# Optimal Resource Allocation Policies for Multi-User Backscatter Communication Systems
## 4 Optimal Resource Allocation of Passive Tags
Theorem 3.2: _Given the reader transmission power of each activated semi-passive tag as \(P^{\max}\), the optimal \(t_{m}^{a*}\) and \(n_{m}^{*}\) have the following structure._

\[t_{m}^{a*}=\min\{\max\{\frac{\eta P^{\max}h_{m}T}{P^{c}+\eta P^{\max}h_{m}}, \frac{\gamma_{m}}{z_{m}^{*}+b_{m}}\},\hat{T}_{m}\},\]

\[n_{m}^{*}=\frac{T}{t_{m}^{a*}}-\frac{P^{c}}{\eta P^{\max}h_{m}},\]

_where \(z_{m}^{*}\) is the solution for \(f(z_{m},b_{m})=0\) with respect to \(z_{m}\), \(f(z_{m},b_{m})=\frac{1}{2}+\frac{1}{2}\text{erf}(\sqrt{z_{m}})-\frac{1}{2\sqrt {\pi}}(z_{m}+b_{m})z_{m}^{-\frac{1}{2}}e^{-z_{m}}\)._

Proof: Please refer to [22]. Theorem 3.2 reveals that the optimal control policy of an activated tag has a threshold-based structure. The thresholds are with respect to the reader transmission power. Normally, an activated semi-passive tag has two working strategies; i.e., _directly active_ or _sleep-then-active_. The decision on whether the activated semi-passive tag first enters the sleep state is characterized in [22]. If \(NP^{\mathrm{ave}}<MP^{\max}\), the method used for Problem P5 cannot solve Problem P4, due to the coupling of \(P_{m}\), \(t_{m}^{a}\) and \(n_{m}\) in the objective function and constraints. However, Problem P4 can be reformulated as Problem P6 based on Lemma 2,

\[(\mathbf{P6}) \max_{\mathbf{P},\mathbf{i}\epsilon} \sum_{m=1}^{M}t_{m}^{a}\left[\frac





## 7 Simulation Results

In this section, simulation results are given to corroborate the superiority of the proposed resource allocation policies. The simulation environment is set as follows. For convenience, we assume that the duration of each tag is \(T=1\) s. The forward and backward channel gains are set as \(h_{i}=g_{i}=10^{-3}d_{i}^{-3}\), where \(d_{i}\) is the distance between the reader and \(i\)-th tag. \(d_{i}\) is randomly generated from 4 m to 6 m. The energy harvesting efficiency and maximum BER are set as \(\eta=0.5\) and \(\epsilon=0.3\), respectively. Moreover, we set the circuit and noise power as \(P^{c}=-20\) dBm and \(\sigma^{2}=-90\) dBm, respectively. In addition, all of the experimental parameters are listed in Table 1. For performance comparison, the equal power allocation policy is considered. All of simulation curves are adopted 1000 Monte Carlo runs. Figures 3 and 4 show the curves of average total goodput versus the average reader transmission power. We can observe that as the average reader transmission power increases, the average total goodput first increases and then becomes invariant after the average reader transmission power exceeds some threshold. This is because the BER of each tag is close to 0 for high signal-to-noise ratio (SNR) when high allocated power is available. For low average reader transmission power, the passive tags cannot work. The performance of the case of semi-passive tags is superior to the case of passive tags due to the adoption of _sleep-then-active_ strategy. Moreover, the proposed schemes yield much larger goodput than the scheme of equal power allocation. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Values** \\ \hline Time duration & \(T=1\) s \\ Distance & 4–6 m \\ Forward channel gains & \(10^{-3}d_{i}^{-3}\) \\ Backward channel gains & \(10^{-3}d_{i}^{-3}\) \\ Energy harvesting efficiency & \(\eta=0.5\) \\ Maximum BER & \(\epsilon=0.3\) \\ Circuit power & \(P^{c}=-20\) dBm \\ Noise power & \(\sigma^{2}=-90\) dBm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 3: Average total goodput vs. average reader transmission power with \(N=3\). ## 8 Conclusions

This work studies resource allocation policies for multi-user BackCom systems by considering cases of both passive and semi-passive tags. To account for link reliability, the optimization problems are formulated to maximize the total system goodput of the two cases. For the passive tags case, we derive the optimal resource allocation policy by exploiting the priority of allocated reader transmission power. The \"activate or not\" decision is further discussed.