## 5 Proposed Model
### Multi-Tier Federated Scheduling
The third-tier scheduling can be formulated as:  
\[\text{3 rd tier}=\text{LBS}_{\text{n}=1,2,...,n}(^{\text{pri},\text{ene}^{ \text{cons}},\text{TTI},\text{CQI},\text{dl}\ )}\rightarrow \begin{cases}\text{Txion} \\ \text{Rxion}\\ \text{Sleep}\end{cases}  \tag{15}\]  
On the whole, the proposed multi-tier hierarchical scheduling reduced the interference and data collision problems in the SDWSN environment. For the second- and third-tier scheduling, the NCFT method is utilized. The detailed explanation of the NCFT method for scheduling for second and third tiers is provided below. Figure 3 represents the second- and third-tier scheduling. Figure 3: NCFT based Multi-Tier Scheduling. NCFT-Based Scheduling (Second and Third Tiers)  
The proposed work frames the multi-tier scheduling as a non-cooperative game problem in which the players involved in the non-cooperative game theory act selfishly based on their strategies. The entities involved in the proposed NCFT game model are:  
\[\text{Players}:\text{M}=\langle\text{pl}_{1},\text{pl}_{2}\rangle\] \[\text{Player Strategy}:\,\text{St}_{\text{pl}_{1}}=\{\aleph_{1} \},\text{St}_{\text{pl}_{2}}=\left\{\aleph_{1}\right\};\,\aleph_{1}\in\{\text{ coll}\}\] \[\text{Profile of Strategy}:\text{St}=\text{St}_{\text{pl}_{1}} \times\text{St}_{\text{pl}_{2}}\] \[\text{Remeneration Function}:\text{R}(\text{St})=\tau_{1}\]  
where \(\langle\text{pl}_{1},\text{pl}_{2}\rangle\) denotes the first and second players, respectively, in which pl\({}_{1}\) implies the entity to be scheduled, and pl\({}_{2}\) implies the entity to perform scheduling. The strategy profile of every player can be denoted as \(\text{St}_{\text{pl}_{1}}\times\text{St}_{\text{pl}_{2}}\) in which \(\aleph_{1}\in\{\text{coll}\}\) is the strategy of every player to minimize the collision possibility, and \(\text{R}(\text{St})\) denotes the remuneration function. To be clearer, in NCFT every player tends to maximize their remuneration function based on the \(\aleph_{1}\). The player response can be determined by framing the linear equation based on Nash equilibrium, which can be formulated as:  
\[\text{r}_{1}\left(\aleph_{1},\aleph_{2}^{*}\right)=\frac{\text{k}}{2}\left( \frac{\aleph_{1}+\aleph_{2}^{*}}{2}+\text{ck}_{1}\aleph_{2}^{*}\right)-\aleph_{1 }^{2} \tag{17}\]  
\[\text{r}_{2}\left(\aleph_{1}^{*},\aleph_{2}\right)=\frac{\text{k}}{2}\left( \frac{\aleph_{1}^{*}+\aleph_{2}}{2}+\text{ck}_{1}^{*}\aleph_{2}\right)-\aleph_{2 }^{2} \tag{18}\]  
The linear equations can be differentiated based on the \(\aleph_{1}\) and can be provided as:  
\[\frac{\partial\text{r}_{1}}{\partial\aleph_{1}}=\frac{\text{k}}{4}+\frac{\text {kc}}{2}\aleph_{2}^{*}-2\aleph_{1}^{*}=0 \tag{19}\]  
\[\frac{\partial\text{r}_{2}}{\partial\aleph_{2}}=\frac{\text{k}}{4}+\frac{\text {kc}}{2}\aleph_{1}^{*}-2\aleph_{2}^{*}=0 \tag{20}\]  
Equations (19) and (20) can be rewritten to obtain the new linear equations that can be provided as:  
\[\frac{\text{k}}{4}+\frac{\text{kc}}{2}\aleph_{2}^{*}=2\aleph_{1}^{*}\]  
\[\text{Players}:\text{M}=\langle\text{pl}_{1},\text{pl}_{2}\rangle\]0  
By solving Equation (21), the Nash equilibrium point can be obtained. Since we adopt a non-cooperative game model, the remuneration of the players is reduced; hence, they must obtain lesser \(\text{r}_{1}\) with minimized \(\aleph_{1}\). Algorithm 3 provides pseudocode for the proposed NCFT-based multi-tier scheduling.