# Interference-Robust Transmission in Wireless Sensor Networks
## 3 Transmission in the Presence of Interference
\tag{13}\]  
Similarly, it can be shown that the collision probability of an ACK packet after successful data packet transmission can be represented as:  
\[p_{m,c}^{ack}=1-\exp\left\{-\tau_{\text{idle}}^{-1}\left(\delta+T_{\text{pkt},m}^{ack}\right)\right\}. \tag{14}\]  
The probability of transmission failure due to the packet collision can be represented as:  
\[p_{m,c}\left(L\right)=1-\left(1-\rho\right)\exp\left(-\frac{L+\alpha_{m}}{R_{ m}\tau_{\text{idle}}}\right) \tag{15}\]  
where \(\alpha_{m}\) is a constant indifferent from the payload size \(L\) and can be determined as:\[\alpha_{m} = {\alpha^{\prime}}_{m}R_{m} \tag{16}\] \[= \left(2T_{\text{base}}+\frac{L_{\text{mbr}}^{data}+L_{\text{mbr}}^{ ac}}{R_{m}}+\delta\right)R_{m}.\]  
Here, \(T_{\text{base}}=\left(L_{\text{shr}}+L_{\text{phr}}\right)\Big{/}R_{\text{base}}\), \(L_{\text{mbr}}^{data}\) and \(L_{\text{mbr}}^{ack}\) respectively denote the MHR bit size of the data and the ACK packet. It can be shown that the data throughput can be represented as:  
\[S_{m}\left(L,\gamma\right)=R_{m}\frac{L}{L+\beta_{m}}\left(1-\rho\right)\exp \left(-\frac{L+a_{m}}{R_{m}\tau_{\text{idle}}}\right)\left(1-b_{\text{base},s} \left(\gamma\right)\right)^{L_{\text{phr}}^{data}+L_{\text{phr}}^{ack}}\left(1 -b_{m,s}\left(\gamma\right)\right)^{L+L_{\text{mbr}}^{data}+L_{\text{mbr}}^{ ac}} \tag{17}\]  
where \(\beta_{m}=\beta_{m}^{\prime}R_{m}=\left({\alpha^{\prime}}_{m}+\delta\right)R _{m}\) and \(b_{\text{base},s}\) denotes the BER of PHR transmission. Assuming that \(b_{\text{base},s}\) and \(b_{m,s}\) are very low with the use of an appropriate transmission rate in the absence of interference, Equation (17) can be approximated as:  
\[S_{m}\left(L,\gamma\right)\approx R_{m}\frac{L}{L+\beta_{m}}\left(1-\rho\right) \exp\left(-\frac{L+a_{m}}{R_{m}\tau_{\text{idle}}}\right). \tag{18}\]  
Taking the derivative of Equation (18) with respect to \(L\), i.e.,  
\[\frac{\partial S_{m}}{\partial L}=-\frac{\tau_{\text{idle}}^{-1}\left(1-\rho \right)}{\left(L+\beta_{m}\right)^{2}}\exp\left(-\frac{L+a_{m}}{R_{m}\tau_{ \text{idle}}}\right)\left(L^{2}+\beta_{m}L-\beta_{m}R_{m}\tau_{\text{idle}}\right) \tag{19}\]  
we can see that there exists a payload size that maximizes the data throughput. This implies a tradeoff between the packet transmission efficiency and the probability of transmission failure. The use of a smaller payload size may improve the robustness to interference, but it may also increase the signaling overhead, deteriorating the overall transmission efficiency. The payload size maximizing the data throughput can be determined as:  
\[L_{m}^{*}\cong-\frac{\beta_{m}}{2}+\sqrt{\left(\frac{\beta_{m}}{2}\right)^{2 }+\beta_{m}R_{m}\tau_{\text{idle}}}. \tag{20}\]  
It can be seen that the payload size depends on the average idle period of interference, \(\tau_{\text{idle}}\), and the transmission rate \(R_{m}\), as well. With \(\beta_{m}=\beta_{m}^{\prime}R_{m}\), Equation (20) can be rewritten as:  
\[L_{m}^{*} = \left[-\frac{\beta_{m}^{\prime}}{2}+\sqrt{\left(\frac{\beta_{m}^{ \prime}}{2}\right)^{2}+\beta^{\prime}{}_{m}\tau_{\text{idle}}}\right]\,R_{m} \tag{21}\] \[= T_{m}^{*}R_{m}\]  
where \(T_{m}^{*}\) denotes the payload transmission time that maximizes the data throughput. It can be shown that:  
\[p_{m,c}\left(L\right)=1-\left(1-\rho\right)\exp\left(-\frac{L+\alpha_{m}}{R_{ m}\tau_{\text{idle}}}\right) \tag{15}\]0  
Since \(R_{m}\) is typically large, Equation (22) implies that the sensitivity of the payload transmission time with respect to the transmission rate is much smaller than that of the payload size. Moreover, \(T_{m}^{*}\) depends on \(\beta_{m}^{\prime}\). The sensitivity of \(\beta_{m}^{\prime}\) with respect to the transmission rate is much smaller than that of \(\beta_{m}\).