## 5 Performance Analysis
### Analytical Model  
In order to analyze and derive the performance measures of the proposed system, we define an analytical model using the continuous-time Markov chain (CTMC) model. The CTMC is used as a tool to model the proposed scheme and evaluate the scenarios being discussed. The proposed scheme is assumed to operate inside the 5G C-RAN component and manage the resource allocation for three different 5G network slices, namely the uRLLC, eMBB, and mMTC slices. The used CTMC model has five dimensions with variables (\(i,j,k\), \(m\), and \(n\)) and is based on organizing the requests of each 5G slice on a priority basis. These five variables represent the state of the proposed system at any given instance, where \(i,j\), and \(k\) represent the number of ongoing uRLLC services, eMBB services, and mMTC services, respectively. The variables \"\(m\)\" and \"\(n\)\" correspond to the current number of queued services of type eMBB and mMTC request, respectively. The number of queued mMTC services includes the interrupted mMTC service requests that were forced to terminate their connections by the arrival of uRLLC requests. The total current requests in the system (C-RAN) at any given time, including the queued requests, is the sum of \(i+j+k+m+n\). Let us represent this total as T such that T = \(i+j+k+m+n\) and 0 \(\leq\) T \(\leq\) (C + M + N). Therefore, the state of the system can be represented by a five-dimensional state (i, j, k, m, and n), and when a new event occurs, the \begin{table}
\begin{tabular}{l l l} \hline uuRLLC Arrival Process: & & eMBB or mMTC Arrival Process: \\ \hline \multicolumn{3}{c}{**Case of eMBB:**} \\
**IF** available BBUs \textgreater{} 0 **THEN** & **IF** available & BBUs \textgreater{} 0 **THEN** \\ \multicolumn{3}{c}{Accept uRLLC} & \multicolumn{1}{c}{} \\
**ELSE** & & \\
**IF** No. of eMBB or mMTC in progress \textgreater{} 0 & & Check eMBB queue \\
**THEN** & & **IF** eMBB queue is not full **THEN** \\
**DO** **one BBU** **Preemption:** & & Insert eMBB in its queue \\ \multicolumn{3}{c}{(in-service mMTC is selected first if any]} \\ \multicolumn{3}{c}{- **IF** interrupted service \(==\) mMTC} \\ \multicolumn{3}{c}{**IF** mMTC queue is not Full} \\ \multicolumn{3}{c}{Insert mMTC in its queue} \\ \multicolumn{3}{c}{**ELSE**} \\ \multicolumn{3}{c}{mMTC Leaves the system } \\ \multicolumn{3}{c}{**- ELSE**} \\ \multicolumn{3}{c}{eMBB Leaves the system } \\ \multicolumn{3}{c}{**- - END**} \\ \multicolumn{3}{c}{Assign the uRLLC to released BBU} \\ \multicolumn{3}{c}{**ELSE**} \\ \multicolumn{3}{c}{Block uRLLC} & \multicolumn{1}{c}{} \\
**END** & & **ELSE** \\
**END** & & **END** \\ \hline \end{tabular}
\end{table} 
Table 2: The pseudocode (algorithm) of the proposed scheme. system will change its current state to another state, and the value of the five variables will change when the system state moves into different states. In order to derive the detailed balance equation of the proposed system and simplify its expression, we will define the variables, notations, and indicators that are used in the analytical model as follows:  
The variable M represents the buffer size for storing new delay-sensitive eMBB requests. The variable N represents the second buffer, and this buffer has a size for storing new and interrupted-delay-tolerant mMTC requests. The variables M and N correspond to the maximum occupancy of 1 QeMBB and 2 QmMTC queues, respectively. The variables \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are used to indicate the arrival rates of uRLLC, eMBB, and mMTC requests, respectively. Moreover, \(\mu_{1}\), \(\mu_{2}\), and \(\mu_{3}\) are used to indicate the departure rates of uRLLC, eMBB, and mMTC requests, respectively. The 5G C-RAN is assumed to have C resource units (BBUs). Table 3 summarizes the main notations and parameter descriptions that are used here.