# Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks
## 5 Numerical and Simulation Results
### Study of PGL for Larger and Different Network Topologies  
In this and the following section, we compare the two algorithms for three network topologies, where we have specified a certain network size in terms of meters for the purpose of simulation. Because of the large number of SNs, we use only the PGL algorithm with \(100\times 100\) pixels for this study. Recall our assumption stated in the Introduction that each node in the network is within one hop of at least one GW and that there are many more SNs than GWs. Because of this, we have assumed that when one packet overlaps another packet, that the power of the weaker packet is so much greater than the noise that we can neglect the noise. Therefore, whether our topologies cover a large or small area, we still assume the weaker packet can be decoded on its own, because it is sufficiently stronger than the noise. In fact, the distances in the key equations, Equations (6) and (10), are in a ratio, so if the topology is scaled larger, the scaling factor cancels out and does not affect our results. We consider a sine-shaped topology in Figure 12, a circle-shaped topology in Figure 13, and a random topology in Figure 14, each in a \(10\) m\(\times\)\(10\) m area. To make the circle topology, 100 SNs are grouped into 10 groups of 10 SNs each. The centers of the groups are equally spaced on a circle of radius 4 m. First, each SN is independently perturbed in angle from the center of its group by a zero mean Gaussian random variable (RV) of standard deviation of 4.5 degrees. From that perturbed location, each SN is again perturbed in both X and Y coordinates by iid zero mean Gaussian RVs with standard deviation 0.2 m. To make the sine topology, the initial X coordinates are distributed randomly over 9 m. The initial Y coordinates are the result of mapping the initial X coordinates through a sine curve that has an amplitude of 4.5 m and period of 9 m. These initial X and Y coordinates are then perturbed by iid zero mean Gaussian RVs with standard deviation of 0.27 m. Only one random  
Figure 11: Average Contention for different number of SNs with one and two GWs, by WBG and PGL algorithms with capture only and capture with IC for different pixel densities. outcome of each type of topology for a given number of SNs is used. Figure 12a shows the locations of 100 SNs deployed by sine shape and three GWs are placed by PGL algorithm. Figure 12b compares the average contention, for different numbers of GWs with capture and IC and with capture only. It is obvious that the average contention decreases significantly for capture and IC, compared to capture only. Figure 12c shows the deployed locations of 100 SNs in sine shape and three GWs by naive placement. In Figure 12d, it is clear that the average contention for 20-100 SN and 1-3 GW placed by PGL is always less than that based on naive GW placement. Figures 13 and 14 show the similar conclusion with the SN deployed separately by circle and random topologies. Specifically, for the circle topology in Figure 13, we note that the naive and optimal GW placements are similar for two GWs, and the corresponding average contentions are close. For the random topology, we observe a larger difference between the naive and PGL placements. Figure 12: Simulation visualizations for three GWs placed by PGL (**a**) and by naive placement (**c**) along a sine wave composed by 100 SNs. (**b**) The average contentions comparing in different number of GWs by capture and IC to capture only separately; and (**d**) the average contentions for capture and IC comparing in different number of GWs placed by PGL and naive. Figure 14: Simulation visualizations for three GWs placed by PGL (**a**) and by naive placement (**c**) in the circumference composed by 100 SNs in a uniformly random distribution. (**b**) The average contentions comparing in different number of GWs by capture and IC to capture only separately; and (**d**) the average contentions for capture and IC comparing in different number of GWs placed by PGL and naive.