# A Semi-Random Elliptical Movement Model for Relay Nodes in Flying Ad Hoc Networks
## 4 Node Distribution in SREM
### Empirical Node Distribution  
The SREM model is based on straight-line movements between waypoints defined along the boundary of an ellipse. While the node distribution can theoretically be described using a probability density function, the complexity introduced by the indicator function and the double integral renders closed-form analytical solutions practically intractable. To address this limitation and gain empirical insight into the actual distribution characteristics, we analyze the distribution tendencies of relay nodes under the SREM model using the Monte Carlo method. Figure 5 visualizes the node distribution results obtained by repeatedly sampling UAV trajectories generated between randomly selected departure points \(\theta_{i}^{(n)}\) and arrival points \(\theta_{i}^{(n+1)}=\theta_{i}^{(n)}+\delta\), where \(\delta\in\left[\frac{\pi}{6},\frac{\pi}{2}\right]\), along the elliptical boundary. In this experiment, the size of the ellipse was fixed to ensure consistency in node scale, and all trajectories were constrained to a single elliptical path. This setup clearly revealed the spatial tendency of node avoidance near the center and concentration toward the boundary. Figure 4: Analytical node distribution in SRCM. Figure 5: Empirical node distribution in SREM based on the Monte Carlo method. To quantitatively analyze the spatial distribution of nodes, we define the midpoint (\(M\)) of each straight-line segment representing the UAV's movement as a representative location. The distance between this midpoint and the center of the ellipse (\(O\)) is then normalized by the semi-major axis length \(a\), resulting in a normalized radial distance variable \(r_{\text{norm}}\), defined as follows:  
\[r_{\text{norm}}=\frac{\|M-O\|}{a},\quad\text{where}\quad M=\frac{1}{2}(P_{i}^{( n)}+P_{i}^{(n+1)}) \tag{5}\]  
This variable serves as a quantitative measure of how far each trajectory lies from the center of the ellipse, and it enables consistent comparison across different elliptical scales. The interval \(r_{\text{norm}}\in[0,1]\) was uniformly divided, and for each subinterval, the number of segments whose midpoints fell within that range was counted to generate a histogram representing the distribution. Although using the midpoint of each segment as a representative node location is a simplified approximation, it is theoretically valid under the assumptions of the SREM model, in which UAVs travel at constant speeds along linear trajectories. Since each segment corresponds to a UAV's position over a fixed period of time, and the velocity is uniform, the segment's spatial contribution serves as an unbiased estimator of node density [37]. Under the assumption of uniform temporal and spatial distribution, this approach is statistically sound. When a sufficiently large number of segments is sampled, the histogram based on the normalized radial distance variable \(r_{\text{norm}}\) functions as a practical statistical indicator that captures the overall bias and concentration tendencies of node distribution. Figure 6 visualizes the distribution of segment midpoints' normalized distances from the ellipse center under varying angular increment ranges \(\delta\) in the SREM model. Each subplot illustrates how the values of \(r_{\text{norm}}\) change depending on the range of \(\delta\) used when UAVs determine their next waypoint. In (a), a relatively large angular increment \(\delta\in[\frac{\pi}{6},\frac{\pi}{2}]\) is applied. Even under this setting, most midpoints do not pass through the center but remain distributed along the ellipse boundary. As the angular increment range narrows--as shown in (b) with \(\delta\in[\frac{\pi}{12},\frac{\pi}{6}]\), and in (c) with \(\delta\in[\frac{\pi}{24},\frac{\pi}{22}]\)--the UAV trajectories increasingly adhere to the elliptical path and avoid the center, favoring the outer regions. In particular, subplot (c) reveals a clear distinction in concentration along the major and minor axes, with the center region remaining largely vacant.