# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 4 Orthogonal Multiple Access
### Latency-Oriented System  
We now examine the relevant KPIs for the latency-oriented case. In this case, intermittent users maintain a queue of up to \(Q\geq 1\) packets, discarding the oldest one when a new packet arrives and the queue is already full. As for the PAoI case, we consider the grouping system, in which the \(U_{I}\) intermittent users are placed in \(G\) groups. The throughput of the broadband user is the same as in the PAoI-oriented system, given by (14). We now focus on intermittent user \(u\): the state of its queue is represented by a Markov chain, whose discrete time instants represent the time just after each slot allocated to it. In the following, we will refer to any slot allocated to the considered user \(u\) as an allocated slot. The elements of the state transition probability matrix \(\mathbf{P}^{(Q)}\) are given by  
\[P_{ij}^{(Q)}= \begin{cases}0,&\text{if }j<i-1;\\ \mathrm{Bin}(j-i+1;GT_{\mathrm{int}},\alpha),&\text{if }i-1\leq j<Q-1;\\ \sum_{k=Q-i+1}^{GT_{\mathrm{int}}}\mathrm{Bin}(k;GT_{\mathrm{int}},\alpha),& \text{if }j=Q-1.\end{cases}  \tag{21}\]  
Using basic [PERSON] theory, the steady-state distribution \(\boldsymbol{\pi}^{(0)}\) is derived as the left-eigenvector of \(P^{(Q)}\) with eigenvalue \(1\), normalized to sum to \(1\). \[ \begin{cases}&\boldsymbol{\pi}^{(0)}(\mathbf{I}-\mathbf{P}^{(Q)})=0;\\ &\sum_{q=0}^{Q}\pi_{q}^{(0)}=1.\end{cases}  \tag{22}\]  
We can now consider the slots between two allocated slots by deriving the steady-state distribution \(n\) slots after the last allocated slot, which we denote as \(\boldsymbol{\pi}^{(n)}\). \[\pi_{i}^{(n)}= \begin{cases}\sum_{j=0}^{i}\pi_{i}^{(0)}\mathrm{Bin}(i-j;n\alpha ),&\text{if }i<Q;\\ \sum_{j=0}^{Q}\sum_{k=Q-j}^{n}\pi_{j}^{(0)}\mathrm{Bin}(k;n,\alpha),&\text{if }i=Q.\end{cases}  \tag{23}\]  
At each allocated slot, the oldest packet in the queue is transmitted. If a new packet is generated when the queue is already full, the oldest packet is dropped from the buffer. Consider a specific packet generated in the \(n\)-th slot after an allocated one: if it finds \(q\) packets in the queue when it is generated, it will be transmitted at the \(q+1\)-th allocated slot after it is generated, unless some packets ahead of it are dropped due to new arrivals. We can then define a generation vector of length \(\ell\), whose \(i\)-th element contains the number of packets generated in the slots between the \(i-1\)-th and \(i\)-th allocated slots after the generation of the considered packet. The first element of the vector contains the number of packets generated between the considered packet's generation and the first allocated slot after it. We then define the set \(\mathcal{G}_{\ell}^{(n)}\), which contains all the generation vectors of length \(\ell\) for a packet generated in the \(n\)-th slot after the last one allocated:  
\[\mathcal{G}_{\ell}^{(n)}=\{0,\ldots,GT_{\mathrm{int}}-n\}\times\{0,\ldots,GT_ {\mathrm{int}}\}^{\ell-1}. \tag{24}\]The probability of each generation vector in the set is then given by:  
\[p_{\text{gen}}(\mathbf{g};\ell,n)=\text{Bin}(g_{1};GT_{\text{int}}-n,\alpha) \prod_{i=2}^{\ell}\text{Bin}(g_{i};GT_{\text{int}},\alpha). \tag{25}\]  
The considered packet is then transmitted by the \(\ell\)-th allocated slot after its generation if \(q+1-\ell\) packets ahead of it are either dropped or transmitted at that point. For a given generation vector \(\mathbf{g}\in\mathcal{G}_{\ell}^{(n)}\), we then formulate condition \(\psi_{\ell}^{(\mathbf{g},q)}\):  
\[\psi_{\ell}^{(\mathbf{g},q)}=\delta\left(\sum_{i=1}^{\ell}\left[q+1-Q+\sum_{j =1}^{i}g_{j}\right]^{+}+\ell-(q+1)\right), \tag{26}\]  
where \(\delta(x)\) is the delta function, which is equal to 1 if \(x=0\) and 0 otherwise, and \([x]^{+}=\max(x,0)\). The condition naturally excludes the cases in which the considered packet is dropped, i.e., when a new packet arrives and finds a full queue, with the considered packet being first in line.