## 2 Network Model  
In this section, a downlink NOMA system with short packet communications and the effective capacity of NOMA networks are presented in the following sections. In a downlink NOMA system with short packet communications, a base station (BS) serves \(M\) users within a finite blocklength of \(n\) symbol periods, as shown in Figure 1.  
The packets of each user are stored temporarily in the memory space, i.e., an individual buffer, and then transmitted in a first-input-first-output process. As a further development, the packets of \(M\) users are superposed at the BS. To simplify the analyses, the BS and users are single antenna nodes, respectively. The wireless communication links from the BS to users are subjected to independent block Rayleigh fading. The corresponding channel coefficients are denoted as \(h_{1}\), \(\cdots\), \(h_{m}\), \(\cdots\), and \(h_{M}\), respectively. To fully exploit the advantage of NOMA with regard to orthogonal multiple access (OMA), it can be assumed that the channels from the BS to the users are distinguished largely and ordered as \(\left|h_{1}\right|^{2}\leq\cdots\leq\left|h_{m}\right|^{2}\leq\cdots\leq\left| h_{M}\right|^{2}\). It is assumed that the BS is capable of obtaining channel state information of wireless links. When detecting the \(m\)-th user's signal \(x_{m}\), the signals of the previous \(m-1\) user will interfere. Therefore, the SIC detection method is adopted. When the signals \(x_{m}\) is detected, the signals of the previous \(m-1\) user is detected and deleted. The \(m\)-th user receives not only its own signal but also the signal of the previous \(m-1\) user. Therefore, the blocklength of the \(x_{m}\) must not be shorter than the blocklength of the \(x_{m-1}\), where \(x_{m}\) is the signal related to \(U_{m}\). Based on the above explanations, the received signal at the \(m\)-th user is  
\[y_{m}=h_{m}\sum\limits_{i=1}^{M}\sqrt{a_{i}P_{s}}x_{i}+n_{m}, \tag{1}\]  
where \(a_{i}\) is a coefficient for power allocation of the \(i\)-th user (\(1\leq i\leq M\)). From the perspective of user fairness, the power coefficients of users satisfy the relationship of \(a_{1}\geq\cdots\geq a_{m}\geq\cdots\geq a_{M}\)[8; 20]. \(P_{s}\) is the transmitting power at the BS. \(n_{m}\sim\mathcal{CN}(0,N_{0})\) denotes the additive Gaussian noise where mean = 0 and variance = \(N_{0}\) at the \(m\)-th user. Using SIC technology, before decoding the \(m\)-th user's information, the \(p\)-th user's information (\(p<m<M\)) is first decoded and then deleted successfully. Hence, the received signal-to-interference-plus-noise ratio (SINR) for the \(m\)-th user can be given by the following formula:  
\[\gamma_{m}=\frac{\rho\left|h_{m}\right|^{2}a_{m}}{\rho\left|h_{m}\right|^{2} \sum\limits_{i=m+1}^{M}a_{i}+1}, \tag{2}\]  
Figure 1: Illustration of a downlink NOMA system with short packet communications. where \(\rho=\frac{R_{i}}{N_{0}}\) stands for the transmitting SNR. After the \(M\)-th user successfully detects and deletes the signals of the previous \(M-1\) users, the SNR of user \(M\) is  
\[\gamma_{M}=\rho|h_{M}|^{2}a_{M}. \tag{3}\]  
According to [PERSON]'s theory, the decoding error probability of the receiver can be ignored when the coding blocklength tends to infinity [21]. On the contrary, the error probability of receiving ends for short packet communications is non-negligible, which mainly works to achieve the goal of low latency. Under these situations, the achievable rates of short packet communications cannot be characterized by the Shannon formula at a given blocklength. As previously mentioned in [11], the transmission rate of the data packet is a function of SNR, block length, and error probability.