## 4 Results
As expected, as the number of receivers increases, the frame length associated to the maximum FSR decreases for all the considered values of \(\sigma\). Note that the fading spread has a twofold impact on the optimal frame length. On the one hand, a larger \(\sigma\) supports a higher diversity gain, and thus a higher probability of multiple packets detection. As a consequence, the optimal frame length decreases as the fading spread increases. On the other hand, a higher \(\sigma\) results in a lower powering probability, thus the reduction of the frame length also results from the reduced average number of powered nodes in a frame. Figure 8: Impact of number of deployed receivers \(M\). The considered scenario consists of nodes uniformly distributed over a horizontal line at the center of the illumination area and receivers located inside the illumination area. (**a**) Expected capture rate. (**b**) Expected frame success rate. Figure 7: Impact of number of deployed receivers \(M\). The considered scenario consists of randomly disseminated nodes and receivers located inside the illumination area. (**a**) Expected capture rate. (**b**) Expected frame success rate. In order to analyze the impact of the illuminator-distance on the optimal frame length, a scenario with receivers located within the illumination area at a distance of \(d_{IR}=6\) m is considered in Figure 10. Along with a higher FSR, which is expected at the considered \(d_{IR}\), we observe that in this case the optimal frame length is less sensible to fading spread variations. Interestingly, when \(M=10\), which corresponds to the case where the number of receivers is equal to the number of nodes, only two slots are required to achieve optimal performance. The theoretical limit of the expected FSR in the considered scenario is achieved when \(L=1\) and each receiver captures a different tag. Finally, we analyze the optimal frame length as a function of the number of backscattering nodes for \(M=1,2,10\) and \(\sigma=2\) dB (Figure 10(a)), \(\sigma=4\) dB (Figure 10(b)), and \(\sigma=6\) dB (Figure 10(c)). The theoretical optimal frame length value of FSA is also reported in order to highlight the reduction of frame duration in presence of capture effect and capture diversity. As expected, the optimal frame length decreases as the number of receivers increases. Furthermore, the fading spread also has substantial impact on the optimal frame duration. Figure 9: Frame success rate as a function of frame length \(L\) for different number of deployed receivers. Receivers are placed at the border of the illumination area at a distance of \(d_{IR}=10\) m from the illuminator. (**a**) \(\sigma=2\) dB. (**b**) \(\sigma=4\) dB. (**c**) \(\sigma=6\) dB. Figure 11: Optimal frame length as a function of \(K\) for different number of deployed receivers. Receivers are placed at the border of the illumination area at a distance of \(d_{IR}=10\) m from the illuminator. (**a**) \(\sigma=2\) dB. (**b**) \(\sigma=4\) dB. (**c**) \(\sigma=6\) dB. Theoretical FSA optimal frame length is also reported for comparison. Figure 10: Frame success rate as a function of frame length \(L\) for different number of deployed receivers. Receivers are placed inside the illumination area at a distance of \(d_{IR}=6\) m from the illuminator. (**a**) \(\sigma=2\) dB. (**b**) \(\sigma=4\) dB. (**c**) \(\sigma=6\) dB.