# Priority Multi-Server Queueing System with Heterogeneous Customers
## 3 Process of the System States  
Let at the time \(t\),  
* \(n_{t}\) be the number of busy servers, \(n_{t}=\overline{0,N}\);
* \(r_{t}\) be the number of servers serving type 1 customers, \(r_{t}=\overline{0,n_{t}}\);
* \(v_{t}\) be the state of the underlying process \(v_{t}=\overline{0,N}\);
* \(m_{t}^{(j,k)}\) be the state of underlying process of service on \(j\)th server servicing type \(k\) customer, \(m_{t}^{(j,1)}=\overline{1,r_{t}},m_{t}^{(j,2)}=\overline{1,n_{t}-r_{t}}\). We assume the following dynamical enumeration of the busy servers. The servers, which provide service to customers of type 2, are located after servers serving customers of type 1. In addition, we assume that the servers serving customers of the \(k\)-th type are numbered in the order of their occupation, i.e. the server that starts the service is numbered by the maximum number among all servers engaged in servicing customers of this type. When some server finishes the service, the corresponding renumbering of servers occurs. The operation of the queue under consideration is described by the Markov chain  
\[\xi_{t}=\{n_{t},r_{t},\
u_{t},m_{t}^{(1,1)},m_{t}^{(2,1)},\ldots,m_{t}^{(r_{t},1)},m_{t}^{(1,2)},m_{t}^{(2,2)},\ldots,m_{t}^{(n_{t}-r_{t},2)}\}\]  
with the state space  
\[\Omega=\{(n,r,\
u,m^{(j,1)},m^{(l,2)}),\,n=\overline{0,N},r=\overline{0,n},\, \
u=\overline{0,W},\,m^{(j,1)}=\overline{1,M_{1}},j=\overline{1,r},\]  
\[m^{(l,1)}=\overline{1,M_{2}},l=\overline{1,n-r}\}.\]  
It can be calculated that cardinality of the set \(\Omega\) is equal to \(K=(W+1)(1+\sum\limits_{n=1}^{N}\sum\limits_{l=0}^{n}M_{1}^{l}M_{2}^{n-l})\). Introduce the following notation:  
\(\mathbf{e}_{n}\) is a column vector of size \(n\), consisting of 1's;  
\(I\) (\(O\)) is an identity (zero) matrix of appropriate dimension. When needed we will identify the dimension of this matrix with suffix;  
\(diag\) {\(A_{I}\), \(l=\overline{1,L}\)} is a diagonal matrix with diagonal blocks \(A_{I}\);  
\(diag^{-}\{A_{I},l=\overline{0,L}\}\) is a sub-diagonal matrix with the sub-diagonal blocks \(A_{I}\);  
\(diag^{+}\{A_{I},l=\overline{0,L}\}\) is an over-diagonal matrix with the over-diagonal blocks \(A_{I}\);  
\(\otimes\) and \(\oplus\) are the symbols of the Kronecker product and sum of matrices, see [30];  
\(A^{\otimes l}=\underbrace{A\otimes\ldots\otimes A}_{l},\,l\geq 1,\,\,A^{\otimes 0 }=1\);  
\(A^{\oplus l}=\sum\limits_{m=0}^{l-1}I_{n^{m}}\otimes A\otimes I_{n^{l}-n-1}\), \(l\geq 1\), for the matrix \(A\) having \(n\) rows;  
\(\tilde{W}=W+1\);  
Let us arrange the states of the chain \(\xi_{t}\) in the lexicographic order and form the matrices \(Q_{n,n^{\prime}}\), \(n,n^{\prime}=\overline{0,N}\), consisting of the rates of the chain transition from the states corresponding to the value \(n\) of the first component to the states corresponding to the value \(n^{\prime}\) of this component. Then the infinitesimal generator of the chain is defined by the following theorem. **Theorem 1**. : _The infinitesimal generator of the Markov chain \(\xi_{t}\) has the following block structure:_  
\[Q=\left( \begin{array}{cccccc}Q_{0,0}&Q_{0,1}&O&\ldots&O&O\\ Q_{1,0}&Q_{1,1}&Q_{1,2}&\ldots&O&O\\ O&Q_{2,1}&Q_{2,2}&\ldots&O&O\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ O&O&O&\ldots&Q_{N-1,N-1}&Q_{N-1,N}\\ O&O&O&\ldots&Q_{N,N-1}&Q_{N,N}\end{array} \right),\]  
_where_  
\[Q_{n,n}=diag\{D_{0}\oplus S_{1}^{\oplus r}\oplus S_{2}^{\oplus n-r},\,r= \overline{0,n}\},\,n=\overline{0,N-1},\]  
\[Q_{N,N}=diag\{(D_{0}+D_{2})\}\oplus S_{1}^{\oplus r}\oplus S_{2}^{\oplus N-r}, \,r=\overline{0,N-1},\,\,D(1)\oplus S_{1}^{\oplus N}\}\]  
\[+diag^{+}\{D_{1}\otimes I_{M_{1}^{r}}\otimes\vec{e}_{M_{2}}\vec{\rho}_{1} \otimes I_{M_{2}^{N-r-1}},r=\overline{0,N-1}\},\]  
\[Q_{n,n-1}=\left( \begin{array}{c}diag\{I_{W}\otimes I_{M_{1}^{r}}\otimes(S_{ 0}^{(2)})^{\oplus n-r},\,r=\overline{0,n-1}\}\\ \frac{O_{\tilde{W}M_{1}^{n}\times\tilde{W}}\stackrel{{ n-1}}{{r =0}}M_{1}^{r}M_{2}^{n-r-1}}\,\,\mid\,I_{W}\otimes(S_{0}^{(1)})^{\oplus n}\\ \end{array}