# IRS-Enabled Ultra-Low-Power Wireless Sensor Networks: Scheduling and Transmission Schemes
## 6 Results
On the one hand, the system throughput is maximized, while on the other fairness may be compromised. To ensure fairness amongst ISNs, a PF scheme can be implemented. PF caters particularly to ISNs with relatively weaker channel quality, as it takes into consideration the throughput of each ISN within a certain window of time \(t_{w}\). By tuning \(t_{w}\), we can enhance fairness, albeit by compromising system throughput. To highlight the fairness aspect, in Figure 7 we plot Jain's fairness index, represented as  
\[J(t)=\frac{[\sum_{l=1}^{L}T_{l}(t)]^{2}}{L\sum_{l=1}^{L}T_{l}(t)^{2}}, \tag{43}\]  
versus the number of ISNs \(L\) in the system, where \(T_{l}\) is the throughput at frame \(t_{f}\) and \(l^{*}\) denotes the index of a scheduled ISN, defined as  
\[T_{l}(t_{f})= \begin{cases}T_{l}(t_{f}-1)+R_{l}(t_{f}),&l=l^{*}\\ T_{l}(t_{f}-1),&l\
eq l^{*},l=1,\ldots,L\end{cases}  \tag{44}\]  
Note that \(\frac{1}{L}\leq J(t)\leq 1\), in which \(1\) indicates best case for fairness when \(T_{l}(t)=T\). It be seen in Figure 7 that RR and PF have the best fairness performance among all other multi-user scheduling algorithms, as the others focus on ISNs with better channel quality and forgo fairness between the sensors. Moreover, it can be seen that the fairness indexes of PF and RR do not decrease with increasing number of ISNs, while those of OS and the benchmark decrease, as more ISNs indicate more competition and unfairness in the system. Furthermore, with a larger time window of \(t_{w}=50\), PF approaches the fairness of RR scheduling and achieves higher fairness than with a smaller time window of \(t_{w}=5\). Finally, it can be seen that the benchmark with OS results in worse fairness performance than OS under a random phase shift matrix. The minimum throughput \(\tilde{T}_{l}(t_{f})\) of the system, defined as  
\[\tilde{T}_{l}(t_{f})=\min_{l=1,l}\ \ T_{l}(t_{f}), \tag{45}\]  
Figure 7: [PERSON]’s fairness index as defined in (43) vs. number of ISNs \(L\). is plotted in Figure 8 to show that PF can be used to ensure fairness; furthermore this fairness can be adjusted by tuning the time window \(t_{w}\) of past transmissions. The benchmark and OS rarely schedule the worst ISN in the system; thus, their minimum throughput is zero. Moreover the uniform RR scheduling technique performs better than PF for the minimum throughput under certain transmitting powers and \(\alpha\), which is the fraction of time reserved for information transmission. SNR feedback for all the ISN links is only necessary for both PF and OS, as these rely on the varying random channels; RR does need such feedback from the user to the BS, meaning that more time can be allocated for information transmission. This is captured through \(\alpha\). Here, \(\alpha\in[0,1]\) is a fraction of the time reserved for actual transmission and scales inversely according to \(L\). For both PF and OS, the fraction of \(\alpha\) is less than 1, as there is indeed time (negligible though it may be according to IEEE 802.11 standards) for SNR feedback from the user for all sensor links involved in the network. On the other hand, for RR, which is a channel-independent scheduling scheme, \(\alpha=1\). In Figure 9, which shows the average throughput of the system versus the transmit power, we assume that \(M\) is sufficiently large to ensure asymptotic favorable propagation (recall Lemma 1). We can observe the impact of the number of ISNs \(L\) on system performance under OS, with multi-user diversity being improved with a larger number of sensors. Moreover, note that for RR the impact of \(L\) is negligible with optimal choice of phases. This is because each sensor takes a turn to transmit, and having more sensors take turns, even if are optimal, plays virtually no role on the average throughput for the system due to the randomness of the channels.