# QoS-Aware Resource Management in 5G and 6G Cloud-Based Architectures with Priorities
## 2 Scheduler Priority Queue
### No Buffer Queue for Pre-Empted q-Type Services  
If the queue length is zero (typical examples where in specific protocol payers there is not any buffer available) then the model is simplified as shown in Figure 7. Pre-emption probability of an already existing class q service into the system equals at least the sum of products \(P_{ \begin{subarray}{c}i,j\\ i,j\end{subarray} } \begin{matrix}i+j=C\\ i\
eq C\end{matrix} \) with the transition probability \(b_{ \begin{subarray}{c}(i,j)\\ i+j=C\end{subarray} }\). The transition probability \(b_{ \begin{subarray}{c}(i,j)\\ i\
eq C\end{subarray} }\) is defined as \(Q_{(i,j)\rightarrow(i+1,j-1)}\), that is the probability that the system goes over from state \((i,j)|i+j=C\) to state \((i+1,j-1)|i+j=C\). This could happen only if the arrival time of p requests \((T_{\lambda\mu},T_{\lambda h})\) is less than the class q serving time \(\left(T_{\mu d}\right)\) and less than the class p serving time \(\left(T_{\mu n}\right)\) in state \((i,\,j)\), that is,  
\[Q_{(i,j)\rightarrow(i+1,j-1)}=P\left(min\left(\left(T_{\mu d},T_{\mu b}\right),T_{\mu b}\right)>T_{\mu n}\right) \tag{6}\]  
Following the same technique as in (5),  
\[Q_{(i,j)\rightarrow(i+1,j-1)}=b_{ \begin{subarray}{c}(i,j)\\ i\
eq C\end{subarray} } \begin{matrix}i+j=C\\ i\
eq C\end{matrix} =\frac{\lambda_{n}+\lambda_{h}}{\lambda_{n}+\lambda_{h}+i \mu_{n}+j\mu_{d}} \tag{7}\]  
Therefore, the blocking probability of the class q connection previously being served, in the case of a model without queue, is calculated to be  
\[P_{B|Q=0}=\sum_{j=1}^{n}\sum_{i=0}^{C-j}b(i,j)\cdot P_{i,j}\]  
Figure 6: Third case of transition from states (2,2) or (2,1) into state (2,0). together with the obvious restrictions  
\[\left(0\leq\sum\limits_{i=0}^{C-1}b(i,C-i)\cdot P_{i,C-i}\leq P_{B|Q=0}\right) \cup\left(\sum\limits_{i=0}^{C-1}\frac{\lambda_{n}+\lambda_{h}}{\lambda_{n}+ \lambda_{h}+i\mu_{n}+(C-i)\mu_{d}}\cdot P_{i,C-i}\leq P_{B|Q=0}\right)\]  
The blocking probability for a pre-empted class q connection in a system without queue is provided by  
\[P_{B,pre-empted|Q=0}=\sum\limits_{i=0}^{C-1}\frac{\lambda_{n}+\lambda_{h}}{ \lambda_{n}+\lambda_{h}+i\mu_{n}+(C-i)\mu_{d}}\cdot P_{i,C-i}. \tag{8}\]  
Considering these bounds, recalling Equation (4), it can be rewritten as  
\[0\leq P_{B|Q\
eq 0}\leq\sum\limits_{i=0}^{C-1}b(i,C-i+1)\cdot P_{Q+ i,n-i}\left| \begin{array}{c}if\ Q+i>n\\ Q+i=n\end{array} \right.\] \[\left.+\sum\limits_{i=1}^{C}\sum\limits_{ \begin{array}{c}q=0\\ q\leq i\end{array} }^{Q-1}b(i,C-i)\cdot\left(1-T_{yout/(i,n-i+q)}\right)P_{i,n -i+q}\right.\]  
where \(b_{(i,j)|}i+j=C\) is the maximum transition probability when the system has all channels occupied, whereas in all others states the case is \(b_{(i,j)|}i+j\leq C\leq\sum\limits_{ \begin{array}{c}(i,j)|\\ i\
eq C\end{array} }^{b}i+j=C\). So, \(i\
eq C\)  
Figure 7: The state diagram (two-dimensional Markov chain) for GSM/GPRs traffic without queue. the blocking probability for a pre-empted packet in a system with queue implementation is calculated as  
\[P_{B,pre-empted|Q\
eq 0}=\sum\limits_{i=0}^{C-1}b(i,C-i+1)\cdot P_{Q+i,n-i} \begin{array}{c}if\begin{array}{c}Q+i>n^{+}\\ Q+i=n\end{array}\\ \sum\limits_{i=1}^{C}\sum\limits_{\begin{subarray}{c}q=0\\ q\leq i\end{subarray}}^{Q-1}b(i,C-i)\cdot\Big{(}1-T_{yout/(i,n-i+q)}\Big{)}P_{ i,n-i+q}\end{array} \]  
or  
\[P_{B,pre-empted|Q\
eq 0}=\sum\limits_{i=0}^{C-1}\frac{\lambda_{n}+\lambda_{ h}}{\lambda_{n}+\lambda_{h}+\mu_{n}+(C-i+1)\mu_{d}}\cdot P_{Q+i,n-i} \begin{array}{c}if\begin{array}{c}Q+i>n^{+}\\ Q+i=n\end{array}\\ +\sum\limits_{i=1}^{C}\sum\limits_{\begin{subarray}{c}q=0\\ q\leq i\end{subarray}}^{Q-1}\frac{\lambda_{n}+\lambda_{h}}{\lambda_{n}+\mu_{n}+ (C-i)\mu_{d}}\cdot\Big{(}1-T_{yout/(i,m-i+q)}\Big{)}P_{i,n-i+q}\end{array}  \tag{9}\]  
The waiting time of a queued pre-empted packet is defined as the time difference between the period of time an arbitrarily selected waiting packet spends in queue to the time it successfully accesses a free channel.