## 5 Performance Evaluation
### Computational Complexity  
The computational complexity of the proposed FCM-OR protocol is expressed as follows:  
Figure 17: MAC layer packet classification of different OR protocols with varying duty cycles in a highly dense area.  
\[\mathcal{O}(N\cdot[C_{\text{FCM}}+C_{\text{FR}}]), \tag{7}\]  
where \(N\) is the total number of nodes in the network, \(C_{\text{FCM}}\) denotes the cost of the fuzzy c-means clustering step, and \(C_{\text{FR}}\) represents the cost of computing the Forwarding Rank (FR) for candidate forwarders.  
The complexity of FCM clustering for a single node is as follows:  
\[C_{\text{FCM}}=\mathcal{O}(n_{c}\cdot c\cdot t), \tag{8}\]  
where \(n_{c}\) is the number of one-hop neighbors, \(c\) is the number of clusters, and \(t\) is the number of iterations until convergence. As both \(c\) and \(t\) are constants in our implementation, this simplifies to the following:  
\[C_{\text{FCM}}=\mathcal{O}(n_{c}). \tag{9}\]  
Similarly, FR computation is performed over the same \(n_{c}\) nodes:  
\[C_{\text{FR}}=\mathcal{O}(n_{c}). \tag{10}\]  
Thus, the total worst-case complexity becomes the following:  
\[\mathcal{O}(N\cdot n_{c}). \tag{11}\]  
In sparse regions where \(n_{c}\) is below the clustering threshold and FCM is skipped, the complexity reduces to the following:  
\[\mathcal{O}(N). \tag{12}\]  
Therefore, the worst-case computational complexity of FCM-OR is \(\mathcal{O}(N\cdot n_{c})\), where \(n_{c}\) is the number of neighbor nodes considered during local clustering and FR computation. In sparse regions where clustering is bypassed, the complexity reduces to \(\mathcal{O}(N)\), making the protocol scalable and lightweight for resource-constrained sensor nodes.  
The overall protocol efficiency is evaluated using key performance metrics, including packet retransmission overhead (Figures 7, 12 and 16), collision rate per packet (Figure 10), and end-to-end packet delivery delay (Figure 15). Simulation results confirm that packet overhead and collision rates are significantly lower in FCM-OR than in ORW and ORR. However, end-to-end delay is slightly higher due to detour routing, which avoids high-density areas to improve energy efficiency and network longevity.  
Simulation results confirm that FCM-OR outperforms ORW and ORR in terms of energy consumption, network lifetime, load balancing, and collision minimization. As demonstrated in Figures 7, 12, and 16, packet delivery ratio (PDR) values exceed one due to multiple packet receptions at the sink, highlighting the impact of redundant forwarding. Future research could extend this comparison to more recent OR protocols. Furthermore, adaptive parameter selection based on application-specific requirements can further optimize performance trade-offs in energy-constrained WSNs.  
Moreover, in this study, we have derived a detailed calculation of computational complexity in Section 5.3, where it is proven that our protocol can explicitly reduce computational overhead, making the protocol scalable and lightweight for resource-constrained sensor nodes. On the other hand, several recent works such as [56, 57, 58, 59] did not consider the computational overhead for the algorithm. Furthermore, compared to the total delay, computation overhead is ignorable.