## 5 Performance Evaluation
### Impact of the Stretch Factor \(\epsilon\)  
In this section, we study the impact of the stretch factor, \(\epsilon\), in the performance of BSMH. To do so, we vary the value of \(\epsilon\) from 0.3 to 4.5 and observe its impact on the routing path stretch and load balance achieved by BSMH. Although the experiments are conducted on all topologies shown in Figures 8 and 9, to facilitate the readability, in what follows, we show only the results on one topology with single hole, i.e., Topology 2 (Figure 8b). Figure 10 depicts the impact of \(\epsilon\) on the routing path stretch. As shown, the average routing path stretch increases gradually when \(\epsilon\) increases. This phenomenon can be explained as follows. In BSMH, packets are forwarded along the Euclidean paths that are the image of the base paths through homothetic transformations. The scale factor of the homothetic transformations is proportional to \(\epsilon\). Therefore, the increase of \(\epsilon\) leads to the enlargement of the Euclidean paths, and increase the routing path stretch, consequently. The impact of \(\epsilon\) on the maximum packet forwarding ratio is shown in Figure 11. As shown, when \(\epsilon\) increases from 0.3 to 0.8, almost all the Euclidean routing paths stay inside the network. \begin{table}
\begin{tabular}{l l} \hline \hline
**Factor** & **Value** \\ \hline MAC type & CSMA/CA \\ Interface queue model & DropTail \\ Transmission of radio & TwoRayGround \\ Antenna type & OmniAntenna \\ Queue length & 50 packets \\ Transmission range & 40 m \\ Node initial energy & 30 J \\ Node idle power & 9.6 mW \\ Node receive power & 45 mW \\ Node transmit power & 88.5 mW \\ Packet sending interval & 10 s \\ Data packet size & 50 bytes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of a sensor node. Figure 10: Impact of \(\epsilon\) on routing path stretch. The blue lines represent the average stretch of all routing paths, the red lines represent the percentage of Euclidean routing paths that exceed the network boundary. Therefore, increasing \(\epsilon\) helps to enlarge the number of sensors participating in forwarding packets, thereby reduce the maximum packet forwarding ratio. When \(\epsilon\) varies from 0.8 to 2.0, several Euclidean routing paths may exceed the network boundary, thus the increase of \(\epsilon\) almost cannot help to improve the load balance (i.e., the maximum forwarding ratio is almost stable when \(\epsilon\) varies in this range). The greater \(\epsilon\), the more Euclidean routing paths exceeding the network boundary. Specifically, when \(\epsilon\) reaches 2.5, more than 25% of Euclidean routing paths staying outside of the network boundary. Hence, increasing \(\epsilon\) cannot enlarge the number of sensors participating in forwarding packets. In the meanwhile, the increase of \(\epsilon\) results in the increase of routing path length (see Figure 10), thus imposes more traffic load on the sensor nodes. Consequently, when \(\epsilon\geq 2.5\), the increase of \(\epsilon\) worsen the load balance. Figure 12 illustrates the impact of \(\epsilon\) on the network lifetime. The trend of the network lifetime is similar to that of the load balance, but they are not exactly the same. As shown, the network lifetime gets the peak at \(\epsilon=1.5\). Before the peak, the increase of \(\epsilon\) extends the network lifetime significantly, but beyond the peak, the network lifetime decreases fast when increasing \(\epsilon\). This phenomenon can be explained as follows. When \(\epsilon\leq 1.5\), the increase of \(\epsilon\) improve the load balance significantly (this can be seen through the big slope in Figure 11), thus the network lifetime is extended. When \(\epsilon\) varies from 1.5 to 2.5, the load balance stays almost the same, but the average routing path stretch of all routing paths increases (as shown in Figure 10). Please note that besides the energy consumed for sending packets, nodes must spend energy for receiving packets from its neighbors.