# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 5 Non-Orthogonal Multiple Access
### Latency-Oriented System
We define \(n(i;o)\) as the slots between the last available allocated slot and slot \(i\):  
\[n(i;o)=i-\max(\mathcal{T}_{o}(i-1)\cup\{o-G\}). \tag{67}\]  
We define the generation set \(\mathcal{J}_{o}(i)\) as  
\[\mathcal{J}_{o}(i)=\{0,\ldots,G\}^{|\mathcal{T}_{o}(i-1)|-1}\times\{0,\ldots, n(i;o)-1\}. \tag{68}\]  
Each vector \(\mathbf{j}\) in the set corresponds to a possible sequence of past events that led to this point. We define the number of queued packets at the \(i\)-th allocated slot for the generation vector \(\mathbf{j}\) for a given starting queue \(q_{0}\), denoted as \(q_{i}(\mathbf{j};q_{0})\), as  
\[q_{i}(\mathbf{j};q_{0})=[q_{i-1}-1]^{+}+j_{i}. \tag{69}\]If we condition the set on the fact that the packet generated in slot \(i\) finds \(q\) packets in the queue, we get  
\[p_{\text{gen}}(\mathbf{j};o,i,q,q_{0})=\delta(q_{|\mathbf{j}|}(\mathbf{j};q_{0}) -q)\text{Bin}(j_{||j};n(i;o),\alpha)\prod_{k=1}^{|\mathbf{j}|-1}\text{Bin}(j_{ k};G,\alpha). \tag{70}\]  
For each initial queue \(q_{0}\), we can then define a set \(\mathcal{C}_{o,q_{0}}^{(i)}(l)\), which contains the generation vectors that cause exactly \(l\) transmission opportunities to be unused. \[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]0  
We then get \(p_{\text{C}}(l;i,q,o)\). \[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]1  
We now repeat the same consideration for transmission opportunities after the transmission of the considered packet. The number of packets in the queue after the transmission \(V(\mathbf{g})\), for a given generation set \(\mathbf{g}\), is  
\[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]2  
There are at least \(V(\mathbf{g})\) occupied transmission opportunities after the transmission of the packets. We can then define the generation set \(\mathcal{H}_{\ell}(V)\), which represents the possible new packet arrivals. \[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]3  
The probability of each vector \(\mathbf{h}\) in the set is given by:  
\[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]4  
We define the number of queued packets at the \(i\)-th allocated slot for the generation vector \(\mathbf{h}\), denoted by \(q_{i}(\mathbf{h})\), as  
\[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]5  
where \(q_{0}=0\). We can then define the set \(\mathcal{F}_{\ell}(f)\), which requires \(f\) transmission opportunities to be unused by the considered user. \[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]6  
The probability of having \(f\) unused transmission opportunities for the user by the \(d\)-th slot in the frame after the packet transmission, given the generation vector \(\mathbf{g}\), is then  
\[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]7  
In the following, we denote \((1-\varepsilon_{B})\) as \(p_{1}\) to simplify the notation. We now compute the probability that \(r\) packets from the broadband user frame are correctly received by slot \(d\), given that there are \(f\) transmission opportunities before it left unused by the considered user:  
\[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]8  
where \(p_{\text{free}}\) is the same as in (29). We can then define the probability that at least \(K\) packets have been received by slot \(  
The results are presented in the form of Pareto frontiers, that capture the best trade-offs between the throughput of the broadband user \(S_{B}\) and the 90 th percentile of the timeliness metric for the intermittent users. The parameter settings are shown in \[S_{B} =\frac{Kp_{s,B}}{N}. \tag{62}\]9 With the selected parameters and if only the broadband user is considered, the optimal source and coded block sizes are \(N=77\) and \(K=64\), where \(K\) is limited to 64 to make the solution practical), respectively, which results a throughput of \(S_{B}=0.8147\) packets per slot. The latter corresponds to the upper bound in throughput for both OMA and NOMA systems evaluated in the following. We first consider PAoI-oriented systems, whose performance has a strong dependence on the aggregate arrival rate \(U_{I}\alpha\). We assume that \(U_{I}=4\), which allows us to explore a wide range of values for \(\alpha\).