## 2 System Model
### Wireless System with Real-World Links  
We consider the receiver at the center of the circular area to be part of a wireless network with \"real-world\" links that use a specific modulation scheme, a specific target BER and constant bandwidth. We assume the absence of any error control coding or any multiuser decoding scheme or any cooperation between transmitters and/or receivers and make the approximation that the interference behaves like white Gaussian noise, which combined with the thermal noise results in a combined total noise power spectral density \(\mathrm{N}_{0}\)/2 in bandwidth BW. In this case there is a linear relationship between the achieved rate and the Signal to Interference plus Noise Ratio (SINR). We assume that the system's bandwidth does not depend on the symbol duration of either the desired signal or the interfering signals. Thus the total power spectral density would be: \(\frac{\mathrm{N}_{0}}{2}=\frac{(\mathrm{P}_{\mathrm{int}}+\mathrm{P}_{ \mathrm{noise}})}{2\mathrm{BW}}\) (where \(\mathrm{P}_{\mathrm{int}}\) and \(\mathrm{P}_{\mathrm{noise}}\) is the total power of interference and surrounding thermal noise respectively) and the energy per symbol for a link would be equal to \(\mathrm{E}_{\mathrm{s}}=\mathrm{P}_{\mathrm{I}}\mathrm{G}_{\mathrm{ii}}/ \mathrm{R}_{\mathrm{s}}\) which results in an energy per symbol to interference-plus-noise density ratio equal to: \(\frac{\mathrm{E}_{\mathrm{s}}}{\mathrm{N}_{0}}=\frac{\mathrm{P}_{\mathrm{i}} \mathrm{G}_{\mathrm{ii}}}{(\mathrm{\sum}_{\mathrm{j}\
eq\mathrm{i}}\mathrm{P}_{ \mathrm{i}}\mathrm{G}_{\mathrm{ij}})+\mathrm{n}_{\mathrm{i}}}\cdot\frac{ \mathrm{BW}}{\mathrm{R}_{\mathrm{s}}}\). As explained before if the SINR criterion is true for a given threshold value \(\gamma\) communication will be successful, and using the formulation from above we have: \(\frac{\mathrm{P}_{\mathrm{i}}\mathrm{\cdot}\mathrm{D}\cdot\mathrm{d}_{(1)}^{- \mathrm{a}}}{(\mathrm{\sum}_{\mathrm{j}\
eq\mathrm{i}}\mathrm{P}_{\mathrm{j} \mathrm{\cdot}}\mathrm{D}\cdot\mathrm{d}_{(1)}^{-\mathrm{a}})+\mathrm{n}_{ \mathrm{i}}}\geq\frac{\mathrm{E}_{\mathrm{s}}}{\mathrm{N}_{0}}\cdot\frac{ \mathrm{R}_{\mathrm{requirements. Based on the interference calculation, as presented earlier, and assuming that (i) all transmitters in the area operate with transmission power P and that (ii) the interference power is much larger than the background noise (I >> n), we see that the \"_interference limited communication range_\" \(\mathrm{d_{ILR}}\), as defined above, is given by:  
\[\mathrm{d_{ILR}}=\sqrt[a]{\frac{P_{I}\cdot D}{\frac{(\sum_{j\
eq i}P_{j}\cdot D \cdot d_{(j,j)}^{-a})+n_{i}}{N_{0}}\cdot\frac{R_{b}}{k\cdot BW}}}\approx\sqrt[a]{ \frac{P\cdot D}{\frac{E_{s}}{N_{0}}\cdot\frac{R_{b}}{k\cdot BW}}}=a]{\frac{PD}{N _{0}}\cdot\frac{R_{b}}{k\cdot BW}} \tag{6}\]  
Again we distinguish two cases:  
\[\mathrm{d_{ILR}}=\left\{\sqrt[a]{\frac{1}{R_{b}N_{t}}\cdot\frac{k\cdot BW}{2 \frac{E_{s}}{N_{0}}\cdot\frac{(r_{max}^{2}-r_{min}^{2})}{(\ln r_{max}-\ln r_{ min})}}},\text{for }a=2\right. \tag{7}\]  
\[\mathrm{d_{ILR}}=\left\{\sqrt[a]{\frac{1}{R_{b}N_{t}}\cdot a}\sqrt[a]{(a-2) \cdot\frac{k\cdot BW\cdot(r_{max}^{2}-r_{min}^{2})}{2\frac{E_{s}}{N_{0}}\left( \frac{1}{r_{min}^{a-2}}-\frac{1}{r_{max}^{a-2}}\right)}},\text{for }a>2\right. \tag{8}\]  
In Figure 3 we see that the theoretical value of the \(\mathrm{d_{ILR}}\) is a decreasing function of the number of interfering nodes and asymptotically falls to zero. When we have a high value for the path loss exponent (\(a>2\)), we see that we obtain lower values for the \(\mathrm{d_{ILR}}\), when the number of transmitting nodes is relatively small. As \(\mathrm{N_{t}}\) increases, the rate of convergence of \(\mathrm{d_{ILR}}\) to zero is: \(\mathrm{(N_{t})}^{-\frac{1}{2}}\) for \(a=2\) and \(\mathrm{(N_{t})}^{-\frac{1}{a}}\) for \(a>2\).  
Figure 3: Interference limited communication range (\(\mathrm{r_{max}}=100\) m).