# Priority Multi-Server Queueing System with Heterogeneous Customers
## 5 Numerical Experiments  
In this section, we present the results of numerical experiments that allow us to estimate the effect of the input rate \(\lambda\) and correlation in the \(MMAP\) on the system performance measures. We consider three \(MMAP\)s with the same arrival rates of customers of both types but different coefficients of correlation. These \(MMAP\)s are defined by the matrix \(D_{0},D_{1},D_{2}\) as follows. For each \(MMAP\), to get the matrices \(D_{1},D_{2}\), we first define a certain matrix \(D\) and then split it into the matrices \(D_{1},D_{2}\) in the proportion \(D_{1}=0.7D,\;D_{2}=0.3D\). The first \(MMAP\) is the superposition of two stationary Poisson flows. In this case the matrices \(D_{0},D\) are defined as follows:  
\[D_{0}=-6.124137,\;D=6.124137.\]  
For this \(MMAP\), the coefficients of variation of inter-arrival times are \(c_{var}^{(1)}=c_{cur}^{(2)}=1\), and the coefficients of correlation of inter-arrival times are \(c_{1}=c_{2}=0\). The second \(MMAP\) is defined by the matrices  
\[D_{0}=\left( \begin{array}{cc}-8.281261&0\\ 0&-0.268743\\ \end{array} \right),\;\;D=\left( \begin{array}{cc}8.226134&0.055127\\ 0.149638&0.119104\\ \end{array} \right).\]  
For this \(MMAP\), the coefficients of variation of inter-arrival times are \(c_{var}^{(1)}=1.693996,\;c_{var}^{(2)}=3.417903,\) and the coefficients of correlation of inter-arrival times are \(c_{1}=0.023423,\;c_{2}=0.187824\). The third \(MMAP\) is defined by the matrices  
\[D_{0}=\left( \begin{array}{cc}-29.668039&0.003450\\ 0.006900&-0.952137\\ \end{array} \right),\;\;D=\left( \begin{array}{cc}29.323061&0.341527\\ 0.068995&0.876242\\ \end{array} \right).\]  
For this \(MMAP\), the coefficients of variation of inter-arrival times are \(c_{var}^{(1)}=2.394561,\)\(c_{var}^{(2)}=3.087863,\) and the coefficients of correlation of inter-arrival times are \(c_{1}=0.205982,\;c_{2}=0.402641\). The number of servers \(N=8\). The service time of a priority customer has Erlang distribution defined by the vector \(\boldsymbol{\beta}_{1}=\left( \begin{array}{cc}1&0\\ \end{array} \right)\) and the matrix \(S_{1}=\left( \begin{array}{cc}-40&40\\ 0&-40\\ \end{array} \right)\). The service time of a non-priority customer has Erlang distribution defined by the vector \(\boldsymbol{\beta}_{1}=\left( \begin{array}{cc}1&0\\ \end{array} \right)\) and the matrix \(S_{1}=\left( \begin{array}{cc}-10&10\\ 0&-10\\ \end{array} \right)\). Experiment 1. In the experiment, we investigate the behavior of loss probabilities associated with the system under consideration. We consider the probabilities \(P_{loss,1}\), \(P_{loss,2}^{input}\) and \(P_{loss,2}^{serv}\) as functions of the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). Figure 1 depicts the dependence of the probability that a priority customer will be lost, \(P_{loss,1}\), on the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). As expected, the value of \(P_{loss,1}\) increases with increasing \(\lambda\). Under the same value of \(\lambda\), this probability is essentially greater for the larger coefficients of correlation in the \(MMAP\). This effect is easily explained intuitively. The positive correlation in the arrival process causes fluctuation of the instantaneous arrival rate. Periods of time when customers arrive frequently (and likely a lot of customers is lost due to the business of all servers) alternate with the periods when customers arrive rarely (and likely starvation of the servers occurs). Figure 2 shows the dependence of the probability of losing non-priority customers due to occupancy of all servers, \(P_{loss,2}^{input}\), on the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). Figure 1: The dependence of the probability \(P_{loss,1}\) on the input rate \(\lambda\) for the marked Markovian arrival processes (\(MMAP\)s) with different coefficients of correlation.