## 3 System Model and Framework
### Energy Expenditure Constraints  
The energy depletion in processing and sensing is muchless as compared to data transmission and reception. The energy depletion in data communication is taken into account according to the Friis space model [64]:  
\[E_{Tx}=\ell\times\text{E}_{\text{elec}}+\ell\times\text{E}_{t}\times d^{4} \tag{2}\]  
\[E_{Rx}=\ell\times\text{E}_{e}+\ell\times\text{E}_{\text{bf}} \tag{3}\]  
\[E_{Tx}=\ell\times\text{E}_{e}+\ell\times\text{E}_{\text{s}}\times d^{2} \tag{4}\]  
\(E_{t}\) is used for amplifier of NN, RN, CH and CC nodes and in the manipulation of the equations they are used as \(\in_{1}\),\(\in_{2}\),\(\in_{3}\),\(\in_{4}\) respectively. Equation (2) is used for long distance communication and Equation (4) computes the energy depletion in the short distance communications. Data is received at relay nodes, CH nodes and cluster coordinator nodes so the energy released in this task is computed by Equation (3). The \begin{table}
\begin{tabular}{l l} \hline \(E_{Tx}\) & Energy Consumption for data transmission \\ \(E_{Rx}\) & Energy Consumption for data reception \\ \(E_{dec}\) & Energy Consumption for radio electronics \\ \(\in_{1},\in_{2},\in_{3},\in_{4}\) & Transmit amplifier of the normal node, relay node, and CH node \\  & and CC node respectively \\ \(x,y,z,p,q\) & Cardinality of NN1, RN1, CH1, CC1 and BS1 \\ \(S_{(u,v)}\) & Data rate from node u to v \\ \(S_{\text{max}}\) & Maximum data rate \\ \(\ell\) & Data length of the packet \\ \(C_{cc}\), \(C_{ch}\), \(C_{m}\), \(C_{bs}\), \(C_{m}\) & Cost of CC, CH, RN, BS and NN \\ \(\left|\right|\) & Cardinality of a set \\ \(F_{0}\) & System budget \\ \(d_{(u,v)}\) & Distance from node u to v \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters used in ME-CBCCP for IoT. data rate for the nodes u to v is the same as the data length \(l\). Energy depleted by each node per unit time is computed by:  
\[E_{u}=\sum_{v=RN}r_{uv}\cdot S_{uv}\cdot(\mathrm{E}_{dec}^{NN}+\mathrm{e}_{i} \cdot d_{u,v}^{2})\ \ \forall\ u\in NN \tag{5}\]  
\[E_{v}=\sum_{u=RN(|CN)}r_{uv}\cdot S_{uv}\cdot\mathrm{E}_{dec}^{NN}+(\mathrm{E }_{dec}^{NN}+\mathrm{e}_{i}\cdot d_{u}^{2})+\sum_{u=RS(|CN)}r_{uv}\cdot S_{uv} \cdot(\mathrm{E}_{dec}^{NN}+\mathrm{e}_{i}\cdot d_{u}^{2})\ \ \forall\ v\in RN \tag{6}\]  
\[E_{v}=\sum_{u=CR(|CN)}r_{uv}\cdot S_{uv}\cdot\mathrm{E}_{dec}^{CH}+(\mathrm{E }_{dec}^{CH}+\mathrm{e}_{i}\cdot d_{u}^{2})+\sum_{u=CR(|UCO)}r_{uv}\cdot S_{uv} \cdot(\mathrm{E}_{dec}^{CH}+\mathrm{e}_{i}\cdot d_{u}^{2})\ \ \forall x\in CH \tag{7}\]  
\[E_{v}=\sum_{u=CR(|UCO)}r_{uv}\cdot S_{uv}\cdot\mathrm{E}_{dec}^{CO}+(\mathrm{ E}_{dec}^{CO}+\mathrm{e}_{i}\cdot d_{u}^{2})+\sum_{u=CR(|CN)}r_{uv}\cdot S_{uv} \cdot(\mathrm{E}_{dec}^{CO}+\mathrm{e}_{i}\cdot d_{u}^{2})\ \ \forall y\in CCO \tag{8}\]  
\[E_{v}=\sum_{y\in CCO}r_{uv}\cdot S_{uv}\cdot\mathrm{E}_{dec}^{BS}\ \ \forall z\in BS \tag{9}\]  
\(E_{u},E_{v},E_{x},E_{y},E_{z}\) denote the energy consumed by sensor nodes, relay nodes, cluster head nodes, cluster coordinator nodes and base stations, respectively, in the transmission and reception processes. The symbols \(E_{elec}^{NN},E_{elec}^{RN},E_{elec}^{CH},E_{elec}^{CCO},E_{elec}^{BS}\) denote the energy consumed by the electronics of a normal node, relay node, cluster head node, cluster coordinator node and base station node. Equation (5), shows the energy depletion in data transmission by a normal node to the relay node within the local cluster. Equation (6) is used to compute the energy consumption in the process of data reception and transmission by the relay node to the CH and Equation (7) shows the energy depletion by CH in data communication with the upper layer cluster coordinator. Equation (8) depicts the energy used for the data communication process by a cluster coordinator with the upper layer cluster coordinator or the BS. Equation (9) depicts the energy consumed by the base stations in the BS layer.