# Distributed Resources Allocation Method for Space-Ground Integrated Mobile Communication System
## 3 System Model
### Problem Formulation  
As is mentioned above, to relieve the core network overhead, we shift some of the work of allocation from the original core network controller to the mobile station side, that is, the mobile station will first determine which physical slice (satellite or cellular network) is the most suitable for itself. Considering that slices are different in the latency and the channel atmosphere, we introduce a parameter C to describe the cost of certain slices to help the MS make the decision. \[\mathbf{C}=[\mathbf{Cost}]=G\big{(}\mathbf{T},\ \mathbf{R}^{\prime},\mathbf{A}(t), \mathbf{P}_{kb}(t)\ \big{)} \tag{1}\]  
\[=\lambda_{1}\mathbf{T}+\lambda_{2}\mathbf{R}^{\prime}+\lambda_{3}\mathbf{A}(t) +\lambda_{4}\mathbf{P}_{kb}(t) \tag{2}\]  
where \(\mathbf{R}^{\prime}\) denotes the actual allocated data rate of users  
\[\mathbf{R}^{\prime}=\mathbf{R}\mathbf{G}_{\mathbf{e}}(\mathbf{R},\mathbf{T})= \mathbf{R}\frac{\mathbf{G}(\mathbf{R},\mathbf{T})}{100} \tag{3}\]  
among which, \(G_{ij}(R,T)\in[0,100]\) is an urgency factor to describe how much content in the package is valuable to be given the first priority, which has been fully investigated in [43; 44; 45]. The resources allocated for user access should be as small as possible under the premise of meeting its priority and data volume, so as to improve the data utilization efficiency of the system. Hence, the allocation procedure can be transformed to an optimization problem. \[\underset{\mathbf{A}(t),\mathbf{P}_{kb}(t),\mathbf{R}^{\prime}}{ \mathbf{minimize}}(\mathbf{C})=\underset{\mathbf{A}(t)}{\mathbf{minimize}} \big{(}G\big{(}\mathbf{T},\mathbf{R}^{\prime},\mathbf{A}(t),\mathbf{P}_{kb}(t) \big{)}\big{)} \tag{4}\] \[\text{s.t.}\ A_{ij}(t)\in[0,1],1\leq i\leq 3,1\leq j\leq N\] (5) \[\sum_{i=1}^{3}\sum_{j=1}^{N}A_{ij}\leq C_{ch} \tag{6}\]  
Here, in our research, we propose three types of slices as mentioned before, thus, \(T=Z^{+},[1,3]\) which is a \(1\times N\) array where N indicates the number of total requests. Parameter \(A=R^{+},[0,1]\) is a \(3\times N\) array aimed to describe the state of every slice and  
Figure 4: Distributed network decision-making process flow. RB (used, crowed, etc.) which is positively correlated with channel allocation cost in the formula. P indicates the density of MS around a particular mobile devices. Sorting through the optimization problems we just listed, we can get the following formula. \[\textbf{P1}:\underset{\textbf{A}(t),\textbf{P}_{kb}(t),\textbf{R}^{ \prime}}{\textbf{minimize}} \left(\lambda_{1}\textbf{T}+\lambda_{2}\textbf{R}^{\prime}+ \lambda_{3}\textbf{A}(t)+\lambda_{4}\textbf{P}_{kb}(t)\right) \tag{7}\] \[s.t. \sum_{i=1}^{3}\sum_{j=1}^{N}A_{ij}\leq C_{ch}\] (8) \[T_{ij}\in\{1,2,3\},1\leq i\leq 3,\ 1\leq j\leq N\] (9) \[=\lambda_{1}\mathbf{T}+\lambda_{2}\mathbf{R}^{\prime}+\lambda_{3}\mathbf{A}(t) +\lambda_{4}\mathbf{P}_{kb}(t) \tag{2}\]0  
The problem is to find the best couple \(\textbf{A},\textbf{P}\) and \(\textbf{R}^{\prime}\) to minimize the cost function. Formula (7) restricts the maximum capacity of the logical channel, while formula (10) defines the allocation state of the slice. Corresponding to the channel slicing problem, the goal of each MS when asking for access to the network is to find the best physical network to switch in reflected by **A**, the transmitting power **P** and proposed data rate \(\textbf{R}^{\prime}\). However, the optimization problem listed here is a NP-hard problem which cannot be worked out by the MS quickly. Thus, we divide the whole problem into two sub-optimization problems: allocated data rate optimization, **P2**, and allocated physical network optimization, **P3**. The first optimization problem **P2** is to find the optimal transmit power and the number of resource blocks to meet the needs of users under the assumption of a physical distribution network. In the previous section, we discussed the goal for the MS in the resource allocation problem **P1** as a minimization problem, where one of the goals is to minimize \(R^{\prime}\).