# Interference-Robust Transmission in Wireless Sensor Networks
## 3 Transmission in the Presence of Interference  
We estimate the probability of transmission failure and data throughput in the presence of interference and then determine the payload size maximizing the data throughput. We assume that a node transmits data packets without consideration of the channel condition and confirm the transmission by receiving an ACK packet. The data transaction time can be represented as:  
\[\hat{T}_{m}=T_{\text{pkt},m}^{data}+T_{\text{pkt},m}^{ack}+2\delta \tag{9}\]  
where \(T_{\text{pkt},m}^{data}=T_{\text{pkt},m}\left(L\right)\), \(T_{\text{pkt},m}^{ack}=T_{\text{pkt},m}\left(0\right)\) and \(\delta\) denotes the time from the transmission to the reception state and vice versa. The probability of transmission failure due to the packet collision can be represented as:  
\[p_{m,c}=1-\left(1-p_{m,c}^{data}\right)\left(1-p_{m,c}^{ack}\right) \tag{10}\]  
where \(p_{m,c}^{data}\) and \(p_{m,c}^{ack}\) denote the collision probability of data and ACK packets, respectively. When a data packet is transmitted in the presence of interference, it may experience the collision with a probability of one (i.e., \(p_{m,c|\text{busy}}=1\)). Even when it is transmitted in the absence of interference, it may probabilistically experience the collision. Although a WLAN device transmits the signal after carrier sensing-based clear channel assessment (CCA) or energy-based CCA, it may not detect the presence of the signal transmitted by low-power WSN devices. The probability of packet collision can be represented as:  
\[p_{m,c}^{data}=\left\{ \begin{array}{ll}\rho;&\text{if int. source can detect data packets}\\ \rho+\left(1-\rho\right)p_{m,c|idle}^{data};&\text{otherwise}\end{array} \right. \tag{11}\]  
where \(p_{m,c|idle}^{data}\) can be derived in what follows. For given idle state length \(T_{\text{idle}}\) and data packet transmission time \(T_{\text{pkt},m}^{data}\), a packet collision may occur if a data packet is transmitted (\(T_{\text{idle}}-T_{\text{pkt},m}^{data}\)) seconds after the beginning of the idle state. Let \(t_{d}\) be the time difference between the beginning of the idle state and the presence of a new data packet (i.e., \(0\leq t_{d}\leq T_{\text{idle}}\)). Assuming that the duration of the idle state is Pareto or exponentially distributed, it can be shown that:  
\[p_{m,c|idle}^{data}\left(t_{d}\right) = \Pr\left[T_{\text{idle}}<t_{d}+T_{\text{pkt},m}^{data}|t_{d}\right] \tag{12}\] \[= \left\{ \begin{array}{ll}1-\left(\frac{t_{d}}{t_{d}+T_{\text{pkt },m}^{data}}\right);&\text{Pareto dist.}\\ 1-\exp\left(-\tau_{\text{idle}}^{-1}T_{\text{pkt},m}^{data}\right);&\text{exp. dist.}\end{array} \right.\]  
where \(\sigma\) denotes the shape parameter of Pareto distribution. It can be seen that the probability depends on \(t_{d}\) when the idle state is Pareto-distributed. When the idle state is exponentially distributed, Equation (11) can be rewritten as:  
\[p_{m,c}^{data}=\left\{ \begin{array}{ll}\rho;&\text{if int. source can detect data packets}\\ 1-\left(1-\rho\right)\exp\left(-\tau_{\text{idle}}^{-1}T_{\text{pkt},m}^{data} \right);&\text{otherwise.}\end{array} \right.