## 5 Simulation Results  
In this section, representative simulation results are presented to evaluate the effectiveness of the auction-based mechanism and the impacts of key factors on the performance of the proposed scheme. For the link between the satellite and the relay \(R_{k}\), the channel gain is modeled as \(G_{SR_{k}}=L_{SR_{k}}G_{r}G_{s}h_{SR_{k}}^{2}\), where \(L_{SR_{k}}\) is the free space loss, \(G_{r}\) and \(G_{s}\) are the terrestrial receiver and satellite transmitter antenna gains, respectively, \(h_{SR_{k}}\) is the small scale of fading, undergoing the Shadowed-Rice distribution [41; 42; 43]. For the ground link, the channel gain is modeled as \(G_{R_{k}n}=d_{R_{k}n}^{-\eta}h_{R_{k}n}\), where \(d\) is the distance between \(R_{k}\) and \(n\), \(n\) can be \(D\) or \(C_{k}\), \(\eta\) is the path loss exponent, \(h_{R_{k}n}\) is the small scale of fading, undergoing the Rayleigh distribution [44]. Detailed system parameters are provided in Table 1.  
We assume the secondary relays follow the uniform distribution around the primary user in square range with the side length of 1000 m. The secondary receiver obeys uniform distribution around its transmitter in a circle with a radius of 300 m. We generate 30 relays and the users, respectively, for example in Figure 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Satellite transmission power & 100 w \\ AWGN power & \(-\)108 dBw \\ Satellite transmitter antenna gain \(G_{s}\) & 20 dB \\ Relay receiver antenna gain \(G_{r}\) & 25 dB \\ Center frequency & 4 GHz \\ Nakagami fading parameter of \(H_{SR_{k}}\) & 19.4 \\ Scatter component of \(H_{SR_{k}}\) & 0.158 \\ Terrestrial path loss exponent & 2 \\ Number of realizations & \(10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation ParametersFigures 4 and 5 reveal the capacity of PU under the maximum satellite-relay and relay-destination scheme, the proposed auction-based algorithm and maximum satellite-relay scheme verse transmit power of relays \(P_{k}\) from 10 W to 60 W for AF and DF protocols, respectively. Besides, all selected the secondary networks are at least satisfied the \(R_{min}\). The optimal capacity is achieved for the maximum satellite-relay and relay-destination scheme, which acquires the globe CSIs and can be seen as the upper boundary, but the overheads are always beyond the satellite processing capability, which would cause large delay. The capacity of PU for proposed auction-based algorithm is slightly lower than the maximum satellite-relay and relay-destination scheme, because the partial CSI is implemented by relay, and the SU obtains more capacity. Although the computation consumption is the same as the maximum satellite-relay link scheme for satellite, a significantly improved capacity can be realized. Figure 4: The capacity of PU under the maximum satellite-relay and relay-destination scheme, the proposed algorithm and maximum satellite-relay scheme verse different transmit power of relays for AF protocols. Figure 3: Uniform random distribution model in the cases of urban and populated scenarios. Figures 6 and 7 show the sum capacity of both PU and SU under the maximum satellite-relay and relay-destination scheme, the proposed auction-based algorithm and maximum satellite-relay scheme verse transmit power of relays \(P_{k}\) from 10 W to 60 W for AF and DF protocols, respectively. The proposed algorithm obtain much more sum capacity than conventional schemes, which is achieved by the auction mechanism for the maximum possible capacity of secondary network. Besides, the satellite avoids global CSI acquisition and large computational complexity. Those facts indicate that the proposed auction-based relay selection scheme is effective, leading to a win-win situation for both the primary network and the secondary network.