## 3 Optimal Transmission Policy Based on POMDP
Therefore, there are three types of threshold policies accordingly: (1) When \(\rho_{1}=\rho_{2}=\rho_{3}\), the optimal policy is a one-threshold policy; (2) When \(\rho_{1}<\rho_{2}=\rho_{3}\), the optimal policy is a two-thresholds policy; (3) When \(\rho_{1}<\rho_{2}<\rho_{3}\), the optimal policy is a three-thresholds policy. The optimal policy for the three-thresholds policy is illustrated in Figure 2, and the interval \([0,1]\) is separated in four regions by the thresholds \(\rho_{1},\rho_{2}\) and \(\rho_{3}\). Intuitively, one would think that there should exist only three regions, i.e., if \(X_{i}\) is small, one should play safe; if \(X_{i}\) is high, one should gamble, and, somewhere in between, sensing is optimal. However, if the transmitter can not obtain the feedback CSI, for some cases, a three-threshold policy is optimal and an example is shown in [28]. However, with the help of the delayed feedback CSI, our POMDP-based transmission policy has only three regions, i.e., \((\rho_{2},\rho_{3})=\emptyset\). Therefore, the necessary conditions that tell if action \(O\) should be selected under a given SIN communication scenario, with data rate \(R_{g}\) and \(R_{b}\) in good and bad states, respectively, two thresholds \(\{\rho_{1},\rho_{2}\}\), and cost of action \(O\) is \(\tau\), are given in the following theorem. Figure 2: Thresholds of the optimal transmission action. **Theorem 1**. : _In terms of the POMDP-based optimal transmission scheme constructed by the Bellman function Equation (6), where \(X_{i}R_{g}\) is the expected return when the risky action \(A\) is taken, \(R_{b}\) bits are transmitted regardless of the channel conditions when action \(C\) is selected, and the expected return when the sensing action \(O\) is taken is \((1-\tau)\big{[}(1-X_{i})R_{b}+X_{i}R_{g}\big{]}\), therefore:_  
_If \(R_{b}/R_{g}<(1-2\tau)/(1-\tau)\) then the optimal policy is a two-thresholds \(\{\rho_{1},\rho_{2}\}\) policy, and the optimal action \(a_{i}\) can be selected from \(\{A,C,O\}\);_  
_Otherwise, if \(R_{b}/R_{g}\geq(1-2\tau)/(1-\tau)\) then the optimal policy is a one-threshold \(\rho\) policy and the optimal action \(a_{i}\) can be selected from \(\{A,C\}\)._  
Proof of Theorem 1.: In our SIN POMDP-based transmission scheme, without loss of generality, assume that the optimal policy has two thresholds \(0<\rho_{1}\leq\rho_{2}<1\). Note that, since \(\rho_{1}\) is the solution of \(V_{\beta,C}(X_{i})=V_{\beta,O}(X_{i})\), and \(\rho_{2}\) is the solution of \(V_{\beta,O}(X_{i})=V_{\beta,A}(X_{i})\), it is easy to establish that  
\[\left\{ \begin{array}{l}V_{\beta,C}(\rho_{1})=V_{\beta,O}(\rho_{1})\\ V_{\beta,A}(\rho_{2})=V_{\beta,O}(\rho_{2})\end{array} \right.\quad. \tag{7}\]  
From Equation (5), we have  
\[\left\{ \begin{array}{l}\rho_{1}=\frac{\tau R_{b}}{(1-\tau)(R_{g}-R_{b})}\\ \rho_{2}=\frac{(1-\tau)R_{b}}{(1-\tau)R_{b}+R_{g}}\end{array} \right.\quad. \tag{8}\]  
If the optimal policy has two thresholds then \(\rho_{1}<\rho_{2}\), and the communication parameters should satisfy  
\[\frac{R_{b}}{R_{g}}<\frac{1-2\tau}{1-\tau}. \tag{9}\]  
Otherwise, if the optimal policy has one threshold \(\rho=(\rho_{1}=\rho_{2})\), then the communication parameters turn to satisfy  
\[\frac{R_{b}}{R_{g}}\geq\frac{1-2\tau}{1-\tau}. \tag{10}\]  
Note that Theorem 1 establishes the structure that tells if action \(O\) should be used, and two types of threshold policies exist depending on the system parameters-in particular, the cost of the sensing action \(\tau\) versus the ratio of \(R_{b}/R_{g}\), and the optimal policy space is partitioned into two regions, which is illustrated in Figure 3.  
Figure 3: The optimal transmission policies thresholds are determined by the SIN communication parameters based on delayed feedback CSI. Figure 3 illustrated that the established two optimal policies regions can be further partitioned into three regions at most. As one should expect, the optimal transmission scheme here is a myopic policy that maximizes the immediate reward.