# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 3 Experimental Analysis
### Optimization Analysis  
One of the main goals of the given generalized analytical model is to get a quick estimate of the performance measures in order to select a threshold value for e-HARQ prediction operation. Keeping in mind the stringent URLLC latency requirements, we might favor systems with the smallest transmission and subcode lengths given the tendencies above, however, we have to take into account false e-HARQ predictions. As was mentioned above, false-positive mispredictions increase the total BLER, whereas false-negative errors mainly cause unnecessary retransmissions, and hence degrade the spectral efficiency. Therefore, to find an optimal threshold parameter, the following cost function has to be minimized:  
\[\min_{\theta}(w_{1}P_{FN}(\theta)+w_{2}T^{*}(\theta)), \tag{41}\]  
where \(w_{1}\) and \(w_{2}\) are weight factors, \(w_{1}+w_{2}=1\) and \(T^{*}=\frac{T}{T_{N}}\)--is a normalized mean spending time until successful delivery. Figure 10 shows dependence between the min cost function and the selected weight factor \(w_{1}\) with a corresponding threshold value \(\theta\) from left to right, respectively. It demonstrates in general higher min cost at lower \(w_{1}\), which can be explained by the fact, that the normalized mean transmission time is significantly larger \(P_{FN}\). So we see a reasonable monotonous decline of the min cost function for a family of curves \(N,d\) when incrementing the weight \(w_{1}\) of \(P_{FN}(\theta)\). This is consistent with expectations in the sense that this decline comes along with the rise of the threshold parameter \(\theta\) as shown in Figure 10 (right), which leads to an unfavorable behavior in terms of the main performance measures (Figures 7 and 8). Therefore, given the target network characteristics, the weighted factor can be tuned; and, as can be seen in Figure 11, which shows cost functions with different weights and SNR values, it might be reasonable to take bigger weight value \(w_{1}\), e.g., \(w=[0.8,0.2]\) in Figure 11a in worse network conditions to combat with higher FN probability. We also visualize a shift of local minimum to the left for lower weight \(w_{1}\) values over a set of SNRs. The cost function in Figure 11d has a strict minimum at \(\theta=1\) due to quite a rapid decline of FN probability to zero, and further shifts the minimum fully to the right at \(\theta=0\) in better network conditions. It is worth mentioning that with the increase of subcode length, the cost function also rises. Figure 9: e-HARQ prediction errors: false negative probability and false positive probability from left to right, respectively. Since we do not know what the actual cost function is, which is usually determined by the physical constraints, we try to find out here the way the system behaves under a range of cost functions. It should be noted that instead of a normalized mean spending time, other performance measures, e.g., blocking probability, could be included into the cost function evaluation. However, for the given initialization parameters, the blocking probability has shown insignificant variation over the whole range of \(\theta\), and therefore is not of practical interest. We further compare the performance measures and find the maximum gain achieved in comparison with the traditional HARQ approach given the constrained FN and FP probabilities, by formalizing and solving the given optimization problem:  
\[\underset{N,d,\theta}{\max}(G(T(N,d,\theta),T_{HARQ}(N,d,\theta))), \tag{42}\]  
with the constraints \(P_{FN}(N,d,\theta)<FN_{\max}\) and \(P_{FP}(N,d,\theta)<FP_{\max}\). Here,  
\[G(x,y)=\frac{\mid x-y\mid}{y}*100\%, \tag{43}\]  
and \(FN_{\max},FP_{\max}\)--are the maximum possible values of false negative and false positive probabilities, correspondingly. This approach allows us selecting optimal values \(N,d,\theta\) for operation in a flexible 5G environment.The optimization problem is solved using direct search.