## 3 eDSA = Efficient Dual Sink Algorithm for Event-Driven WSNs by using SS and MS
### Energy Model in General
#### 3.3.2 Round System (RS)  
For simplicity, discussion is carried out at \(d\)\(<\) d\({}_{0}\) and Event Node (EN) denotes the active nodes. */ Time relationship:  
\[\mathcal{T}_{\mathcal{R}_{\mathrm{j}}}\)=\(\mathcal{T}_{\mathcal{EE}}\)\(+\)\(\mathcal{T}_{\mathrm{SSA}}\)\(+\)\(\mathcal{T}_{\mathrm{RD}}\)\(+\)\(\mathcal{T}_{\mathrm{MSmov}}\)\(+\)\(\mathcal{T}_{\mathrm{MSA}}\) (6)  
Energy relationship:  
\[E_{\mathcal{T}_{\mathcal{R}_{\mathrm{j}}}}\)=\(E_{\mathcal{T}_{\mathcal{EE}}}\)\(+\)\(E_{\mathcal{T}_{\mathrm{SA}}}\)\(+\)\(E_{\mathcal{T}_{\mathrm{RD}}}\)\(+\)\(E_{\mathcal{T}_{\mathrm{MSmov}}}\)\(+\)\(E_{\mathcal{T}_{\mathrm{MSA}}}\) (7)  
(1) Energy consumption for EE:  
\[E_{\mathcal{T}_{\mathcal{EE}}}\)=\(\sum_{\ell=0}^{\mathrm{N}}\)( E\({}_{\mathrm{elec}}\)\(\ast\)\(\delta_{\mathrm{EN}_{\ell}^{\mathrm{J}}}\)\(+\)\(\mathcal{E}_{\mathrm{amp}}\)\(\ast\)\(\delta_{\mathrm{EN}_{\ell}^{\mathrm{J}}}\)\(\ast\)(\(d_{\mathrm{SS-EN}_{\ell}}^{\mathrm{J}}\))\({}^{2}\))\(+\)E\({}_{\mathrm{elec}}\)\(\ast\)\(\delta_{\mathrm{EN}_{\ell}^{\mathrm{J}}}\)\(\ast\)N (8)  
where:  
\(\ell\) is the sensor index ( \(i\)\(\in\)\(\{\)1\(\sim\)N\({}_{\mathrm{SS}}^{\mathrm{J}}\)\(\}\))  
j is the round index  
Figure 3: Processing map based on time coordinate.  
\(\delta_{\text{EN}_{i}^{\text{j}}}\) shows corresponding size of the message that be used during the communication between EN\({}_{i}^{\text{j}}\) and SS.  
\(d_{\text{SS}-\text{EN}_{i}}^{\text{j}}\) is the distance between SS and EN\({}_{i}^{\text{j}}\) in the current round.  
(2) Energy consumption for SSA:  
\[E_{\mathcal{T}_{\text{SSA}}}=\sum_{i=0}^{\text{N}_{\text{SS}}^{\text{j}}}( \text{N}_{\text{SS}}^{\text{j}}*\overline{d_{\text{SS}}^{\text{j}}})*\mathcal{ C}_{\text{SS}}^{\text{j}} \tag{9}\]  
where:  
\(\text{N}_{\text{SS}}^{\text{j}}\) is the total number of ENs under SS's control. \(\sum_{i=0}^{\text{N}_{\text{SS}}^{\text{j}}}\text{EN}_{i}^{\text{j}}\) is a set of ENs under SS's controls. \(\overline{d_{\text{SS}}^{\text{j}}}\) is the average distance between \(\text{N}_{\text{SS}}^{\text{j}}\) and EN\({}_{i}^{\text{j}}\). \(\mathcal{C}_{\text{SS}}^{\text{j}}\) shows the capacity that indicates how much energy does SS cost to manage a unit distance(\(\mathcal{d}_{\text{unit}}\)) data communication. The capability is determined in a priority. And the value of \(\mathcal{C}_{\text{SS}}^{\text{j}}\) is proportional to its residual energy \(E_{\text{SS}}^{\text{j}}\) and radio frequency ( V\({}_{\text{SS}-\text{fre}}^{\text{j}}\)) at \(\mathcal{R}_{\text{j}}\):  
\[\mathcal{C}_{\text{SS}}^{\text{j}}=f(E_{\text{SS}}^{\text{j}},\text{V}_{ \text{SS}-\text{fre}}^{\text{j}})=\text{\mu}_{\text{SS}}^{\text{E}}\text{E}_ {\text{SS}}^{\text{j}}+\text{\mu}_{\text{SS}}^{\text{V}}\text{V}_{\text{SS}- \text{fre}}^{\text{j}}+\text{\eta}_{\text{SS}} \tag{10}\]  
where, \(E_{\text{SS}}^{\text{j}}\) is the residual energy of SS at \(\mathcal{R}_{\text{j}}\). \(\text{\mu}_{\text{SS}}^{\text{E}}\), \(\text{\mu}_{\text{SS}}^{\text{V}}\)_and_\(\text{\eta}_{\text{SS}}\) are the parameters. (3) Energy consumption for RD. \[E_{\mathcal{T}_{\text{RD}}}=\sum_{\mathcal{K}=0}^{\text{N}_{\text{MS}}^{\text{ N}_{\text{MS}}^{\text{j}}}}(\text{ E}_{\text{elec}}*\delta_{\text{EN}_{\text{$X$}}^{\text{j}}}+\text{ \mathcal{E}}_{\text{amp}}*\delta_{\text{EN}_{\text{$X$}}^{\text{j}}}*(d_{\text {SS}-\text{MS}}^{\text{j}})^{2})+\text{E}_{\text{elec}}*\delta_{\text{EN}_{ \text{$X$}}^{\text{j}}}*\text{N} \tag{11}\]  
where:  
\[\text{N}=\text{N}_{\text{SS}}^{\text{j}}+\text{N}_{\text{MS}}^{\text{j}} \tag{12}\]  
\(\mathcal{K}\in\{1\text{$\sim$}\text{N}_{\text{MS}}^{\text{j}}\}\)  
\(\text{N}_{\text{MS}}^{\text{j}}\) is the total number of ENs under MS's control. \(\sum_{\mathcal{K}=0}^{\text{N}_{\text{MS}}^{\text{j}}}\text{EN}_{\mathcal{K}} ^{\text{j}}\) is a set of ENs under MS's control. (4) Energy consumption for MSmov:  
For the movement to the destination (\(\mathcal{X}_{MS}^{\text{j}}\), \(\mathcal{Y}_{MS}^{\text{j}}\)) from current (\(\mathcal{X}_{MS}^{\text{j}-1}\), \(\mathcal{Y}_{MS}^{\text{j}-1}\)).