## 4 Analysis of the Model with the _MAP_, Simultaneous Service Process and Retrials
The vectors \(\boldsymbol{\pi}_{i},\ i\geq 0\), are calculated as:  
\[\hat{I}=I_{W\hat{K}_{N}}-\mathrm{diag}\{O_{W\hat{K}_{N-1}},I_{WK_{N}}\};\]7where the vectors \(\mathbf{\varphi}_{i}\) are computed from the recursion  
\[\hat{I}=I_{W\hat{K}_{N}}-\mathrm{diag}\{O_{W\hat{K}_{N-1}},I_{WK_{N}}\};\]8  
with the initial vector \(\mathbf{\varphi}_{0}\) given as the unique solution to the system  
\[\hat{I}=I_{W\hat{K}_{N}}-\mathrm{diag}\{O_{W\hat{K}_{N-1}},I_{WK_{N}}\};\]9  
where the stochastic matrices \(\mathbf{G}_{i}\) are computed from the backward recursion  
\[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]0  
with the terminal condition  
\[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]1  
where the matrix \(\mathbf{G}\) is the stochastic solution to the matrix equation  
\[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]2  
solution of which can be found using different variants of the iterative method. The numerical stability of this algorithm is provided via operation only with the non-negative matrices. More details relating to the realization of this algorithm can be found in [46; 47; 48]. Analysis of the Model with the _Map_, Simultaneous Service Process, and Impatience of the Waiting Customers  
Let us return to the model of Section 3 and modify it as follows. We assume that customers waiting in the buffer are impatient. More specifically, we assume that when \(i\) customers reside in the buffer, each of them, independently of others, may depart from the system without service. The importance of account of customer impatience phenomenon in real-world systems is explained, e.g., in [51; 52; 53; 54; 55; 56; 57; 58; 59; 60]. The total departure rate due to impatience is assumed to be equal to \(\sigma_{i},\ i\geq 1\), when \(i\) customers are waiting in the buffer. The most common in the literature case is when \(\sigma_{i}=i\sigma,\ i\geq 1\), where \(\sigma\) is the positive constant defining the individual impatience rate of a customer. In this case, the behavior of the system is described by the \(MC\)\(\xi_{t}^{(imp)}\) that has the same state space as the \(MC\)\(\xi_{t}\) analyzed in Section 3. The generator of the \(MC\)\(\xi_{t}^{(imp)}\) is practically the same as the generator \(\mathbf{Q}\) defined by Lemma 1 with the following two modifications: the blocks \(\mathbf{Q}_{i,i}\) and \(\mathbf{Q}_{i,i-1}\) now have the form  
\[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]3  
\[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]4  
These modifications imply that, in contrast to the \(MC\)\(\xi_{t}\) that is level-independent \(QBD\), the \(MC\)\(\xi_{t}^{(imp)}\) is level-dependent and belongs to the class of \(AQTMC\). If the total impatience rate \(\sigma_{i}\) tends to infinity when \(i\to\infty\), it follows from [61] that the \(MC\)\(\xi_{t}^{(imp)}\) is ergodic for any values of the system parameters. The stationary probabilities of the \(MC\)\(\xi_{t}^{(imp)}\) can be computed using the numerically stable algorithm described in the previous section. It is worth noting that account of possible impatience of customers staying in the orbit can also be easily implemented.