## 3 Process of System States and Its Stationary Distribution
Now, we have to explain the form of the block \(Q_{i,i},\ i>1\). This is the diagonal block of the generator. Therefore, all its diagonal entries are negative, and the modules of these entries define the intensities of the exit of the Markov chain \(\xi_{t}\) from the corresponding states. The exit of the Markov chain \(\xi_{t}\) from the state having value \(i\) of the first component is possible in the following ways:  
1. Underlying processes \(\
u_{t}\) of arrivals or \(m_{t}\) of service leave their current states. Corresponding transition intensities of the two-dimensional process \((\
u_{t},m_{t})\) are defined, up to the sign, by the diagonal entries of the matrix \(D_{0}\oplus S=D_{0}\otimes I_{M}+I_{W}\otimes S\). 2. Any customer leaves Storage 1 and transits to the buffer or departs from the system due to impatience. Corresponding rates of the exit of the process \(\xi_{t}\) from its states are given by the matrix \((\alpha_{1}+\gamma_{1})C_{K+1}\otimes I_{(R+1)WM}\). 3. Any customer leaves Storage 2 and transits to the buffer or departs from the system due to impatience. Corresponding rates of the exit of the process \(\xi_{t}\) from its states are given by the matrix \((\alpha_{2}+\gamma_{2})I_{K+1}\otimes C_{R+1}\otimes I_{WM}\). The non-diagonal entries of the matrix \(Q_{i,i}\) define the intensities of the transitions of the Markov chain \(\xi_{t}\) without the change of the value \(i\) of the first component. These transitions are defined by:1. non-diagonal entries of the matrix \(I_{(K+1)(R+1)}\otimes(D_{0}\oplus S)\) when one of the processes \(\
u_{t}\) or \(m_{t}\) makes a transition without customer arrival or service completion. 2. entries of the matrix \(E_{K+1}^{+}\otimes I_{R+1}\otimes D_{1}\otimes I_{M}\) when a new Type-1 customer arrives and occupies the place in Storage 1. 3. entries of the matrix \(I_{K+1}\otimes E_{R+1}^{+}\otimes D_{2}\otimes I_{M}\) when a new Type-2 customer arrives and occupies the place in Storage 2. 4. entries of the matrix \(\alpha_{1}C_{K+1}E_{K+1}^{-}\otimes I_{(R+1)WM}\) when a Type-1 customer departs Storage 1 due to impatience. 5. entries of the matrix \(\alpha_{2}I_{K+1}\otimes C_{R+1}E_{R+1}^{-}\otimes I_{WM}\) when a Type-2 customer departs Storage 2 due to impatience. As a result of these derivations, we prove the form of the blocks \(Q_{i,i},\ i>1\), presented above. The derivation of the formula for the block \(Q_{1,1}\) is similar. One can pay attention to the additional summand \((E_{K+1}^{-}\otimes I_{R+1}+\hat{I}_{K+1}\otimes E_{R+1}^{-})\otimes I_{W}  
chain \(\xi_{t},\ t\geq 0,\) belongs to the class \(AQTMC\). As follows from [24], a sufficient condition for the ergodicity of the Markov chain \(\xi_{t}\) is the fulfillment of the inequality:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]1  
where the vector \(\mathbf{y}\) is the unique solution to the system:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]2  
Because \(Y_{0}=I,\ Y_{1}=O,\ Y_{2}=O,\) it is easily observed that Condition (3) holds true for all possible values of the system parameters. (2) Let us consider the case \(\varphi=0.\) In this case, the blocks of the generator for \(i\geq 2\) have the following form:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]3  
and:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]4  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]5  
Since the blocks of the generator do not depend on the variable \(i\) when \(i\geq 2,\) the Markov chain \(\xi_{t},\ t\geq 0,\) belongs to the class of continuous-time quasi-Toeplitz-Markov chains (\(QTMC\)) or \(M/G/1\)-type Markov chains; see [21].