## 4 Design and Method
### Description of the mAximum Data Collection Rate  
In R-WSNs, the data collection rate of a sensor is affected directly by the amount of remaining energy extracted from ambient conditions. More energy supplement means that more packets will be collected. According to the Equations (7) and (10), for a sensor \(i\), the maximum data collection rate per unit of time is formulated as:  
\[g(i)= \frac{E_{i}-(e_{idle}+e_{r}\sum_{j\in\vec{S}_{i}}(\lambda_{j} \varphi(j,i)+g_{j}))}{e_{g}}-\frac{e_{l}+e_{t}\sum_{j\in\vec{S}_{i}}(\lambda_{ i}\varphi(i,j)+g_{i})}{e_{g}} \tag{11}\]  
where, \(e_{l}\) is used for a sensor to listen channel before data delivery with a fixed value for each time transmission. Hence, it is affected significantly by data flow, i.e., a sensor does not need to broadcast its  
Figure 6: The aggregated transit traffic from a sensor to another sensor. existence when the channel is idle if there are no packets to deliver. Now, we let \(e_{l}=k\times\sum_{i,j\in V_{j}\in S_{i}}f_{i,j}\). The energy depletion for listening and transmission can be expressed as:  
\[e_{l}+e_{t}(i,j)=(k+e_{t})\times\sum_{i,j\in V_{j}\in S_{i}}f_{i,j} \tag{12}\]  
Since \(k\) is fixed and \(e_{t}\) is variable, we can let \(e_{tr}\) represents the \(k+e_{t}\). In fact, the energy utilization in idle state is a fixed value and small with less than 1 mW in per unit of time, which is only about \(1/100\) that of transmission and neglected in our work. Hence, the Equation (11) can be rewritten as:  
\[g(i)= \frac{E_{i}-e_{r}\sum_{j\in\widehat{S_{i}}}(\lambda_{j}\varphi(j,i)+g_{j} )}{e_{g}}-\frac{e_{tr}\sum_{j\in\widehat{S_{i}}}(\lambda_{i}\varphi(i,j)+g_{i })}{e_{g}} \tag{13}\]  
From the Equation (13), we can observe the data collection rate for a sensor is affected significantly by the amount of its received data. The available energy is mainly consumed for at least two parts: receiving and transmitting. When more packets are accepted by a sensor, its data generation rate declines due to limited power supplement, which is an objective and obvious phenomenon in WSNs and R-WSNs. However, completely distinct strategies are adapted in them and a sample is provided for elaborating different routing schemes, as shown in Figure 7.  
We assume node \(j\) and node \(i\) are source nodes and generate packets by sensing the environment periodically. There are at least two transmission schemes for delivering packets to the sink \(s\). One of them is that packets can be forwarded along the link path \(\{j,i,k,s\}\), where sensor \(i\) aggregates the packets from an upstream neighbor \(j\) with its self generated data and forwards it to the next-hop adjacent sensor until reaching the sink \(s\). This scheme is widely and recognized and used in traditional WSNs, which can reduce the amount of packets by removing redundant information and reducing energy consumption, effectively contributing to extended network lifetime. Another transmission scheme is that data from sensor \(j\) and \(i\) are delivered along paths \(\{j,a,b,s\}\) and \(\{i,k,s\}\), respectively. Both raw data generated by node \(j\) and \(i\) will be transmitted to destination without considering data aggregation. The main differences to the two schemes can be summarized separately. For the first strategy, the advantage is energy consumption reduced with fewer packets by aggregating data. However, on the one hand, the data generation rate of node \(i\) declines. On the other hand, nodes \(a\), \(b\) do not participate in the data transmission procedure, although their power is replenished repeatedly. For the second strategy, even though total energy consumption increases, the data collection rate of sensor \(i\) and sensors usage efficiency for node \(a\), \(b\) are improved. Therefore, for improving data collection rate, the latter data delivery method is more suitable to R-WSNs with better performance compared to the first method.