# Interference-Robust Transmission in Wireless Sensor Networks
## 2 System Model  
As illustrated in Figure 2, we consider a star-topology WSN comprising a network coordinator and its child nodes located in an operation range of WLANs. The WSN employs a periodic frame structure for synchronous network operation and data communications, where the period of the frame and the length of the data communication interval are \(T_{\text{period}}\) and \(T_{\text{comm}}\), respectively. The network coordinator transmits a beacon at the beginning of each frame for synchronized network operation. It allocates communication resources to a target node using a handshaking protocol during the network operation interval [35]. Non-target nodes may stay in sleep mode to minimize power consumption. We assume that a transmitter node generates \(L_{\text{bulk}}\)-bit data at each transaction. The \(L_{\text{bulk}}\)-bit data is fragmented by a number of data packets each of which comprises a \(L\)-bit data payload (\(L_{\text{min}}\leq L\leq L_{\text{max}}\)) and signaling bits (e.g., packet header). The receiver confirms the packet reception by sending an ACK packet. The transmitter retransmits the data packet if it does not receive an ACK packet. We also assume that the transmission rate is adjustable according to the channel condition. Then, the packet transmission time with transmission mode \(m\) can be represented as:  
\[T_{\text{pkt},m}\left(L\right)=\frac{L_{\text{shr}}+L_{\text{phr}}}{R_{\text{ base}}}+\frac{L_{\text{mhr}}+L}{R_{m}} \tag{1}\]  
where \(L_{\text{shr}}\), \(L_{\text{phr}}\) and \(L_{\text{mhr}}\) are respectively the bit size of the synchronization header (SHR), the physical layer packet header (PHR) and the medium access control (MAC) layer packet header (MHR), \(R_{m}\left(\in\left\{R_{1},R_{2},...,R_{M}\right\}\equiv\Pi\right)\) denotes the transmission rate of MHR and data payload of transmission mode \(m\) and \(R_{\text{base}}\) is the transmission rate of SHR and PHR (\(R_{\text{base}}\in\Pi\)). We assume that \(R_{1}<R_{2}<\cdots<R_{M}\), and the ACK packet has no payload (i.e., \(L=0\)). A node with transmission mode \(m\) may experience packet loss when the received SNR, denoted by \(\gamma\), is lower than a threshold \(\hat{\gamma}_{m}\) and/or when the packet has collision with the interference signal. We assume that the SNR is unchanged during each packet transmission and randomly varies between the packet transmissions [36]. Then, the probability of transmission failure can be represented as:  
\[\hat{p}_{m}\left(L,\gamma\right)=1-\left(1-p_{m,c}\left(L\right)\right)\left(1 -p_{m,s}\left(L,\gamma\right)\right) \tag{2}\]  
where \(p_{m,c}\) and \(p_{m,s}\) denote the probability of transmission failure due to the packet collision and low SNR, respectively. The probability of transmission failure due to SNR can be represented as:  
\[p_{m,s}=1-\left(1-p_{\text{fail}}^{\text{sync}}\left(\gamma,L_{\text{shr}} \right)\right)\left(1-b_{m,s}\left(\gamma\right)\right)^{L_{\text{phr}}+L_{ \text{mhr}}+L} \tag{3}\]  
Figure 2: A star-topology WSN in a synchronized operation mode. where \(p_{\rm fail}^{\rm sync}\left(\gamma,L_{\rm shr}\right)\) denotes the probability of synchronization failure at an SNR of \(\gamma\) when the SHR length is \(L_{\rm shr}\) (e.g., \(p_{\rm fail}^{\rm sync}\left(\gamma,L_{\rm shr}\right)\) for IEEE 802.15.4 communications is referred to in [37]) and \(b_{m,s}\left(\gamma\right)\) denotes the bit error rate (BER) of transmission mode \(m\) at an SNR of \(\gamma\). Note that \(\tilde{p}_{m}\) is an upper bound of the packet error rate (refer to Appendix B). The probability of transmission failure due to the packet collision can be calculated in terms of the channel occupancy of the interference signal. The channel occupancy of the WLAN signal can be characterized using a semi-Markov model [38; 39]. The channel has two simple operation states, the busy and idle state, whose duration is described by the probability density function (PDF) \(f_{T_{\rm busy}}\left(t\right)\) and \(f_{T_{\rm idle}}\left(t\right)\), respectively.