## 4 Analysis of the Model with the _MAP_, Simultaneous Service Process and Retrials
Let us briefly discuss the relation of the generators \(\mathbf{Q}\) and \(\tilde{\mathbf{Q}}\) of the \(MC\)\(\xi_{t}\) describing the system with a buffer and the \(MC\)\(\chi_{t}\) describing the analogous system with retrials. The generator \(\mathbf{Q}\) of infinite size has the following non-zero blocks: the blocks \(\mathbf{Q}_{i,i}\) are of size \((WK_{i}\times WK_{i}),\ i=\overline{0,N}\), the blocks \(\mathbf{Q}_{i,i+1}\) are of size \((WK_{i}\times WK_{i+1}),\ i=\overline{0,N-1}\), the blocks \(\mathbf{Q}_{i,i-1}\) are of size \((WK_{i}\times WK_{i-1}),\ i=\overline{1,N}\), the blocks \(\mathbf{Q}^{0},\mathbf{Q}^{+},\mathbf{Q}^{-}\) are square matrices of size \(WK_{N}\). Correspondingly, the procedure for computation of the stationary probability vectors given in Theorem 2 includes the iterative solution of the square matrix equation for the matrix \(\mathbf{R}\) of size \(WK_{N}\), solution of the system of the linear algebraic equations for the vector \((\boldsymbol{\pi}_{0},\boldsymbol{\pi}_{1},\ldots,\boldsymbol{\pi}_{N})\) of size \(WK_{N}+1\) and the recursive computation of the vectors \(\boldsymbol{\pi}_{i},\ i\geq N\), via the formula \(\boldsymbol{\pi}_{i+1}=\boldsymbol{\pi}_{i}\mathbf{R},\ i\geq N\). The generator \(\tilde{\mathbf{Q}}\) has infinitely many large non-zero blocks \(\tilde{\mathbf{Q}}_{i,j},\ i,j\geq 0,\ |i-j|\leq 1\), all of size \(W\hat{K}_{N}\). Therefore, in general, the check of the existence of the stationary distribution of the \(MC\)\(\chi_{t}\) and computation of this distribution require more computer memory. Until now, we assumed a general dependence of the total retrial rate \(\alpha_{i}\) on the number \(i\) of customers in the orbit. A quite popular in the literature and easier for the study case is when \(\alpha_{i}\) is equal to some constant \(\gamma\) for all \(i,\ i>0\) (the so-called constant retrial policy, see, e.g., [40; 41; 42]). This corresponds to the possible scenarios when all \(i\) customers staying in the orbit are allowed to make retrials independently of each other with the rate \(\frac{\gamma}{t}\) or only one customer, e.g., the eldest one, is granted the possibility to retry with the rate \(\gamma\). It is evident that in this case, the \(MC\)\(\chi_{t}\) is, as the \(MC\)\(\xi_{t}\), the level-independent \(QBD\) and modifications of Theorem 1 and Assertion 1 are applicable for obtaining the ergodicity condition and computing the stationary distribution. In particular, the following statement holds good. Theorem 2: _The criterion for ergodicity of the \(MC\)\(\chi_{t}\) is the fulfillment of the inequality_  
\[\bar{\mathbf{y}}\bar{\mathbf{Q}}^{-}\mathbf{e}>\bar{\mathbf{y}}\bar{\mathbf{Q} }^{+}\mathbf{e}\]  
_where the row vector \(\bar{\mathbf{y}}\) is the unique solution to the system of equations_  
\[\bar{\mathbf{y}}(\bar{\mathbf{Q}}^{0}+\bar{\mathbf{Q}}^{-}+\bar{\mathbf{Q}}^{+ }),\ \bar{\mathbf{y}}\mathbf{e}=1\]  
_where the matrices \(\bar{\mathbf{Q}}^{0},\ \bar{\mathbf{Q}}^{-},\ \bar{\mathbf{Q}}^{+}\) have the form:_  
\[\bar{\mathbf{Q}}^{0}=\mathbf{\Omega}-\gamma\hat{I},\]  
\[\bar{\mathbf{Q}}^{-}=\gamma\text{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n= \overline{0,N-1}\},\]  
\[\bar{\mathbf{Q}}^{+}=\text{diag}\{O_{W\hat{K}_{N-1}},D_{1}\otimes I_{K_{N}}\}.\]  
The vectors of the stationary probabilities are defined by the modification of Assertion 1, where all symbols \(\mathbf{Q}\) are replaced with the symbols \(\bar{\mathbf{Q}}\). Therefore, the analysis of the retrial queue with the constant retrial policy is similar to the analysis of the system with an infinite buffer. Only computer implementation of the check of the ergodicity condition and computation of the stationary distribution becomes more complicated due to the more strict requirements for the computer memory and longer run time. In particular, the check of the ergodicity condition fulfillment for the system with an infinite buffer assumes the solution of system (2) for the vector \(\mathbf{y}\) of size \(WK_{N}\).