# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 3 Experimental Analysis
### Optimization Analysis
Figure 12 demonstrates the maximum achievable gain in percentage in terms of a mean spending time over the traditional HARQ approach for optimized parameters \(N,d,\theta\). It shows the gain curves for a number of FN limits and SNR equal to 12 dB. It is worth stressing that the higher is the \(FN_{\max}\) probability, the more exponential is the rise of the gain curve over \(FP_{\max}\). It can be seen that by constraining the FP probability to 0.15, we can achieve 20% of gain at FN = 0.1 and around 11% at FN = 0.001 in terms of a mean number of microslots required for successful delivery. Table 3 shows the gain in terms of a probability of a system being idle for a number of SNRs and \(FN_{\max}\) values given a number of constrained \(FP_{\max}\) values. In general, we see here a qualitatively consistent picture. By relaxing the \(FN_{\max}\) constraint, the gain rises for all the cases of the constrained \(FP_{\max}\). Figure 10: Min cost function versus weight factor, and corresponding \(\theta\) for SNR = 8, \([N,d]\) from left to right, respectively. It is worth mentioning that at small \(FN_{\text{max}}\) values, e.g., \(FN_{\text{max}}=10^{-4}\) the highest gain equal to 12.36% is achieved in network scenario of 10 dB, and in general rises in better SNRs up to a certain limit, after which it tends to lower down, which can be explained by the higher accuracy of the predictor and less erroneous transmission process. The latter behavior can be also visualized when relaxing the conditions of \(FN_{\text{max}}\). Also, the better channel conditions are, the faster the gain curve reaches its maximum. All in all, these results clearly demonstrate the positive gain of the e-HARQ prediction operation over the whole range of the FN and FP probabilities. In this paper we investigated a downlink network scenario with only one UE in terms of analytical modelling. In a general case, we assume that the gNB schedules the resources among the UEs without collisions, which brings us to consideration that the collective gain scales up given the number of users. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & \(FP_{\text{max}}=\mathbf{0.05}\) & & & \(FP_{\text{max}}=\mathbf{0.1}\) & & & & \(FP_{\text{max}}=\mathbf{0.2}\) & \\  & & \(FN_{\text{max}}\) & & & & \(FN_{\text{max}}\) & & & & \(FN_{\text{max}}\) & \\
**SNR** & \(\mathbf{10^{-4}}\) & \(\mathbf{10^{-3}}\) & \(\mathbf{10^{-2}}\) & \(\mathbf{10^{-1}}\) & \(\mathbf{10^{-4}}\) & \(\mathbf{10^{-3}}\) & \(\mathbf{10^{-2}}\) & \(\mathbf{10^{-1}}\) & \(\mathbf{10^{-4}}\) & \(\mathbf{10^{-3}}\) & \(\mathbf{10^{-2}}\) & \(\mathbf{10^{-1}}\) \\ \hline
5 & 1.32 & 4.41 & 5.23 & 6.17 & 1.60 & 5.62 & 8.17 & 40.80 & 2.48 & 9.52 & 28.51 & 40.80 \\
6 & 1.12 & 2.51 & 5.68 & 6.86 & 1.97 & 3.58 & 7.8 & 39.6 & 3.23 & 9.26 & 39.67 \\
7 & 1.27 & 3.12 & 5.72 & 6.76 & 1.98 & 4.65 & 15.04 & 39.09 & 4.75 & 9.89 & 23.48 & 39.09 \\
8 & 0.65 & 2.43 & 4.62 & 5.81 & 2.45 & 5.2 & 21.58 & 37.43 & 5.77 & 8.41 & 26.43 & 37.43 \\
9 & 0.55 & 1.67 & 3.38 & 6.02 & 2.87 & 8.78 & 18.07 & 32.33 & 6.4 & 18.70 & 23.70 & 32.33 \\
10 & 0.3 & 1.47 & 2.87 & 2.87 & 3.3 & 7.51 & 21.3 & 27.37 & 12.36 & 16.46 & 21.3 & 27.37 \\
11 & 0.71 & 1.42 & 2.08 & 9.85 & 4.73 & 8.10 & 18.14 & 22.78 & 9.84 & 13.63 & 18.14 & 22.78 \\
12 & 0.68 & 2.29 & 7.62 & 7.62 & 5.32 & 7.15 & 15.51 & 19.53 & 8.35 & 11.40 & 15.51 & 19.53 \\ \hline \end{tabular}
\end{table} 
Table 3: Gain in % in terms of a probability of a system being idle.