## 6 Performance Analysis
### Communication Cost  
The communication cost is measured by the total number of packet transmissions [4,9]. The communication cost of both SLP and SLP-E is the result of four phases, _i.e._, the broadcast by the base station during the network initialization, the \(h\)-hops limited flooding initialized from the source, the \(h\)-directed routing, and the shortest path routing. As the communication cost of SLP and SLP-E resulting from the base station broadcast is the same as that of the existing work [3,4,9,13], we focus on studying the communication cost resulting from the last three phases. In addition, since \(h<\!\!<\!\sqrt{S/\pi}\), where \(S\) denotes the area of the network, the communication cost for the limited flooding phase is extremely low. Specifically, the communication overhead of limited hop flooding is \(C_{{}_{L}}=\pi h^{2}\rho\), where \(\rho\) denotes the node density. We also have the \(n=S\rho\). Then, we obtain \(C_{{}_{L}}=(\pi n\,/\,S)h^{2}\). Since \(h<\!\!<\!\sqrt{S/\pi}\), then we have \(C_{{}_{L}}=(\pi n\,/\,S)h^{2}<<n\). Since the communication cost of limited flooding is extremely low. Therefore, we don't consider the communication cost resulted from this phase as well. According to the above analysis, we will only discuss the communication cost from the \(h\)-directed routing and the shortest path routing phases. Below, we will discuss the communication cost of SLP and SLP-E, respectively. (1) Communication Cost for SLP  
Figure 2 shows the idea of SLP using a circle \(I\) with a center point \(s\) and a radius \(h\). In SLP, a packet is first transmitted to a phantom location (e.g., \(p_{4}\in I\)) after \(h\)-directed routing. Then, the packet is transmitted from \(p_{4}\) to the base station by the shortest path routing. Hence, the communication cost of SLP is \(\mathit{Hop}_{s,p4}+\mathit{Hop}_{p4,b}\), where \(\mathit{Hop}_{s,p4}=h\). According to the cosine theorem, we also have \(\mathit{Hop}_{\mathit{p}\mathit{+}\mathit{,b}}=\sqrt{h^{2}+H^{2}\cdot 2 hH\cos \alpha}\) for the triangle with \(\mathit{s}\), \(\mathit{p}\mathit{\ast}\) and \(\mathit{b}\). Therefore, we obtain the communication of SLP as follows:  
\[\mathit{h}\,+\,\sqrt{h^{2}+H^{2}\cdot 2 hH\cos\alpha} \tag{2}\]  
where \(\alpha\in(0,\,2\pi]\). According to Equation (2), the communication cost of SLP achieves the maximum value of \(H+2h\) when \(\alpha=\pi\). Figure 2 shows that in this case, the packet first reaches \(\mathit{p}_{5}\) through the \(\mathit{h}\)-directed routing, and then the base station through the shortest path routing. Similarly, the communication cost for SLP achieves the minimum value of \(H\) when \(\alpha=2\pi\). Under this situation, the packet first reaches \(\mathit{p}_{6}\) through the \(\mathit{h}\)-directed routing, and then the base station through the shortest path routing. To form a more general case, we assume that \(\mathit{p}_{3}\) is a random point on circle \(\mathit{I}\) (as shown in Figure 2). We calculate the average hops from \(\mathit{p}_{3}\) to the base station for the more general case as:  
\[\int\limits_{0}^{\pi}\sqrt{h^{2}+H^{2}\cdot 2 hH\cos\alpha}\,/\,\pi d\alpha \tag{3}\]  
The average communication cost of SLP is thus obtained by:  
\[\mathit{h}+\int\limits_{0}^{\pi}\sqrt{h^{2}+H^{2}\cdot 2 hH\cos\alpha}\,/\,\pi d\alpha \tag{4}\]  
(2) Communication Cost for SLP-E  
The communication cost of SLP-E is calculated for the following two situations:  
(a) If the phantom location, namely \(\mathit{p}_{4}\), is on arc \(\mathit{p}_{i}\mathit{Cp}_{2}\), the communication cost of SLP-E is the same as that of SLP:  
\[\mathit{Hop}_{\mathit{s}\mathit{,p}\mathit{+}\mathit{+}\mathit{ Hop}_{\mathit{p}\mathit{+}\mathit{,b}}}=\mathit{h}+\sqrt{h^{2}+H^{2}\cdot 2 hH\cos\alpha}\.\]  
(b) If the phantom location, namely \(\mathit{p}_{3}\), is on arc \(\mathit{p}_{i}\mathit{p}_{5}\mathit{p}_{2}\), the packet will be first routed to E and then to F along the arc \(\mathit{EF}\).