# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 5 Non-Orthogonal Multiple Access
### PAoI-Oriented System  
As in the OMA case, we first consider a PAoI-oriented system, in which \(Q=1\) and preemptive scheduling are used for all intermittent users. Since all intermittent users have the same arrival rate \(\alpha\), we can easily compute the success probability of the broadband user:  
\[p_{\text{s,B}}=\sum_{r=K}^{N}\text{Bin}(r;N,(1-\varepsilon_{B})(1-\alpha)^{U_{ l}}). \tag{33}\]  
The throughput for the broadband user is then:  
\[S_{B}=\frac{Kp_{\text{s,B}}}{N}. \tag{34}\]  
We now turn to computing the value of \(\Delta_{90}\). We consider a specific intermittent user \(u\), whose probability of generating at least one packet before the next allocated slot is  
\[\rho=1-(1-\alpha)^{G}. \tag{35}\]  
If a packet from an intermittent user is transmitted, the probability of success (without considering the interference from the broadband user) is  
\[p_{\text{s,I}}=(1-\varepsilon_{I})(1-\rho)^{\frac{U_{l}}{\alpha}-1}. \tag{36}\]  
As we did in the OMA case, we can divide the PAoI in three parts:  
\[\Delta=W+Y+Z, \tag{37}\]  
where, as above, \(W\) is the waiting time from the packet generation to its transmission and \(Z\) is the inter-transmission time. \(Y\) is the decoding latency, i.e., the number of slots from the transmission until its successful decoding, which is 0 for OMA (in that case, packets are either decoded immediately or lost due to erasure or collision), but can be non-zero for NOMA if the packet is recovered later with SIC. The distribution of \(W\) is simple to derive:  
\[p_{W}(w)=\frac{\alpha(1-\alpha)^{w}}{\rho},\;w\in\{0,\ldots,G-1\}. \tag{38}\]  
We can now compute the pmfs of \(Y\) and \(Z\), but to do so we first compute some auxiliary functions. We define the offset \(o\) as the index of the slot that represents the first allocated slot for the considered user in the frame. Denote by \(\mathcal{T}_{o}(d)\) the set of transmission opportunities for the user from the beginning of the frame to slot \(d\), whose first element is \(o\):  
\[\mathcal{T}_{o}(d)=\{i\in\{o,\ldots,d\}:\text{mod}(i-o,G)=0\}. \tag{39}\]  
The probability that the user will transmit \(m\) packets by slot \(d\) for a given offset \(o\) is  
\[p_{\text{rc}}(m;d,o)=\text{Bin}(m;|\mathcal{T}_{o}(d)|,\rho). \tag{40}\]  
We now derive the probability that the first packet from the intermittent user to be decoded in a frame is correctly received in slot \(d\). This only happens if three conditions are met:  
1. The interference from the broadband user can be successfully removed by SIC; i.e., \(K\) packets from it have been received and decoded in the current frame. 2. There is no interference from other intermittent users. 3. There are no channel errors. The second and third conditions are easy to compute, and are summarized by (36). To consider the third one, we consider the two cases in which the packet is transmitted and decoded in the same slot (denoted as \(\mathcal{A}\)) and the one in which it is decoded later (denoted as \(\mathcal{B}\)). In the former case, at least \(K\) packets from the broadband user have already arrived before \(d\), and SIC is performed immediately; in the latter case, the intermittent user packet is retroactively decoded when the \(K\)-th broadband user packet is decoded. We start with the first one:  
\[ \begin{split} p_{F}(d;o|\mathcal{A})=\rho p_{s,I}\sum_{m=0}^{| \mathcal{T}_{o}(d)|}p_{\text{rc}}(m;d,o)\text{Bin}(0;m,p_{s,I})\sum_{r_{o}=0}^ {d-|\mathcal{T}_{o}(d)|}\text{Bin}\bigg{(}r_{o};|\mathcal{T}_{o}(d)|-m,(1- \varepsilon_{B})(1-\rho)^{\frac{U_{I}}{C}}\bigg{)}\\ \times\sum_{r_{t}=K-r_{o}}^{|\mathcal{T}_{o}(d)|-m}\text{Bin} \bigg{(}r_{t};|\mathcal{T}_{o}(d)|-m,(1-\varepsilon_{B})(1-\rho)^{\frac{U_{I}} {C}-1}\bigg{)}.\end{split}  \tag{41}\]  
In case \(\mathcal{A}\), the decoding delay is always \(0\), i.e., \(Y=0\).