## 5 Performance Evaluation  
### Classical Approach  
First, we discuss the classical clustering performance evaluation. We are interested in optimizing the transmission range, which depends highly on node density. In a WSN, the nodes are stationary after deployment, but it is necessary to deploy sensor nodes wisely because, in our network model, we introduce the network connectivity equation as a function of the transmission range, which is related to node density and the number of sensor nodes in each cluster. From Table 5, we can see that the relationship between the expected number of clusters and the transmission range is reciprocal. The transmission range \(r_{tx}\) needs to be higher if the expected number of clusters decreases. If we can change the network area, for example A = 100 m \(\times\) 100 m in Table 5, and if the number of nodes is 100, the node density will be 0.01 m\({}^{-2}\), which is an increase from the previous node density. According to the calculation, the transmission range has decreased from 34 m to 17 m, but the number of clusters remains the same. This illustration, along with data from Table 5, demonstrates \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Angular Frequency \(\omega\)** & **Phase Angle \(\varphi\)** & **Iteration Steps Via Equation (28) \(t_{0}\)** & **Iteration Steps Via Equation (33)** \\ \hline
0.6908 & 0.3454 & 1.77\(\sim\)2 & 1.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Iterations for finding the target nodes. how the transmission range will decrease as node density increases. Figure 5a shows the relationship between total number of sensor node and transmission range. When the number of clusters increases, transmission range decreases that is depicted in Figure 5b. In a practical sense, the selection of the optimum number of clusters should be energy efficient and consume low level of power. In order to select energy-efficient cluster heads, it is essential to calculate the energy consumption between the CH and its member nodes, which is referred to as intra-cluster communications, and between the CH and the BS, which is inter-cluster communications. In the next step, we must characterize intra- and inter-cluster energy consumption, and determine how much energy is saved for different cluster formations through the energy model discussed in Section 3.2. It is worth depicting how, in a cluster, the node consuming the most energy is the CH because it needs to gather information from its member nodes, aggregate their data, and then send them directly to the BS (in our model, we assume that there is single-hop communication with the BS). For this reason, we show the energy dissipation of each member node and the CH separately. In our energy calculation, the power levels of data from the Berkeley CC2420 chip set [49] are used. Before starting the computation, some parameters (showed in Table 6) need to be set (\(E_{elce}=50\) nJ/bit, \(e_{fs}=10\) pJ/bit/m\({}^{2}\) and \(e_{mp}=0.0013\) pJ/bit/m\({}^{4}\), \(E_{DA}=5\) pJ/bit per signal) from [35]. Figure 5: (**a**) The relationship between the number of sensor node and transmission range. And (**b**) the number of clusters and transmission range. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Area (A)** & **No. of Sensor Nodes (N)** & **Node Density (\(\rho\)) m\({}^{-2}\)** & **Expected Number of Clusters** & **Transmission Range (\(r_{\text{rx}}\)) m** \\ \hline \multirow{6}{*}{200 m \(\times\) 200 m} & 500 & 0.0125 & 46 & 16.59 \\ \cline{2-4}  & 450 & 0.01125 & 42 & 17.40 \\ \cline{2-4}  & 400 & 0.01 & 38 & 18.36 \\ \cline{2-4}  & 350 & 0.00875 & 33 & 19.51 \\ \cline{2-4}  & 300 & 0.0075 & 29 & 21 \\ \cline{2-4}  & 250 & 0.00625 & 25 & 22.70 \\ \cline{2-4}  & 200 & 0.005 & 20 & 25 \\ \cline{2-4}  & 150 & 0.00375 & 16 & 29 \\ \cline{2-4}  & 100 & 0.0025 & 11 & 34 \\ \cline{2-4}  & 50 & 0.0013 & 6 & 47 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Transmission range for a fixed network area and different node densities. (Probability of connectivity is 0.99).