# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 2 System Model
### Model Description  
Let us consider a downlink 5G network scenario, in which a UE firstly predicts a feedback respond based on the partially received from the gNB codeword using the e-HARQ scheme at the first phase, and then process it with a traditional HARQ scheme at the second phase. We assume packets to be of the same transmission length, equal to \(N\) OFDM symbols. Taking into account all retransmissions required for the successful delivery, the packet has to be transmitted to the UE in \(T_{N}\) microslots, otherwise the packet is assumed to be lost. The functioning of the system is given in discrete time with the length \(n\) equal to 1 OFDM symbol. In comparison with our previous work [17], where we studied only one codeword length used for prediction(Figure 1a) and performed analytical analysis on a time slot level equal to an actual transmission length; here, we have to operate on a microslot level of 1 OFDM symbol in order to achieve a fair comparison of the general systems with various \(N,d\) values, which substantially complicates the logic of mathematical expressions. We suppose that all the changes in the system occur at time moments \(nh,h=1,2,\dots\)  
A packet arrives in the system with the probability \(a,0<a\leq 1\). If the eFB, generated by the e-HARQ predictor at the first phase, is ACK, then the packet moves to the second phase with the probability \(b_{1}(l)\), where \(l\) is a current number of deliveries of the given packet. Otherwise, when the eFB is NACK, the packet is to be retransmitted with the probability \(\overline{b}_{1}(l)\). However, the processing of the previous negatively acknowledged packet based on the e-HARQ predictor is still being done at the second phase with a deterministic probability equal to 1. By doing this, we achieve parallel operation of e-HARQ and HARQ, but keep priority on the HARQ decision in order to avoid unnecessary retransmissions generated by the e-HARQ misprediction. The found regular HARQ FeedBack (rFB) at the second phase allows terminating the delivery process in case of ACK with the probability \(b_{i}(l)\) or continuing the retransmission process in case of NACK with the probability \(\overline{b}_{i}(l),i=2,3\). Here and further, \(\overline{a,b}:=\{a,a+1,\dots,b\}\) with \(b>a\) is an interval of natural numbers between \(a\) and \(b\). We clarify the difference between \(b_{2}(l)\) and \(b_{3}(l)\) that can be expressed in terms of the feedback Random Variables (RVs) \(\mathrm{eFB},\mathrm{rFB}\in\{\mathrm{ACK},\mathrm{NACK}\}\) as,  
\[b_{2}(l) := P(\mathrm{rFB}_{l}=\mathrm{ACK}|\mathrm{eFB}_{l}=\mathrm{ACK}), \tag{1}\] \[b_{3}(l) := P(\mathrm{rFB}_{l}=\mathrm{ACK}|\mathrm{eFB}_{l}=\mathrm{NACK}). \tag{2}\]  
By setting the dependence between the feedback generated at the first phase and HARQ's decision at the second phase in terms of \(b_{2}(l)\) and \(b_{3}(l)\), we achieve a system model that can be used for analysis of existing e-HARQ prediction schemes. If the packet reaches its delay constraint in terms of a maximum number of microslots \(T_{N}\) or a maximum number of transmissions \(T_{c}=\lfloor\frac{T_{N}}{N}\rfloor\), it is assumed to be lost. It should be noted that a new packet may arrive only when the previous packet has been successfully transmitted or at an empty system. Figure 2 presents the structure of the proposed analytical model. The functioning of the system is described by the homogeneous Markov chain \(\zeta_{n}\) at time moments \(nh,h=1,2\ldots\) with the state space \(X=X_{m}^{*}\cup X_{t}\), where \(X_{m}^{*}\) defines the set of major states, at which the active events in the model take place and which will be further noted with the star, as  
\[X_{m}^{*}=\{(0,0,0,0)^{*},(1,1,1,t)^{*},(s,l,v,t)^{*}:s\in\{1,2, 3\},\\ l=\overline{\lceil\frac{v}{2}\rceil+u(s,v),v-1},v=\overline{2,T_{c}}, t\leq T_{N}\}, \tag{3}\]  
where \(v\) is a time slot number, and  
\[u(s,v)= \begin{cases}1,&\text{if }\{s=1\}\cap\{v=2n,n=2,3,\ldots\}.\\ 0,&\text{otherwise}.\end{cases}  \tag{4}\]  
\[t(l,v)=