# Clustering for Lifetime Enhancement in Wireless Sensor Networks
## 3 Advanced Clustering Approach
### Clustering Efficiency: Analysis Study  
Clustering divides the WSN into several sets, either equal or unequal. The number of nodes that construct a cluster represents its density. Therefore, cluster density is a fundamental parameter that affects the clustering efficiency. According to [PERSON], the optimal density of a cluster in a WSN composed of 100 nodes distributed in an area of 100 m \(\times\) 100 m is 10 (one is a cluster head, and the remaining nine nodes are cluster members). To check the efficiency of the cluster density, firstly, we should extract the equation that represents a cluster's consumed energy, which is the sum of the energy consumed by the CH and the CMs. Equations (5)-(7) give the energy by consumed a cluster head, by members, and their sum, which represents the energy consumed by a cluster, respectively. \[E_{CH}(CH_{d2 BS},\ M)=E_{Tx}(CH_{d2 BS})+E_{Rx}(M)+E_{DA}(M+1) \tag{5}\]  
\[E_{CM}=\sum_{i=1}^{n}E_{Tx}(CH_{d2 CH}(i))_{CH} \tag{6}\]  
\[E_{C}=\ E_{CH}+E_{CM} \tag{7}\]  
where:  
\(E_{CH}(CH_{d2 BS},\ M)\) is the consumed energy by a CHdistant from the BS by \(d2 BS\) with \(M\) member nodes. \(E_{CM}\) represents the consumed energy of a node as a member in a cluster with a density of \(n\) nodes with \(CH_{d2 CH}(i)\) as the distance to each node in the cluster where it is a CH. \(E_{C}\) is the total consumed energy by a cluster. Our aim is to reach a conclusion that helps any researcher in the field know how to deal with the clustering approach to develop more efficient routing protocols. According to the literature, the standard network used to simulate WSNs is a WSN composed of 100 nodes distributed randomly in a 100 m \(\times\) 100 m area, where the BS position is in the center of it, as shown in Figure 1. Therefore, based on this topology, the min distance to the BS could be farther than 0 m and less than the Euclidean distance between the BS (50, 50) and the furthest point in the area, which are the four vertices' points (i.e., (0, 0), (0, 100), (100, 0), and (100, 100)). This distance equals 70.7107 m. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(L\) & Packet length (4000 bits). \\ \hline \(E_{dec}\) & Energy dissipated to run the transmitter or receiver circuitry (50 nJ/bit). \\ \hline \(\varepsilon_{fs}\) & Transmitter amplifier, energy-free space model (10 pJ/bit/m\({}^{2}\)). \\ \hline \(\varepsilon_{mp}\) & Transmitter amplifier, energy \\  & multipath model (0.0013 pJ/bit/m\({}^{4}\)). \\ \hline \(d\) & Distance (m). \\ \hline \(d_{0}\) & Approximately 87.7 m. \\ \hline \(n_{p}\) & Number of packets. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LEACH energy model parameters. The second parameter is the number of nodes (i.e., the cluster's density). To name a cluster, we need at least two nodes, one as CH and one as CM. However, a cluster's density could surpass the optimal value considered by Leach (i.e., 10 nodes). In that case, we determined that the max number of members should be double this value, which is 20 nodes. The third parameter is the distance between a CM and their CH. It is farther than 0 m and limited by a range of 20 m. To this point, we fixed the ranges of the three essential parameters in a clustering approach based on the LEACH hypothesis, which is clustering with one-hop communication either within a cluster or between a CH and its BS, which is efficient to prolong a WSN's performance. Now, to check this hypothesis, we compare it against the basic manner, which is nodes sending their packets directly towards the BS. To perform this comparison in the right way, we must utilize the same conditions. Our main performance parameters are the consumed energy/node lifetime, node throughput, first, half, and last dead nodes. It is obvious that defining the same conditions means considering the three parameters for the clustering approach; we need a cluster where all nodes should be in the range of the rest of the nodes (i.e., they represent a complete graph).