## 3 Process of System States and Its Stationary Distribution
As follows from [21], the necessary and sufficient condition for the ergodicity of the \(QTMC\) is the fulfillment of:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]6  
where the vector \(\mathbf{z}\) is the unique solution to the system:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]7  
Let the row vector \(\boldsymbol{\eta}\) be the unique solution to the system:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]8  
It is easy to check that the solution of this system is given by:  
\[Q_{0,1}=\mathbf{a}_{K+1}\otimes\mathbf{a}_{R+1}\otimes(D_{1}+D_{2})\otimes\mathbf{\beta},\]9  
By direct substitution into (5) and using the so-called mixed product rule for the Kronecker product of matrices, it is possible to check that the vector \(\mathbf{z}\) can be represented in the form:  
\[Q_{1,1}=I_{(K+1)(R+1)}\otimes(D_{0}\oplus S)+E_{K+1}^{+}\otimes I_{R+1}\otimes D _{1}\otimes I_{M}+I_{K+1}\otimes E_{R+1}^{+}\otimes D_{2}\otimes I_{M}-\]0  
where the vector \(\mathbf{u}\) is the solution of System (2). Taking into account (6), we easily obtain that the right-hand side of Inequality (4) is equal to \(\mu.\) The left-hand side of Inequality (4) is equal to the left-hand side of inequality (1). Theorem 1 is proven. **Remark 1**. : _The vector \(\mathbf{u}\) defines the joint distribution of the number of customers in the storages and the underlying process of the \(MMAP\) during the time intervals when the system is overloaded, i.e., the number of customers in the buffer is huge. Correspondingly, the expression in the left-hand side of Inequality (1) defines the average rate of customers' arrival to the buffer when the system is overloaded. Therefore, the meaning of (1) is that the mean arrival rate is less than the mean service rate when the system is overloaded._  
**Remark 2**. : _It can be verified that the vector \(\mathbf{u}\) defines the joint distribution of the number of customers and the underlying process of MMAP in the queueing system with the MMAP arrival process, two parallel stations containing \(K\) and \(R\) servers, correspondingly, no buffers, and the exponential service time distribution having the intensity \(\alpha_{r}+\gamma_{r},\ r=1,2.\) The marginal distribution of the number of busy servers and the state of the underlying process of arrivals at the rth station coincide with such a distribution for the [PERSON] loss model of the \(MAP/M/N_{r}/N_{r}\) type where \(N_{1}=K\) and \(N_{2}=R\), the service time at the rth station having the exponential distribution with the rate \(\alpha_{r}+\gamma_{r}\) and MAP defined by the matrices \(\tilde{D}_{0}^{(r)}=D_{0}+D_{r}\) and \(\tilde{D}_{1}^{(r)}=D_{r},\ r,\bar{r}=1,2,r\
eq\bar{r}.\)_  
In this case, if the arrival process is the mixture of two independent stationary Poisson processes with the rates \(\lambda_{1}\) and \(\lambda_{2}\), correspondingly, these marginal distributions are defined by the probabilities:  
\[Q_{1,1}=I_{(K+1)(R+1)}\otimes(D_{0}\oplus S)+E_{K+1}^{+}\otimes I_{R+1}\otimes D _{1}\otimes I_{M}+I_{K+1}\otimes E_{R+1}^{+}\otimes D_{2}\otimes I_{M}-\]1  
where \(\delta_{r}=\frac{\lambda_{r}}{\alpha_{r}+\gamma_{r}}\). The joint distribution of two stations' states is defined here by the product of the marginal distributions. **Remark 3**. : _It is worth making the following observation. In the majority of queueing models with the MMAP and its partial case MAP (Markov Arrival Process), the ergodicity condition includes only the average arrival rates and does not depend, e.g., on the correlation of successive inter-arrival times and their variance. This is easily explained by the fact that the stability condition imposes restrictions on the system parameters in the situation when the system is overloaded. In such a situation, the queue length (or the number of customers in the orbit in retrial queues) is very large, and the concrete pattern of the arrival process does not matter.