# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 4 Orthogonal Multiple Access
### PAoI-Oriented System  
In PAoI-oriented OMA system the transmission queue size is \(Q=1\) and preemptive scheduling is used, i.e., a newly arrived packets replaces the one stored in the buffer. In this case, the KPIs are given by \(\Delta_{90}\) (the 90 th percentile of the PAoI) for the intermittent users, and the throughput \(S_{B}\) for the broadband user. We consider the grouping model, in which the \(U_{I}\) intermittent users are divided into \(G\) groups. As we mentioned earlier, the scheme applies TDMA between groups. Users in the same group contend for the channel in the same slots. The ALOHA and TDMA systems are extreme cases of the grouping scheme, with \(G=1\) and \(G=U_{I}\), respectively. The OMA grouping scheme is represented in Figure 2, along with the two extreme cases.  
Denote the probability of successfully decoding the broadband users frame (i.e., the \(N\) packets contained in it) as \(p_{s,B}\). The throughput \(S_{B}\) is  
\[S_{B}=p_{s,B}\frac{(T_{\text{int}}-1)K}{T_{\text{int}}N}. \tag{14}\]  
As the broadband user can only use \(T_{\text{int}}-1\) slots out of every \(T_{\text{int}}\), setting up more frequent transmission opportunities for the intermittent users reduces the broadband user's throughput. Probability \(p_{s,B}\) is easy to compute in this case, as orthogonal access prevents collisions with the intermittent users.  
\[p_{s,B}=\sum_{r=K}^{N}\text{Bin}(r:N,1-\varepsilon_{B}). \tag{15}\]  
In order to compute the success probability for intermittent users, we first consider the probability \(\rho\) that an intermittent user accesses the channel in the next intermittent slot that is allocated to it, i.e., the probability that at least one packet is generated in a \(GT_{\text{int}}\) interval.  
\[\rho=1-(1-\alpha)^{GT_{\text{int}}}. \tag{16}\]  
The probability \(p_{s,I}\) that a packet from an intermittent user is decoded successfully is then given by two components. First, all other intermittent users in the same group must not have any packets to send in that slot, and second, there must not be a channel erasure.  
\[p_{s,I}=(1-\varepsilon_{I})(1-\rho)^{\frac{U_{I}}{\alpha}-1}. \tag{17}\]  
The PAoI is then \(\Delta=W+Z\), given by the sum of two components: the first, \(W\), is the waiting time between the generation of a packet and its successful transmission. The second, \(Z\), is the inter-transmission time between the slot when the packet is transmitted and the slot in which the next successful packet from the same user is decoded.  
The pmf of the waiting delay \(W\) of a successful transmission in PAoI-oriented OMA is then given by:  
\[p_{W}(w)=\frac{\alpha(1-\alpha)^{w}}{1-(1-\alpha)^{GT_{\text{int}}}},\ w\in\{0,\dots,GT_{\text{int}}-1\}. \tag{18}\]  
Since transmission opportunities for the intermittent users in a given group are scheduled in one slot every \(GT_{\text{int}}\), \(Z\) is \(GT_{\text{int}}\) times the number of reserved slots between consecutive transmissions. This is a geometric random variable, whose parameter is \(\rho p_{s,I}\). The pmf of \(Z\) is then given by  
\[p_{Z}(z)=(1-\rho p_{s,I})^{\frac{z}{C_{\mathrm{int}}}-1}\rho p_{s,I},\ \forall z\in\mathbb{Z}^{+}\setminus\{0\}:\mathrm{mod}(z, GT_{\mathrm{int}})=0. \tag{19}\]  
The pmf of the PAoI \(\Delta\) is now easy to find by convolving the distributions of \(W\) and \(Z\). Since \(W\)'s support is \(\{0,\ldots,GT_{\mathrm{int}}-1\), and \(Z\)'s support is \(GT_{\mathrm{int}}\times\mathbb{Z}^{+}\), the convolution is reduced to a simple multiplication:  
\[p_{\Delta}(\tau)=p_{W}(\mathrm{mod}(\tau,GT_{\mathrm{int}}))p_{Z}(\tau- \mathrm{mod}(\tau,GT_{\mathrm{int}})),\forall\tau\in\mathbb{Z}^{+}. \tag{20}\]  
We can now easily derive the KPI \(\Delta_{90}\) by applying (11).