## 4 Analysis of the Model with the _MAP_, Simultaneous Service Process and Retrials
For the retrial queueing system, the size of the vector \(\tilde{\mathbf{y}}\) is \(WK_{N}\). Another popular in the literature and much more difficult for analysis dependencies of the total retrial rate \(\alpha_{i}\) on the number \(i\) are of the forms \(\alpha_{i}=i\alpha\) (the so-called classical retrial policy) and \(\alpha_{i}=i\alpha+\gamma\) (the so-called linear retrial policy, see [43]) where \(\alpha\) and \(\gamma\) are the known constants. The use of these simple dependencies leads to the possibility of derivation of the differential equations for the vector-generating function of the stationary probabilities. However, these equations can be solved analytically only when \(N=1\) and the cardinality of the state space of the components \(\{v_{t},\eta_{t}\}\) is equal to \(1\). This means that the arrival flow should be the stationary Poisson and the service time distribution be exponential. Because we intend to consider a more general case, we would not obtain any benefit from assuming such simple dependence of \(\alpha_{i}\) on \(i\). Thus, we suppose this dependence is arbitrary with the restriction that the limit \(\lim\limits_{i\to\infty}\alpha_{i}\) exists and is equal to infinity. It is evident that if \(\alpha_{i}\) depends on \(i\), then the \(MC\)\(\chi_{t}\) belongs to the class of level-**dependent**\(QBD\) processes. Therefore, the famous results by [PERSON], see [1], obtained for level-**independent**\(QBD\) processes are not applicable here. Instead, we will show that the constructed \(MC\)\(\chi_{t}\) belongs to the class of asymptotically quasi-Toeplitz \(MCs\) (\(AQTMC\)s), see [44], and exploit this fact. According to the definition given in [44], the \(MC\) with a generator \(\tilde{\mathbf{Q}}\) having the blocks \(\tilde{\mathbf{Q}}_{i,ij}\) belongs to the class of \(AQTMC\)s if there exist the limits  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]0  
and the matrix \(\sum\limits_{k=0}^{2}\mathbf{Y}_{k}\) is stochastic. Here, \(\mathcal{F}_{i}\) is a diagonal matrix with positive diagonal elements defined as the moduli of the corresponding diagonal elements of matrix \(\tilde{\mathbf{Q}}_{i,i}\). Let us denote  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]1  
where \(\circ\) denotes Hadamard's product of matrices, see [45]. It is possible to directly check that the limits \(\mathbf{Y}_{k},\ k=0,1,2\), indeed exist and are defined as follows:  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]2  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]3  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]4  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]5  
It is easy to check that \(\sum\limits_{k=0}^{2}\mathbf{Y}_{k}\mathbf{e}=\mathbf{e}\), i.e., the matrix \(\sum\limits_{k=0}^{2}\mathbf{Y}_{k}\) is the stochastic one. Therefore, the \(MC\)\(\chi_{t}\) indeed belongs to the class of \(AQTMC\). According to [44], the sufficient condition for ergodicity of the \(MC\)\(\chi_{t}\) is the fulfillment of the inequality  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]6  
where \(\mathbf{w}\) is the unique solution to the system of equations  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]7  
The condition  
\[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\]8  
is sufficient for the \(AQTMC\) to be non-ergodic. Because each of the first \(N-1\) block rows of the matrix \(\mathbf{Y}_{0}+\mathbf{Y}_{1}+\mathbf{Y}_{2}\) has only one non-zero stochastic block, it is easy to see that the sub-vectors \(\mathbf{w}_{n}\) in representation \(\mathbf{w}\) in the form \((\mathbf{w}_{0},\mathbf{w}_{1},\ldots,\mathbf{w}_{N})\) are zero vectors for \(n=\overline{0,N-2}\).