## 5 End-to-End Performance Estimation
### End-to-End Performance Estimates Based on Per-Hop Statistics  
In this section, we will investigate how to estimate such end-to-end performances using per-hop statistics such as the \(m\)-th moments of \(D\) and \(P_{out}\) in the previous sections. First, the average end-to-end hop count \(\eta\) is estimated as  
\[\eta\approx\frac{L}{\mathbb{E}\{X|K\geq 1\}}=\frac{L\Phi}{2\sin\left(\frac{\Phi}{ 2}\right)\mathbb{E}\{D|K\geq 1\}}, \tag{20}\]  
where \(L\) is the distance between the source and the destination as illustrated in Figure 4. It is noted that \(X\) is the hop distance \(D\) projected onto the direction from \"_the transmitting node_\" to the destination.  
Because the actual route is a squiggly line between the source and the destination, this approximation is in fact the lower bound on the actual end-to-end hop count as shown in the example case in Figure 4. Let \(H\) be the hop distance \(D\) projected onto the direction from \"_the source_\" to the destination. In the figure, the progress in the second hop projected onto the transmitter (Relay\({}_{1}\))-Destination axis \(x_{2}=d_{2}\cos\phi_{2}\) is greater than the progress along the source-destination axis, \(h_{2}\). Thus, in general, \(\mathbb{E}\{X\}\geq\mathbb{E}\{H\}\), which gives \(\frac{L}{\mathbb{E}\{X\}}\leq\frac{L}{\mathbb{E}\{H\}}\) for \(K\geq 1\). However, as indicated in [13], for a enough large \(L\), \(x_{i}\approx l_{i}\) for most of hops \(i\) because the angle between the two axes \"Relay\({}_{i}\)-Destination\" and \"Source-Destination\" is almost zero, which makes the proposed approximation in Equation (20) becomes tighter.  
Using the approximate end-to-end hop count \(\eta\), the end-to-end outage rate can be estimated as  
\[P_{out:EE}\approx 1-(1-P_{out})^{\eta}=1-(1-P_{out})^{\frac{L}{ \mathbb{E}\{X,k\geq 1\}}}\] \[=1-\left[1-e^{-\frac{\Phi\lambda}{\rho^{2}}\left(1-(1+\beta R)e^{ -\beta R}\right)}\right]^{\frac{L}{\mathbb{E}\{X,k\geq 1\}}}, \tag{21}\]  
where \(P_{out}\) is the per-hop outage in Equation (8).  
Lastly, the total transmit energy of the multi-hop transmissions normalized by that of the one-hop transmission (i.e., direct transmission from the source to the destination) is approximately given by  
\[\mathrm{Y}\approx\frac{\eta\cdot\mathbb{E}\{D^{\alpha}|K\geq 1\}}{L^{ \alpha}}=\frac{\Phi\,\mathbb{E}\{D^{\alpha}|K\geq 1\}}{2L^{\alpha-1}\sin \left(\frac{\Phi}{2}\right)\mathbb{E}\{D|K\geq 1\}}, \tag{22}\]  
\(\alpha\) is the path-loss exponent, which is assumed to be two for the LoS condition in the following sections. This normalization is a common way to compare different routing schemes with the common reference by the direct (or one-hop) transmission [54]. Moreover, we note that \(\eta\) and \(\mathrm{Y}\) are defined only if the end-to-end multi-hop transmission is successful.  
It is noted that, if comparing the energy consumptions in the RF circuitry to transmit and receive a packet, the reception energy may be comparable to the transmit energy depending on applications and hardware platforms, as in [55; 56]. However, in this paper, following the framework in [13], we do not consider the reception energy. It is expected that the energy consumption for data reception and the corresponding total energy consumption would increase, as the end-to-end hop count increases. However, the exact analysis incorporated with the reception energy consumption will be covered in the future.