## 3 Results and Discussion
### Simulations for Chain-Based Network
total Transmissions (\(N\)) considering only information frame/packet (without acknowledgments) transfer will be as shown in Equation (38),  
Figure 4.5  
where \(\lambda_{k}\) is a constant. Using Equation (38), an upper bound on number of total transmissions can be derived as depicted in Equation (39) for the cases when \(\lambda_{k}{P_{e}}^{k}=1\). \[ \begin{split} N=&(n)*(k-1)*(1+1+1\ldots+1+1)\\ =&(n)*(k-1)*(L+1)\end{split}  \tag{39}\]  
As proved in the preceding section, \(L=0,1,3\) in the proposed method, EBDKF, and Average fusion methods, respectively, to achieve a network MSE less than 0.3. It is evident that \(n=1\) in proposed method and average method as a node will send only one fused data frame/packet in a link. As there are two \(\hat{x_{ind}}\), \((\hat{x_{ind}})*((p_{ind}^{curr})^{-1})\), \(x_{loc}\) for each link in EBDKF per data gathering cycle, \(n=6\). Therefore, by substituting in Equation (39) for upper bound, \(N_{proposed}=(k-1)\), \(N_{AVG}=(4)*(K-1)\), \(N_{EBDKF}=(12)*(K-1)\). Therefore, when the network MSE is less than 0.3, total retransmissions will be lesser than the upper bound. In order to obtain only retransmission upper bound, \(L+1\) in Equation (38) will have to be replaced by \(L\). When acknowledgment frames/packets are also considered, the total transmissions will be even more than the values calculated in preceding sections. Although, the information retransmissions have been shown as a solution to improve the accuracy of the information fusion with AVG and EBDKF methods as it was stated in preceding section, they consume additional energy which is a major disadvantage in WSN s. To get the exact number of total retransmissions when BER is 0.05, Figure 9 was obtained to visualize the total retransmission attempts for data frames/packets utilized in the network when the maximum retransmission attempts are limited to one, two, three, and four among EBDKF and AVERAGE fusion techniques. We do not observe the total retransmissions of proposed method, as we came to conclusion that frame/packet retransmissions very slightly improve the accuracy of the estimate generated by proposed method. It can be clearly observed from Figure 9 that all values are within the upper bound for retransmission of 157 and 294 for AVG and EBDKF when \(L=3\) and \(L=1\), respectively. As observed in Figure 9, approximately total of 24 retransmission attempts are used in AVG method and 93 in EBDKF to achieve a network MSE less than 0.3. These additional retransmissions consume excessive energy for information retransmissions. Therefore proposed method can effectively save transmission energy compared to other fusion algorithms. 3.1.2 Simulations for Data Accuracy of a Deterministic Dynamic Parameter Having Bounded Sensor Noise  
The actual value of the system parameter measured by sensor nodes is not a constant value all the time; it can vary over time. For example, when sensor nodes are used to monitor the transmission lines in a smart grid environment [3; 4], voltage or frequency fluctuations can occur. Therefore, it is necessary to evaluate the performance of the proposed method for dynamic states to check whether the estimates can track the varying parameter. Here, we simulate the system parameter having a mean of 1.80 and having intermittent sinusoidal fluctuation representing the dynamic nature. All other values set for simulation are same as for deterministic parameter case. Here, we record the mean value of estimates of 50 sensors as a vector in OMNeT++ for 600 data gathering cycles. The dynamic parameter was modeled as a discrete sinusoidal parameter given by \(sin(k/20)\) where k is the sensor network's \(k\)th time step of measurement or in other words \(k\)th data gathering cycle.