# Analytical Model and Feedback Predictor Optimization for Combined Early-HARQ and HARQ
## 2 System Model
### Balance Equations  
If \(a,0<a\leq 1\) the Markov chain \(\xi_{n}\) is aperiodic, and there exists a stationary probability distribution \(\left[s,l,v,t\right],\left(s,l,v,t\right)\in X\) which is found from the balance equations:  
\[a[0,0,0,0]^{*}=\overline{u}(\sum_{s=2}^{3}\sum_{(s,l,v,t)\in X_{m}^{*}\setminus X _{\text{end}}}b_{s}(l)[s,l,v,t]^{*}+\sum_{(s,l,v,t)\in X_{\text{end}}}[s,l,v,t]^ {*}), \tag{9}\]  
\[[0,0,0,1]^{*}=a([0,0,0,0]+\sum_{s=2}^{3}\sum_{(s,l,v,t)\in X_{\Xi _{n}}\setminus X_{\text{end}}}b_{s}(l)[s,l,v,t]^{*}+\\ \sum_{(s,l,v,t)\in X_{\text{end}}}[s,l,v,t]^{*}), \tag{10}\]  
\[[1,1,t]^{*}=[0,0,0,t-1]=\ldots=[0,0,0,1], \tag{11}\]  
\[[1,l,v,t]^{*}=[0,l-1,v-1,t-1]=[0,l-1,v-1,t-2]=\ldots\\ =[0,l-1,v-1,t-(N+d-1)]=\overline{b}_{2}(l-1)\\ [2,l-1,v-1,t-(N+d)]^{*},v=\overline{3,7_{c}}, \tag{12}\]  
\[[s,l,v,t]^{*}=\gamma(l,v)[s,l,v-1,t-1]+\delta(l,v)[s+2,l-1,v-1,t-1], \tag{13}\]  
where \(s=\overline{2,3}\) and  
\[[2,l,v-1,t-1]=[2,l,v-1,t-2]=\ldots=[2,l,v-1,t-\\ (N-d-1)]=b_{1}(l)[1,l,v-1,t-(N-d)]^{*}, \tag{14}\]  
\[[3,l,v-1,t-1]=[3,l,v-1,t-2]=\ldots=[3,l,v-1,t-\\ (N-d-1)]=\overline{b}_{1}(l)[1,l,v-1,t-(N-d)]^{*}, \tag{15}\]  
\[[4,l-1,v-1,t-1]=[4,l-1,v-1,t-2]=\ldots\\ =[4,l-1,v-1,t-(N+d-1)]=b_{1}(l)\overline{b}_{3}(l-1)\\ \delta(l,v)[3,l-1,v-1,t-(N+d)]^{*}, \tag{16}\]  
and  
\[[5,l-1,v-1,t-1]=[5,l-1,v-1,t-2]=\ldots\\ =[5,l-1,v-1,t-(N+d-1)]=\overline{b}_{1}(l)\overline{b}_{3}(l-1)\\ \delta(l,v)[3,l-1,v-1,t-(N+d)]^{*}, \tag{17}\]  
where \((s,l,v,t)\in X\).  
\[X_{\text{end}}=X_{\text{end}}(1)\cup X_{\text{end}}(2)\cup X_{\text{end}}(3) \subset X_{m}^{*}, \tag{18}\]  
presents a set of termination system states, from which we have to leave the system due to the reaching the \(T_{N}\) limit, where  
\[X_{\text{end}}(1)=\{\mu(t(l,v+1)-T_{N})\cdot(1,l,v,t(l,v))^{*}\}, \tag{19}\]\[X_{\text{end}}(2) =\{\mu(t(l+1,v+1)-T_{N})\cdot(2,l,v,t(l,v))^{*}\}, \tag{20}\] \[X_{\text{end}}(3) =\{\mu(t(l+1,v+1)-T_{N})\cdot(3,l,v,t(l,v))^{*}\},\] (21) \[\mu(x)=\begin{cases}1,&\text{if }x>0.\\ 0,&\text{otherwise}.\end{cases} \tag{22}\]  
The normalizing equation is defined as  
\[\sum_{(s,l,v,t)\in X}\left[s,l,v,t\right]=1. \tag{23}\]  
One of the typical challenges of the systems that model technical systems in discrete time is to properly define the state space. In its turn, the cardinality of the state space has a direct impact on the feasibility of finding the stationary probability distribution. The cardinality for the given model's state space is equal to \(\mid X\mid=\mid X_{m}^{*}\mid+\mid X_{t}\mid\), where \(\mid X_{m}^{*}\mid\) --is an overall number of all the major states, and is defined as  
\[\mid X_{m}^{*}\mid=\mid\left[0,0,0,0\right]+\left[1,1,1,t(1,1)\right]+\\ \sum_{v=3}^{\lfloor\frac{T_{N}}{2}\rfloor}\sum_{l=\lceil\frac{v}{2 }\rceil+u(1,v)}^{v-1}(1-\mu(t(l,v+1)-T_{N}))*\left[1,l,v,t(l,v)\right]+\\ \sum_{s=2}^{3}\sum_{v=2}^{\lceil\frac{T_{N}}{2}\rceil}\sum_{l= \lceil\frac{v}{2}\rceil}^{v-1}(1-\mu(t(l+1,v+1)-T_{N}))[s,l,v,t(l,v)]\mid, \tag{24}\]  
and \(\mid X_{t}\mid\) --is an overall number of all the transition states, and is computed as  
\[\mid X_{t}\mid=\mid\sum_{(1,l,v,t)\in X_{m}^{*}}\sum_{j=1}^{N+d-1 }\left[0,l-1,v-1,t-j\right]+\\ \sum_{s=2}^{3}\sum_{(s,l,v,t)\in X_{m}^{*}}\sum_{j=1}^{N-d-1} \gamma(l,v)[s,l,v-1,t-i]+\\ \sum_{s=2}^{3}\sum_{(s,l,v,t)\in X_{m}^{*}}\sum_{j=1}^{N+d-1} \delta(l,v)[s+2,l-1,v-1,t-j]\mid. \tag{25}\]  
For example, for \(N=4,d=1\) and \(T_{N}=28\) the cardinality set is equal to 122 states, out of which 26 belong to the major ones, and 96 states to the transition ones.