## 5 Simulation Results
### Parameter Settings  
In this section, we investigate the impact of routing factor \(\alpha\) and clustering factor \(\beta\) on the network lifetime and provide a guideline for setting these parameters. In the default setting of the simulation, there are 400 nodes randomly deployed in a 200 \(\times\) 200 m\({}^{2}\) area. The clustering range \(R_{c}\) is set as \(70\) m.  
At first, we evaluate the network lifetime when \(\alpha\) and \(\beta\) vary from \(0\) to \(1\). As shown in Figure 4, the optimal value of \(\beta\) is \(0.2\), and it is irrelevant with respect to the value of \(\alpha\). According to Equation (8), the \(\beta\) determines how to select the CH for each CM, and it will impact the scale of clusters in \(\mathcal{R}(1)\). This \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Application} & Initial energy & 2 J \\  & Data packet size (\(l\)) & 125 Bytes \\  & Sink location & \((0,0)\) \\  & Round & 20 TDMA frames \\ \hline \multirow{4}{*}{Radio model} & \(E_{elec}\) & 50 nJ/bit \\  & \(\epsilon_{fs}\) & 10 pJ/bit/m\({}^{2}\) \\ \cline{1-1}  & \(E_{DA}\) & 5 nJ/bit/signal \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. indicates that the setting of \(\beta=0.2\) is optimal to reach the load balance of all clusters. Therefore, the \(\beta\) is set as \(0.2\) in the rest of the simulations. On the other hand, the optimal \(\alpha\) varies from \(0.5\) to \(1\) with different values of \(\beta\). Nevertheless, since the optimal \(\beta\) is fixed at \(0.2\), the optimal setting of the parameters can be obtained by \(\alpha=0.6\) and \(\beta=0.2\) from Figure 4. In HCR-1, the routing factor \(\alpha\) plays an important role in selecting reasonable relays. Thus, in the rest of this section, we investigate how to set the optimal value of \(\alpha\) in different network scenarios. Figure 5 shows the relations between \(\alpha\) and network lifetime when \(R_{c}\) grows from \(30\) m to \(70\) m. The optimal \(\alpha\) varies from \(0.6\) to \(0.9\) with respect to different \(R_{c}\), and the optimal \(\alpha\) gets larger with smaller \(R_{c}\). Generally, when \(R_{c}\) gets smaller, the node will receive less AC messages before its backoff timer \(T_{b}\) terminates. Thus, it will increase the number of CHs due to the growth of the probability that the JOIN list is empty. In this case, selecting a closer relay is helpful to improve the energy efficiency. This is the reason that the optimal \(\alpha\) gets larger with smaller \(R_{c}\). The optimal network lifetime is approached with \(R_{c}=70\) m and \(\alpha=0.6\). The clustering range \(R_{c}\) has been proven to have great impacts on the network lifetime [19, 25]. Figure 5 shows that, with the constraint of \(R_{max}=R_{t}=70\) m to keep the network connectivity, the growth of \(R_{c}\) is helpful to reduce the inter-cluster traffic load and prolong the network lifetime. Then, we study the optimal \(\alpha\) when the number of nodes \(N\) is set as \(200\), \(400\) and \(800\). The network area is fixed at 200 \(\times\) 200 m\({}^{2}\), and the clustering range \(R_{c}=70\) m. Figure 6 shows that the network lifetime becomes longer with a larger number of nodes. This is because the distance between nodes is shorter, and there are more nodes that can be used to balance the energy consumption. The optimal \(\alpha\) is \(0.6\) when \(N=400\), while the optimal \(\alpha\) is \(0.8\) with \(N=800\). This is because the number of relay candidates in \(\mathcal{R}(1)\) increases with more neighbor nodes. Therefore, the CHs with low energy tend to select a closer relay to avoid early failure, which makes the optimal \(\alpha\) larger. Figure 4: Network lifetime with different routing factor \(\alpha\) and clustering factor \(\beta\). It is worth noting that the \(\alpha\) does not have an optimum value when \(N=200\). When the nodes are not dense enough, there are few CHs that can serve as relays.