## 2 Modeling and Formulation
### Field Partitioning and Sector Area Calculation  
Sensors and a stationary \(BS\) are deployed over the sensing field. We find this to be an appropriate and effective mechanism for dividing the R-WSN field into convergent partitions in space, but with different numbers of nodes under the geographical constraint of the R-WSN field. Therefore, we assume that this \(BS\) is located at the center of the network field and has no energy constraint. The \(BS\) in the network is equipped with a power-controlled capability directional antenna, which can reach any part of the network [22]. It helps when dividing the R-WSN field (using an antenna beam width (\(\theta\)) to define the RN and SN). The same scan order of CBS in [23] is considered in this study. We examine one quarter of the network for simplification. We consider the field to be distributed into four main quarters, and each quarter is divided into a given number of sectors by varying its transmission power level (\(R\)) or radius, and beam width (\(\theta\)). The signal sent by the \(BS\) to define the radius \(R_{m}\) of the \(m\)th ring is calculated using the following equation:  
\[R_{m}=m\times R_{1},\forall_{m}=1,2,3,\ldots \tag{1}\]  
where \(R_{1}\) is the radius of the first ring, which is used to obtain the angle of beam width (\(\theta_{m}\)) for the \(m\)th ring in the following equation:  
\[\theta_{m}=90/((2\times m)-1),\forall_{m}=1,2,3,\ldots \tag{2}\]  
From (1) and (2), the area \(A_{m}\) of all sectors in the \(m\)th ring is obtained, as shown in the following equation:  
\[A_{m}=\pi R_{m}^{2}\ /4,\forall_{m}=1,2,3,\ldots \tag{3}\]  
where \(A_{m}\) is the area of the \(m\)th ring with radius \(R_{m}\). The sector areas on the same ring are equal, and the area of one sector is calculated as  
\[AS_{m}=\frac{\pi R_{m}^{2}-\pi R_{m-1}^{2}}{SN},\forall_{m}=1,2,3,\ldots \tag{4}\]  
The area for a specific sector \(AS_{m}\) is determined. \(SN\) represents the number of sectors, and \(R_{m}^{2}\) is the radius of the last ring. The network density is calculated as \(\rho=N/\pi_{R}^{2}\), where \(N\) is the number of deployed sensors. The partition mechanism of the network process is illustrated in Figure 1. The first quarter shows the first phase of the network-dividing process, in which the network field is divided into a set of rings with different transmission power levels. The corresponding sectors are represented with their  
Figure 1: Network design technique with \(j\)th rings of \(i\)th sectors. Each ring has an equal sector area. These equally spaced sectors are used to form even-sized clusters in the network. rings. In the network initialization phase, we assume that each node \(n_{k}\in N\) needs to send its location information to the \(BS\) with an auto hi message (\(AHM\)). This message contains the node number (\(n_{k}\)), residual energy (\(RE_{k}\)), and message release time (\(rt_{k}\)), as shown as follows:  
\[AHM_{k}=(n_{k},RE_{k},rt_{k}). \tag{5}\]  
This message is the first that every node must send to the \(BS\). The nodes have not chosen the \(CH\) for each part (sector) yet to contact with the \(BS\). Therefore, the nodes can use the normal aggregation style to send these messages to the \(BS\). The \(BS\) starts to collect and analyze these data to determine the node location and geographical relations among themselves [17]. After the R-WSN field is divided into rings and sectors, the \(BS\) begins the next phase by selecting the \(CH\) for sectors, depending on their residual energy and location [26]. The \(CH\) should have more energy than the other nodes and be directly connected with all nodes in the sector. \(BS\) estimates the residual energy from the collected energy information. It finds which node energy level is higher than the average energy level, and selects those nodes as \(CH\) (more than one \(CH\) may exist in one sector). We have divided the field into SN sectors. Each sector has one \(CH\); if there is a large sector then we assume these large sectors will have more than one \(CH\).