## 2 System Model
In the following sections we describe our approach for interference modeling when the transmitting nodes are positioned within the networking area using a random uniform distribution. We assume, as illustrated in Figure 2, that a receiver of interest is placed at the center of each circular area and \(\mathrm{N_{t}}\) transmitters-from all other wireless body/personal area networks, with equal transmission powers \(\mathrm{P}\), are placed randomly with uniform distribution inside an area bounded by the circles with radius \(\mathrm{r_{max}}\) and \(\mathrm{r_{min}}\) (\(\mathrm{r_{min}}\) is the radius of the exclusion region around the receiver where no transmitters are allowed to operate). Inside the ring from r to r + dr there are on average \(\mathrm{N(r)}=\frac{\mathrm{N_{t}}\cdot 2\pi rdr}{\pi(r_{max}^{2}-r_{min}^{2})}= \left(\frac{2\mathrm{N_{t}r}}{r_{max}^{2}-r_{min}^{2}}\right)dr\) transmitters causing interference with power equal to:  
\[\mathrm{I(r)}=\sum_{\mathrm{N(r)}}\mathrm{P_{j}}\mathrm{G_{ij}}=\mathrm{PD} \left(\frac{\left(\frac{2\mathrm{N_{t}r}}{r_{\max}^{2}-r_{min}^{2}}\right)}{ \mathrm{r^{a}}}\right)dr\Longrightarrow\mathrm{I(r)}=\left(\frac{2\mathrm{ PDN_{t}}}{\left(\mathrm{r_{max}^{2}-r_{min}^{2}}\right)\mathrm{r^{a}-1}} \right)dr \tag{2}\]  
The total mean interference at the center of the circle then would be equal to:  
\[\mathrm{I}=\int_{\mathrm{r_{min}}}^{\mathrm{r_{max}}}\left(\frac{2\mathrm{ PDN_{t}}}{\left(\mathrm{r_{max}^{2}-r_{min}^{2}}\right)\mathrm{r^{a}-1}} \right)dr\Longrightarrow\mathrm{I}=\frac{2\mathrm{PDN_{t}}}{\mathrm{r_{max}^{ 2}-r_{min}^{2}}}\int_{\mathrm{r_{min}}}^{\mathrm{r_{max}}}\frac{dr}{\mathrm{ r^{a-1}}} \tag{3}\]  
Figure 2: Uniform random node placement. Receiver at the center of the networking area. In [32] the authors performed a survey on wireless body communication and have found that the estimation of path loss exponent for body area wireless communication has be shown to be between 3 and 4, or in non-line-of-site situations the measured range is between 5 and 6. In a similar work in [33] the derived data are shown to be around 2.4 to 2.57 and 3.1 to 3.9. In our work thus to cover the general case we distinguish two different cases, for the calculation of the integral, depending on the path loss exponent \(a\) both of which we analyze below. For path loss exponent \(a\) equal to 2 we get:  
\[\mathrm{I}=\frac{2\mathrm{PDN}_{\mathrm{t}}}{r_{\mathrm{max}}^{2}-r_{\mathrm{ min}}^{2}}(\mathrm{ln}\ r_{\mathrm{max}}-\mathrm{ln}\ r_{\mathrm{min}}) \tag{4}\]  
where for \(\mathrm{r}_{\mathrm{max}}\gg\mathrm{r}_{\mathrm{min}}\) we have \(\mathrm{I}\approx\frac{2\mathrm{PDN}_{\mathrm{t}}}{r_{\mathrm{max}}^{2}}( \mathrm{ln}\ r_{\mathrm{max}}-\mathrm{ax}\ r_{\mathrm{min}})\), that is the interference has a strong dependence on \(\mathrm{r}_{\mathrm{max}}\) (and consequently on the transmitter density \(\frac{\mathrm{N}_{\mathrm{t}}}{\mathrm{nr}_{\mathrm{max}}^{2}}\)) and a much weaker (logarithmic) dependence on \(\mathrm{r}_{\mathrm{min}}\) (the size of the exclusion region). For path loss exponent \(a\) greater than 2 we get:  
\[\mathrm{I}=\frac{2\mathrm{PDN}_{\mathrm{t}}}{(\mathrm{a}-2)(r_{\mathrm{max}}^{ 2}-r_{\mathrm{min}}^{2})}\left(\frac{1}{r_{\mathrm{min}}^{\mathrm{a}-2}}- \frac{1}{r_{\mathrm{max}}^{\mathrm{a}-2}}\right) \tag{5}\]  
where for \(\mathrm{r}_{\mathrm{max}}\gg\mathrm{r}_{\mathrm{min}}\) we have: \(\mathrm{I}\approx\frac{2\mathrm{PDN}_{\mathrm{t}}}{(\mathrm{a}-2)\ r_{\mathrm{max}}^{2}}\left(\frac{1}{r_{ \mathrm{min}}^{\mathrm{a}-2}}\right)\), the interference has a strong dependence on \(\mathrm{r}_{\mathrm{max}}\) (and consequently on the transmitter density \(\frac{\mathrm{N}_{\mathrm{t}}}{\mathrm{nr}_{\mathrm{max}}^{2}}\)) and polynomial dependence on \(\mathrm{r}_{\mathrm{min}}\). We distinguish two different cases, which are described below for realistic \"real-world\" links and Shannon capacity links.