## 3 The Analysis
### Meeting the Delay Requirement  
In order to analyze the delay and reliability guarantee of the aforementioned system, a suitable analysis model is needed. In this work, network calculus will be relied on to establish a corresponding mathematical model for the considered system. Network calculus is a queueing theory for QoS analysis [14], which has been widely used for performance evaluation in various networks. For the consider URLLC traffic model and wireless channel model described in Section 2, an equivalent analysis model for single link transmission can be abstracted as Figure 2 shows. The arrival process, denoted by \(A(t)\), is composed of all information packets generated from a URLLC information source; while the service process, denoted by \(S_{ch}(t)\), is the service provided by the wireless channel. In network calculus analysis, arrival curve and service curve are defined to \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(M\) & \(k_{M}\) & \(a_{1}^{(M)}\) & \(a_{2}^{(M)}\) & \(a_{3}^{(M)}\) & \(a_{4}^{(M)}\) & \(b_{1}^{(M)}\) & \(b_{2}^{(M)}\) & \(b_{3}^{(M)}\) & \(b_{4}^{(M)}\) \\ \hline
256 & 4 & 0.228768 & 0.229083 & 0.118223 & 0.423927 & 0.183242 & 0.038011 & 0.994472 & 0.006911 \\
64 & 4 & 0.198324 & 0.512831 & 0.209086 & 0.079759 & 0.408618 & 0.027517 & 0.120616 & 1.46718 \\
16 & 3 & 0.658747 & 0.117219 & 0.224034 & – & 0.115521 & 1.467927 & 0.482023 & – \\
4 & 2 & 0.143281 & 0.856719 & – & – & 1.557531 & 0.57239 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting coefficients for M-QAM. describe the characteristics of (cumulative) arrival process and (cumulative) service process, respectively, as follows. **Definition 1** (**Arrival Curve**). : _An arrival process \(A(t)\) is said to have an arrival curve \(\alpha(t)\) if for all \(0\leq s\leq t\)[14],_  
\[A(t)-A(s)\leq\alpha(t-s), \tag{9}\]  
_or equivalently \(A(t-s)\leq\alpha(t-s)\), where \(A(t)\) is the amount of traffic arrived during period \([0,t)\)._  
**Definition 2** (**Service Curve**). : _Consider a system \(S\) with input process \(A(t)\) and output process \(A^{*}(t)\). The system is said to have a service curve \(\beta(t)\) if for all \(t\geq 0\)[14],_  
\[A^{*}(t)\geq A\otimes\beta(t), \tag{10}\]  
_where \"\(\otimes\)\" is the min-plus convolution, and \(A\otimes\beta(t)=\inf_{0\leq s\leq t}[A(s)+\beta(t-s)]\)._  
For the considered URLLC traffic, it is assumed to be a periodical process with interval of \(\tau\) and information bit length of \(k\), then its arrival curve \(\alpha_{u}(t)\) can be expressed as  
\[\alpha_{u}(t)=\frac{k}{\tau}t+k. \tag{11}\]  
For the considered single wireless channel with finite coding length and finite constellation, when SNR \(P\), coding rate \(r_{c}\), error probability \(\epsilon\) and modulation order \(M\) are given, and when the bandwidth \(W\) for transmission is allocated, the service process \(S_{ch}(t)\) has service curve \(\beta_{ch}(t)\) as:  
\[\beta_{ch}(t)=R_{2}(n,P,\epsilon,M)\cdot W\cdot t. \tag{12}\]  
Then, the delay of an information packet transmitted via a single link can be derived and summarized in Lemma 2. Detailed proof is given in Theorem 2.19 in [14]. **Lemma 2**.