# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 3 System Model  
In the following, we denote random variables with capital letters (e.g., \(X\)) and their values with the corresponding lowercase letters (e.g., \(x\)). Sets are denoted in calligraphic font (e.g., \(\mathcal{U}\)), and the corresponding standard capital letters denote their cardinality (e.g., \(U\)). Vectors are denoted with bold lowercase letters (e.g., \(\mathbf{x}\)), and matrices with bold capital letters (e.g., \(\mathbf{X}\)). probability mass functions (pmfs) are denoted with a lowercase \(p\) and Cumulative Distribution Functions (CDFs) with a capital \(P\). Table 1 provides a quick reference for the most important notation used in the rest of the paper. We define the outcome of the user's activity in a slot as an event, which happens with probability \(p\) and is mutually exclusive with other outcomes. The outcome vector \(\mathbf{k}\) then corresponds to the composite event in which the \(i\)-th outcome is observed \(k_{i}\) times, and the probability vector \(\mathbf{p}\) contains the probability of each outcome (which does not necessarily sum to 1 as we consider that none of the outcomes might occur). We can then define the \begin{table}
\begin{tabular}{c l c l} \hline \hline
**Symbol** & **Meaning** & **Symbol** & **Meaning** \\ \hline \(\mathcal{U}\) & Set of users & \(U\) & Cardinality of \(\mathcal{U}\setminus\{1\}\) \\ \(K\) & Information packets in a frame & \(N\) & Total packets in a frame \\ \(\mathcal{A}_{t}\) & Set of users that can transmit in \(t\) & \(a_{u,t}\) & Indicator of user \(u\)’s activity in \(t\) \\ \(\varepsilon_{B}\) & Broadband user channel error & \(\varepsilon_{I}\) & Intermittent user channel error \\ \(\Delta\) & Peak Age of Information (PAoI) & \(T\) & Latency-reliability \\ \(\Delta_{\text{Ag0}}\) & 90 th per. of PAoI & \(T_{\text{Ag0}}\) & 90 th per. of latency-reliability \\ Mult(\(\mathbf{k};N,\mathbf{p}\)) & Multinomial function & Bin\((k;N,p)\) & Binomial function \\ \(G\) & Number of groups & \(\alpha\) & Intermittent user activation rate \\ \(S_{B}\) & Broadband user throughput & \(p_{\text{s},B}\) & Frame decoding probability \\ \(T_{\text{int}}\) & OMA intermittent slot period & \(\rho\) & Intermittent slot activation rate \\ \(p_{\text{s},I}\) & Intermittent user success prob. & \(W\) & Waiting delay \\ \(Z\) & Inter-transmission time & \(Q\) & Intermittent user queue size \\ \(\mathbf{P}^{(Q)}\) & Queue state transition matrix & \(\boldsymbol{\pi}^{(n)}\) & \(n\)-th slot steady-state distribution \\ \(\mathcal{G}_{\ell}^{(n)}\) & \(n\)-th slot generation set & \(p_{\text{gen}}(\mathbf{g};\ell,n)\) & \(n\)-th slot generation probability \\ \(\psi_{\ell}^{Eq}\) & Transmission condition for \(\mathbf{g}\) & \(\mathcal{H}_{\ell}^{(n,t)}\) & Valid transmission vector set \\ \(\mathcal{T}_{o}(d)\) & NOMA transmission slot set & \(p_{\text{tx}}(m;d,o)\) & Prob. of transmitting \(m\) packets \\ \(F\) & First intermittent decoding & \(\varnothing\) & No int. decoding in a frame \\ \(R\) & Int. decoding in a given slot & \(E\) & A given int. decoding is the first \\ \(L\) & A given int. decoding is the last & \(\omega\) & Offset of the next frame \\ \(M\) & Consecutive empty frames & \(\mathcal{O}\) & Set of possible offsets \\ \(O\) & Decoding with a given offset & \(D\) & A packet is dropped \\ \(\mathcal{J}_{o}(i)\) & Past generation set & \(\mathcal{C}_{\phi\phi}^{(j)}(i)\) & Future generation set \\ \(C\) & Unused future slots & \(V\) & Number of packets left in queue \\ \(\mathcal{H}_{\ell}(V(\mathbf{g}))\) & Future arrival set & \(\mathcal{F}_{\ell}(f)\) & Set of unused past slots \\ \(Y\) & Decoding delay & \(p_{\text{free}}\) & Free channel probability \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main notation used in the paper. multinomial function \(\text{Mult}(\mathbf{k};n,\mathbf{p})\), which corresponds to the probability of outcome vector \(\mathbf{k}\) being observed over \(n\) slots.