## 5 Proposed Work
### Computation of the Optimal Number of DCVs  
This section presents an analytical approach to determine the optimal number of data collector vehicles (DCVs) needed to guarantee a continuous network operation. As stated earlier, in LS-WSNs scenarios, the DCVs may experience a very long traveling path for data gathering because many DCPs need to be traversed. Data collection in this case will incur large delays with a single DCV, especially for time-sensitive applications including waste management. For this reason, we propose the use of multiple DCVs since a single DCV is infeasible in an LS-WSN scenario. We aim to determine the optimal number of DCVs while ensuring network stability and timeliness. For the purpose of analysis, we plan to achieve this objective under a given time constraint (\(T_{c}\)). However, the approach can also be applied for delay-tolerant applications since we are using opportunistic mobile DCVs. If \(T_{d}\) denotes the total time spent by a DCV in one trip of data gathering, then \(T_{d}\) equals the total movement time on a fixed path plus the stopping time at optimal points for data collection. Thus,  
\[T_{d}=\sum_{k=1}^{s}T_{k}+\sum_{k=1}^{s-1}d(OP_{k}-OP_{k+1})/v \tag{6}\]  
where \(T_{k}\) denotes the stopping time at the \(k\)th optimal point, and \(d(OP_{k}-OP_{k+1})\) is the Euclidean distance between two optimal points. Thus, the stopping time at an OP (\(T_{so}\)) is calculated as:  
\[T_{so}=\sum_{k=1}^{|L(OP_{j})|}\sum_{i=1}^{|N(l_{k})|}D_{k}/v \tag{7}\]  
where \(D_{k}\) denotes the data size of the CH (\(l_{k}\)), while \(L(OP_{j})\) and \(N(l_{k})\) denote the set of CHs and set of sensors within the \(OP_{j}\) and CH \(l_{k}\), respectively. The DCV moves at a speed of \(v\) (m/s). The stopping time of the DCV is negligible compared to the time spent for data collection. The time spent for data collection depends on the data size of the CH transmitting the data at every DCP. \(r^{\prime}\) represents data collection rate of the DCV and \(L_{DREQ}\) denotes the data size (i.e., length of the data request (DREQ) in bits). By inputting Equation (7) into (6), the DCV's traveled time (i.e., the total time spent visiting all optimal points in a region) can be modified as follows:  
\[T_{d}=\sum_{k=1}^{s}\left(\sum_{j=1}^{|L(OP_{k})|}\sum_{i=1}^{|s(l_{j})|}L_{ DREQ}/r^{\prime}\right)+\sum_{k=1}^{s-1}d(OP_{k}-OP_{k+1})/v \tag{8}\]  
Let \(T_{c}\) denote the deadline time constraint for data collection, and let \(V\) represent the optimal number of DCVs in deadline time constraint. Thus, the optimal time (\(T_{opt}\)) using \(V\) number of DCVs is given by  
\[T_{opt}=\frac{1}{V}\left[\sum_{k=1}^{s}\left(\sum_{j=1}^{|L(OP_{k})|}\sum_{i= 1}^{|s(l_{j})|}\frac{L_{DREQ}}{r^{\prime}}\right)+\sum_{k=1}^{s-1}\frac{d(OP_{ k}-OP_{k+1})}{v}\right] \tag{9}\]  
For a deadline time constraint, the value is  
\[T_{opt}\leq T_{c} \tag{10}\]  
Combining Equations (8)-(10), we obtain:  
\[V\geq\left[\frac{\sum_{k=1}^{s}\left(\sum_{j=1}^{|L(OP_{k})|}\sum_{i=1}^{|s( l_{j})|}\frac{L_{DREQ}}{r^{\prime}}\right)+\sum_{k=1}^{s-1}\frac{d(OP_{k}-OP_{k+1})}{v}}{ d_{TC}}\right] \tag{11}\]Hence, Equation (11) can be used to determine the optimal number of DCVs deployed in the network for data collection for SC waste management operations. We note that the initial position of each MCV depends on the limit of the deadline time of data collection and the path length. Based on the above parameters, the path length traveled by each \(DCV_{i}\) (\(L_{path(\theta_{i})}\)) is calculated as:  
\[L_{path(\theta_{i})}=T_{c}\times v \tag{12}\]  
To provide further clarification, if we consider a network containing 8 OPs and 13 CHs (i.e., leader nodes), as shown in Figures 10 and 11 for a single DCV and multiple DCVs, respectively, and assume that each sensor relays 5 packets with a data size of 512 bytes, then CHs 1-13 have 50, 55, 65, 65, 60, 55, 65, 65, 100, 60, 50, 65, and 60 data packets to relay, respectively.