## 3 System Model and Problem Formulation
### System Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Description** \\ \hline \(K\) & The set of base stations \\ \(k\) & The index of base stations \\ \(W_{\text{k}}\) & Bandwidth resources owned by base station \(k\) \\ \(N\) & The number of slices \\ \(n\) & The index of slice \\ \(I\) & The set of users \\ \(i\) & The index of users \\ \(\mathcal{U}_{n}\) & The number of users with service \(n\) \\ \(SNR_{m}\) & The signal-to-noise ratio of the user connected to the MBS \\ \(SINR_{k}\) & The signal-to-interference-noise ratio of the user connected to the SBS \\ \(P_{m}\) & Transmit power of the MBS \\ \(P_{k}\) & Transmit power of the SBS \\ \(G_{i,m}\) & Downlink channel gains from the MBS \\ \(G_{i,k}\) & Downlink channel gains from the SBS \\ \(\sigma^{2}\) & The average background noise power \\ \(r_{i,m}^{n}\) & Downlink transmission rate of user \(i\) connected to MBS \\ \(r_{i,k}^{n}\) & Downlink transmission rate of user \(i\) connected to SBS \\ \(R_{i}\) & Total downlink transmission rate of user \(i\) \\ \(x_{i,k}\) & Binary variable used to indicate which BS the user \(i\) is connected to \\ \(y_{\alpha_{i}}\) & Binary variable that indicates whether the packet was successfully transmitted \\ \(|\mathcal{Q}_{u_{i}}|\) & Total number of data packets transmitted by the BS to user \(u_{i}\) \\ \(\lambda_{i}\) & A binary variable that indicates whether user \(i\) uses DC \\ \(\phi\) & The fixed cost consumed per user using DC \\ \(\varphi\) & Additional consumption of dual-connected users \\ \(P_{n}\) & Rate limit of service \(n\) \\ \(l_{n}\) & Delay limit of service \(n\) \\ \(V\) & Number of cities \\ \(V_{\min}\) & Least costly route solution \\ \(c_{l,\rho}\) & The cost of traveling from city \(l\) to city \(o\) \\ \(d_{l,\rho}\) & Used to indicate whether the traveler departs from \(V_{l}\) to \(V_{o}\) \\ \(P_{t}\) & Total cost \(P_{t}\) when passing through all cities \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and descriptions. \[r_{i,m}^{n}=B_{i,m}^{n}\log_{2}\left(1+\frac{P_{h}G_{i,k}}{\sigma^{2}}\right) \tag{3}\]  
\[r_{i,k}^{n}=B_{i,k}^{n}\log_{2}\left(1+\frac{P_{k}G_{i,k}}{\sum\limits_{j=\{1,2,\ldots,n\},j\
eq k}P_{j}G_{i,j}+\sigma^{2}}\right),\;k=\{1,2\ldots,K\}; \tag{4}\]  
thus, we have the total transmission rate of the \(i\)th user, that is,  
\[R_{i}=x_{i,m}r_{i,m}^{n}+\sum_{k=1}^{K}x_{i,k}r_{i,k}^{n}=\sum_{k=0}^{K}x_{i,k }r_{i,k}^{n} \tag{5}\]  
where the binary variable \(x_{i,k}\) is the indicator that user \(i\) is connected to BS \(k\). If user \(i\) is connected to BS \(k\), \(x_{i,k}\) is equal to 1, otherwise it is 0. \(B_{i,k}^{n}\) is the bandwidth resource obtained by user \(i\) of service \(n\) from the BS \(k\). We use the transmission success rate of data packets [11] to represent the QoE of users served by the slice \(n\) and it can be given by  
\[QoE_{n}=\frac{\sum\limits_{u_{i}\in\mathcal{U}_{u}}\sum\limits_{q_{n}\in \mathcal{Q}_{u}}Y_{q_{n}}}{\sum\limits_{u_{i}\in\mathcal{U}_{u}}|\mathcal{Q}_{ u_{i}}|}. \tag{6}\]  
where \(|\mathcal{Q}_{u_{i}}|\) is defined as the total number of data packets transmitted by the BS to user \(u_{i}\). The binary variable \(\mathrm{y}_{q_{n}}\) denotes whether the packets are successfully transmitted. For each service, the \(\mathrm{y}_{q_{n}}=1\) only when the rate limit and the delay limit are met at the same time, otherwise it is equal to 0. The rate limit and delay limit here are set according to the service level agreement (SLA) of the 5G network slicing technology. Since the more users use dual connectivity, the heavier the burden on the system, we set the cost consumed by dual-connected users, where \(\lambda_{i}\) is an indicator for dual-connected users and \(\phi\) is the fixed cost consumed per user using DC. We define \(\varphi\) as the additional consumption of dual-connected users in the system and it can be expressed by  
\[\varphi=\phi\sum\limits_{i=0}^{N}\lambda_{i} \tag{7}\]