## 7 Numerical and Results  
In this section, we validate the analytical results and test the tightness of the end-to-end performance estimation based on per-hop statistics by comparing with simulation results. We assume \(\alpha=2\) to model the LoS channel. In addition, we set \(R_{0}=\frac{50}{9}\approx 5.6\) m, which corresponds to the transmission range for the omni-directional antenna \(\Phi=2\pi\). In other words, the beamwidth is \(360^{\circ}\), and the maximum possible per-hop transmission range is \(R_{0}\approx 5.6\) m assuming battery-powered devices with limited transmission range. Of course, this value of \(R_{0}\) can be changed depending on the system parameters and applications. As the beamwidth \(\Phi\) becomes narrower, the transmission range \(R\) increases based on Equation (2). The simulation results are averaged over \(10^{7}\) iterations with random topology. In each trial, the locations of nodes are uniformly distributed over a 1 km\({}^{2}\) square region, where the number of nodes follows Poisson distribution with \(\lambda=0.015\) nodes/m\({}^{2}\). For a given \(L\), the source is located at the center of the 1 km\({}^{2}\) space. In each hop, the next-hop node is selected following FN or NN. We declare an end-to-end outage of a trial, if an outage occurs in any intermediate hop.  
Assuming the end-to-end distance \(L\) of 125 m, Figures 5a, 6a and 7a show how the end-to-end outage probability \(P_{out:EE}\), the average end-to-end hop count \(\eta\), and the total transmit (radiated) energy consumption Y change, respectively, as the blockage parameter varies over \(0.01\leq\beta\leq 0.04\). In the figures, we compare the theoretical results (i.e., the approximation proposed in Section 5) with the end-to-end simulation (exact) results for \(\Phi=40^{\circ}\) and \(60^{\circ}\), which give the transmission ranges of \(R=R_{0}\times\frac{360^{\circ}}{40^{\circ}}=50\) m and \(R_{0}\times\frac{360^{\circ}}{60^{\circ}}\approx 33.3\) m, respectively. In the three figures, the solid lines with the 'x'- and '+'-markers represent the furthest neighbor (FN) routing with \(\Phi=40^{\circ}\) and \(60^{\circ}\), respectively. On the other hand, the solid lines with the dot and asterisk markers indicate the nearest neighbor (NN) routing with \(\Phi=40^{\circ}\) and \(60^{\circ}\), respectively. The circles, triangles, squares, and diamonds correspond to the end-to-end simulation results for FN with \(\Phi=40^{\circ}\), FN with \(\Phi=60^{\circ}\), NN with \(\Phi=40^{\circ}\), and NN with \(\Phi=60^{\circ}\), respectively. Moreover, the \"I-shape\" error bars overlaid on the simulation markers indicate 95% confidence intervals computed using [PERSON]'s formula [57]. In Figures 6a and 7a, the baseline cases with \(\beta=0\), which are indicated by the dotted lines, are illustrated to highlight the impact of the blockage effects. In Figures 5b, 6b and 7b, we perform the same set of simulations for a different end-to-end separation \(L=250\) m and compare the results with the numerical calculationsbased on the per-hop statistics. To quantify the impact of the blockage on three different performance metrics, the baseline cases without blockage are provided in Tables 1-3.