## 6 Decision Engine for Multi-User Systems
### Relay  
This subsection simulates the method described in Section 5. As a reminder, a relay between the transmitter and receiver has to decide between AF and DF, using the same policies as in the single link scenario when failures occurred. The costs \(c_{DF}\) and \(c_{AF}\), defined in Equations (23) and (26), are computed offline and compared to obtain the decision map. The relay estimates the SNR of both channels, finds the code rate of the received bundle, and looks up the decision map for whether or not to decode it. Figure 7 shows the decision map for \(R_{1}=2/3\). It can be observed that AF is preferred when both \(\mathit{SNR}_{1}\) and \(\mathit{SNR}_{2}\) are high enough, since the resulting \(\mathit{SNR}_{AF}\) is high and so AF removes the decoding cost at the relay, offsetting the small additional risk of decoding failure at the end user. Especially when \(\mathit{SNR}_{2}>4.5\) dB, AF is the better choice regardless of \(\mathit{SNR}_{1}\). The simulations also show that the AF region shifts to the right as the code rate \(R_{1}\) increases. This makes sense because for higher code rates, the \(\mathit{SNR}\) must be increased correspondingly so that the risk of decoding failure is maintained at a low level for AF to prevail as discussed earlier.  
Monte Carlo simulations also verify that the proposed relay HARQ strategy provides higher throughput than existing ones. Again, we could interpret the cost as delay, and so the information bits delivered per unit cost would measure the average throughput. The simu  
Figure 6: Throughput of different incremental redundancy (IR) schemes for a single link.  
Figure 7: Relay decision map, shown for \(R_{1}=2/3\).  
channel with deterministic gain to show that the relay decision map in Figure 7 indeed chooses the forwarding scheme with a higher throughput. We then introduce stochastic channel gains to simulate a more practical scenario. Although the relay decision engine was derived based on the assumption of AWGN channels, we show that the smart relay using our proposed policy based on the measured channel side information (CSI) is also suitable in this scenario and outperforms a fixed AF or DF relay.  
In order to perform a fair comparison all relays use the same HARQ strategy described earlier when it comes to the single-link regime. Figure 8 shows the average throughput using different relay strategies as a function of \(\textit{SNR}_{2}\), given a fixed \(\textit{SNR}_{1}=4\) dB and \(R_{1}=2/3\). The relay decision map in Figure 7 predicts that AF is the better choice if \(\textit{SNR}_{2}>3\) dB, and indeed we see in the figure that AF results in higher throughput than DF when \(\textit{SNR}_{1}>3\) dB. The smart relay is programmed to take the strategy with higher throughput.  
Using the decision map should provide an advantage against channel variations because the relay can measure the SNR of its received signal and adopt the appropriate strategy accordingly, whereas a relay with fixed forwarding scheme will fail to adapt to the time-varying channel. The received signal is modeled as \(y=gx+n\) where we assume unit transmit power (\(E[x^{2}]=1\)) and additive Gaussian noise \(n\sim\mathcal{N}(0,\sigma^{2})\). The channel gain \(g\) is uniformly distributed over the range \([0.75,1.25]\), remaining constant within each bundle but changing across different bundles and links. Figure 9 shows the average throughput of the different relay strategies in the fading scenario as a function of \(\textit{SNR}_{2}\) for fixed \(\textit{SNR}_{1}=4\) dB and \(R_{1}=2/3\). The smart relay exhibits a noticeable gain in throughput compared to AF or DF only relays. The gain is especially prominent in the region where AF and DF have similar performance, because our proposed hybrid relay strategy combines the advantages of both when neither of them significantly dominates the other.  
Figure 8: Average throughput of different relay strategies in AWGN channel.