## 2 Radio Resource Management
### Power, Beamwidth and Bandwidth Flexibility  
We assume a GEO high-throughput satellite system cp, comprising a single GEO multi-beam satellite providing coverage to a wide region of Earth via \(B\) spot-beams. We focus on the forward link and assume a total number of \(K\) single-antenna User Terminals (UTs) distributed across the overall satellite coverage area. We assume that the considered payload can flexibly manage three resource settings, i.e., power, bandwidth and beamwidth, similarly to [37; 38]. From a practical perspective, the flexible power allocation can be obtained by using a TWTA (Traveling Wave Tube Amplifiers) by adapting IBO (Input Back-off). At the same time, the flexible payload must be able to separate the signals into frequency blocks and then reorder them to obtain a flexible bandwidth. This process requires a channelizer onboard the satellite, as mentioned by the authors in [39], to identify the frequency plan (color assignment by frequency and polarization). Finally, to achieve beamwidth flexibility, the OMUX (Output Multiplexer) of the traditional payload should be replaced by reconfigurable BFNs (Beamforming Networks). The change of BFN configuration is halfway between the possibility of synthesizing any beam and choosing from a set of configurations for the same coverage [40]. The heterogeneous traffic demand distribution over the satellite beams and the variations of such traffic demand during the satellite lifetime are the main motivations behind the recent emergence of dynamic radio resource management [8]. In order to adapt to actual demands, the proposed system manages radio resources. The payload manager receives input data from gateways and user beams and then generates optimal control by using the ML model to reconfigure the satellite. The goal of RRM is to manage the available resources to minimize the error between the offered capacity (\(C_{t}^{b}\)) and the requested capacity (\(R_{t}^{b}\)) in \(b\)th beam at time slot \(t\). It may seem feasible to achieve a solution using optimization techniques at a larger scale. However, the number of resources to be managed, the constraints coming from the system and the huge number of traffic demand situations may result in a problem that conventional techniques cannot solve. On the other hand, the resource management problem in SatComs is, in most cases, nonlinear and nonconvex due to the logarithmic function as well as the nonlinear dependencies of \(CINR\) in the optimization [8]. This is due to the fact that \(C_{t}^{b}\) can be calculated as \(C_{t}^{b}=BW_{t}^{bc}SE_{t}^{b}\), where \(SE_{t}^{b}\) is the spectral efficiency of the modulation and coding scheme of a commercial reference modern used in the \(b\)th beam over \(t\)[41]. \(SE_{t}^{b}\) depends on \(CINR\) in the \(b\)th beam and in turn \(CINR\) depends on the power, bandwidth and beamwidth allocated to each beam (\(P_{t}^{b}\), \(BW_{t}^{bc}\) and \(\theta_{t}^{b}\), respectively). In this sense, a generic RRM cost function is defined as follows [33]:  
\[\min_{P_{t}^{b},BW_{t}^{b},\theta_{t}^{b}}\frac{\beta_{1}}{B}\sum_{b=1}^{B}|C_ {t}^{b}-R_{t}^{b}|-\frac{\beta_{2}}{B}\sum_{b=1}^{B}P_{t}^{b}-\frac{\beta_{3} }{B}\sum_{nc=1}^{N_{c}}\sum_{bc=1}^{B_{c}}BW_{t}^{bc} \tag{1}\]  
such that the following is the case. \[ \begin{cases}C_{t}^{b}\geq R_{t}^{b}&P_{t}^{b}<P_{max,b}\,\ \theta_{t}^{b}>\theta_{min,b}\, BW_{t}^{bc}<BW_{max,b}\\ C_{t}^{b}=C_{max,b}&P_{t}^{b}=P_{max,b}\,\ \theta_{t}^{b}=\theta_{min,b}\, BW_{t}^{bc}=BW_{max,b}\end{cases}  \tag{2}\]  
\[\sum_{b=1}^{B}P_{t}^{b}\leq P_{max,T} \tag{3}\]  
\[\sum_{bc=1}^{B_{c}}BW_{t}^{bc}\leq BW_{max,c} \tag{4}\]  
\[\theta_{t}^{b}\in\{\theta_{1},\theta_{2},...,\theta_{N}\} \tag{5}\]  
The cost function aims to minimize three parameters for each time instant, \(t\). The first parameter is the error between the offered capacity (\(C_{t}^{b}\)) and the required capacity (\(R_{t}^{b}\)), where \(\beta_{1}\) (in s/bit) is the weight of the error in the cost function.