# Cooperative Admission Control with Network Coding in 5G Underlying D2D-Satellite Communication
## Problem Formulation  
### Joint Admission Control-ATSPSN  
The formulated problem consists of multiple constraints and multiple objective functions. The objective function defined as  
\[\Delta(R,P^{l}_{d},P^{l}_{c},P^{l}_{s})=\Delta_{s}(R)\sum_{l\in R}\Delta_{T}(R,P^{l}_{d},P^{l}_{c},P^{l}_{s}). \tag{10}\]  
where \(\Delta_{s}=\frac{|X|}{|R|}\) and \(\Delta_{T}(R,P^{l}_{d},P^{l}_{c},P^{l}_{s})=(1+v^{l})*(1-v^{l})C^{l}_{X_{d}}+1/ 2*(1+v^{l})*(v^{l})C^{l}_{X_{c}}+(1/2)*v^{l}*(v^{l}-1)C^{l}_{X_{s}}\). The utility function in (10) ensures that the user is selected only in one mode, i.e., SAT, D2D, or cellular. The utility function is either a sum-rate maximization problem or a power minimization problem. We will use the first approach as it provides more flexibility to determine the optimum power from the complete range of power available with users and sensors. The objective function, along with the constraints equation, is given as:  
\[max_{(R,P^{l}_{d},P^{l}_{c},P^{l}_{s})}\Delta(R,P^{l}_{d},P^{l}_{c},P^{l}_{s}) \tag{11}\]  
**subject to**  
**C1**: \(\Delta_{T}(R,P^{l}_{d},P^{l}_{c},P^{l}_{s})\geq C^{min}_{l}\), \(\lor\)\(leR\)
**C2**: \(P^{l}_{l_{d}}\leq r^{\eta}*(1+v^{l})*(1-v^{l})P^{min}_{l_{d}}\), \(\lor\)\(leR\)
**C3**: \(P^{l}_{l_{s}}\leq r^{\eta}*v^{l}*(v^{l}-1)P^{min}_{l_{s}}\), \(\lor\)\(leR\)
**C4**: \(P^{u}_{l}\leq(1+v^{l})*(v^{l})P^{max}_{l,u}\), \(\lor\)\(leR\)
**C5**: \(\sum_{l\in R}*(1+v^{l})(1-v^{l})P^{l}_{l(l,u)}\leq p^{max}_{bs}\)
**C6**: \(\sum_{l\in R}v^{l}(v^{l}-1)P^{l}_{l(l,u,SAT)}\leq p^{max}_{SAT}\)
**C7**: \(P^{l}_{l_{d}}\geq 0,P^{l}_{l}\geq 0,P^{l}_{l_{d}}\geq 0,P^{l}_{l_{s}}\geq 0, \lor\)\(leR\)
**C8**: \(C^{l}_{l_{s(l)}}-C^{l}_{l_{d(l)}}\geq 0,\lor\)\(leR\)
**C9**: \(X_{d}\cap X_{c}\cap X_{s}=R\). The objective function in (11) defines the optimum power allocation and user admission along with the appropriate mode assignment. Constraint C1 defines that the capacity of the device selected is greater than the minimum rate requirement. C2 to C6 defines the power boundaries for the algorithm. C2 and C3 ensures that power allocated to mobile user/ sensors less than minimum power \((P_{ls}^{min}/P_{ld}^{min})\) beyond radius (\(r\)) to avoid interference in other cell. C4 ensures that the terrestrial users' uplink power should always be less than or equal to the devices' maximum operating power, i.e., \(P_{Lut}^{max}\). The summation of all terrestrial (\(P_{l_{(d,ls)}}\)) and satellite (\(P_{l_{(d,SAT)}}^{l}\)) downlinks power should be less than maximum power of base station and satellite, respectively (refer to C5 and C6) i.e., \(P_{bs}^{max}\) and \(P_{SAT}^{max}\). The power allocated cannot be zero, as ensured by C7. The C8 ensures to keep penalizing cost at low. If the capacity of link i in D2D mode is greater than or equal to SAT mode, D2D mode is selected. Terrestrial-D2D communication framework is the primary means of communication, and terrestrial-SAT is a secondary means of communication to enhance the network's capacity and user admission. C9 ensures that users are selected in only one mode at one time. The formulated problem is mix integer non-linear problem (MINLP). User selection is binary, while power and mode selection is an integer variable. The problem is NP-hard and relaxed to the linear problem by using outer approximation. Convex linear problem is formulated by fixing integer to upper bounds and solved using outer approximation. GAMS Branch-cut uses search space, and NOMAD uses refined directions to decrease the processing time to find a global optimum solution. By introducing a new variable \(z_{k}\), the objective function in (11) in \(j\)th iteration, mathematically represents as:  
\[max_{(z_{k},R,P_{d}^{l},P_{c}^{l},P_{s}^{l})}\Delta(R,P_{d}^{l},P_{c}^{l},P_{s} ^{l}) \tag{12}\]  
subject to C1 to C9 (11). C10; \(C_{l}^{u}\geqslant z_{k}\lor leR\)  
C11; \(C_{l}^{d}\geqslant z_{k}\lor leR\)  
C13; \(C_{l}^{s}\geqslant z_{k}\lor leR\).