## 3 Results and Discussion
### Interference Measurements  
Our interference simulation experiments are presented in Figure 5. We compare our simulation results with the theoretical values calculated using Equations (4) and (5). The number of the interfering nodes increases from 1 to 100. All nodes are randomly placed (using uniform distribution) inside a circular area bounded by the circles with \(\mathrm{r_{min}}\) and \(\mathrm{r_{max}}=100\) m. The receiver of interest is positioned at the center of the area where the highest amount of interference is exhibited. The transmission power is assumed the same for all transmitting nodes and is equal to 1 mW. In the simulations we calculate the distance of each transmitting node from the center of the networking area where the receiver of interest is located. Path losses due to fading and shadowing are not considered. The path loss exponent \(a\) varies from 2 (free space) to 4 [35]. The channel bandwidth is assumed constant and equal to \(10^{7}\mathrm{Hz}\) and all transmitting nodes employ a BPSK modulation scheme (thus k in Equations (7) and (8) is equal to 1).  
The results from the proposed theoretical model are in a very good agreement with the values of the mean interference levels obtained by simulations. Figure 5a shows the effect of the path loss exponent on interference and Figure 5b shows the effect of \(\mathrm{r_{min}}\) (the distance around the receiver where a transmitter cannot be placed, _i.e._, the exclusion region). As expected a higher value for the path loss exponent results in lower levels of interference, and a larger \(\mathrm{r_{min}}\) leads to lower interference.  
Figure 5: Interference power comparison for (**a**) different path loss exponents; and (**b**) different \(\mathrm{r_{min}}\).  
In Figure 6, we present the mean interference levels, using simulations, for path loss exponent equal to 2 and 4 (plot (a) and plot (b), respectively) when we relax our assumption on the receiver's position and assume that it is placed at distance (\(x\)) from the center of the networking area, where the offset x \(\in\) [0, r\({}_{\text{max}}\)]. When a receiver is close to the borders of the networking area the interference level decreases. This given that the number of the nearby interfering transmitters decreases near the borders, is natural since the separation distance of interfering transmitters becomes larger than r\({}_{\text{max}}\) and thus their interference diminishes.  
Figure 5: _Cont._  
Figure 6: Calculated interference at the center (**black** line) _vs._ simulation results **(light blue** lines) for different offset positions of the receiving node (r\({}_{\text{max}}\) = 100 m, path loss exponent a = 2 (**a**) and 4 (**b**), r\({}_{\text{min}}\) = ranges from 0.4 to 2 m).  
The difference, between the interference levels at the center of the networking area, calculated with Equations (4) and (5), and at a quite large offset from the center is remarkably small as is evident in Figure 6. In fact the approximation is quite good with even a 60% or 70% (for a = 2) and 90% (for a = 4) offset from the center of the circle.  
The interference level on the border of the circular area (100% offset) is approximately half of the level at the center. Therefore we can assume that, for the largest part of the networking area, the interference level as given by Equations (4) and (5) is a quite accurate approximation. The same holds true for the values of the d\({}_{\mathrm{LIR}}\), given by Equations (7) and (8). If more accuracy is desired it is possible to calculate the interference at any offset (x) using the method described in [24], but in this case a much more complicated analysis would be required.