## 3 Cross-Layer Design for Finite Blocklength Coding
### Embedded Markov Chain Model  
In this finite blocklength coding system, we model the packet and symbol queues as a two-dimensional Markov chain, denoting the system state as \((q_{1}[t],q_{2}[t])=(q_{1},q_{2})\). The transition probability of the two-dimensional Markov chain is derived from Equations (7) and (10). Fortunately, we observe that the evolution of the symbol queue, i.e., the transmission process of encoded symbols, is determined by the packet queue length, i.e., \(q_{1}[t]=\zeta\), when a new variable-length encoding process starts. Consequently, the two-dimensional Markov chain is transformed into an embedded Markov chain. In this embedded Markov chain, the packet queue length at the beginning of an encoding process becomes the sole system state, with the transmission process of encoded symbols embedded into the transition probabilities. In our previous work [37], we utilized a two-dimensional Markov chain as it was necessary to calculate the average delay of the system. However, in a system with a hard delay constraint, we only need to ensure that the maximum delay does not exceed the threshold. Therefore, we utilized a one-dimensional embedded Markov chain. To facilitate understanding, Figure 3 illustrates the process of the embedded Markov chain. From a global perspective, incorporating all time slots, the system operates in a two-dimensional paradigm, constituted by the packet queue and the symbol queue. In Figure 3, the blue line represents the change in the packet queue, while the red line represents the change in the symbol queue. However, our focus is selectively directed towards time slots where the symbol queue is empty, as indicated by the red arrow. Then, we treat the packet queue as an embedded Markov chain. With the derived one-dimensional Markov chain, we can significantly reduce the complexity for the steady-state analysis and system performance optimization. **Theorem 2**. : _The packet queue \(q_{1}[t]\) having a maximum queue length \(K<\infty\) and a delay-violation probability \(\xi=0\) is a necessary and sufficient condition for satisfying the hard delay constraint in the system, defined as \(D=2K\)._  
Proof. : Firstly, we demonstrate the sufficiency of the theorem. The given maximum length \(K\) of the packet queue \(q_{1}[t]\) and the delay-violation probability \(\xi=0\) ensure that data packets can wait for at most \(K\) time slots in the packet queue. Additionally, the condition \(\xi=0\) in the packet queue guarantees the transmission of symbols in the symbol queue within the specified delay constraint. Consequently, the hard delay constraint is satisfied, affirming the sufficiency of the theorem. Subsequently, we establish the necessity of this through a proof by contradiction. To derive a contradiction, let us assume that the transmission of the symbol queue requires  
Figure 3: Transition diagram of \((q_{1}[t],q_{2}[t])=(K,0)\). The other queue states are similar to the transition diagram. The diagram also depicts the evolution of the system state over time slots. In the embedded Markov chain, we focus exclusively on the time slots indicated by the red arrows. \(K+1\) time slots while adhering to the hard delay constraint \(D\). In this scenario, the accumulation of data packets in the packet queue surpasses the prescribed maximum length \(K\), resulting in packet loss and a violation of the hard delay constraint. This contradiction highlights the necessity of the theorem. Therefore, the theorem is proved. Given the policy \(\mathbf{\mathcal{F}}=\{g_{th}(\zeta,k,l)\}\), we have \(g_{th}(\zeta,k,l)\geq 0\).