# An Energy-Efficient Mobile Sink-Based Unequal Clustering Mechanism for WSNs
## 4 Proposed Algorithm
### Related Concepts  
#### 4.1.1 Balancing the Energy Consumption Ratio  
Based on the definition of network lifespan proposed by [PERSON] and [PERSON] [25] the network lifespan is the time when the first CH exhausts its energy supply. Since the nodes are uniformly distributed in the network, the number of nodes in \(i\)th corona (C\({}_{i}\)) and the total number of nodes in the network can be obtained respectively as follows [43]:  
\[N_{i}=(2i-1)N_{1} \tag{3}\]  
\[M=\sum_{i=1}^{k}(2i-1)N_{1}=k^{2}N_{1} \tag{4}\]  
where \(k\) is the number of coronas in the network. Since the nodes are uniformly distributed throughout the network, the density of nodes is equal in every part of the network as follows:  
\[\rho=\frac{M}{S} \tag{5}\]\(\rho\) and \(S\) denote the density of nodes and the area of the network, respectively. If the area of the innermost corona is \(S_{1}=\pi r^{2}\) which \(r\) denotes the width of coronas then the area of \(i\)th corona can be calculated via the area of the inner most corona [43]:  
\[S_{i}=S_{1}(2i-1),\ S_{1}< S_{2}<S_{3}<\ldots<S_{k} \tag{6}\]  
The energy usage of CHs for aggregating, receiving and transmitting the data packets of local member nodes is measured as follows:  
\[ECHintra_{i}=l\big{(}(N_{i}-1)(E_{rx})+(N_{i})\big{(}E_{agg}\big{)}+(N_{i} \cdot\beta)(E_{tx})\big{)} \tag{7}\]  
\(\beta\) denotes the compression ratio and can be in the range \([\frac{1}{N_{i}},1]\), that \(\beta=\frac{1}{N_{i}}\) denotes the perfect aggregation and \(\beta=N_{i}\) denotes the CH does not perform any aggregation [25]. We considered the perfect aggregation when every CHs compresses their received data packets from its cluster into one outgoing packet. In the multi-hop data transmission model, CHs located in \(i\)th corona have to relay the data packets from \(j\)th corona \(\{\text{C}_{j}\ |\ i+1\leq j\leq k\}\). Thus, the energy consumption of CH in CCWSN can be written as follows:  
\[ECH_{i}=l\bigg{(}\bigg{(}\bigg{(}N_{1}\bigg{(}k^{2}-\sum\limits_{j=1}^{i}(2j-1 )\bigg{)}\bigg{)}\cdot\beta\bigg{)}(E_{rx})+\bigg{(}\bigg{(}N_{1}\bigg{(}k^{2} -\sum\limits_{j=1}^{i}(2j-1)\bigg{)}\bigg{)}\cdot\beta\bigg{)}(E_{tx})\bigg{)}+ ECHintra_{i} \tag{8}\]  
For CH election, we used the model proposed in [12]. In their model, a centroid region with radius '\(\lambda\)' is considered in clusters. The distance between nodes and centroid points of clusters will be estimated. If the distance is less than the radius of centroid regions then these nodes will be recognized as centroid region nodes or CHs candidates. In each round, the nodes which have the maximum residual energy and minimum distance to centroid point will be elected as CH.  
Since there is a monotonically increasing sequence relationship between the areas of adjacent clusters in the same sector and different coronas, the area of centroid region would increase similarly. Furthermore, this value will be obtained with respect to the area of innermost centroid region:  
\[Area_{i}=(2i-1)Area_{1} \tag{9}\]  
Since the uniform node distribution has been applied, then the number of nodes in centroid region of clusters in innermost corona with respect of density of nodes will be calculated as follows:  
\[NCH_{1}=\rho\times Area_{1} \tag{10}\]  
Thus, the number of centroid region nodes in the clusters of \(i\)th corona will be estimated as follows:  
\[NCH_{i}=(2i-1)NCH_{1} \tag{11}\]