# Multiservice Loss Models in Single or Multi-Cluster C-RAN Supporting Quasi-Random Traffic
## 2 The f-MC-SC Model
### TC Probabilities via the Proposed Convolution Algorithm  
The algorithm of this section consists of three steps (presented below as steps A to C) and takes advantage of the fact that the analytical model has a PFS.  
Step A  
In this step, the occupancy distribution (o.d.) of the RRHs is calculated. To this end, we first calculate the o.d. for each service class \(w,s\) calls, \(q_{\text{f},w,s}(j)\), assuming that the \(w\)-th RRH serves only service class \(s\) calls:  
\[q_{\text{f},w,s}(j)=\begin{cases}1,&\text{for }j=0\\ \binom{(N_{w,s})}{i}\alpha_{w,s,\text{idle}}^{i}q_{\text{f},w,s}(0),&\text{for }1 \leq i\leq\lfloor\frac{\text{C}}{\beta_{w,s}}\rfloor,j=ib_{w,s}^{\text{r}},\\ 0,&\text{otherwise}\end{cases} \tag{8}\]  
where \(i\) expresses the number of in-service service class \(s\) calls while \(j\) expresses the occupied RRUs. The values of \(q_{\text{f},w,s}(j)\) are normalized via \(G_{\text{f},w,s}=\sum_{j=0}^{\text{C}}q_{\text{f},w,s}(j)\) and are denoted as \(q^{\prime}_{\text{f},w,s}(j)=q_{\text{f},w,s}(j)/G_{\text{f},w,s}\).  
Having determined \(q^{\prime}_{\text{f},w,s}(j)\), one should calculate, without considering the first service class, the aggregated o.d. of the \(w\)-th RRH \(Q^{w}_{\text{f},(-1)}\):  
\[Q^{w}_{\text{f},(-1)}=q^{\prime}_{\text{f},w,2}*\ldots*q^{\prime}_{\text{f},w, s}*\ldots*q^{\prime}_{\text{f},w,S_{w}}, \tag{9}\]  
where the convolution operation between \(q^{\prime}_{\text{f},w,x}\equiv q^{\prime}_{\text{f},a}\) and \(q^{\prime}_{\text{f},w,y}\equiv q^{\prime}_{\text{f},b}\) can be expressed via:  
\[q^{\prime}_{\text{f},a}*q^{\prime}_{\text{f},b}=\Bigg{\{}q^{\prime}_{\text{f},a}(0)\cdot q^{\prime}_{\text{f},b}(0),\sum_{i=0}^{1}q^{\prime}_{\text{f},a}(i )\cdot q^{\prime}_{\text{f},b}(1-i),\ldots,\sum_{i=0}^{\text{C}}q^{\prime}_{ \text{f},a}(i)\cdot q^{\prime}_{\text{f},b}(C-i)\Bigg{\}}. \tag{10}\]  
Finally, to compute the normalized o.d. of the \(w\)-th RRH, \(q^{\prime}_{\text{f},w}\), one should compute \(q_{\text{f},w}=Q^{w}_{\text{f},(-1)}*q^{\prime}_{\text{f},w,1}\) as well as the normalization \(q^{\prime}_{\text{f},w}(j)=q_{\text{f},w}(j)/G_{\text{f},w}\) where \(G_{\text{f},w}=\sum_{j=0}^{\text{C}}q_{\text{f},w}(j)\).  
Step B  
Herein, we compute, for all RRHs, the aggregated o.d. excluding the \(w\)-th one, via:  
\[Q_{\text{f},(-w)}=q^{\prime}_{\text{f},1}*\ldots*q^{\prime}_{\text{f},w-1}*q^ {\prime}_{\text{f},w+1}*\ldots*q^{\prime}_{\text{f},W}. \tag{11}\]  
The convolution operation between \(q^{\prime}_{\text{f},x}\) and \(q^{\prime}_{\text{f},y}\) is the following:  
\[q^{\prime}_{\text{f},x}*q^{\prime}_{\text{f},y}=\Bigg{\{}q^{\prime}_{  
Step C  
In this step, we consider the operation \(Q^{\prime}_{\ell,(-w)}*q^{\prime}_{\ell,w}\) which results to the un-normalized values of \(Q_{\ell,w}(j)\) which can be normalized via \(G^{*}_{\ell,w}=\sum_{j=0}^{T}Q_{\ell,w}(j)\):  
\[Q^{\prime}_{\ell}(j)=\frac{Q_{\ell,w}(j)}{G^{*}_{\ell,w}}. \tag{13}\]  
To obtain the computational o.d. \(Q^{\prime}_{\ell}(j)\), one may choose any of the \(W\) RRHs since all RRHs are exactly the same.  
Based on (13), we calculate the TC probabilities due to the unavailability of CRUs and RRUs according to (14) and (15), respectively:  
\[B^{\text{TC}}_{\epsilon,w,s}=\sum_{j=-T-b^{*}_{w,s}+1}^{T}Q^{\prime}_{\ell}(j), \tag{14}\]  
\[B^{\text{TC}}_{\epsilon,w,s}=\frac{1}{G^{*}_{\ell,w}}\sum_{x=C-b^{*}_{w,s}+1}^ {C}q^{\prime}_{\ell,w}(x)\sum_{y=x}^{T-b^{*}_{w,s}}Q^{\prime}_{\ell,(-w)}(T-b^ {\text{r}}_{w,s}-y). \tag{15}\]