# On Connectivity and Energy Efficiency for Sleeping-Schedule-Based Wireless Sensor Networks
## 4 System Simulation
### The Simulation Results of the Communication Radius on the Connecting Performance  
The simulation is carried out in two different conditions, (1) when the node communication radius \(R\) is determined, the functional relationship between the node connectivity \(K\) and the connectivity probability \(P_{r}\), and (2) when the network node connectivity \(K=8\), the functional relationship between the node communication radius \(R\) and the connectivity probability \(P_{r}\). In order to guarantee the accuracy of the results, every 50 sprinklers were divided into a group, up to 100 groups, a total of 5000 times [22]. This experiment is mainly to determine the statistics of these 100 groups of data, then take any node in each set of experiments, count the number of its neighbor nodes according to the given communication radius \(R\). Then we calculate the connectivity probability \(P_{r}\), where the range of connectivity \(K\) is 0-10 [24], the communication radius \(R\) is, in turn, \(R=9\), \(R=14\), \(R=16\), and \(R=18\). The simulations are shown in Figure 3.  
Figure 3 shows the function curve of the network node connectivity \(K\) and the connectivity probability \(P_{r}\), when the node communication radius is \(R=9\), \(R=14\), \(R=16\), and \(R=18\), respectively. Among them, the simulation curve is drawn based on the results of 100 groups of experiments, and the theoretical curve is derived from the calculation formula of Poisson distribution [25]. As can be seen from Figure 3, the simulation curve is close to the theoretical curve, that is to say, the error is relatively small. We can prove that the number of sensor nodes randomly sprinkled in the monitoring area can be approximately regarded as the Poisson distribution. Thus, the network node connectivity \(K\) also conforms to the principle of Poisson distribution.  
Figure 2: The relationship between the network connectivity and node communication radius.  
In addition, we analyze the above four Figures one by one, in Figure 3a, when the node communication radius \(R=9\), and the connectivity \(K=2\), the probability is the largest. In Figure 3b, when the node communication radius \(R=14\), and the connectivity \(K=6\), the probability is the largest. In Figure 3c, when the node communication radius \(R=16\), and the connectivity \(K=7\) or \(K=8\), the probability is the largest, at this point the network is firmly connected. Figure 3d when the node communication radius \(R=18\), the connectivity probability is the largest, and the connectivity \(K\) has exceeded the given range. It can be concluded that, when the node communication radius \(R\) is larger, the expected network connectivity \(K\) is greater. In other words, in order for the greater network connectivity \(K\), the node communication radius \(R\) must be larger [26]. According to this discipline combined with the new network connectivity and energy efficiency algorithm, the algorithm chooses the optimal sleep nodes to wake up, to make the network to the desired connectivity, at the same time, the energy consumption can be reduced to the minimum.