## 3 Analysis of the Model with the \(Map\), Simultaneous Service Process, and Infinite Buffer
For future use, let us denote  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]  
\[\mathrm{diag}^{-}\{\mathbf{Q}_{i,i-1},\ i=\overline{1,N}\}=\]  
\[\mathrm{diag}\{D_{0}\oplus H_{i}^{0},\ i=\overline{0,N}\}+\mathrm{diag}^{+} \{D_{1}\otimes H_{i}^{+},\ i=\overline{0,N-1}\}+\]  
\[\mathrm{diag}^{-}\{I_{W}\otimes H_{i}^{-},\ i=\overline{1,N}\}\]  
where \(\mathrm{diag}\{\dots\},\ \mathrm{diag}^{+}\{\dots\}\), and \(\mathrm{diag}^{-}\It follows from Lemma 1 that the \(MC\)\(\xi_{t}\) belongs to the class of level-independent \(QBD\) processes, see [1]. Using the result of [1], it is possible to prove the following statement. Theorem 1: _The MC \(\xi_{t}\) is ergodic if and only if the following inequality holds good:_  
\[\lambda<\mathbf{u}H_{N}^{-}\mathbf{e}\]  
_where the row vector \(\mathbf{u}\) is the unique solution to the system_  
\[\mathbf{u}(H_{N}^{0}+H_{N}^{-}H_{N-1}^{+})=\mathbf{0},\ \mathbf{ue}=1.\]  
Proof: According to [1], the criterion for ergodicity of the \(MC\)\(\xi_{t}\) is the fulfillment of the inequality  
\[\mathbf{y}\mathbf{Q}^{-}\mathbf{e}>\mathbf{y}\mathbf{Q}^{+}\mathbf{e}\]  
where the row vector \(\mathbf{y}\) is the unique solution to the system  
\[\mathbf{y}(\mathbf{Q}^{0}+\mathbf{Q}^{-}+\mathbf{Q}^{+})=\mathbf{0},\ \mathbf{ye}=1.\]  
Using the so-called mixed product rule for Kronecker's product of matrices, it can be verified that the vector \(\mathbf{y}\) that is the solution of (4) can be represented in the form  
\[\mathbf{y}=\boldsymbol{\theta}\otimes\mathbf{u}\]  
where \(\boldsymbol{\theta}\) is the invariant probability vector of the underlying \(MC\) of the \(MAP\) and the vector \(\mathbf{u}\) is the solution of system (2). By substituting this expression for the vector \(\mathbf{y}\), one can make clear that inequality (3) is transformed to the form (1). Theorem 1 is proven. The left-hand side of (1) defines the mean arrival rate. The right-hand side of (1) defines the mean rate of customers' departure from the overloaded system when the number of simultaneously processed customers is permanently equal to \(N\). Let inequality (1) hold true. Then the stationary probabilities \(\pi(i,\
u,\eta)\) of the states of the \(MC\)\(\xi_{t}\) exist. Denote by \(\pi_{i}\) the row vector of these stationary probabilities listed in the lexicographic order, \(i\geq 0\). **Assertion 1**. : _The vectors of the stationary probabilities \(\pi_{i},i\geq 0\), of the MC \(\xi_{t}\) are calculated as follows:_  
_the vectors \(\pi_{i},\ i\geq N+1\), are computed as:_  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]0  
_where the matrix \(\mathbf{R}\) is the minimal non-negative solution of the matrix equation_  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]1  
_and the vector \((\pi_{0},\pi_{1},\dots,\pi_{N})\) is the unique solution to the following system of linear algebraic equations_  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]2  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]3  
Proof of Asssertion 1 immediately follows from [1]. The statement of this assertion can also be easily verified using the substitution of the presented expressions for the row vectors \(\pi_{i},\ i\geq 0\), into the system of equilibrium (Chapman-Kolmogorov) equations:  
\[\mathbf{\Omega}=\mathrm{diag}\{\mathbf{Q}_{i,i},\ i=\overline{0,N}\}+ \mathrm{diag}^{+}\{\mathbf{Q}_{i,i+1},\ i=\overline{0,N-1}\}+\]4The statements of Theorem 1 and Assertion 1 define an evident algorithm for verification of the existence of the stationary distribution of the system states under the fixed set of the system parameters and the algorithm for computation of this distribution if it exists.