# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 2 System Model
### Model Description
\begin{cases}vN+2d(l-\lceil\frac{v}{2}\rceil),&\text{if }v=2n,n=1,2,\ldots\\ vN+d(2(l-\lceil\frac{v}{2}\rceil)+1),&\text{otherwise}.\end{cases}  \tag{5}\]  
However, since we are considering the model in microslots of the length \(n\) equal to a duration of 1 OFDM symbol, we have to define a set of transition states \(X_{t}\), which allow reaching the major states  
\[X_{t}=\{\{(s-1,l-1,v-1,t-j):s=1\},\{\gamma(l,v)(s,l,v-1,\\ t-i):s\in\{2,3\}\},\{\delta(l,v)(s+2,l-1,v-1,t-j):s\in\{2,3\}\},\\ (s,l,v,t)\in X_{m}^{*},i=\overline{1,N-d-1},j=\overline{1,N+d-1}\}. \tag{6}\]  
Here,  
\[\gamma(l,v)= \begin{cases}1,&\text{if }\{l<v-1\}\cup\{v=2\}.\\ 0,&\text{otherwise}.\end{cases}  \tag{7}\]  
Figure 2: Structure of the model with two phases, corresponding to e-HARQ and HARQ, respectively. Here \(i=\overline{2,3}\). \[\delta(l,v)= \begin{cases}1,&\text{if }\{l>\lceil\frac{v}{2}\rceil,v=2n\}\cup\{v=2n+1\},n= 1,2,\ldots.\\ 0,&\text{otherwise}.\end{cases}  \tag{8}\]  
It can be seen in (3) and (6) that we utilize two parameters for time slot definition: \(v\) as a time slot number, which is measured in packet's transmission lengths, and \(t\)--a microslot number measured in OFDM symbols. We have to keep track of both of these parameters in order to compose a proper state space with the correct number of transmissions. Otherwise, by leaving out e.g., a time slot number \(v\) we may end up in the scenario where the states with the same number of microslots but varying number of transmissions will be erroneously presented as identical state. Moreover, the given system state definition allows clearer differentiation between the major states of the system. The processing at the phases modeled by the parameter \(s\) can be considered in a binary case for better understanding, which is presented for the major state space set \(X_{m}^{*}\) in Table 1. Here, the position of one in a binary representation means the activity of the given phase. The phase states description in Table 1 refers only to major states, in which the actual change in the system takes place. However, the state path clarifies, which and how many transition states should we go through to reach the major ones: e.g., in order to get to \(s=1\), the packet has to go through \(N+d-1\) transition 0- states. We draw a graph shown in Figure 3 that clearly demonstrates a whole picture of transition probabilities between the major phase states. \begin{table}
\begin{tabular}{c c c} \hline \hline \(s\) & **Description** & **States Path** \\ \hline \(0\) & Empty system & \\ (0,0) & & \\ \hline \(1\) & \multirow{2}{*}{The partial codeword is being processed at phase 1} & \(0(N+d-1)\) \\ (1,0) & & \\ \hline \(2\) & \multirow{2}{*}{The packet is processed at phase 2 due to ACK at phase 1} & \(2(N-d-1)\), \\ (0,1) & & \(4(N+d-1)\) \\ \hline \(3\) & \multirow{2}{*}{The packet is being retransm. At phase 1 because of NACK, and previous negatively acknowledged packet is being processed at phase 2} & \(3(N-d-1)\) \\ \end{tabular}
\end{table} 
Table 1: Description of the phase states. Figure 3: The graph of transition probabilities, taking into account only the major states. The case \(b_{1}(l)=1\) corresponds to regular HARQ operation, since no retransmissions based on e-HARQ prediction take place.