## 5 Performance Analysis
### Analytical Model
Hence, the state space of S\({}_{2}\) can be written as:  
\[S_{2}=\{(i,j,k,m,n)/0<i+j+k<C;m=0;n=0\} \tag{3}\]  
Considering the possible states of S\({}_{2}\), the detailed balanced equation for S\({}_{2}\) can be represented as:  
\[(i\mu_{1}+\lambda_{1}+j\mu_{2}+\lambda_{2}+k\mu_{3}+\lambda_{3})\pi_{i,j,k,0,0 }=\lambda_{1}\pi_{i-1,j,k,0,0}+(i+1)\mu_{1}\pi_{i+1,j,k,0,0}+\lambda_{2}\pi_{ i,j-1,k,0,0}+ \tag{4}\]  
Substate 3 (S\({}_{3}\)): These substates represent the system states when \(i+j+k\) = C and m = \(n\) = 0. In this case, the whole system resource BBUs at this moment are fully utilized, while both queues 1 QeMBB and 2 QmMTC are still empty. Thus, the balanced equation for S\({}_{3}\) can be represented as:  
\[(i\mu_{1}+\alpha_{1}\lambda_{1}+j\mu_{2}+\lambda_{2}+k\mu_{3}+\lambda_{3})\pi_ {i,j,k,0,0}=\alpha_{3}\lambda_{1}\pi_{i-1,j,k,0,0}+(j\mu_{2}+k\mu_{3})\pi_{i, j,k,1,0}+\lambda_{2}\pi_{i,j-1,k,0,0}+ \tag{5}\]  
Substates 4 (S\({}_{4}\)): These substates represent all the remaining feasible states of the system when 0 < T < C where 0 < m < = M and 0 < n < = N. In this case, clearly all BBUs are occupied, and the arrival of both eMBB and mMTC services has to wait in their own queues if the corresponding queue is not full. Thus, the balanced equation for S\({}_{3}\) can be represented as:  
\[ \begin{pmatrix}\alpha_{6}i\mu_{1}+\alpha_{1}\lambda_{1}+\alpha_{7}(1-\alpha_ {6})i\mu_{1}+\\ \alpha_{6}j\mu_{2}+\alpha_{7}(1-\alpha_{6})j\mu_{2}+\alpha_{8}k\lambda_{1}+\\ \alpha_{6}k\mu_{3}+\alpha_{7}(1-\alpha_{6})k\mu_{3}+\alpha_{9}\lambda_{3}\\ \alpha_{3}\lambda_{1}\pi_{i,j,k,m-1,n}+\alpha_{3}\lambda_{1}\pi_{i,j,k,m,n-1} \end{pmatrix} \pi_{i,j,k,m,n}=\alpha_{3}\lambda_{1}\pi_{i-1,j+1,k,m-1,n}+\alpha _{3}\lambda_{1}\pi_{i-1,j+1,k,m,n-1}+\\ \alpha_{3}\lambda_{1}\pi_{i,j,k,m-1,n}+\alpha_{3}\lambda_{1}\pi_{i,j,k,m,n-1} \end{pmatrix} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Boolean Indicators** & **Condition** & **Boolean Indicators** & **Condition** \\ \hline \multirow{2}{*}{\(\mathbf{\alpha_{1}}\)} & _1, for i \(<\) C_ & & _1, for m \(>\) 0_ \\  & _0, for i = 0_ & _0, for m \(\leq\) 0_ \\  & _1, for k = 0_ & _1, for n \(>\) 0_ \\  & _0, for k \(<\) 0_ & _0, for n \(\leq\) 0_ \\  & _1, for i \(>\) 0_ & _1, for m \(<\) 0_ \\  & _0, for l = 0_ & _0, for m \(=\) 0_ \\  & _1, for k \(>\) 0_ & _1, for n \(<\) 0_ \\  & _0, for k \(\leq\)Using all the above substates and assuming that all other remaining states that are not covered by those substates are empty, we can derive the steady state probability of the system [23; 24]. We assume that the Markov chain is memoryless and has a finite number of states. Within this memoryless feature, the constraint is that the work is performed at this moment or the current time (now), which is aperiodic, so it does not repeat itself. The Markov chain process can be described by an individual, time-independent matrix \(\pi_{\text{ij,k,m,n}}\). In that regard, \(\pi_{\text{ij,k,m,n}}\) will be assumed to be zero for non-feasible states when the steady-state probability is \(\pi_{\text{ij,k,m,n}}\). Therefore, we define the value of the matrix \(\pi_{\text{ij,k,m,n}}=0\) for the situation where (\(i\),\(j\),\(k\),\(m\),\(n\)) \(\
otin\) S.  
Accordingly, having the defined system state, S, the transition matrix \(\mathbf{Q}\), and the vector \(\pi\) representing the steady-state probability satisfied the following condition:  
\[\sum\limits_{ \begin{array}{c}\forall S\\ 0<\pi_{i,j,k,m,n}<1\end{array} }\pi=1 \tag{5a}\]  
Based on this condition, the system global balance equations and the normalization equation can be derived using the following two equations:  
\[\pi Q=0;\text{ }\text{and }\sum\limits_{\forall S}\pi_{i,j,k,m,n=1} \tag{6}\]  
The CMC model that is used to analyze the proposed system is assumed to be irreducible and has a finite state chain. Accordingly, the CMC is ergodic since it is irreducible and positive-recurrent [23]. Therefore, all the state probabilities \(\pi_{\text{ij,k,m,n}}\) can be derived.