## 1 Introduction
Once this random number is less than a particular threshold value \(T(n)\), which is given by Equation (1), then the node qualifies as a cluster head for the current round. After the CH election, non CH nodes select a CH based on the head's signal strength to be part of its cluster. These non CH then transmit the captured data through single-hop communication technique to the CH for aggregation and onward submission to the BS. \[T(n)= \begin{cases}\frac{P}{1-P(mod(\frac{1}{p}))}&\text{ if }n\in G\\ 0&\text{ otherwise}\end{cases}  \tag{1}\]  
where \(r\) is the current round, \(G\) is the set of nodes that are not selected as cluster head, and \(P\) is the desired percentage of cluster head. The authors in [22] explained a probability-based clustering algorithm called Distributed Energy-Efficient Clustering algorithm (DEEC). The algorithm elects cluster heads based on the ratio between the residual energy of each node and the average energy of the network. However, the single-hop communication approach is used in sending data to the BS. This makes the distant nodes dissipate a huge amount of energy, thereby affecting the lifetime of the network. In [23], [PERSON] and [PERSON] analyzed the Low Energy Adaptive Clustering Hierarchy (LEACH) protocol, which is a homogeneous system, and then studied the impact of heterogeneity. The authors then proposed a LEACH heterogeneous system which seeks to compare two systems; the heterogeneous and homogeneous systems. Simulation results using MATLAB show that the proposed LEACH heterogeneous system significantly reduces energy consumption and increases the total lifetime of the wireless sensor network. The main problem identified in this scheme is that the cluster heads (CHs)are not chosen based on their residual energy, and this affects the lifetime of the network. A model called Developed Distributed Energy-Efficient Clustering (DDEEC) for heterogeneous wireless sensor networks has been proposed in [24]. It is the enhanced version of DEEC. This scheme resolves the penalizing effect in the DEEC protocol. However, the DDEEC has a similar problem as DEEC, where a single-hop communication approach is used in sending data to the BS in the network. This makes the distant nodes dissipate a huge amount of energy, thereby affecting the lifetime of the network. An Enhanced Distributed Energy Efficient Clustering Scheme (EDEEC) for heterogeneous WSN has been described in [25]. This is an extended version of DEEC with normal, advanced, and super-node classification based on the node's energy. The problem identified in this protocol is similar to that of DEEC, SEP, and DDEEC in which no appropriate communication approach was introduced to reduce the energy depletion of the distant nodes deployed in the network. In [26], the authors proposed a new optimization scheme. The new algorithm modified the average probability of advanced nodes whose residual energy is less than the Threv (threshold residual energy value) to now depend on the average distance of the nodes from the Base station rather than the average energy of the network. The scheme further implemented TEEN and different amplification energy levels in the protocol to conserve energy in the network. Results showed that the proposed protocol performed better than the existing scheme in terms of throughputs, residual energy, and network lifetime. However, the single-hop communication technique used affected the lifetime of the distant nodes. An improved form of E-DEEC has been proposed in [27]. iE-DEEC improved the election probability of the protocol in [8] by taking into account the distance of super-nodes and the average distance of all the nodes to the BS in selecting CHs. The scheme has also introduced different amplification energy levels to minimize the energy consumption during the communications between the CHs and BS and also within inter and intra clusters. MATLAB R2017a was used for simulation to evaluate the effectiveness of the scheme.