# User-Perceived Capacity: Theory, Computation, and Achievable Policies
## 6 Simulation Results  
In this section, we will validate the accuracy and effectiveness of our theoretical results through simulation. We assume that the large scale path loss is calculated as \(\rho=28+22\log_{10}d+20\log_{10}f_{c}\,\text{dB}\) with the carrier frequency \(f_{c}\) as \(6\,\text{GHz}\) and the distance \(d\) as \(100\,\text{m}\), grounded in a typical channel model in the 5G NR standard [67]. Next, we calculate the transmission powers over the AWGN channel as \(P(0)=0\) W, \(P(1)=5.19\,\text{mW}\), and \(P(2)=10.38\,\text{mW}\) by using BPSK and QPSK modulations when transmission rate \(s[t]=1\) and 2, respectively. In particular, the occupied bandwidth is 1 MHz and the one-sided noise power spectral density \(N_{0}\) is \(-150\,\text{dBm}/\text{Hz}\). Additionally, we model the fading channel with a three-state block-fading distribution, with states \([1,4,9]\) and equal probabilities. The power consumption over fading channels is equal to \(P(s,l)=\frac{P(s)}{l^{2}}\). Figure 3 demonstrates the performance of the policy in the fading channel scenario. Since we adopt discrete time-slots, only the points shown in the figure represent the operating points of our proposed policy. The horizontal axis represents the period size, which is the maximum number of service time-slots for the periodically arriving data packets. The vertical axis represents the average power consumption. We demonstrate the performance of cross-layer scheduling with \(B=6\), \(10\), and \(14\), where the frame sizes are quantized by the minimum transmission unit \(\delta=5\times 10^{3}\) bits. Meanwhile, we set  
\(T\in\{7,\dots,30\}\), with the timeslot duration as \(0.5\,\mathrm{ms}\). We observe that the theoretical results of the proposed policy align well with the simulation results, providing a cross-layer scheduling policy that meets the continuous reading requirement in the fading channel scenario. We also notice that, when the periodically arriving data packets are fixed, the average power consumption decreases as period \(T\) increases. There are two reasons for this. First, similar to the AWGN scenario, as the period increases, the average power consumption naturally decreases. Second, a larger period means more time-slots are available for transmission, allowing for the possibility to wait for better channel conditions, thereby reducing power consumption. Figure 4 illustrates the variation of user-perceived capacity with respect to the average power consumption in a fading channel scenario. The horizontal axis represents the average power consumption, while the vertical axis denotes the user-perceived capacity. We present performance variations for the cases of \(T=15,20,30\). It is observed that there is good agreement between theoretical and simulated results. Additionally, it is noted that as the average power increases, the user-perceived capacity continuously increases. However, the rate of increase in user-perceived capacity slows down with the increase in average power. This is consistent with the law of channel capacity, where the marginal effect of the capacity increase diminishes as the average power increases. We next compare the optimal rate and power adaptation scheme with a heuristic method, referred to as the fixed rate transmission scheme. Specifically, this benchmark serves the user with a fixed rate B/T in each timeslot. In Figure 5a, we set \(T=20\) and evolve the user-perceived capacity under the two schemes. It is evident that the user-perceived capacity achievable by our policy is higher than the fixed rate transmission scheme. Similarly, Figure 5b presents the average power under different values of \(T\) with \(B=20\). Since the benchmark dose not consider the time-varying channel state, it suffers from a larger amount of power consumption compared to the optimal scheme. Figure 6 demonstrates the influence of the imperfect channel state information to the performance of the proposed transmission scheme, where we set \(B=20\) and \(T\in\{10,20\}\).