## 2 Related Works
Mathematically, T(n) is expressed as follows:  
\[\text{T(n)}= \begin{cases}\frac{p}{1-p\left(mod\left(r,\frac{1}{p}\right) \right)},&if\ n\in G\\ 0,&else\end{cases} , \tag{1}\]where \(p\) is the percentage of CHs in all network systems, \(r\) is the chosen round number, \(mod\left(r,\frac{1}{p}\right)\) represents the quantity of selected CHs before the current round, and G is the collection of sensor nodes that have not been chosen as the CH previously. At \(r=0\), the chance that every node becomes CH is \(p\). If any node becomes the CH in the \(r\)-th round, it will not be reappointed as CH in any \(\left(\frac{1}{p-r}\right)^{th}\) later round. This enhances the chances of different sensor nodes becoming CHs. After the \(\left(\frac{1}{p}\right)^{th}\) round, all nodes have a possibility \(p\) of becoming a CH again [12]. The length of \(r\) is chosen prior, with each node being expected to perform a CH task once during its lifetime. Hence, CH selection process is repeating round by round until getting a minimum energy consumption. For the choice of the CH, a reformist calculation has been proposed [14, 15, 16]. In [14], reforming in CH selection process for LEACH is introduced by incorporating the remaining energy of the nodes and the distance to the BS. The enhanced protocol ensures that a node with a higher residual energy has a higher probability of becoming CH. Nodes close to BS are more likely to be selected as CHs since the protocol minimizes the energy required for data transmission. Similarly, CH selection is reformed for LEACH based on minimum distance to the BS by targeting the inefficiency in [15]. These methodologies separate nodes into clusters with a CH that is similar with the original LEACH protocol. Toward the start of each round, every node not previously picked as a CH settles on a free choice through a randomized calculation of whether to expect a CH task. Nodes that decide to be a CH declare their status to the remainder of the network system. The distance between CHs can be obtained from the ad message that CHs send to all nodes in the network [29]. Afterward, each node makes a list of the chosen CHs and picks the CH with the lowest distance. The assessment for determining the CH involves the residual energy and distance. The CH is turned on in each round by processing both assessment boundaries. LEACH-E is an improved version of LEACH that computes the choice variable (\(d_{ij}\)), which is the decision value of node \(i\) for CH \(j\), as follows:  
\[d_{ij}=\sqrt{\left(\alpha s_{ij}\right)^{2}+\left(E_{i}-E_{m}\right)^{2}}, \tag{2}\]  
where the sensitivity parameter, \(\alpha=\frac{\Delta E_{avg}}{\Delta S_{avg}}\), is the ratio between the average residual energy and the average distance from each node to the CH within the same cluster. Furthermore, \(s_{ij}\) is the estimated distance between node \(i\) and CH \(j\), \(E_{i}\) is the initial energy for node \(i\), \(E_{m}\) is the remaining energy of CH \(j\), and \(\left(E_{i}-E_{m}\right)\) is defined as the first-order difference in the consumed energy. The calculation in [25] has some improvement over LEACH-E, in terms of average delay and number of alive nodes. Each round starts with the remaining energy of every neighbor in its group range for every node. I-LEACH is an improved version of LEACH that select CH regarding the residual energy, the number of adjoining nodes, and the node's position with respect to the BS. This method of selection of CH in I-LEACH has a deformity, which prompts an extreme CH determination that clusters a non-uniform number of nodes and, thus, a non-uniform energy level in the network at the same time. The choice variable (\(d_{ij}\)) is determined as follows:  
\[d_{ij}=\sqrt{\left(\alpha s_{ij}\right)^{2}+\beta\left(E_{i}-E_{m}\right)^{2}}, \tag{3}\]  
where \(\alpha=\Delta E_{avg}\) and \(\beta=\Delta S_{avg}\).