# Multiservice Loss Models in Single or Multi-Cluster C-RAN Supporting Quasi-Random Traffic
## 2 The f-MC-SC Model
### The Analytical Model
The system's state space, \(\mathbf{\Omega}\), expresses all possible states and can be described via:  
\[\mathbf{\Omega}=\Bigg{\{}\mathbf{n}:n_{w,s}\geq 0,\sum_{s=1}^{S_{\text{ex}}}n_{w,s}b_{ w,s}^{t}\leq\text{C},\sum_{w=1}^{W}\sum_{s=1}^{S_{\text{ex}}}n_{w,s}b_{w,s}^{c} \leq T\Bigg{\}}. \tag{1}\]  
To determine the probability the system is in steady-state \(\mathbf{n}\), \(P_{\text{fin}}(\mathbf{n})\), it is necessary to define the adjacent vectors \(\mathbf{n}_{w,s}^{-}=(n_{1,1},\ldots,n_{1,S_{1}},\ldots,n_{w,1},\ldots,n_{w,s}-1, \ldots,n_{w,S_{w}},\ldots,\allowbreak n_{W,1},\ldots,n_{W,S_{W}})\) and \(\mathbf{n}_{w,s}^{+}=(n_{1,1},\ldots,n_{1,S_{1}},\ldots,n_{w,1},\ldots,n_{w,s}+1, \ldots,n_{w,S_{w}},\ldots,n_{W,1},\ldots,\allowbreak n_{W,S_{W}})\) as well as the corresponding steady-state probability distributions \(P_{\text{fin}}(\mathbf{n}_{w,s}^{-})\) and \(P_{\text{fin}}(\mathbf{n}_{w,s}^{+})\), respectively. Assuming that \(\mathbf{n}_{w,s}^{-},\mathbf{n},\mathbf{n}_{w,s}^{+}\in\mathbf{\Omega}\), Figure 2 presents the corresponding state transition diagram for service class \(s\) calls in the (serving) \(w\)-th RRH. Figure 1: The f-MC-SC model. Figure 2: Transition diagram for calls of service class \(s\) in the \(w\)-th RRH. Based on Figure 2 and the reversibility of the corresponding Markov chain, we have a local balance equation, for the adjacent states \(\mathbf{n}_{w,s}^{-}\) and \(\mathbf{n}\):  
\[(N_{w,s}-n_{w,s}+1)v_{w,s}P_{\text{fin}}(\mathbf{n}_{w,s}^{-})=n_{w,s}u_{w,s}P_{ \text{fin}}(\mathbf{n}) \tag{2}\]  
and an additional one for states \(\mathbf{n}\) and \(\mathbf{n}_{w,s}^{+}\):  
\[(N_{w,s}-n_{w,s})v_{w,s}P_{\text{fin}}(\mathbf{n})=(n_{w,s}+1)u_{w,s}P_{\text{fin}} (\mathbf{n}_{w,s}^{+}). \tag{3}\]  
Equations (2) and (3) are satisfied by the PFS according to (4), when \(\mathbf{n}\in\mathbf{\Omega}\):  
\[P_{\text{fin}}(\mathbf{n})=\frac{1}{G}\prod_{w=1}^{W}\prod_{s=1}^{S_{w}}\binom{N_{ w,s}}{n_{w,s}}a_{w,s,\text{idle}}^{n_{w,s}}, \tag{4}\]  
where \(G=\sum_{\mathbf{n}\in\mathbf{\Omega}}\prod_{w=1}^{W}\prod_{s=1}^{S_{w}}\binom{N_{w,s} }{n_{w,s}}a_{w,s,\text{idle}}^{n_{w,s}}\) refers to the normalization constant. In a system that accommodates calls generated via a finite number of traffic sources (users), we distinguish CBP in time congestion (TC) and call congestion (CC) probabilities. The latter refers to CBP, while the former refers to the proportion of time the system is under congestion. Assuming that the number of traffic sources (i.e., the number of MUs) is high, then CC probabilities are slightly lower than TC probabilities in quasi-random single or multi-rate loss models [29; 30]. Assuming Poisson arrivals and taking into consideration the PASTA property [29; 30], it is easy to verify that TC and CC probabilities lead to the same results. The total TC probability for the service class \(w,s\) calls, \(B_{\text{tot},w,s}^{\text{TC}}\), can be calculated via:  
\[B_{\text{tot},w,s}^{\text{TC}}=B_{\text{r},w,s}^{\text{TC}}+B_{\text{c},w,s}^ {\text{TC}} \tag{5}\]  
where \(B_{\text{r},w,s}^{\text{TC}}\), \(B_{\text{c},w,s}^{\text{TC}}\) are the TC probabilities caused as the result of insufficient RRUs and CRUs for the service class \(w,s\) calls, respectively. In order to sum up \(B_{\text{r},w,s}^{\text{TC}}\) and \(B_{\text{c},w,s}^{\text{TC}}\) the sets of steady-states that form them should be disjoint. When the steady-state has both insufficient CRUs and RRUs, it is accounted only to the \(B_{\text{c},\text{m},s}^{\text{TC}}\). Regarding the CC probabilities for service class \(s\) calls, they are determined assuming \(N_{w,s}-1\) sources. The values of \(B_{\text{tot},w,s}^{\text{TC}}\) can be computed using either a BF method or a convolution algorithm (which are described in Sections 2.2 and 2.3, respectively).