## 5 Performance Analysis
### Performance Measurements  
Using the derived steady-state probability matrix \(\pi\), we can define and derive the performance measures that can be used to compare and study the QoS metrics for both eMBB and mMTC services, as they have a lower priority than uRLLC. The queuing analysis methods presented in [26; 27] were followed. These performance measures include the forced termination probability (PF), the service completion rate (SCR), and the average queuing delay (D). In addition, the system resource utilization for both eMBB and mMTC services is measured. These performance measures are derived and defined as follows:  
1. **eMBB PF**: the forced termination probability of the eMBB is the probability that an uRLLC request with higher priority arrives and the in-service eMBB is forced to terminate its in-service connection before its normal end. The in-service eMBB is forced to release its occupied BBUs and assign them to the arriving uRLLC when the system resources are fully utilized, there are no available BUU resources, and the in-service mMTC connections are zero. The released resources are then given to the newly arrived uRLLC. In this case, the forced termination probability can be derived as the average rate of terminated eMBB services divided by the admitted rate of eMBB services. Therefore, we can denote PF\({}_{1}\) as the forced termination probability of the requests of type eMBB, which can be derived as follows: \[PF_{1}=\frac{\text{the average rate of terminated eMBB}}{\text{the eMBB admitted rate}}\] (6a)Using the derived balance equations, the average number of eMBBs forced to terminate is given by:  
\[\sum_{ \begin{subarray}{c}\forall S\\ i+j+k=Candk=0\end{subarray} }\frac{\lambda_{1}j}{C-j}\pi_{i,j,k,m,n} \tag{6b}\]  
The total eMBB admitted rate is designed as \(\lambda_{2}(1-PB_{2})\). Hence, the PF\({}_{1}\) can be rewritten as:  
\[PF_{1}=\sum_{ \begin{subarray}{c}\forall S\\ i+j+k=Candk=0\end{subarray} }\frac{\lambda_{1}j}{(C-j)\lambda_{2}(1-PB_{2})} \tag{7}\]
2. **mMTC PF**: This probability can be derived following the procedures used for eMBB. The mMTC is forced to terminate its connection by the arrival of a higher-priority request (in this case, uRLLC and eMBB) when all BBU resources are busy and its queue is full. Similar to (7), the forced termination probability for mMTC is defined as: \[PF_{2}=\sum_{ \begin{subarray}{c}\forall S\\ i+j+k=Candn=N\end{subarray} }\frac{\lambda_{1}k}{(C-j)\lambda_{3}(1-PB_{3})}\] (8)
3. The service completion rate (**SCR**): The rate of service completion of both eMBB and mMTC is measured as the total rate of service completion of each type by unit time. Using the derived balance equations, we can derive this rate. Let us assume that SCR\({}_{1}\) and SCR\({}_{2}\) are the service completion rates for eMBB and mMTC, respectively. The service completion rate of eMBB and mMTC services is derived as shown in (9) and (10), respectively. \[SCR_{1}=\sum_{\forall S}j\mu_{2}\pi_{i,j,k,m,n}\] (9)  
This equation represents the service completion rate for eMBB. This rate is computed by adding the service completion rates (departure rate) of all states that represent an eMBB departure, where j represents the number of in-service eMBB, \(\mu_{2}\) the service rate of eMBB, and \(\pi_{i,j,k,m,n}\) the probability that we are in state (\(i\),\(j\),\(k\),\(m\),\(n\)). Using the steady state characteristics, we can find the competition rate by adding all the states that have an eMBB departure. \[SCR_{2}=\sum_{\forall S}k\mu_{3}\pi_{i,j,k,m,n} \tag{10}\]
4. C-RAN resource utilization (**U**) by both eMBB and mMTC services: this metric indicates the efficiency of the proposed system in terms of increasing BBU utilization by allowing both eMBB and mMTC services to be queued and reattempted. Using the system state space S, we can notice that at each state space (\(i\),\(j\),\(k\),\(m\), and \(n\)), the system has C resource units (BBUs), and (\(i+j+k\)) BBUs are occupied by either uRLLC, eMBB, or mMTC services.