## 2 Modeling and Formulation
### Air-R
Our working environment is a dynamic sensing activity, in which each sensor randomly sends its recharging requests through the \(CH\) to the \(BS\). The \(BS\) checks and analyzes the incoming requests from nodes to find any abnormal recharging requests and determine the node sector location. Then, the \(BS\) sends the Air-R in \(RT\) within a given trip time \(T\). The analysis is performed for values of \(T\in 30\), 60, 90, 120, 150,..., 350 min. We examine the monitoring field twice to show the network scalability and sustainability. We assume that the Air-R flies at a constant speed of 6 m/s, and that the constant replacing time for each sensor is 3 s. Every esteem in the figures is the mean of the outcomes (achieved by applying the proposed technique to 20 different scans of the field for the same network size). We first examine the performance and sustainability of the network without the proposed SBR technique and then apply the proposed SBR in small networks (by varying the network size from 10 sensors to 100 sensors within \(T\)). Figure 5 shows that applying the SBR technique significantly improves R-WSN performance and network sustainability for a small-sized network. The figure shows that the average of the network sustainability (using the SBR) starts from 100%, and becomes sustainable at 80% with the increase of the nodes; whereas, it is sustainable at 42% without SBR. The sustainability average gap between them is high, particularly if the network size is small. A few damaged nodes are determined, which may affect network efficiency and sustainability. Hence, the SBR technique can make the network more sustainable. Similarly, when the network size is greater than 200 sensors, network sustainability keeps the ratio at 78% with SBR Figure 6. Thus, the proposed technique keeps the sensors alive with time. Without applying the proposed SBR, the ratio is less than 37%. Therefore, the performance and sustainability of the network increase with SBR. Clearly, the proposed technique can work effectively in large networks due to the early and rapid detection of failed nodes. This technique successfullydiscovers the failure by studying and analyzing the recharging requests from the nodes to the \(BS\) (as well as the characteristics of the Air-R) in order to reach the exact sector and replace the failed nodes at the right time. When utilizing the SBR technique in our analysis, we focus on the following key performance parameters:

* _Network efficiency and sustainability average:_ This parameter demonstrates the number of times the objective of keeping all sensors alive is reached when the simulation runs several times. In the proposed analysis, the solution is compared in terms of R-WSN sustainability in a small or large network. The obtained result is plotted in Figure 4:0 For the adopted simulation settings, the replacement techniques perform better than recharging failure nodes. This result is attributed to the fact that the time required for node replacement is significantly lower than the time needed to continuously recharge the failed nodes if failure occurs, which is a reasonable assumption. Figure 7a,b shows that the average of detecting and expecting the failure node is high in both network sizes when the SBR algorithm is applied (i.e., 87% for small R-WSN at the end of time and 36% without applying the SBR). For a large-sized network, the average is 81% with SBR and 30% without applying SBR. Therefore, the SBR technique utilizing Air-R always succeeds in keeping the network alive with high efficiency and sustainability. Figure 4:1 Network sustainability for a small network from 10 to 100 nodes. Figure 4:2 Network sustainability for a large network from 100 to 1000 nodes. * _Fast detection and the number of replaced failed nodes:_ These parameters give a quantitative value of the number of failed nodes each time when Air-R takes a _RT_ (in which the SBR technique succeeds in keeping the network alive).