## 2 System Model
### Communication Model  
As depicted in Figure 1, we consider a 5G TDD system that is enabled with network slicing and SUL transmission. The SUL band has a larger coverage than the NR band [5]. Network slicing enables the BS to provide services with different QoS requirements, and it maintains isolation between services. The whole bandwidth is divided into two parts, i.e., NR band with bandwidth \(W^{nr}\) for both the downlink and uplink transmission in the TDD pattern, and the SUL band with bandwidth \(W^{sul}\) only for SUL transmission. UEs are distributed in the coverage of the BS randomly. Let \(\mathcal{M}=\{1,2,\cdots,m,\cdots,M\}\) denote the slice set, and \(\mathcal{N}_{m}\) denote the set of UEs belonging to slice \(m\). It is assumed that each UE belongs to only one slice, that is, one UE uses only one service. UEs in different slices with different QoS metrics are marked with different colors in Figure 1. This paper focuses on a scenario where each slice has different downlink and uplink rate requirements. Key notations that are used in this paper are presented in Table 1. To reasonably allocate the resources of the SUL band, we first define a SUL admission identifier \(x\), based on the reference signal received power(RSRP), as below:  
\[x= \begin{cases}nr&RSRP\geq\delta\\ sul&RSRP<\delta\end{cases}  \tag{1}\]  
where \(\delta\) is the RSRP threshold to decide the SUL band admission. In other words, UEs with \(x=sil\) are classified as edge UEs of the cell, using the SUL band for uplink transmission. Obviously, the downlink communication can only use the NR band for transmission. Note that UE may be scheduled to transmit either on the SUL or on NR-UL, but not on both at the same time [5]. Furthermore, the switching time between the NR UL carrier and the SUL carrier is 0 is [14]. We denote \(P_{m,n}^{dl,x}\) and \(P_{m,n}^{ul,x}\) to represent the transmission power in downlink and uplink, respectively. \(P_{m,n}^{dl,x}\) and \(P_{m,n}^{ul,x}\) represent the downlink and uplink path loss between UE \(n\) of slice \(m\) and the BS, respectively. For the sake of simplicity, the communication direction set is denoted as \(\mathcal{D}=\{d|d=dl\ or\ d=ul\}\), where \(d=dl\) and \(d=ul\) represent the downlink \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(W^{nr}\) & The bandwidth of the NR band. \\ \(W^{sul}\) & The bandwidth of the SUL band. \\ \(x\) & The identifier for SUL admission. \\ \(\mathcal{M}\) & The set of slices provided by the BS. \\ \(\mathcal{N}_{m}\) & The set of UE belonging to slice \(m\). \\ \(\gamma_{m,n}^{dl}\) & The SNR of UE \(n\) of slice \(m\) in direction \(d\). \\ \(r_{m,n}^{dl}\) & The achievable rate of UE \(n\) under \(\gamma_{m,n}^{dl}\) on one RB. \\ \(R_{m,n}^{dl}\) & The achievable rate of UE \(n\) in the period \(T\). \\ \(\delta\) & The RSRP threshold to decide SUL band admission \\ \(\lambda\) & TDD pattern \\ \(\tau\) & The duration of a TDD pattern \\ \(T\) & The period where \(\lambda\) remains the same \\ \(u_{m}^{dl}\) & The bandwidth allocated to slice \(m\) in direction \(d\). \\ \(\eta_{m,n^{r}}^{nl^{r}}\) & The load of slice \(m\) in direction \(d\) of NR band \\ \(\eta_{sul}^{sul}\) & The load of SUL band \\ \(d_{m,n}^{dl}\) & The UE-level QoS satisfaction rate of UE \(n\) of slice \(m\) in direction \(d\). \\ \(q_{m}^{dl}\) & The slice-level QoS satisfaction rate of slice \(m\) in direction \(d\). \\ \(q\) & The BS-level QoS satisfaction rate. \\ \(\zeta\) & The variance of slice load \\ \(\alpha_{m}^{dl}\) & The weight of slice \(m\) in direction \(d\) \\ \(\beta\) & The weight of \(\zeta\) to make the trade-off with \(q\) \\ \(R_{m}^{dl}\) & The throughput requirement of slice \(m\) in direction \(d\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of main notations. and uplink, respectively.