# Markov Chain Model-Based Optimal Cluster Heads Selection for Wireless Sensor Networks
## 1 Introduction
The sensor nodes are deployed in the sensing area through a distributed algorithm [7; 11]. Consequently, the distribution of nodes in the sensing field is not even. The current clustering protocols [7; 8; 9; 10; 11; 12] utilize the distributed algorithm for CH selection that increase the computational overhead and also causes the resources to drain very quickly. We use Markov Bi-directional Chain Model (MBCM) [21; 26] to equally divide the sensing area into clusters, and to examine the behavior of the cluster formation process. We also used this formulation [21; 26] for optimal CH selection and to analyze the stochastic characteristics of the MOCHs in the sense of the mean value, the probability mass function (PMF), the standard deviation (SD), and the coefficient of variation (COV) for the optimal selection of CHs. Through this formulation [21; 26], the obtained CHs are optimal in number and well-distributed all across the network, which leads to higher energy efficiency, better fairness among nodes, and prolonged the network lifetime. In our proposed clustering protocol, each sensor node consumes energy equally by revolving the CH's responsibility between all the sensor nodes. In MOCHs, the CHs are centrally selected by the BS based on the residual energy and distance of each node from the BS. The optimal number of CHs is selected by the BS because the BS is more reliable and equipped with high-speed processors than a CH and root nodes [27]. Only a defined number of \(C_{MNs}\) are joined with a CH to equally distribute the energy and data load over the network. The main contributions of our method are defined as follows:  
* When the CH positioned backward compared to its direction of transmission towards the BS, back transmission occurs. This type of data transmission contributes to extending the general path length traveled by the locally collected data, which leads to a decrease the lifetime of the network. We define a simple but effective strategy for the node association to its CHs to tackle the problem of backward transmission. * The recently designed clustering protocols [8; 9; 10; 11; 12], randomly select the CHs through a distributed algorithm which generates unequal clusters with uneven cluster size in the network; these unstable clusters create an unbalanced energy situation that leads to the end the network lifetime being much earlier. The proposed model centrally selects the optimal number of CHs by restricting the number of \(C_{\textit{MNs}}\) in the cluster and guarantees the longer network lifetime. * The proposed model also deals with non-associated \(C_{\textit{MNs}}\) which affect the network stability and also drain the available power resources. The remainder of the paper is organized as follows: Section 2 describes the related work and background. We discussed the methodology, the details of Markov bi-directional chain model, and motivations in Section 3. The network model, node deployment strategy and the working of the proposed clustering protocol with phases in a round are discussed in detail in Section 4. Evaluation measures and simulation results for the lifetime of the network, the network stability, the messages towards the BS, the CH selection, and the transmission time of the network are discussed in Section 5. Finally, the conclusion is drawn in Section 6.