# Markov Chain Model-Based Optimal Cluster Heads Selection for Wireless Sensor Networks
## 2 Related Work and Background
### Cluster Size  
Numerous existing algorithms have presented different schemes to get an optimal cluster size in a network, where all the member nodes are distributed evenly. In [34], the authors argue that cluster sizes and intra-transmission ranges should be different in a WSN. The distance between the concerned CH and the BS is one constraint that affects the cluster size. The small size clusters are preferred at the lesser distance from the BS because the CHs nearer to the BS will transmit more data. Arranging Cluster sizes and Transmission ranges (ACT) for WSNs aims to use an equal amount of energy for each cluster by regulating the cluster size that decreases gradually from the furthest cluster to the nearest one. ACT tries to control the cluster size by regulating the radius of clusters. Another way to control the cluster size is by restricting the member nodes in the clusters. The LEACH [7; 35] tries to fulfill this requirement through adjusting the number of CHs in a network. The authors define the probability value \"\(p\)\" for each node to become a CH. Therefore, the expected strength of member nodes in a cluster is defined as \"\(1/p\)\".  
In [36], the authors analyze the network performance through different parameters like network radius, round numbers, the length of the data packet, and consumption of the network energy. By assuming the uniform sensing nodes in the network, they compared LEACH with Optimal Energy Consumption Model (OECM) [36] and found that the CHs selection process of OECM is not stable. They considered a connection among the optimal number of CHs and assorted parameters. This method forms clusters in every round and restricts the cluster numbers by regulating the CHs in every round. However, in OECM [36] cluster formation is performed at the start of every round and information cannot be forwarded when cluster formation is in the process. So, in this way more transmission delay will occur.  
To overcome the issue of longer transmission, a strategy is defined in [29]. This protocol uses multi-hop routing with a flexible transmission range of communication between clusters to overcome this issue. It selects CHs stochastically, for which rotating epoch and election probability is related to the residual energy. Nodes with higher residual energy have a higher probability of becoming CHs. However, the hot spot becomes a major issue in [29], as CHs closer to the BS will transmit more data packets. A well-balanced payload network to control data overhead through route detection and energy consumption for data transmission should be taken into account. To cope with this kind of problem, energy-efficient clustering is suggested in [13; 36]. These protocols consider appropriate sizes for different clusters by looking at their hop counts towards the BS. Energy-efficient Clustering (EC) [13] successfully controls the cluster size through an adjustable probability of nodes to become a CH and to equally distribute the energy load in the network.  
Our proposed protocol uses a random and an optimal CH selection scheme to manage the clusters in the sensing area. In the setup phase, the nodes estimate the location of BS and forward the sensed data at the end of each round. In previous clustering protocols [9; 10; 11; 37; 12], some distant and overpopulated CHs face heavy traffic problems and exhaust their power resources very quickly. However, the MOCHs cluster sizes are controlled by the CHs and only a restricted number of \(C_{MNs}\) can join a cluster. The BS adjusts the cluster size in a manner that the more congested sensing area contains more CHs. Consequently, our designed protocol equally divides the energy load among all the CHs by dynamically rotating the role of CHs.