## 3 Process of System States and Its Stationary Distribution  
Let, during the epoch \(t,\ t\geq 0\),  
* \(i_{t},\,i_{t}\geq 0\), be the number of customers in the infinite buffer and on the server,
* \(k_{t},\,k_{t}=\overline{0,K}\), be the number of customers in Storage 1,
* \(r_{t},\,r_{t}=\overline{0,R}\), be the number of customers in Storage 2,
* \(v_{t},\,v  
* \(\mathbf{a}_{l}\) is the row vector of size \(l\) with all zero entries except the entry \((\mathbf{a}_{l})_{0}\), which is equal to one, \(l=K+1,\ R+1\);
* \(\otimes\) and \(\oplus\) are the symbols of the Kronecker product and the sum of matrices; see, e.g., [23]. Let us enumerate the states of the Markov chain \(\xi_{t}\) in the lexicographic order and refer to the set of states of the chain having value \(i\) of the first component of the Markov chain as level \(i,\ i\geq 0\). Let \(Q\) be the generator of the Markov chain \(\xi_{t},\ t\geq 0\). **Lemma 1**. : _The generator \(Q\) has the following block-tridiagonal structure:_  
\[Q=\left( \begin{array}{cccccc}Q_{0,0}&Q_{0,1}&O&O&O&O&\ldots\\ Q_{1,0}&Q_{1,1}&Q^{+}&O&O&O&\ldots\\ O&Q_{2,1}&Q_{2,2}&Q^{+}&O&O&\ldots\\ O&O&Q_{3,2}&Q_{3,3}&Q^{+}&O&\ldots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array} \right).\]  
_The non-zero blocks \(Q_{i,j}\), \(i,j\geq 0\), containing the intensities of the transitions from level \(i\) to level \(j\) have the following form:_  
\[Q_{0,0}=D_{0},\]  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]  
\[Q_{1,1}=I_{(K+1)(R+1)}\otimes(D_{0}\oplus S)+E_{K+1}^{+}\otimes I_{R+1}\otimes D _{1}\otimes I_{M}+I_{K+1}\otimes E_{R+1}^{+}\otimes D_{2}\otimes I_{M}-\]  
\[(\alpha_{1}+\gamma_{1})C_{K+1}\otimes I_{(R+1)WM}-(\alpha_{2}+\gamma_{2})I_{K+ 1}\otimes C_{R+1}\otimes I_{WM}+\]  
\[\alpha_{1}C_{K+1}E_{K+1}^{-}\otimes I_{(R+1)WM}+\alpha_{2}I_{K+1}\otimes C_{R +1}E_{R+1}^{-}\otimes I_{WM}+(E_{K+1}^{-}\otimes I_{R+1}+\hat{I}_{K+1}\otimes E _{R+1}^{-})\otimes I_{W}\otimes S_{0}\mathbf{\beta},\]  
\[Q_{i,i}=I_{(K+1)(R+1)}\otimes(D_{0}\oplus S)+E_{K+1}^{+}\otimes I_{R+1}\otimes D _{1}\otimes I_{M}+I_{K+1}\otimes E_{R+1}^{+}\otimes D_{2}\otimes I_{M}-(i-1) \varphi l_{(K+1)(R+1)WM}-\]  
\[(\alpha_{1}+\gamma_{1})C_{K+1}\otimes I_{(R+1)WM}-(\alpha_{2}+\gamma_{2})I_{K +1}\otimes C_{R+1}\otimes I_{WM}+\]  
\[\alpha_{1}C_{K+1}E_{K+1}^{-}\otimes I_{(R+1)WM}+\alpha_{2}I_{K+1}\otimes C_{R +1}E_{R+1}^{-}\otimes I_{WM},\ i\geq 2,\]  
\[Q_{i,i+1}=Q^{+}=\gamma_{1}C_{K+1}E_{K+1}^{-}\otimes I_{(R+1)WM}+\gamma_{2}I_{K +1}\otimes C_{R+1}E_{R+1}^{-}\otimes I_{WM},\ i\geq 1,\]  
\[Q_{1,0}=(\mathbf{a}_{K+1})^{T}\otimes(\mathbf{a}_{R+1})^{T}\otimes I_{W}\otimes\mathbf{S}_ {0},\]  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]0  
The proof of the lemma is performed by means of the analysis of the intensities of all possible transitions of the Markov chain \(\xi_{t}\) during the time interval having infinitesimal length. Its brief outline is as follows. The block-tridiagonal form of the generator \(Q\) is easily explained by the fact that customers of both types arrive at the system and depart from it (due to service completion or impatience) one by one. When the server is idle (by default, the buffer and both storages are empty), the dynamics of the Markov chain \(\xi_{t}\) is defined only by the process \(\
u_{t}\). The intensities of its transitions to another states are defined by the non-diagonal entries of the matrix \(D_{0}\), and the intensities of the exit from the corresponding states are defined, up to the sign, by the diagonal entries of this matrix. Therefore, \(Q_{0,0}=D_{0}\). The form of the block \(Q_{0,1}\) is explained as follows. This block contains the rates of transition of the Markov chain \(\xi_{t}\) when the number of customers in the system increases from zero to one. This happens when the customer of any type arrives. The intensities of the transition of the underlying process \(\
u_{t}\) at the arrival moment are defined by the entries of the matrix \(D_{1}+D_{2}\).