## 3 Mathematical Model and Steady State Solution  
### Mathematical Model  
A mathematical model of two-way communication retrial queue subject to disaster and synchronous working vacation is constructed. Let \(n_{1}(t)\) (\(n_{2}(t)\)) denote the number of incoming (outgoing) calls under service at time \(t\), let \(L(t)\) be the size of the retrial queue at time \(t\), and \(X(t)\) represents the server state at time \(t\). The possible values for \(X(t)\) are as follows:  
\[X(t)=\left\{ \begin{array}{l}0,\text{ servers are on vacation on time }t,\\ 1,\text{ servers are in regular service period at time }t\\ 2,\text{ server are being repaired at time }t.\end{array} \right.,\]  
Obviously, \(\{n_{1}(t),n_{2}(t),L(t),X(t)\}\) is a Markov process with a state space of \(\Omega=\{(i,0,n,k),i=0,1,\ldots,c,n\geq 0,k=0,2\}\cup\{(i,j,n,1),0\leq i+j \leq c,n\geq 0\}\). For the systems with a large or possibly infinite number of states represented in block matrix form, only the matrix geometric method can be used to derive exact solutions. If the steady-state probability for state \(i\) is known, then due to repeated behavior, the steady-state probability for state \(i\)+1 can be established. This helps to find the recursive solution of the stationary probability. According to the lexicographical sequence for the states, we have the state transition matrix with the block-tridiagonal form, as follows:  
\[\mathbf{Q}= \begin{bmatrix}\mathbf{A}^{0}&\mathbf{A}^{+}\\ \mathbf{A}^{-}&\mathbf{A}&\mathbf{A}^{+}\\ \mathbf{A}^{-}&\mathbf{A}&\mathbf{A}^{+}\\ &\ddots&\ddots\end{bmatrix}  \tag{1}\]First, we define \(\mathbf{0}\) to denote a matrix of appropriate dimension with all the elements equal to zero, \(\mathbf{e}_{i}\) is the column-vector of dimension \(i\), consisting of \(1\)'s, and \(\mathbf{I}_{i}\) is the identity matrix of dimension \(i\times i\). For the concerned model, these matrices are represented as  
\[\mathbf{A}^{+}= \begin{bmatrix}\mathbf{A}_{v}^{+}&&&\\ &\mathbf{A}_{0}^{+}&&\\ &&\ddots&&\\ &&&\mathbf{A}_{c}^{+}&\\ &&&\mathbf{A}_{r}^{+}\end{bmatrix} ,\]  
\[\mathbf{A}^{-}= \begin{bmatrix}\mathbf{A}_{v}^{-}&&&\\ &\mathbf{A}_{0}^{-}&&\\ &&\ddots&&\\ &&&\mathbf{A}_{c}^{-}&\\ &&&\mathbf{A}_{r}^{-}\end{bmatrix} ,\]  
\[\mathbf{A}= \begin{bmatrix}\mathbf{A}_{v}&\eta\mathbf{I}_{c}&&0\\ 0&\mathbf{A}_{0}&\mathbf{A}_{0}^{u}&&\mathbf{B}_{c+1}\\ &\mathbf{A}_{1}^{\prime}&\mathbf{A}_{1}&\mathbf{A}_{1}^{u}&\mathbf{B}_{c}\\ &\ddots&\ddots&\ddots&\vdots\\ &&&\mathbf{A}_{c}^{l}&\mathbf{A}_{c}&\mathbf{B}_{1}\\ 0&\gamma\mathbf{I}_{c}&0&\cdots&0&\mathbf{A}_{r}\end{bmatrix} ,\]  
where \(\mathbf{A}_{i}^{+}\), \(\mathbf{A}_{i}^{-}\), \(\mathbf{A}_{i}(i=v,r,0,1,\ldots,c)\), \(\mathbf{A}_{i}^{l}(i=0,1,\ldots,c-1)\), \(\mathbf{A}_{i}^{l}(i=1,2,\ldots,c)\), and \(\mathbf{B}_{i}(i=1,2,\ldots,c+1)\) are \((c-i+1)\times(c-i+1)\), \((c-i+1)\times(c-i+1)\), \((c-i+1)\times(c-i+1)\), \((c-i+1)\times(c-i+1)\), \((c-i+1)\times(c-i)\), \((c-i+1)\times(c-i+2)\), and \(i\times(c+1)\) matrices, respectively, with entries given by  
\[\mathbf{A}_{i}^{+}\big{[}k,k^{\prime}\big{]}= \begin{cases}b_{b}\lambda,\ k^{ \prime}=k=c-i+1,\\ b_{b}\lambda,\ k^{\prime}=k=c+1,\ i=v,\\ b_{b}\lambda,\ k^{\prime}=k=c+1,\ i=r,\\ 0,\quad\text{o.w.}\end{cases} ,\]  
\[\mathbf{A}_{i}\big{[}k,k^{\prime}\big{]}= \begin{cases}\lambda,\ k^{\prime}=k +1,k=0,1,2,\ldots,c-i-1,\\ k\mu_{1},\ k^{\prime}=k-1,k=0,1,2,\ldots,c-i,\\ -\varphi_{i,k},\ k^{\prime}=k,k=0,1,2,\ldots,c-i,\\ 0,\quad\text{o.w.}\end{cases} ,\]  
\[\mathbf{A}_{i}^{u}\big{[}k,k^{\prime}\big{]}= \begin{cases}(c-i-k)\alpha,\ k^{ \prime}=k,k=1,2,\ldots,c-i-1,\\ 0,\quad\text{o.w.}\end{cases} ,\]  
\[\mathbf{A}_{i}^{l}\big{[}k,k^{\prime}\big{]}= \begin{cases}k\mu_{2},\ k^{ \prime}=k,k=1,2,\ldots,c-i+1,\\ 0,\quad\text{o.w.}\end{cases} ,\]  
\[\mathbf{A}_{v}\big{[}k,k^{\prime}\big{]}= \begin{cases}\lambda,\ k^{\prime}=k +1,k=0,1,2,\ldots,c-i-1,\\ k\mu_{v},\ k^{\prime}=k-1,k=0,1,2,\ldots,c-i,\\ -\Delta_{v,k},\ k^{\prime}=k,k=0,1,2,\ldots,c-i,\\ 0,\quad\text{o.w.}\end{cases}