## 4 Simulation and Results  
To evaluation our proposed POMDP-based optimal transmission policy, we start by comparing the transmission actions with different setups, each leading to a different optimal policy. We choose the parameters below in order to illustrate that, in theory, the optimal policy is determined by the communication scenario parameters, such as data rate \(R_{b}\), \(R_{g}\) and \(\tau\) which is affected by the round trip time and the duration of the transmission time slot as in Theorem 1. The first set of parameters considered is \(\tau=0.4\), \(R_{g}=2\), \(R_{b}=1\), \(\lambda_{1}=0.9\), \(\lambda_{0}=0.2\), and \(\beta=0.5\). Note that from Theorem 1, \(\tau=0.4>1/3\) represents the action of _betting opportunistically_, which could not be used under this scenario, thus the one-threshold policy is optimal as the numerical result shows in Figure 5a. Furthermore, from Theorem 2, the threshold in Figure 5a is \(\rho=0.5\). Therefore, if \(p_{i}<\rho\), the optimal action is \(a_{i}=C\), else if \(p_{i}\geq\rho\), the optimal action is \(a_{i}=A\), and _betting opportunistically_ is unfeasible in this scenario. If we keep all the parameter values fixed and diminish the cost of sensing to \(\tau=0.15\), then from Theorem 1 we can compute that the optimal policy is the two-thresholds policy, shown in Figure 5b. From Figure 5b we can see that the one-threshold policy gives suboptimal values, and the two thresholds in this scenario are \(\rho_{1}=0.176\) and \(\rho_{2}=0.739\) by using Theorem 3. If \(p_{i}<0.176\), the optimal transmission action is _betting conservatively_ (\(a_{i}=C\)); else if \(0.176\leq p_{i}\leq 0.739\), the optimal transmission action is _betting opportunistically_ (\(a_{i}=O\)), which can achieve a better reward than \(a_{i}=C\) or \(a_{i}=A\) unless \(p_{i}>0.739\), the optimal action is _betting aggressively_ (\(a_{i}=A\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Conditions** & **Corresponding Functions** & **Closed Form Computation Expressions** \\ \hline \multirow{2}{*}{\(\rho_{2}\leq\lambda_{0}\)} & \(V_{\beta}(\lambda_{0})=V_{\beta,A}(X_{i}=\lambda_{0})\) & \(V_{\beta}(\lambda_{0})=\frac{\lambda_{0}R_{g}}{(1-\beta)(1-a\beta)}\) \\  & \(V_{\beta}(\lambda_{1})=V_{\beta,A}(X_{i}=\lambda_{1})\) & \(V_{\beta}(\lambda_{1})=\frac{(\lambda_{1}-a\beta)R_{g}}{(1-\beta)(1-a\beta)}\) \\ \hline \multirow{2}{*}{\(\rho_{1}<\lambda_{0}<\rho_{2}\leq\lambda_{1}\)} & \(V_{\beta}(\lambda_{0})=V_{\beta,O}(X_{i}=\lambda_{0})\) & \(V_{\beta}(\lambda_{0})=\frac{(1-\tau)(1-\beta\lambda_{1}+\beta\lambda_{0})(1 \beta R+1-\tau)(1-\lambda)R_{g}(R_{g}-R_{g})+\tau\beta\lambda_{0}\lambda_{1}R_{ g}}{(1-\beta)(1-a\beta)}\) \\  & \(V_{\beta}(\lambda_{1})=V_{\beta,A}(X_{i}=\lambda_{1})\) & \(V_{\beta}(\lambda_{1})=\frac{(1-\tau)\beta(1-\lambda_{0})(1-\lambda_{1})R_{g}+( 1-\lambda_{1}-a\beta-\tau\beta\lambda_{0}+\tau\beta\lambda_{0}\lambda_{1})R_{ g}}{(1-\beta)(1-a\beta)}\) \\ \Next, we compare the long-term effect of the expected throughput of our adaptive data transmission scheme with conventional fixed-rate schemes, to validate the optimality of our POMDP-based transmission policy over Ka-band channels for SIN communications. Let \(w\) denote the number of the transmission time slots \(W=\{w_{1},w_{2},...,w_{n}\}\), and \(\mathcal{V}=\sum_{i=1}^{w}V_{\beta}(X_{i})\) denote the accumulated expected values in Figure 6.  
The system parameters in these scenarios are as follows: \(R_{g}=2\), \(R_{b}=1\), \(\lambda_{1}=0.9\), \(\lambda_{0}=0.2\), and \(\beta=0.99\). With these parameters, the one-threshold policy is optimal for \(\tau\in(0.333,1]\), and beyond these critical values, the two-thresholds policy will become optimal. We have \(\tau_{1}=0.4\) in Figure 6a, and \(\tau_{2}=0.1\) in Figure 6b, respectively. As expected, at the beginning of several transmission time slots, _betting conservatively_ can archive the same throughput as the adaptive transmission scheme does as is illustrated in Figure 6a.