# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 2 System Model
### Performance Measures  
The stationary probability distribution \(\left[s,l,v,t\right],(s,l,v,t)\in X\) is computed from the balance equations along with the normalizing condition by solving a set of linear equations. In order to derive the main performance measures, we firstly define \(X_{\text{quit}}\) as the set of states, in which a packet can quit the system either successfully or unsuccessfully. Hence, it is defined as  
\[X_{\text{quit}}:=\{(s,l,v,t)^{*}:s\in\{2,3\},l=\lceil\frac{v}{2 }\rceil,v-1,v=2,\overline{T_{c}},\\ t\leq T_{N}\}\cup\{(1,l,T_{c},t):l=\lceil\frac{T_{c}}{2}\rceil+u(1,T_{c }),T_{c}-1,t\leq T_{N}\}. \tag{26}\]  
Then a blocking probability, which is known to be a probability of a packet not being serviced during the predefined period \(T_{N}\), is given as  
\[\pi=\sum_{x\in X_{\text{quit}}}P_{\text{quit}}(x)(1-B(x)), \tag{27}\]where \(B(x),x\in X_{\text{quit}}\) is defined as  
\[B(x):= \begin{cases}0,&\text{if }s_{x}=1.\\ b_{2}(l_{x}),&\text{if }s_{x}=2\text{ and }x\in X_{\text{end}}(2).\\ b_{3}(l_{x}),&\text{if }s_{x}=3\text{ and }x\in X_{\text{end}}(3).\\ 1,&\text{otherwise}.\end{cases}  \tag{28}\]  
\(P_{\text{quit}}:X_{\text{quit}}\to[0,1]\)--is a conditional probability that a packet no matter how, successfully or not, quits the system at state \(x,x\in X_{\text{quit}}\):  
\[P_{\text{quit}}(x):=\frac{b_{\text{quit}}(x)[x]}{\sum_{x\in X_{\text{quit}}}b _{\text{quit}}(x)[x]}\,, \tag{29}\]  
where  
\[b_{\text{quit}}(x):= \begin{cases}b_{2}(l_{x}),&\text{if }s_{x}=2\text{ and }x\
otin X_{\text{end}}.\\ b_{3}(l_{x}),&\text{if }s_{x}=3\text{ and }x\
otin X_{\text{end}}.\\ 1,&\text{otherwise}.\end{cases}  \tag{30}\]  
The mean number of deliveries is calculated analogously  
\[M=\sum_{x\in X_{\text{quit}}}P_{\text{quit}}(x)L(x), \tag{31}\]  
where \(L(x)\) the number of transmissions at state \(x\) is presented as  
\[L(x):= \begin{cases}l_{x}-1&\text{if }s_{x}=1.\\ l_{x}&\text{if }s_{x}=2\text{ or }s_{x}=3.\\ \end{cases}  \tag{32}\]  
In its turn, the mean spending time of a packet in the system, which has been successfully transmitted, is calculated as  
\[T=\sum_{x\in X_{\text{quit}}}P_{\text{success}}(x)v_{x}\,, \tag{33}\]  
where \(P_{\text{success}}(x):X_{\text{quit}}\to[0,1]\) is defined as \(P_{\text{quit}}(x)\) with \(b_{\text{quit}}\) being replaced by \(b_{\text{success}}\):  
\[b_{\text{success}}(x):= \begin{cases}b_{2}(l_{x}),&\text{if }s_{x}=2.\\ b_{3}(l_{x}),&\text{if }s_{x}=3.\\ 0,&\text{otherwise}.\\ \end{cases}  \tag{34}\]  
The probability of the system being idle is:  
\[P_{0}=[0,0,0,0]. \tag{35}\]  
The e-HARQ prediction schemes are characterized by two types of errors: False Negative (FN) errors (e-HARQ predicts NACK, whereas HARQ sends ACK), and False Positive (FP) ones (e-HARQ predicts ACK, whereas HARQ sends NACK respond). FN decisions lead to redundant transmissions, which may result in spectral efficiency degradation but have no impact on latency and Block Error Rate (BLER). Hence, FN errors can be tolerated up to a certain limit. In its turn, FP errors indicate predictor's failure. As was already mentioned, common e-HARQ predictors use a threshold value \(\theta\) to classify between ACKand NACK. Obviously, the selection of \(\theta\) has a direct relation to performance of the e-HARQ mechanism. The FN probability of an e-HARQ predictor can be found as  
\[P_{FN}=\sum_{v=2}^{T_{c}}\sum_{l=\lceil\frac{v}{2}\rceil}^{v-1}b_{3}(l)\left[3,l, v,t\right], \tag{36}\]  
whereas the FP probability of an e-HARQ predictor is computed as  
\[P_{FP}=\sum_{v=2}^{T_{c}}\sum_{l=\lceil\frac{v}{2}\rceil}^{v-1}\overline{b}_{ 2}(l)\left[2,l,v,t\right]. \tag{37}\]  
Figure 4 demonstrates a block diagram of a studied communication system on a high-level perspective, including e-HARQ prediction, HARQ and configuration of the e-HARQ prediction based on the proposed analytical model.