## 3 Cross-Layer Design for Finite Blocklength Coding
### Embedded Markov Chain Model
The steady-state distribution of this Markov chain is denoted as \(\mathbf{\pi}\), which is expressed as  
\[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]  
To ensure the normalization condition, the following equation holds:  
\[\mathbf{1}^{T}\mathbf{\pi^{\mathcal{F}}}=1, \tag{32}\]  
where \(\mathbf{\pi^{\mathcal{F}}}\) signifies the steady-state probability distribution of the embedded Markov chain and \(\mathbf{1}\) represents a \((K+1)\)-dimensional vector with all entries equal to one. Subsequently, we will derive the balance equation. **Lemma 3**. : _At the beginning of the k-th time slot, the probability distribution of the number of symbol blocks already transmitted, denoted by \(l\), which are coded by \(\zeta\) packets, is expressed as_  
\[p(l|k,\zeta)=\sum_{c\in C(k-1,l)}\prod_{i=1}^{k-1}\left(1-c_{i}+(-1)^{1-c_{i}} \left(\Gamma\left(\zeta,i,\sum_{j=1}^{i-1}c_{j}\right)\right)\right), \tag{33}\]  
_where we have_  
\[\Gamma(\zeta,k,l)=1-\left(1-e^{-g_{th}(\zeta,k,l)}\right)^{M}. \tag{34}\]  
_Additionally, \(\mathcal{C}(k,l)\) is defined as the set that contains all vectors obtained by selecting \(l\) positions in a \(k\)-dimensional all-zero vector and setting them to \(1\)._  
Proof. : This proof is conducted through induction on \(k\). The primary approach involves deriving the probability distribution for each time slot sequentially, followed by mathematical induction to establish a general expression. We define \(\Gamma(\zeta,k,l)\) as the transmission probability in the \(k\)-th time slot, which represents the probability of successfully transmitting \(l\) symbols before the \(k\)-th time slot. These symbols are encoded from a set of \(\zeta\) data packets. The transmission probability is given by  
\[\Gamma(\zeta,k,l)=1-\left(1-e^{-g_{th}(\zeta,k,l)}\right)^{M}. \tag{35}\]  
We now proceed by induction. In the first time slot, we have  
\[p(0|1,\zeta)=1. \tag{36}\]  
In the second time slot, we have  
\[p(0|2,\zeta)=1-\Gamma(\zeta,1,0). \tag{37}\] \[p(1|2,\zeta)=\Gamma(\zeta,1,0). \tag{38}\]  
In the third time slot, we have  
\[p(0|3,\zeta)=(1-\Gamma(\zeta,1,0))(1-\Gamma(\zeta,2,0)). \tag{39}\] \[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]0 (40) \[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]1We define \(\mathcal{C}(k,l)\) as the set that contains all vectors obtained by selecting \(l\) positions in a \(k\)-dimensional all-zero vector and setting them to \(1\), assuming the following equation holds for \(k-1\):  
\[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]2  
Based on Equation (42), we can derive the following expression  
\[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]3  
At this point, the mathematical induction proof is complete. As such, this lemma is proved. **Theorem 3**. : _The transition probability is defined as_  
\[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]4  
_Then, the transition probability is calculated as_  
\[\mathbf{\pi^{\mathcal{F}}}=\left[\pi^{\mathcal{F}}(\zeta)\right]_{(K+1)\times 1}. \tag{31}\]5  
Proof. : In a finite blocklength coding system, \(\zeta\) data packets are encoded into \(L(\zeta)\) symbol blocks. The probability of having transmitted \(L(\zeta)-1\) symbol blocks within \(\zeta^{\prime}-1\) time slots is expressed as \(p(L(\zeta)-1|\zeta^{\prime}-1,\zeta)\). The probability of transmitting the last symbol block in the \(\zeta\)-th time slot is expressed as \(1-\left(1-e^{-St_{th}(\zeta,\mathcal{T}^{\prime},L(\zeta)-1)}\right)^{M}\). By multiplying these two probabilities, the lemma is then proven. Based on the transition probability presented in Theorem 3, we can derive the corresponding balance equation for the packet queue state \(\zeta\in\mathcal{Q}\).