## 3 Analysis of the Model with the \(Map\), Simultaneous Service Process, and Infinite Buffer  
Let us analyze the characteristics of the queueing system with the service process described in the previous section. We assume that the arrival process of customers is defined by the \(MAP\). Arrivals can occur only at the epochs of transitions of an underlying continuous-time \(MC\)\(v_{t}\), \(t\geq 0\). This \(MC\) is irreducible and has the finite state space \(\{1,2,\dots,W\}\) and the generator \(D(1)\), which is the sum of two matrices, \(D_{0}\) and \(D_{1}\), the first of which is the sub-generator. The elements of the non-negative matrix \(D_{1}\) define the rates of transitions of \(MC\)\(v_{t}\) that cause customer arrival. The invariant probability vector \(\theta\) of \(MC\)\(v_{t}\) is calculated as a solution to the system \(\theta D(1)=\mathbf{0},\ \theta\mathbf{e}=1\). The fundamental rate \(\lambda\) of the \(MAP\) is given by the formula \(\lambda=\theta D_{1}\mathbf{e}\). Formulas for calculation of the higher moments of inter-arrival times and the coefficient of their correlation can be found, e.g., in [16; 20]. The maximum number of customers that can be processed in the system simultaneously is \(N\). If the number of customers in the systems at the epoch of a customer arrival is less than \(N\), this customer joins the set of customers receiving service. If the number of customers in the systems at the epoch of a customer arrival is equal to \(N\), this customer joins the buffer of an unlimited size. The process of customer service is defined in Section 2. Let us analyze the described queueing system. The behavior of the system is determined by the three-dimensional process  
\[\xi_{t}=\{i_{t},v_{t},\eta_{t}\},\ i_{t}\geq 0,\ v_{t}=\overline{1,W},\ \eta_{t}= \overline{1,K_{\min\{i_{t},N\}}},\ t\geq 0,\]  
where \(i_{t}\) is the number of customers in the system (in the buffer and in service), and \(v_{t}\) and \(\eta_{t}\) are the states of the underlying processes of arrivals and service at the moment \(t,\ t\geq 0\). It follows from definitions of arrival and service processes that the random process \(\xi_{t}\) is the continuous-time \(MC\). Let us arrange the states of the \(MC\)\(\xi_{t}\) in the lexicographic order. The set of the states having value \(i\) of the component \(i_{t}\) is called level \(i,\ i\geq 0\). Let the matrix \(\mathbf{Q}_{i,j}\) contain transition rates from the states that belong to level \(i\) to the states that belong to level \(j,\ i,j\geq 0,|i-j|\leq 1\). **Lemma 1**. : _The generator \(\mathbf{Q}=(\mathbf{Q}_{i,j})_{i,j\geq 0,|i-j|\leq 1}\) of the \(MC\)\(\xi_{t}\) contains the following non-zero blocks:_  
* \(\mathbf{Q}_{i,i}=D_{0}\oplus H_{i}^{0},\ i=\overline{0,N}\)_,_
* \(\mathbf{Q}_{i,i}=D_{0}\oplus H_{i}^{0}:=\mathbf{Q}_{0},\ i\geq N+1\)_,_
* \(\mathbf{Q}_{i,i+1}=D_{1}\otimes H_{i}^{+},\ i=\overline{0,N-1}\)_,_
* \(\mathbf{Q}_{i,i+1}=D_{1}\otimes I_{K_{N}}:=\mathbf{Q}^{+},\ i\geq N\)_,_
* \(\mathbf{Q}_{i,i-1}=I_{W}\otimes H_{i}^{-},\ i=\overline{1,N}\)_,_
* \(\mathbf{Q}_{i,i-1}=I_{W}\otimes H_{N}^{-}H_{N-1}^{+}:=\mathbf{Q}^{-},\ i>N\)_._  
The proof of the statement of Lemma 1 accounts for the probabilistic meaning of the matrices \(D_{0},\ D_{1},\ \mathrm{characterizing}\) the customer arrival process and of the matrices \(H_{n}^{0}\), \(H_{n}^{-}\), and \(H_{n}^{+}\) defining the service process, as well as the definition and properties of [PERSON]'s product and sum of matrices.