## 7 Simulation Results  
In this section, we first evaluated the performance of our proposed E-DDTMD algorithm based on ADMM. Then, we simulated and compared the performance of the proposed optimal mechanism and the following two mechanisms. Finally, the systematic simulation results demonstrated our mechanism is effective in improving the energy efficiency of UWSNs, meanwhile balancing different depths SNs' energy. (1) Transmission after calculation (TAC): for TAC, each SN first computes and processes the data, and then transmits the results to the BS;  
(2) Direct transmission to BS (TDBS): For TDBS, each SN transmits data directly to the BS. We perform the simulation in MATLAB 2018b. In this simulation, there are J = 6 SNs in the assisting layer (the maximum data size accepted by each SN in the assisting layer is limited to 21 KB in each round of transmission). The \(l_{i,B}\) is randomly set from 800m to 1200 m, and the \(l_{i,j}\) is randomly set from 100 m to 300 m. In addition, we assume that the transmit powers \(P_{i,B}\) and \(P_{i,j}\) are 150 mW and 60 mW, respectively, and the channel bandwidth and carrier frequency are 1 kHz and 20 kHz. For the data task of each SN in the optimization layer, we consider that the data size is 10 KB, the number of CPU cycles required to calculate and process the data is 30 M cycles, and the size of the processed result data is set as 0.1 KB. Moreover, the computing power of each SN in the optimization layer is assumed to be 10 Mcycles/s, and the simulation parameters are summarized in Table 2.  
The convergence performance of the proposed E-DDTMD algorithm and centralized decision scheme (CDS) [35] is demonstrated in Figure 4. In Figure 4, it can be seen that in the first 10 iterations, the curve of the proposed E-DDTMD algorithm rapidly drops close to the CDS. Until the 25 th iteration, the algorithm converges to the CDS and then stabilizes. This shows that the proposed distributed algorithm has better convergence performance. In Figure 5, we compare the total overhead of 'The Optimal Mechanism' with the total overhead of 'TAC' and 'TDBS'. It can be observed that the total overhead increases with the number of SNs in the optimization layer in the figure, because increasing the number of SNs in the optimization layer will increase the total data size accordingly. Compared with the two mechanisms of 'TAC' and 'TDBS', the total overhead of 'The Optimal Mechanism' increases more slowly, which proves that 'The Optimal Mechanism' can effectively reduce the overhead of the optimization layer. (The number of SNs selected for mode \(a\), \(b\), \(c\), and \(d\) are 2, 7, 4, and 5, respectively). Figure 6 compares the results of 'The Optimal Mechanism' and the two mechanisms, 'TAC' and 'TDBS', when the size of each SN's data size in the optimization layer varies. In this figure, the number of SNs in the optimization layer is set to 15. As shown in the figure, as each SN's data size increases, the overhead of 'TDBS' increases linearly. The overhead of 'The Optimal Mechanism' and 'TAC' increases slightly (caused by the corresponding increase in the data size after calculating and processing the results), because when the amount of data increases, the overhead of processing data in mode \(a\) and mode \(b\) increases significantly, so 'The Optimal Mechanism' will choose mode \(c\) and mode \(d\) more. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Definition** \\ \hline Bandwidth (\(B_{i}\)) & 1 kHz \\ Carrier frequency (\(f_{i}\)) & 20 kHz \\ Transmit power to BS (\(P_{i,B}\)) & 150 mW \\ Transmit power to assisted layer SN (\(P_{i,f_{i}}\)) & 60 mW \\ Size of computation task (\(D_{i}\)) & 10 KB \\ Required CPU cycles of computation task (\(X_{i}\)) & 30 Mycycles/s \\ Computation capability to each SN (\(f_{i}^{L}\)) & 10 Mycycles/s \\ Size of the calculated data (\(D_{i}^{L}\)) & 0.1 KB \\ The delay constraint (\(T_{i}\)) & 2 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters. Figure 4: Convergence performance of the proposed E-DDTMD algorithm.