## 5 The Proposed Protocol
### Communication Overhead of the Proposed Protocol  
In this subsection, we aim to provide readers with a comprehensive demonstration of how our network model manages energy dissipation. In essence, the radio and channel propagation models for energy consumption are applied in this paper, which were previously described in [47; 48]. The energy used in the proposed protocol is broken down into two phases: setup energy and steady-state energy. * The first round, which involves both setup and steady-state phases, proceeds as follows:
* Setup phase:
* The sink node sends a request information message to each SN. Thus, the energy dissipated by each SN to receive the request information message is as follows: \[E_{RSNs}(NL_{C},d_{SN\tosink})=E_{dec}\times NL_{C},\] (34) where \(NL_{C}\) represents the length of control message. * The energy dissipated by each SN in order to send their information message to the sink node. \[E_{TSNs}(NL_{C},d_{SN\tosink})= \begin{cases}NL_{C}\times E_{dec}+NL_{C} \times\varepsilon_{f_{S}}\times d_{SN\tosink^{\prime}}^{2}&if\ d_{SN\tosink}<d _{o^{\prime}}\\ NL_{C}\times E_{dec}+NL_{C}\times\varepsilon_{emp}\times d_{SN\tosink^{ \prime}}^{4}&if\ d_{SN\tosink}\geq d_{o}.\end{cases} \] (35)
* The energy dissipated by each SN to receive the sink table message. \[E_{RSNs}(NL_{C},d_{SN\tosink})=E_{dec}\times NL_{C}.\] (36) Thus, the energy dissipated in the setup phase (\(E_{Setup}\)) is as follows: \[E_{Setup}=2\times NE_{RSNs}+NE_{TSNs}.\] (37)
* Steady-state phase:
* The energy dissipated by SNs, within a DFG, to send their data to a PDFH. \[E_{TSNs}(NL_{D},d_{SN\to PDFH})= \begin{cases}NL_{D}\times E_{dec}+NL_{D} \times\varepsilon_{f_{S}}\times d_{SN\to PDFH}^{2},&if\ d_{SN\to PDFH}<d_{o},\\ NL_{D}\times E_{dec}+NL_{D}\times\varepsilon_{emp}\times d_{SN\to PDFH}^{4},&if \ d_{SN\to PDFH}\geq d_{o},\end{cases} \] (38) where \(NL_{D}\) represents the length of data message. * The energy dissipated by a PDFH to receive the data from DFGMs, aggregate all the data, including PDFH, and then send them back to the sink node. \[E_{RPDFH}(NL_{D},d_{SN\to PDFH})=(N_{c}-1)\times NL_{D}\times E_{dec}.\] (39) \[E_{DAPDFH}(NL_{D},d)=N_{c}\times NL_{D}\times E_{fusion}.\] (40) \[E_{TPDFH}(NL_{D},d_{PDFH\tosink})= \begin{cases}NL_{D}\times E_{dec}+NL_{D} \times\varepsilon_{f_{S}}\times d_{DPFH\tosink^{\prime}}^{2},&if\ d_{DPH\to sink }<d_{o},\\ NL_{D}\times E_{dec}+NL_{D}\times\varepsilon_{emp}\times d_{DPH\tosink^{ \prime}}^{4},&if\ d_{DPH\tosink}\geq d_{o}.\end{cases} \] (41) Accordingly, the total energy dissipated by all PDFHs is denoted as follows: \[E_{PDFHs}=C\times(E_{RPDFH}+E_{DAPDFH}+E_{TPDFH}).\] (42)
* In the subsequent rounds, if the energy of a PDFH is higher than the energy of its DFG, only the steady-state phase is in effect. However, if the energy of a PDFH falls belowa DFG's average energy, regrouping is initiated. During regrouping, the sink node retains information about the SNs' locations but lacks real-time data on their current energy levels. The sources of energy dissipation in this scenario include the following:  
1. The energy dissipated by all PDFHs to send disjoin messages to the sink node. 2. The energy dissipated by a PDFH to send disjoin messages to the sink node. 3. The energy dissipated by a PDFH to send disjoin messages to SNs. \[E_{TCH}(NL_{C},d_{SDFH\to\sink})= \begin{cases}NL_{C}\times E_{elec}+NL_{C} \times\varepsilon_{fs}\times d_{PDFH\to\sink}^{2},&if\ d_{PDFH\to\sink}<d_{o}, \\ NL_{C}\times E_{elec}+NL_{C}\times\varepsilon_{mp}\times d_{PDFH\to\sink}^{4},&if \ d_{PDFH\to\sink}\geq d_{o}.\end{cases}  \tag{43}\]
4. The energy dissipated by a PDFH to send disjoin messages to SNs. \[E_{TCH}(NL_{C},d_{SN\to PDFH})= \begin{cases}NL_{C}\times E_{elec}+NL_{C} \times\varepsilon_{fs}\times d_{SN\to PDFH}^{2},&if\ d_{SN\to PDFH}<d_{o},\\ NL_{C}\times E_{elec}+NL_{C}\times\varepsilon_{mp}\times d_{SN\to PDFH}^{4},&if \ d_{SN\to PDFH}\geq d_{o}.\end{cases}  \tag{44}\]
4. The energy dissipated by SNs to receive disjoin messages from a PDFH.