## 4 Time-Frequency-Space Resource Allocation Method in Sharing Network
### Mathematical Modeling
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Detailed Explanation** \\ \hline \(C\) & Channel capacity. \\ \(B\) & Channel bandwidth. \\ \(S\) & Average signal power. \\ \(N\) & Average noise power. \\ \hline \(b\) & Index for frequency-domain resource. \\ \(t\) & Index for time-domain resource. \\ \(k\) & Index for user. \\ \(i\) & Index for carrier. \\ \(n\) & Index for beam. \\ \hline _AllocFreq_ & The number of copies of the allocated bandwidth resource. \\ _SchTime_ & Total scheduling duration. \\ _UserNum_ & The total number of scheduled users. \\ _CarrierNum_ & The number of carriers configured on the network side. \\ _Beam1D_ & The ID of the beams. \\ \hline \(BW_{b}\left(t\right)\) & \(b\)-th spectrum resource allocated at time \(t\). \\ \(BW_{k,b}\left(t\right)\) & \(b\)-th spectrum resource allocated for user \(k\) at time \(t\). \\ \(BW_{k,j,n,b}\left(t\right)\) & \(b\)-th spectrum resource allocated for user \(k\) on carrier \(i\) in beam \(n\) at time \(t\). \\ \(\alpha_{k,b}(t)\in\left\{0,1\right\}\) & Whether the \(b\)-th spectrum resources are scheduled at time \(t\) by user \(k\). \\ \(\alpha_{k,j,n,b}(t)\in\left\{0,1\right\}\) & Whether the \(b\)-th spectrum resources in beam \(n\) on carrier \(i\) are scheduled at time \(t\) by user \(k\). \\ \(S_{k,b}\) & Signal strength of user \(k\) on the \(b\)-th spectrum resource at time \(t\). \\ \(I_{k,t,b}\) & Interference strength of user \(k\) on the \(b\)-th spectrum resource at time \(t\). \\ \(N_{k,t,b}\) & Noise strength of user \(k\) on the \(b\)-th spectrum resource at time \(t\). \\ \(S_{k,i,t,b,n}\) & Signal strength of user \(k\) on carrier \(i\) and \(b\)-th spectrum resource in beam \(n\) at time \(t\). \\ \(I_{k,j,t,b,n}\) & Interference strength of user \(k\) on carrier \(i\) and \(b\)-th spectrum resource in beam \(n\) at time \(t\). \\ \(N_{k,i,t,b,n}\) & Noise strength of user \(k\) on carrier \(i\) and \(b\)-th spectrum resource in beam \(n\) at time \(t\). \\ \(S_{k,i,t,b,n}\) & Spectral efficiency of user \(k\) on \(b\)-th spectrum resource and \(n\)-th beam of carrier \(i\) at time \(t\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nomenclature. where \(S/N\) is equivalent to \(\frac{S_{k,k,l}}{I_{k,l,k}+N_{k,l,k}}\), we can obtain  
\[C=\underset{\alpha_{k,k}(t),BW_{k,b}(t)}{argmax}\sum_{t=0.5}^{SchTime}\sum_{k=1}^ {UserNum}\sum_{b=1}^{AllocFreq}\left(\alpha_{k,b}(t)\bullet BW_{k,b}(t)\bullet log \left(1+\frac{S_{k,t,b}}{I_{k,t,b}+N_{k,t,b}}\right)\right) \tag{4}\]  
The implications of each variable in the formula are as follows:  
* \(\alpha_{k,b}(t)\in\{0,1\}\): whether the \(b\)-th spectrum resources are scheduled at time \(t\) by user \(k\). * \(BW_{k,b}\left(t\right)\): the \(b\)-th spectrum resource allocated to user \(k\) at time \(t\). * \(S_{k,t,b}\), \(I_{k,t,b}\), \(N_{k,t,b}\) : the signal strength, interference strength, and noise strength of user \(k\) on the \(b\)-th spectrum resource at time \(t\), respectively. * _UserNum_: the total number of scheduled users (integer). The following constraints exist in the formula:  
* \(\alpha_{k,b}(t)\in\{0,1\}\), map user \(k\) whether schedule the \(b\)-th spectrum resource at time \(t\) or not to two states: \(0\) and \(1\). * \(\sum\limits_{k=1}^{UserNum}\sum\limits_{b=1}^{AllocFreq}\left(BW_{k,b}\left(t \right)\right)\leq Total\)_spectrum resource_, the sum of spectrum resources allocated at a time cannot exceed the total spectrum resources. The above formula aims to optimize the allocation of time, frequency, and spatial resources to achieve the highest collective network experience rate for all users (\(UserNum\)) within a given scheduling duration (\(SchTime\)) and on a specific allocated frequency domain resource (\(AllocFreq\)). This process results in determining values for \(\alpha_{k}(t)\in\{0,1\}\) and \(BW_{k,b}\left(t\right)\). Specifically, for each user, whether the spectrum resource can be scheduled at a particular time \(t\), and if so, how many frequency bandwidth resources can be allocated.