## 3 Process of System States and Its Stationary Distribution
The arrived customer starts service. The initial state of the underlying process of service is selected according to the probabilities given by the entries of the vector \(\mathbf{\beta}\). Because the server is not idle after the customer arrival, generally speaking, the storages may be non-empty, and we have to start accounting for the number of customers in both storages. Indeed, the number of customers in both servers remains equal to zero, but we have to declare the start of accounting formally. The matrix \(\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\) defines that after the customer arrival, both storages remain empty. The symbol \(\otimes\) of the Kronecker product of matrices is very useful to define the transition probabilities or intensities of several independent Markov processes. Summarizing our analysis, we conclude that the block \(Q_{0,1}\) has the form \(Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta}\). Now, let us explain the form of the block \(Q_{i,i+1}=Q^{+},\ i\geq 1\). The increase of the number of customers in the buffer (including the one in service) from the value \(i\) to the value \(i+1\) can happen when some customer from Storage 1 or 2 transits to the infinite buffer. The matrix \(\gamma_{1}C_{K+1}\) defines the intensities of the transition of the customers from Storage 1 to the buffer. Here, the scalar \(\gamma_{1}\) defines the rate of transition of an arbitrary Type-1 customer from the storage to the buffer. The diagonal matrix \(C_{K+1}\) accounts for the fact that the total transition rate from Storage 1 to the buffer is proportional to the current number of customers in Storage 1. At the moment of transition of an arbitrary Type-1 customer from the storage to the buffer, the number of customers in Storage 1 decreases by one. This transition of the number of customers in Storage 1 is described by the matrix \(E_{K+1}^{-}\). Because transition from Storage 1 to the buffer happened, transitions of any other components during the infinitesimal interval are not possible, i.e., that components remain in their previous states. Thus, transition rates of the Markov chain \(\xi_{t}\) from the state having value \(i\) of the first component of the chain to the state having value \(i+1\) of that component when the customer from Storage 1 transits to the buffer are given by the matrix \(\gamma_{1}C_{K+1}E_{K+1}^{-}\otimes I_{(R+1)WM}\). It is easy to understand that the corresponding transition rates when the customer from Storage 2 transits to the buffer are given by the matrix \(\gamma_{2}I_{K+1}\otimes C_{R+1}E_{R+1}^{-}\otimes I_{WM}\). As the result, we obtain the formula presented above for the block \(Q_{i,i+1}\). Now, let us explain the form of the block \(Q_{i,i-1},\ i>1\). The transition of the Markov chain \(\xi_{t}\) from the state having value \(i\) of the first component of the chain to the state having value \(i-1\) of that component is possible when the current service is completed and new service starts (the corresponding transition rates of the chain are given by the matrix \(I_{(K+1)(R+1)W}\otimes\mathbf{S}_{0}\mathbf{\beta}\)) or one of \(i-1\) customers waiting in the buffer departs from the system due to impatience (the corresponding transition rates of the chain are given by the matrix \((i-1)\varphi I_{(K+1)(R+1)WM}\)). As a result of this analysis, we obtain the formula presented above for the block \(Q_{i,i-1},\ i>1\). When \(i=1\), because the customer was the only one in the system, we have to take into account that:  
1. After service completion, new service does not start. Thus, instead of the matrix \(\mathbf{S}_{0}\mathbf{\beta}\), we have the column-vector \(\mathbf{S}_{0}\). 2. The server becomes idle, the buffer empty, and by default, the storages empty. Therefore, we have to postpone the monitoring of the number of customers in the storages. This can be done by means of using the multiplier \((\mathbf{a}_{K+1})^{T}\otimes(\mathbf{a}_{R+1})^{T}\) in the transition probability block \(Q_{1,0}\).