# Multiservice Loss Models in Single or Multi-Cluster C-RAN Supporting Quasi-Random Traffic
## 4 The Generalized f-MC-MC Model
### The Analytical Model  
Consider Figure 10 where \(U\) clusters of RRHs are presented and are separated from the V-BBU whose capacity remains the same as in the f-MC-SC model (i.e., \(T\) CRUs). Cluster \(u\) (\(u=1,\ldots,U\)) consists of \(W_{u}\) RRHs. Each of these RRHs has capacity of \(C_{u}\) RRUs. The \(w\)-th RRH of cluster \(u\) (\(w=1,\ldots,W_{u}\)) accommodates calls from \(S_{u,w}\) different service classes. A call of service class \(s\) (\(s=1,\ldots,S_{u,w}\)) arrives to the \(u\), \(w\)-th RRH according to a quasi-random process with arrival rate \(A_{u,w,s}=(N_{u,w,s}-n_{u,w,s})v_{u,w,s}\), where \(N_{u,w,s}\) is the population of MUs, \(n_{u,w,s}\) is the in-service calls of service class \(s\) calls in the \(u\), \(w\)-th RRH, and \(v_{u,w,s}\) is the call arrival rate per idle MU of service class \(s\) in the \(u\), \(w\)-th RRH. A call of service class \(u\), \(w,s\) requires \(b_{u,w,s}^{r}\) RRUs and \(b_{u,w,s}^{c}\) CRUs, assuming that \(b_{u,w,s}^{r}=b_{u,w,s}^{c}\). An accepted call remains in the \(u\), \(w\)-th RRH for a service time whose distribution can be general with mean \(\mu_{u,w,s}^{-1}\) if the required RUs are available when the call arrives in the serving RRH, i.e., if the occupied RRUs in the \(u\), \(w\)-th RRH do not exceed the value of \(C_{u}-b_{u,w,s}^{r}\) and the occupied CRUs are not more than \(T-b_{u,w,s}^{c}\). In the opposite case, call blocking occurs. Let \(n_{u,w,s}\geq 0\) be the in-service calls of service class \(u\), \(w,s\) (\(u=1,\ldots,U,w=1,\ldots,W_{u}\), \(s=1,\ldots,S_{u,w}\)). Then, we can denote both the vector \(\mathbf{n}=(n_{1,1,1},\ldots,n_{1,1,S_{1,1}},\ldots,n_{u,w,1},\ldots,\)\(n_{u,w,s  
To calculate the steady-state probability distribution \(P_{\text{g,fin}}(\mathbf{n})\), the following vectors are necessary: \(\mathbf{n}^{-}_{u,u,s}=(n_{1,1,1},\dots,n_{1,1,S_{1,1}},\dots,n_{u,u,w,1},\dots,n_{ u,w,s}-1,\dots,n_{u,w,S_{u,u}},\dots,\)\(n_{U,W_{u,1},\dots},n_{U,W_{u,S},w_{U}})\) and \(\mathbf{n}^{+}_{u,w,s}=(n_{1,1,1},\dots,n_{1,1,S_{1,1}},\dots,n_{u,w,1},\dots,n_{ u,w,s}+1,\dots,\)\(n_{u,w,S_{u,u}},\dots,n_{U,W_{u,1}},\dots,n_{U,W_{u,S},w_{U}})\). Furthermore, let \(P_{\text{g,fin}}(\mathbf{n}^{-}_{u,w,s})\), \(P_{\text{g,fin}}(\mathbf{n}^{+}_{u,w,s})\) be the corresponding probability distributions. By further considering that \(\mathbf{n}^{-}_{u,w,s},\mathbf{n}^{+}_{u,w,s}\) belong to \(\mathbf{\Omega}\), Figure 11 presents the state transition diagram for calls of service class \(s\) in the \(u,w\)-th RRH. According to Figure 11 and due to the fact that the Markov chain for service class \(u,w,s\) calls is reversible, we have the following local balance equations, for the adjacent states (a) \(\mathbf{n}^{-}_{u,w,s}\) and \(\mathbf{n}\) (see (17)) and (b) \(\mathbf{n}\) and \(\mathbf{n}^{+}_{u,w,s}\) (see (18)):  
\[(N_{u,w,s}-n_{u,w,s}+1)v_{u,w,s}P_{\text{g,fin}}(\mathbf{n}^{-}_{u,w,s})=n_{u,w,s} \mu_{u,w,s}P_{\text{g,fin}}(\mathbf{n}) \tag{17}\]  
\[(N_{u,w,s}-n_{u,w,s})v_{u,w,s}P_{\text{g,fin}}(\mathbf{n})=(n_{u,w,s}+1)\mu_{u,w,s} P_{\text{g,fin}}(\mathbf{n}^{+}_{u,w,s}). \tag{18}\]  
The PFS of (19) satisfies both (17) and (18) for \(\mathbf{n}\in\mathbf{\Omega}\):  
\[P_{\text{g,fin}}(\mathbf{n})=\frac{1}{G}\prod_{u=1}^{U}\prod_{w=1}^{W_{u}}\prod_{ s=1}^{S_{u,w}}\left(\frac{N_{u,w,s}}{n_{u,w,s}}\right)\alpha_{u,w,s,\text{idle}}^{ n_{u,w,s}}, \tag{19}\]  
where \(G=\sum_{u\in\mathbf{\Omega}}\prod_{u=1}^{U}\prod_{w=1}^{W_{u}}\prod_{s=1}^{S_{u,w }}\left(\frac{N_{u,w,s}}{n_{u,w,s}}\right)\alpha_{u,w,s,\text{idle}}^{n_{u,w, s}}\) and \(\alpha_{u,w,s,\text{idle}}=v_{u,w,s}/\mu_{u,w,s}\) is the offered traffic-load per idle MU for the \(u,w,s\) service class calls. Figure 11: Transition diagram for calls of service class \(s\) in the \(u,w\)-th RRH. Figure 10: The generalized f-MC-MC model.