## 3 Analysis and Numerical Results
### Numerical Results  
\(P\)s are the power levels of the nodes obtained through the analysis described in Subsection 3.1. The graphs shown in Figures 5-7 are obtained for cases where \(M\) is 10, which is the number of nodes to the left of the sink node. In the graphs, \(C_{W,avg}\) is the bandwidth efficiency, which is the transmission rate normalized to the bandwidth, and the system bandwidth efficiency is \(2 MCW,avg\) because there are \(M\) nodes each on the left and right of the sink node. Figures 5-7 correspond to Schemes 1-3, respectively, and the largest \(C_{W,avg}\) value indicates the upper bound of transmission rates. In Scheme 1 (Figure 5), if \(C_{W,avg}\) is higher than the upper bound, the SINR  
values (Equations (4) and (5)) of the boundary \(M\)th and (\(M+1\))th nodes will not satisfy the conditions under Equation (2). In Scheme 2 shown in Figure 6, when \(P_{M\text{-}1}\) increases infinitely in Equation (8), the upper bound of \(SINR_{M}\) is \(3^{1.5}\). Therefore, if \(M=10\) in Equation (2), the maximum value of \(C_{W}\)will be determined to be 0.2924. Accordingly, the maximum value of \(C_{W,\text{mag}}\) (=\(\
icefrac{{1}}{{2}}C_{W}\)) will become 0.1462. Similarly, in Scheme 3 shown in Figure 7, the same \(C_{W}\) bound of 0.2924 is obtained using Equations (12) and (2). On the other hand, because the average \(C_{W,\text{mag}}\) is calculated to be 1/3 of the \(C_{W}\) bound, the actual bound is 0.0975. In Figures 6 and 7, \(P_{9}\) and \(P_{10}\) become absurdly large near the upper bound of \(C_{W,\text{mag}}\) because \(P_{9}\) (=\(P_{M^{-1}}\)) should be much larger than the other terms in Equations (8) and (12) for a high SINR near the upper bound in Equation (2). Accordingly, \(P_{10}\) (=\(P_{M}\)) should be also large in Equations (9) and (13) because of the large \(P_{9}\). Although the \(C_{W,\text{mag}}\) value appears to be small, \(C_{W,\text{mag}}\) is the transmission rate only at the first node and the transmission rate should be accumulated so that the transmission rate at the \(M\)th node, which is the last node should be \(MCw,\text{mag}\). In addition, the power required by the nodes is generally larger in the nodes closer to the sink node. Figure 8 shows \(\overline{P}\), the average of \(P_{i}\)s obtained while changing the transmission rate \(C_{W,avg}\) when \(M=10\) compared to the noise power (\(\overline{P}=\dfrac{\sum P_{i}}{M}/P_{{}_{N}}\)). Figure 8a shows that the range of changes in the average power levels according to the changes in \(C_{W,avg}\) values is very large, amounting to approximately \(10^{7}\). To review the superiority among the Schemes, Scheme 2 is always the most excellent and the superiority of Schemes 1 and 3 changes according to \(C_{W,avg}\). Although the differences do not appear to be large in Figure 8a, if the differences are seen in the expanded figure, Figure 8b, Scheme 2 guarantees almost 50% higher transmission rates. The high \(C_{W,avg}\) part in Figure 8b is expanded because this paper focuses on high rate environments. Figure 8: _Cont._  
Figure 7: Power levels according to the node in Scheme 3. Figure 9 shows the values of \(\overrightarrow{P}\) obtained while increasing the number of nodes when the transmission rate \(C_{W,\text{arg}}=0.025\). The average power level changes according to changes in the number of nodes by almost \(10^{4}\) times. This is because as the number of nodes (\(M\)) increases, the amount of data to be transmitted by the nodes close to the sink node increases in proportion to \(M\), leading to very high required SINRs. From the figure, Scheme 2 is the best and Schemes 1 and 3 cannot support high \(M\) values. For a fixed \(C_{W,\text{arg}}\), \(M\) is limited due to the limited \(SINR_{M}\) in Equation (2). As shown in Figure 9, as the number of nodes increases, the necessary average power rises rapidly. If the entire nodes are split into several groups and a sink node is deployed in each group, the necessary power can be reduced considerably.