# Cooperative Admission Control with Network Coding in 5G Underlying D2D-Satellite Communication
## Appendix D Theorem IV  
Proof. : First, the sets \(X_{c}\), \(X_{d}\), and \(X_{s}\) are constructed using the Algorithm 2 for 1 st transmission slot. Then \(T_{tx,c}\),\(T_{tx,s}\) and \(T_{tx,d}\) are the average time packet completion for the source (base station or satellite) to transmit the packets from \(X_{c}\), \(X_{d}\), and \(X_{s}\). respectively, while \(T_{s,s}\) is average packet time for packet completion via D2D in the sets of \(X_{s}\) and \(X_{d}\), as shown  
\[T_{tx,c}=\frac{|Xc|}{1-\max_{n\in R}\theta_{n}}\] (A4)  
\[T_{tx,d}=\sum_{p\in X_{d}}\frac{1}{1-\max_{n|(W_{l}[n]\
eq 0}\theta_{n}}\] (A5)  
\[T_{s,s}=\sum_{p\in X_{s}}\Bigg{(}\frac{1}{1-\max_{n|(W_{l}[n]\
eq W_{l}[x]} \tau_{x,n}}+\frac{1}{1-\max_{n|(W_{l}[n]=W_{l}[x]}\tau_{x,n}}\Bigg{)}.\] (A6)Here the x transmits the partial packet from the \(X_{s}\) set of the packets. Now x is elected in such a way that it maximizes the average receivers in the network as per Algorithm 4. \(x^{\prime}\) is the device that transmits the remaining packet p, which is an uncoded packet from vector \(W_{l}[x]\). \[T_{s,d}=\sum_{p\in X_{d}}\frac{1}{1-\max_{n|(W_{l}[n]\
eq 0}\tau_{x,n}}. \tag{10}\]  
where x transmits packet \(peX_{d}\), and x is selected such a way that it maximizes the number of average receivers such that \(x^{\prime}=\arg\max\limits_{i|(W_{l}[i]\
eq W_{l}[x])}\sum\limits_{n|(W_{l}[n] =W_{l}[x])}(1-\tau_{i,n})\). Initially, the set of \(X_{c}\), \(X_{d}\), and \(X_{s}\) is considered for the transmission of the linear combination of the packets in 1 st timeslot. After that, the number of time slots required from the transmission such that \(|D_{n}|=0\) defines the upper bound on time (T). The total number of the timeslots required for the transmission to recover the data decreases by updating the \(X_{c}\), \(X_{d}\), and \(X_{s}\) after each transmission slot as compared to fixed \(X_{c}\), \(X_{d}\), and \(X_{s}\) after 1 st timeslot. First, the packet completion from the source/satellite determined for \(X_{c}\), \(X_{d}\), and \(X_{s}\), sets which is denoted by \(T_{tx,c}\),\(T_{tx,s}\) and \(T_{tx,d}\) respectively. Similarly, the average time via the D2D link for \(X_{d}\), and \(X_{s}\)is given as \(T_{s,s}\) and \(T_{s,d}\) respectively. The average transmission required from the source/satellite is \((1-\vartheta_{n})\) where \(\vartheta_{n}\) is lost from the source in case of the base station and satellite in case all the packets available with the satellite after first stage. It takes an average of \(\frac{1}{1-\max\limits_{n\in R}\vartheta_{n}}\) transmission slots to transfer the packets partially. The other part of the packets is taken \(\frac{1}{1-\max\limits_{n|(W_{l}[i]=W_{l}[i]]}\tau_{x,n}}\) timeslots to transmits these residual packets. Let's consider the following 03 cases. * \(T_{tx,c}\geq(T_{s,s}+T_{s,d})\): The base station/ satellite transmits the packet in \(X_{c}\) and takes \(T_{tx,c}\) the average time for the transmission of the packets. While packets in \(X_{t}=X_{d}\cup X_{s}\) are transmitted by the local devices via D2D links and take around \((T_{s,s}+T_{s,d})\) time for transmission. After this, the average of the \(|X_{c}|-|X_{c}|(T_{s,s}+T_{s,d})/\,T_{tx,c}\ =|X_{c}|(T_{tx}-T_{s,s}-T_{s,d})/\,T_{tx,c}\) remaining from \(X_{c}\). The source/ SAT requires an average transmission slot as \((|X_{c}|(T_{tx,c}-T_{s,s}-T_{s,d})/\,T_{tx,c})\ \times T_{tx,c}/|X_{c}|\ =\,T_{tx,c}-T_{s,s}-T_{s,d}\). The average packet completion time becomes as \(T_{a}=T_{tx,c}\). * \(T_{tx,c}\leq(T_{s,s}+T_{s,d})\): After transmission of the packet from the source and via D2D link, the average packets \((|X_{d}|+|X_{s}|-(T_{tx,c}(|X_{s}|+X_{d})/\,T_{s,s}+T_{s,d})\ =(T_{s,s}+T_{s,d}-T_{tx,c})(|X_{s}|+X_{d})/\,T_{s,s}+T_{s,d}\) left in \(X_{n}\).