup nodes as 3.  
We simulated two schemes using MATLAB [25], a well-known simulator for verifying numerical models. We implemented our checkpointing scheme based on a clustering-based routing protocol. As shown in Figure 10, when the number of member nodes in a cluster increases in a range from 10 to 50, the average energy consumption of our checkpointing scheme is less than that of the previous scheme  
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline \(N\) & \(10\leq N\leq 50\) \\ \(N\) & 3 \\ \(H\) & 50 \(m\) \\ \(I\) & 0.061 \(dB\) \\ \(T_{\
u}\) & 280 \(ms\) \\ \(K\) & 64 \(bytes\) \\ \(\Lambda\) & 1.35 \(\times 10^{-6}\) (\(0<\lambda<1.0\)) \\ \(M\) & 2.7 \(\times 10^{-6}\) (\(0<\lambda<1.0\)) \\ \(P\) & 0.5 (\(\lambda/\mu\)) \\ \(I_{ckpt}\) & 25 s \\ \(\delta\) & 500 ms \(\leq\delta\leq\) 1,000 ms \\ \(d_{max}\) & 15 km (the maximum transmission range of acoustic modem = about 20 km) \\ \hline \end{tabular}
\end{table}
Table 2: Parameters for the simulation.  
(without checkpointing). Our scheme is also superior to the previous scheme in terms of scalability, as the energy consumption of the proposed scheme steadily increases with the number of nodes in a cluster.  
The energy consumption of the scheme without checkpointing is higher than that of our scheme, and the difference between the two schemes steadily increases as the number of nodes in a cluster increases. With this extra energy, our scheme reduces the checkpointing interval and increases the reliability of the underwater sensor network. In this case, we derived optimal checkpointing intervals of between about 160 s and about 30 s when the number of sensor nodes ranged from 10 to 50 in Figure 11. The results show that as the number of sensor nodes increase, the amount of extra energy (\(E_{pre}-E_{cbp}\)) increases, as does the amount of checkpointing messages. In summary, the optimal checkpointing interval approaches 20 s as the number of sensor nodes in a cluster increases.  
To verify the proposed energy consumption model, we simulated our scheme under the uniform distribution of a node failure rate. When the number of nodes is 10 and 20 respectively, the energy consumption of each scheme is shown as Figure 12. In case of previous scheme, without checkpointing, as the number of nodes increases, the energy consumption increase exponentially and the variation in  
Figure 10: Comparison of the energy consumption values.  
Figure 9: The steady-state availability.  
each round is also fluctuating, but in the case of the proposed scheme, with checkpointing, the affection of the number of nodes is limited and the variation in each round is also small.