# Beacon Success Rate versus Gateway Density in Sub-GHz Sensor Networks
## 6 Numerical Results  
This section presents numerical results derived from a simulation study using NS-3. Figures 17 and 18 depict the beacon collision probability and the expected number of  
Figure 16: Diagram illustrating the NS-3 Simulator Architecture, spanning from the application layer to the physical (PHY) layer. devices under collision, respectively, versus the network size, where the network size refers to the number of GWs and ENs. The collision probability represents the scenario where two or more packets with an infinitesimal overlap collide. Consequently, the collision probability serves as an upper bound for the packet error rates. The packet error rate depends on the product of the collision probability and the probability of having a C/I that is unfavorable for a successful reception. Further analysis of the PER's dependency on the sensitivity is given in Equation (10). Figure 17: The beacon collision probability in relation to the number of Gateways, \(N\). Figure 18: Expected number of collisions in relation to the number of Gateways. The system parameters used to derive the performance evaluation results are described below. We assume the presence of 160 sub-slots for UL contention windows, a beacon duration of 152 ms, and a beacon interval of 10.08 s. There are 69 channels available for FHSS. While a higher number of channels has the potential to improve performance, a careful balance is required to weigh the benefits against the increased complexity and the power consumption demands, particularly for battery-operated ENs. Statistics for beacon acquisition latency, derived from Monte Carlo simulations using the NS-3 model, are depicted in Figure 19. On each box in the figure, the central red line indicates the median, and the bottom and top edges of the box indicate the 25 th and 75 th percentiles of the samples obtained from the simulation results, respectively. The samples, i.e., the data points marked with + symbols in this plot represent outliers within the dataset. Outliers are observations that deviate significantly from the overall pattern of the data. The green 'x' symbols on the data points represent the dataset's mean values obtained through simulations for a specified number of GWs. Opting for a longer beacon interval, such as 10.08 s, results in a mean beacon acquisition latency of approximately \(\mu=10.6\) s when 50 ENs are attempting to discover a single GW. Conversely, beacon acquisition latency diminishes exponentially with an increasing number of GWs available to serve the WAN in the neighborhood. Thus, choosing an extended beacon interval, along with a frequency-hopping system featuring multiple channels, proves to be a favorable option for a narrow band WAN, especially in scenarios where multiple GWs can serve the ENs. In Figure 20, the Complimentary Cumulative Distribution (CCDF) illustrates field measurements for PSR in relation to the distance from the nearest GW, using 50 kbps GFSK modulation. Within this graph, the 85 th percentile of the PSR, denoted as TP-85, is highlighted. Observations from various locations indicate that a coverage range of approximately 130 m can be attained with PSR values of \(\geq 0.85\). In other words, the TP-85 distance covered is 130 m with PSR values greater than or equal to 85%. These measurements were conducted in Westchester, Los Angeles, an urban area. Multiple measurements were taken at various locations equidistant from the GW to showcase the statistics and impact of interference in a real-life scenario. The results are compared to the NS-3 simulation outcomes, where a fixed additional loss is presumed for the environmental interference. The NS3 model sets the path loss exponent to the mean value predicted by experimental results, while in the field, variations in the terra  
Figure 19: Latency in beacon acquisition relative to the number of Gateways with 50 ENs. to differences in the measurement outcomes.