# Beacon Success Rate versus Gateway Density in Sub-GHz Sensor Networks
## 2 System Model
### Path loss Model  
In this section, the path loss model is presented for the US 902-928 MHz ISM band in an outdoor urban area. The path-loss (in dB) is defined as the total propagation loss between the transmitter and the receiver and can be defined as: \(PL(d,f)\equiv TxP-RSS(d),[dB]\), where the term \(RSS(d)\) in dBm represents the received signal strength at a distance, \(d\) [m]. The term \(TxP\) [dBm] represents the transmitted power at the transmitter. The term Received Signal Strength Indicator (RSSI) [dBm] represents the estimate of RSS, as reported by the receiver. Additional insertion losses in [dB] due to analog RF hardware losses can be added to this formula, depending on where the \(RSS(d)\) refers to. This is relevant if the \(RSS\) is reported for the antenna port of the receiver, whereas the signal is processed at the chipset port. The path loss depends on various factors such as the channel frequency, distance between the transmitter and receiver, and random shadow fading variations caused by the environment [20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. In most of the models published to date, the path loss can be modeled as follows:  
\[PL(d,f)=PL_{0}(d_{0},f_{0})+10\times n\times log_{10}\bigg{(}\frac{d}{d_{0}} \bigg{)}+20 log10\bigg{(}\frac{f}{f_{0}}\bigg{)}+X(0,\sigma),\ [\text{dB}]. \tag{4}\]  
where the term \(PL_{0}(d_{0},f_{0})\) represents the path loss at reference distance \(d_{0}\), and on reference channel frequency, \(f_{0}\). The term \(n\) represents the path loss exponent, the term \(d\) is the distance between the GW and the EN, and \(f\) is the channel frequency in Hz. The term \(X(0,\sigma)\) (in dB) is a random variable that represents the large-scale shadow fading effects, where its mean is zero, and its standard deviation is \(\sigma\), expressed in decibels. The field measurements show that it has a log-normal distribution. Figure 5: The model encompassing path-loss, receiver sensitivity, and co-channel interference. The carrier frequency of the radio waves affects the overall path-loss. Typically, higher frequencies exhibit higher path-loss compared to lower frequencies. This is attributed to smaller wavelengths associated with higher carrier frequencies, making the signal more susceptible to diffraction and scattering, thereby reducing the signal strength. The path-loss exponent, \(n\), can vary based on terrain type and the presence of non-uniform buildings. In practical scenarios, the path-loss exponent is not consistent across a wide area, as assumed in Equation (4). Its variation can be significant and is influenced by the surrounding environment and its dimensions. Several factors contribute to this variability. Firstly, the path-loss exponent tends to be higher in urban areas compared to rural areas. Urban environments, with more buildings and obstacles, lead to increased scattering and absorption of radio waves. Additionally, the presence of non-uniform buildings plays a role in influencing the path-loss exponent. For instance, an area with numerous tall buildings will have a higher path-loss exponent than an area with mostly shorter buildings. This is because tall buildings can obstruct radio waves, resulting in a decreased signal strength. The path-loss exponent serves as a crucial factor for determining the range of a wireless communication system. If the path-loss exponent is excessively high, the signal strength might be below the sensitivity of the receiver. This can lead to dropped connections or poor PSR. To address the variations in the path-loss exponent, a more intricate path-loss model that considers terrain type and the presence of structures such as buildings and tall trees can be devised. This might involve incorporating a geographical map. Instead of opting for such a complex approach, we utilize a statistical method to model the mean path-loss exponent. This involves collecting data on the path-loss exponent from different locations and leveraging these data to generate an estimate for the expected path-loss exponent.