## 4 Design and Method
### System Performance Comparison  
In order to further understand of the performance of our algorithm for Maximizing data collection rate (Mdcr) under network settings, in this section, we provide two schemes for performance comparison, including an optimal distributed lexicographic rate assignment (Dlex) designed for R-WSNs [53] and an Energy-efficient data aggregation tree (Edat) applied in WSNs. In [53], the authors propose distributed algorithms for determining the routing structure and amount of flows on each link without considering data fusion. It involves updated rate computation using optimal lexicographic rate assignment. In this method, nodes first compute their maximum rate using an initial rate procedure. Subsequently, they send a control packet containing the flow id and the maximum achievable rate to their next hop nodes. In [71], Edat is constructed for energy efficiency, in which energy management is the main task and data aggregation is a secondary aim for energy efficiency in routing design. There is limited literature on maximum data generation rate routing based on data aggregation in R-WSNs. At the same time, the two methods were widely recognized and used for maximum data collection rate in R-WSNs and data fusion in WSNs, which can be attributed to the decision to select them for performance comparison in our work. We first compare the data generation rate between our algorithm and other two schemes under the distinct number of sensors and only one sink, where the average node's duty-cycles are \(1\%,10\%,30\%\), respectively, as shown in Figures 12-14. From Figure 12, we can observe the data collection rates increase for all algorithms and that of our algorithm is slightly higher than that of other algorithms with the sensor density being improved for all different node's duty-cycles. Our algorithm is utilized for calculating the upper bound of data flow under ideal conditions rather than establishing a detailed  
Figure 12: Data generation rate with number of nodes increasing when duty cycle is \(1\%\). routing path. In actual application, the network throughput will be lower than the result value of our method. From Figure 12, we can see that the data generation rate of our algorithm is about 9% and 15% higher than that of the Dlex and Edat scheme, respectively. Since high sensor density indicates that more packets will be sensed and forwarded to the destination, the data collection rate will be improved, and power expenditure will be balanced between all sensors if we append more sensor nodes to the monitoring field in a reasonable number of sensors. Despite this, sensors usually keep in a low duty cycle and collision is scarce: if too many sensors are deployed in a smaller area, the collision will be critical and retransmission is more frequent, which causes serious energy waste. In this work, we only consider an ideal condition that a reasonable number of sensors were deployed outdoors and collision was ignored. Comparing Figure 12 with Figure 14, when the duty cycles exceed 30 times in Figure 14 compared with that in Figure 12, we can see that the data collection rate is improved significantly and it is about 25 times. Higher data collection rate is obtained with improved duty cycles, which indicates that a sensor will be in an active state more times and data transmission occurs more frequently. The deeper reason for higher duty cycle is that more energy is extracted from the environment. Therefore,  
Figure 14: Data generation rate with number of nodes increasing when duty cycle is 30%. Figure 13: Data generation rate with number of nodes increasing when duty cycle is 10%. a sensor will generate more packets utilizing extra energy, from which we observe that more data will be collected on a sunny day than in cloudy weather. We next analyze the data collection rate for all sensors under three data correlation settings (\(\alpha=0.001\), \(0.005\) and \(0.01\)), as shown in Figures 15-17.