# Spectrum Slicing for Multiple Access Channels with Heterogeneous Services
## 3 System Model
### Channel Model  
We consider a quasi-static block fading channel, where the received signal by the BS at any slot \(t\) is given as  
\[y_{t}=\sum_{u\in\mathcal{U}}h_{u,t}\,a_{u,t}\,x_{u,t}+z_{t}, \tag{3}\]  
Figure 2: Frame structure for the considered access schemes with \(K=4\), \(N=6\), and \(U=4\).  
where \(h_{u,t}\) is the random fading coefficient for user \(u\) at slot \(t\) and \(z_{t}\) is an Additive White Gaussian Noise (AWGN) noise with variance \(\sigma^{2}\). The random variable \(a_{u,t}\in\{0,1\}\) models user's activity, being equal to \(1\) if the user is active in that slot and \(0\) otherwise. A user is active only if it is allowed to transmit; i.e., if \(u\in\mathcal{A}_{t}\), and if its packet queue \(q_{u,t}\) is not empty:  
\[a_{u,t}=I(u\in\mathcal{A}_{t})I(q_{u,t}>0), \tag{4}\]  
where \(I(\times)\) is the indicator function, equal to \(1\) if the condition is true and \(0\) otherwise. Let \(P_{u}\) be the fixed transmission power of user \(u\), which can be different for each user. The Signal to Noise Ratio (SNR) of user \(u\) at time slot \(t\) is given by:  
\[\text{SNR}(u,t)=\frac{|h_{u,t}|^{2}P_{u}\,a_{u,t}}{|z_{t}|^{2}}, \tag{5}\]  
whereas the Signal to Interference plus Noise Ratio (SINR) of user \(u\) at time slot \(t\) is given by  
\[\text{SINR}(u,t)=\frac{|h_{u,t}|^{2}P_{u}\,a_{u,t}}{|z_{t}|^{2}+\sum_{v\in \mathcal{U}\setminus\{u\}}|h_{v,t}|^{2}P_{v}\,a_{v,t}}, \tag{6}\]  
where \(\mathcal{U}\setminus u\) is the set of users except user \(u\). We can also simply divide the SINR by the noise power \(|z_{t}|^{2}\), giving  
\[\text{SINR}(u,t)=\frac{\text{SNR}(u,t)}{1+\sum_{v\in\mathcal{U}\setminus\{u \}}\text{SNR}(v,t)}. \tag{7}\]  
Hence, the SINR is equal to the SNR in the absence of interference. Next, we define \(\gamma\) as the threshold in the SNR to decode a packet. That is, \(\gamma\) defines the erasure probability of a binary erasure channel (BEC) as  
\[\varepsilon_{u}=\text{Pr}[\text{SNR}(u,t)<\gamma]\qquad\forall t,u:a_{u,t}=1. \tag{8}\]  
Further, we consider a simple collision model, so that packets cannot be decoded in the presence of interference (i.e., collisions). Hence, a packet from user \(u\) can be decoded, with probability \((1-\varepsilon_{u})\) if and only if \(\text{SNR}(u,t)=\text{SINR}(u,t)\). This model neither allows for capture, nor for potential subsequent application SIC within slots containing more than one transmission (i.e., intra-collision SIC), representing the worst-case scenario for schemes that rely on MPR, such as power-domain NOMA. Instead, SIC can be only performed after decoding the broadband user, regeneration of all its \(N\) coded packets, and removing them from the slots that also contain transmissions from the intermittent users (i.e., extra-collision SIC). In slots without a collision, we assume a constant erasure probability for each user, denoted as \(\varepsilon_{B}\) for the broadband user and \(\varepsilon_{I}\) for the intermittent user. Our assumption is that the erasure probability after the interference is canceled is the same as for a free channel, which is a simplification. However, the use of parity checks on all the packets in a frame means that the probability of erroneous packet decodings is very low, and modeling the precise performance of SIC schemes is beyond the scope of this paper, and this is a common assumption in the coded slotted ALOHA literature, which assumes a similar setting [35]. The model provides a general view on the lower bound on performance of the OMA and NOMA schemes that is independent of the underlying channel model.