## 2 System Model  
According to the previous studies [24; 25; 26], we can select an appropriate threshold of the noise temperature \(T_{th}\) to capture the channel capacity that randomly ranges from _good_ to _bad_ state. Then the time-varying rain attenuation at the Ka-band channel is modeled to a two-state GE channel according to the noise temperature \(T\). If the noise temperature satisfies \(T\leq T_{th}\), the channel is on _good_ state, where channel bit error rate (BER) is as low as (\(10^{-8}\)\(\sim\)\(10^{-5}\)); if the noise temperature satisfies \(T>T_{th}\), the channel is on _bad_ state, and the channel BER is as high as (\(10^{-4}\)\(\sim\)\(10^{-3}\)). We denote the transition probability matrix \(\mathbf{G}\) of the two-state GE channel as  
\[\mathbf{G}=\left[ \begin{array}{cc}Pr(g|g)&Pr(b|g)\\ Pr(g|b)&Pr(b|b)\end{array} \right]=\left[ \begin{array}{cc}\lambda_{1}&1- \lambda_{1}\\ \lambda_{0}&1-\lambda_{0}\end{array} \right], \tag{1}\]  
where \(Pr(g|g)=\lambda_{1}\) is the probability that the Ka-band channel is holding on _good_ state, \(Pr(g|b)=\lambda_{0}\) is the probability that the channel state changes from _bad_ to _good_. Without loss of generality, we assume \(1>\lambda_{1}>\lambda_{0}>0\). The transmission time slots can be expressed as \(W=\{w_{1},w_{2},...,w_{n}\}\), the duration of a transmission time slot is a constant \(D\), and and the corresponding states series of the GE channel can be expressed as \(S=\{s_{1},s_{2},...,s_{n}\}\). The proposed POMDP-based transmission scheme with delayed CSI is shown in Figure 1.  
The transmitter can thus obtain the delayed CSI through belief probability \(p\), e.g., if the previous state is in the good state, then the receiver feedback of a single bit information is 1, otherwise 0. Therefore, there are three transmission actions for the transmitter that can be chosen at the beginning of each transmission time slot \(w_{i}\), and each action is explained in detail as follows. _Betting aggressively_ (action \(A\)): When the transmitter believes that the channel has a high chance in a good state, the transmitter decides to \"gamble\" and transmits a high number \(R_{g}\) of data bits. _Betting conservatively_ (action \(C\)): When the transmitter believes the channel is in a bad state, and decides to \"play safe\" and transmits a low number \(R_{b}\) of data bits. _Betting opportunistically_ (action \(O\)): For this action, the transmitter adopts to sense the channel state at the beginning of the slot by sending a control/probing bit. The cost of sensing is a fraction \(\tau\) of the slot, which is the time spent sensing the channel, defined as \(\tau=d_{RTT}/D\), where \(d_{RTT}\) is the round trip time, and \(D\) is the (constant) duration of a transmission time slot. Then the transmitter selects the appropriate transmission action (\(A\) or \(C\)) according to the sensing outcome, and \((1-\tau)R_{g}\) data bits will send if the channel was found to be in the good state or \((1-\tau)R_{b}\) data bits if otherwise. Therefore, at the beginning of \(i\)-th transmission time slot \(w_{i}\), the transmitter needs to decide an optimal action \(a_{i}\) from the three actions above, i.e., \(a_{i}\in\{A,C,O\}\) to maximize the expected throughput of our proposed POMDP-based transmission scheme. Because the transmitter only can have delayed feedback CSI or even no feedback, this is a POMDP problem. Let \(X_{i}\) denote the channel belief, which is the conditional probability that the channel is in the good state at the beginning of the \(i\)-th transmission time slot, from the past channel history of actions and accumulated delay CSI \(H_{i+t}\), thus \(X_{i+t}=Pr[s_{i+t}=1|H_{i+t}]\). Define a policy \(\pi\) as a map from the belief at a particular time \(t\) to an action in the action space.