## 2 System Model
A specific transmission \(i\) over a wireless link is considered successful when condition (1), known as the _SINR criterion_, is true for a given threshold value \(\gamma_{(i)}\) and for a positive vector \(P=(P_{1}\),..., \(P_{k})\) of transmission powers. \[\frac{P_{i}G_{ii}}{\left(\sum_{j\
eq i}P_{j}G_{ij}\right)+n_{i}}\geq\gamma_{(i)} \tag{1}\]  
For each transmission link \(i\), the value of the SINR threshold \(\gamma_{(i)}\) depends on many design parameters and properties of the telecommunication system, such as the target BER, the modulation scheme, the error correction coding employed and the desired transmission bit-rate for each link. A set of simultaneous transmissions is considered successful when condition in Equation (1) is true for all the transmissions of this set. Given the telecommunication system and the application specific characteristics, each transmission might require different values for the SINR threshold \(\gamma\) for successful communication. In this section we study the interference exhibited at the center of a circular networking area when interfering nodes (with equal transmitting power) are randomly distributed. Going a step further from previous works, we define the _interference limited communication range_ (\(d_{ILR}\)) regions for biomedical body/personal area networks to be the critical communication region around a receiver, surrounded by a large number of randomly placed interfering nodes, within which if a transmitter is present, a successful communication link can be established. The value of \(d_{ILR}\), as we will show, depends on a number of parameters: the number and the density of the surrounding interfering transmitters, the exclusion region around each receiver, the interference power level, and most importantly the selected transmission rate and mode of operation of the receiver. We illustrate the generalized concept in Figure 1. The receivers of interest _(pentagons)_ are placed at the center of the networking area, for each wireless body area network, around which many interfering transmitters are randomly deployed, belonging to the same or different wireless body/personal area networks. Given the amount of interference exhibited at the receiver's site the interference limited communication range \(d_{ILR}\) is shown (the light green shaded disk). In this example the receiver acting as control/sink node in WBAN 1 experiences interference from all other transmitters deployed in the same vicinity (even if they belong to the same or different WBAN) transmitting simultaneously in the same channel. At the same time all the transmitters/sensors of WBAN 1 are causing interference to all other receivers. So only a portion of the sensors in WBAN 1 will be able to successfully transmit their data to the receiver of interest. These transmitters (_colored circles_) are located within this critical region \(\mathrm{d_{ILR}}\) (_shaded light green area around each receiver_) inside which they can successfully communicate with the receiver of interest despite all the other interfering transmissions. The sensors/node that are not able to communicate are shown with uncolored dotted circles. In the proposed model we make the approximation that interference behaves like white Gaussian noise which is equivalent to the applicability of the SINR criterion, and is equal to the sum of the received power, from all active transmitters \(\mathrm{I_{i}}=\left(\sum_{j\
eq i}\mathrm{P_{j}}\mathrm{G_{ij}}\right)\), at the receiver of interest \(i\). \(\mathrm{P_{j}}\) is the transmitted power and the path loss from transmitter \(j\) to receiver \(i\) is \(\mathrm{G_{ij}}=(\lambda/4\pi)^{2}\cdot\pi\mathrm{{}^{-i}_{(i,j)}}=\mathrm{D} \cdot\mathrm{D_{(i,j)}^{-i}}\), where \(\lambda\) is the wavelength, \(a\) is the path loss exponent and has typical values from 2 to 5 and \(\mathrm{d_{(i,j)}}\) is the distance between nodes \(i\) and \(j\).