## 3 Mathematical Model and Steady State Solution
### Mathematical Model
,\]  
where \(P_{B}=\sum_{j=0}^{c}y_{c-j,j,1}\) is the blocking probability that an arriving incoming call will be blocked in the retail queue when the servers are in busy mode, \(a_{i}=((i+1)\mu_{w}a_{i+1}/(\lambda+\
u))+(\gamma\Sigma_{k=i+1}^{c}a_{k}/( \lambda+\
u))\) for \(i=0,1,\ldots,c-1\) and \(a_{c}=1\). Below, a simple explicitness is obtained from Equation (4) for the special case of \(c=1\). For the matrices, we have  
\[\mathbf{A}^{+}= \begin{bmatrix}0&0&0&0&0&0&0\\ 0&b_{v}\lambda&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&b_{p}\lambda&0&0&0\\ 0&0&0&0&b_{p}\lambda&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&b_{r}\lambda\end{bmatrix} ,\]  
\[\mathbf{A}^{-}= \begin{bmatrix}0&\
u&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&\
u&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&v\\ 0&0&0&0&0&0&0\end{bmatrix} ,\]  
\[\mathbf{A}= \begin{bmatrix}-\#_{1}&\lambda&\eta&0&0&0&0\\ \mu_{w}&-\#_{2}&0&\eta&0&0&0\\ 0&0&-\#_{3}&\lambda&\alpha&\delta&0\\ 0&0&\mu_{1}&-\#_{4}&0&\delta&0\\ 0&0&\mu_{2}&0&-\#_{5}&\delta&0\\ 0&0&\gamma&0&0&-\#_{6}&\lambda\\ 0&0&0&\gamma&0&\mu_{w}&-\#_{7}\end{bmatrix} ,\]  
\[\mathbf{A}^{+}+\mathbf{A}+\mathbf{A}^{-}= \begin{bmatrix}-\#_{1}&\lambda+\
u& \eta&0&0&0\\ \mu_{w}&-\Delta_{2}&0&\eta&0&0&0\\ 0&0&-\#_{3}&\lambda+\
u&\alpha&\delta&0\\ 0&0&\mu_{1}&-\Delta_{4}&0&\delta&0\\ 0&0&\mu_{2}&0&-\Delta_{5}&\delta&0\\ 0&0&\gamma&0&0&-\#_{6}&\lambda+\
u\\ 0&0&0&\gamma&0&\mu_{w}&-\Delta_{7}\end{bmatrix} ,\]  
where \(\#_{1}=\lambda+\eta+\
u\), \(\#_{2}=b_{v}\lambda+\mu_{w}+\eta\), \(\#_{3}=\lambda+\alpha+\delta+\
u\), \(\#_{4}=b_{v}\lambda+\mu_{1}+\delta\), \(\#_{5}=b_{b}\lambda+\mu_{2}+\delta\), \(\#_{6}=\lambda+\gamma+\
u\), \(\#_{7}=b_{r}\lambda+\mu_{w}+\gamma\), \(\Delta_{2}=\mu_{w}+\eta\), \(\Delta_{4}=\mu_{1}+\delta\), \(\Delta_{5}=\mu_{2}+\delta\) and \(\Delta_{7}=\mu_{w}+\gamma\). Then, we have the following results:  
\[y_{0,0,0}=y_{1,0,0}=0,\]  
\[y_{0,1,1}=\frac{\alpha}{\mu_{2}+\delta}y_{0,0,1},\]  
\[y_{1,0,1}=\frac{\delta}{\gamma+\delta}\frac{\gamma(\lambda+\
u)}{(\mu_{1}+ \delta)(\lambda+\mu_{w}+\
u+\gamma)}+\frac{\lambda+\
u}{\mu_{1}+\delta}y_{0,0,1},\]  
\[y_{0,0,2}=\frac{\delta}{\gamma+\delta}\frac{\mu_{w}+\gamma}{\lambda+\mu_{w}+ \
u+\gamma},\]\[y_{1,0,2}=\frac{\delta}{\gamma+\delta}\frac{\lambda+\
u}{\lambda+\mu_{w}+\
u+\gamma},\]  
where  
\[y_{0,0,1}=\frac{\gamma}{\gamma+\delta}\frac{(\mu_{2}+\delta)\{\mu_{1}(\lambda+ \
u)+(\mu_{1}+\delta)(\mu_{w}+\gamma)\}}{(\lambda+\mu_{w}+\
u+\gamma)\{( \lambda+\mu_{1}+\
u+\delta)(\mu_{2}+\delta)+\alpha(\mu_{1}+\delta)\}}.\]  
Hence, the ergodicity condition is  
\[b_{b}\lambda\bigg{(}\frac{\lambda+\
u}{\mu_{1}+\delta}+\frac{\alpha}{\mu_{2}+ \delta}\bigg{)}<\
u+\frac{\delta}{\gamma}\theta,\]  
where  
\[\theta=\frac{\
u(\mu_{1}+\delta)(\mu_{w}+\gamma)-b_{r}\lambda(\lambda+\
u)( \mu_{1}+\delta)-b_{r}\lambda\gamma(\lambda+\
u)}{\mu_{1}(\lambda+\mu_{w}+\
u+ \gamma)+\delta(\mu_{r}+\gamma)}\bigg{(}1+\frac{\lambda+\
u}{\mu_{1}+\delta}+ \frac{\alpha}{\mu_{2}+\delta}\bigg{)}.\]  
If we set \(b_{b}=b_{r}=1\) and \(\delta=0\), then the ergodicity condition is reduced as follows  
\[\lambda\bigg{(}\frac{\lambda+\
u}{\mu_{1}}+\frac{\alpha}{\mu_{2}}\bigg{)}<\
u,\]  
which is consistent with Equation (14) in [33].