# Priority Multi-Server Queueing System with Heterogeneous Customers
## 4 Stationary Distribution. Performance Measures  
The Markov chain \(\xi_{t}\) is irreducible and admits the values in a finite state space. Therefore, a unique stationary distribution of this chain exists for any values of the system parameters. Let \(\mathbf{p}\) be the row vector of the steady state (stationary) probabilities of the states of the chain enumerated in the lexicographic order. It is well known that the vector \(\mathbf{p}\) is defined as the unique solution of the Chapman-Kolmogorov (equilibrium or balance) equations  
\[\mathbf{p}Q=\mathbf{0},\ \ \mathbf{p}\mathbf{e}=1.\]  
This system can be solved by any of the well-known methods for solving the finite system of linear algebraic equations. However, in the case of a large dimension of this system, the solution of this system can be not trivial due to existing restrictions on the computer memory and computation speed. Therefore, for solution of this system it is advisable to use a special stable algorithm proposed in [31] and based on the idea of substituting this system of equations by an alternative system derived via consideration of a sequence of specially constructed so called censored Markov chains. As the result of computations, we obtain the partitioned vector \(\mathbf{p}=(\mathbf{p}_{0},\mathbf{p}_{1},\ldots,\mathbf{p}_{N})\) where \(\mathbf{p}_{n}\) is a row vector of steady state probabilities corresponding to the value \(n\) of the first component of the Markov chain, \(n=\overline{0,N}\). Having the vectors \(\mathbf{p}_{n}\) be calculated, we are able to calculate a number of stationary performance measures of the queue under consideration. * The distribution of the number of busy servers at an arbitrary time \(p_{n}=\mathbf{p}_{n}\mathbf{e}\), \(n=\overline{0,N}\). * The average number of busy servers \(N_{busy}=\sum\limits_{n=1}^{N}np_{n}\). * The distribution of the number of servers providing service to priority customers \[q_{r}^{(1)}=\sum\limits_{n=r}^{N}\mathbf{p}_{n}\mathbf{u}(r,n-r),\ r=\overline{0,N},\] (1) where \[\ \ \mathbf{u}(r,n-r)=\left( \begin{array}{c}\mathbf{0}^{T}_{W}\sum \limits_{l=0}^{r-1}M_{1}^{l}\mathbf{M}_{2}^{n-l}\\ \mathbf{e}_{W\mathbf{M}_{1}^{n}\mathbf{M}_{2}^{n-r}}\\ \mathbf{0}^{T}_{W}\sum\limits_{l=r+1}^{R}M_{1}^{l}\mathbf{M}_{2}^{n-l}\end{array}  \right).\] To make clear Formula (1), we note that, multiplying the vector \(\mathbf{p}_{n}\) by the vector \(\mathbf{u}(r,n-r)\), we select and sum up the entries of the probability vector \(\mathbf{p}_{n}\) which correspond to the states with \(n\) busy servers of which \(r\) servers are busy with priority customers. Summing the results over \(n\), we obtain the probability \(q_{r}^{(1)}\). * The average number of servers providing service to priority customers \(N_{busy}^{(1)}=\sum\limits_{r=1}^{N}rq_{r}^{(1)}\). * The distribution of the number of servers providing service to non-priority customers \[q_{m}^{(2)}=\sum\limits_{n=m}^{N}\mathbf{p}_{n}\mathbf{u}(n-m,m),\ m=\overline{0, N}.\] (2)Formula (2) is explained similarly to Formula (1). * The average number of servers providing service to non-priority customers \(N_{busy}^{(2)}=\sum\limits_{m=1}^{N}mq_{m}^{(2)}\). Evidently, \(N_{busy}=N_{busy}^{(1)}+N_{busy}^{(2)}\) and this relation can be used for control of accuracy of computations. * The probability that a priority customer will be lost \[P_{loss,1}=\frac{1}{\lambda_{1}}\mathbf{p}_{N}\left( \begin{array}{c}O_{\tilde {W}}\sum\limits_{r=0}^{N-1}M_{1}^{r}M_{2}^{N-r}\times\tilde{W}\\ I_{\tilde{W}}\otimes\mathbf{e}_{M_{1}^{N}}\end{array} \right)D_{1}\mathbf{e}.\] (3) The brief explanation of Formula (3) can be done as follows. A priority customer will be lost if at the moment of his/her arrival all servers are occupied with priority customers. The \(\
u\)th entry of the vector \(\mathbf{p}_{N}\left( \begin{array}{c}O_{\tilde{W}}\sum\limits_{r=0}^{N-1}M_ {1}^{r}M_{2}^{N-r}\times\tilde{W}\\ I_{\tilde{W}}\otimes\mathbf{e}_{M_{1}^{N}}\end{array}