## 2 Modeling and Formulation
### Air-R
** The Air-R has insufficient nodes (\(Air-R_{\textit{cargo}}\)) to replace all failed nodes in \(NR\)s, as expressed in (12):  
\[Air-R_{\textit{cargo}}<NRs \tag{12}\]  
**Situation 3. ** The time available to replace the failed nodes \(T_{\textit{replace}}\) is not enough to replace them all, which is expressed in (13):  
\[(T-T_{\textit{replace}}(NRs)<(NRs*T_{\textit{replace}}) \tag{13}\]  
In all previous situations, the \(BS\) starts analyzing and repeats adding to \(NR\)s until the \(PN_{\max}\) conditions are no longer met. Specifically, the node with the highest \(PN_{\max}\) is removed from the set of nodes \(N\), which means that it is a failed node. In this emergency condition, the proposed solution is implemented with the expectation that the failed nodes will somehow still recharge until replaced. **Situation 4. ** The time available to replace the failed nodes is adequate to replace all of them. The number of available replacement nodes is also sufficient, and the energy needed to cover the complete round trip distance is more than that available in the Air-R. These conditions are expressed in (14):  
\[((T-T_{\textit{replace}}(NRs)\geq NRs*T_{\textit{replace}})\wedge(Air_{\textit{ }}R_{\textit{cargo}}\geq NRs)\wedge(E_{Air-R}\geq E_{RT}^{\max})) \tag{14}\]  
In **Situation 4**, all nodes in \(NR\)s are replaced. The Air-R flies back to the \(BS\) for the next \(RT\), or it checks if it can perform another replacement \(RT\). The SRA describes the behavior of the \(BS\), Air-R, and nodes; it schedules and selects the optimal situation for Air-R to replace every failed node in \(NR\)s.  
An optimal path is determined for Air-R to visit and replace the failed nodes in the sector, which have the largest priority and are provided with a new fully functional one (with a minimum travel distance that is an important phase in the proposed network; this problem can be demonstrated as a well-known traveling salesman problem (TSP)) [30]. The travel distance should be minimized to save the time and energy of the Air-R. Therefore, we proposed the schedule replacement algorithm to solve the problem, as in Algorithm 2. ``` Data: Set of sensor nodes \(NR\)s at time \(T\) Result: Replacement on \(NR\)s in every round trip (RT) RT within \(T\)
1. Travel Method \(\leftarrow\) TSP
2. Nodes replaced \(\leftarrow\) FALSE for\(\forall n_{k}\in N\)do
3. BS \(\leftarrow\) Check
4. if\(PN_{\max}\) is max then
5. Add \(n_{k}\) to NRs
6. end
7. end
8. while Nodes replaced = FALSE do
9. if\((T_{\textit{replace}}(NRs)\geq T)\vee(E_{Air-R}<E_{RT}^{\max})\) \[\vee(Air-R_{\textit{cargo}}<NRs)\vee\] \[(T-T_{\textit{replace}}(NRs)<(NRs*T_{\textit{replace}})\]

## 3 Performance Evaluation

In this section, we evaluate the performance of the proposed SBR technique for R-WSN through an experimental simulation. We investigate the network sustainability and the Air-R status (efficiency and effort). A set of simulations are run to experimentally determine the optimal value of period \(T\) and the appropriate solution for Air-R to replace the failed node described in previous sections. ### Simulation Results

Two different network sizes are examined in our experiments: namely, large and small networks. We examine the influence of SBR on both networks to show the effectiveness of our proposed technique. We investigate a small network consisting of 10-100 nodes that are randomly deployed in the monitoring field with radius \(R=100\) m and a large network consisting of 100-1000 nodes with radius \(R=600\) m.

The \(BS\) is located at the field center and at the corner of each quartet of the R-WSN monitoring area. As explained in Section 2, the \(BS\) divides the network field into a number of rings and sectors utilizing the directional antenna with power control capabilities. The \(BS\) helps select the \(CH\) for each sector after receiving information from sensors and then automatically informs all nodes in the sectors about their _CHs_.