# Priority Multi-Server Queueing System with Heterogeneous Customers
## 5 Numerical Experiments
Figure 2: The dependence of the probability \(P_{loss,2}^{input}\) on the input rate \(\lambda\) for the \(MMAP\)s with different coefficients of correlation. Comparing the Figures 1 and 2, we see that the behavior of the curves is similar, but with the same \(\lambda\), the probability \(P_{loss,2}^{input}\) is greater than the probability \(P_{loss,1}\). This is because priority customers are blocked only when all servers are occupied with priority customers, while non-priority customers are blocked if all servers are busy with any customers, priority and non-priority. Figure 3 shows the dependence of the probability that an arriving priority customer will push out a non-priority customer from the server, \(P_{loss,2}^{serv}\), on the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). It is seen from the figure that the curves for different \(MMAP\)s first increase, and then decrease. The reason for this behavior of the curves lies in the following. When the input rate is relatively small, an arriving priority customer often finds at least one free server and it does not have a need to push out the non-priority customer from the service. As the input rate increases, the system becomes more crowded, and priority customers are forced to push out a non-priority customer to get service. Therefore, the probability \(P_{loss,2}^{serv}\) increases and reaches maximum at some point \(\lambda_{max}\). The further decreasing of this probability is explained by the fact that with \(\lambda\) increasing the most servers become busy with priority customers. Then the number of non-priority customers in the service decreases (they are mostly lost upon arrival to the system) and the probability that a priority customer removes a non-priority customer from the service decreases. Experiment 2. In this experiment, we investigate the mean number of servers providing service to the priority and non-priority customers as functions of the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). Figure 4 shows the dependence of the mean number of servers providing service to priority customers, \(N_{busy,1}\), on the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). Figure 3: The dependence of the probability \(P_{loss,2}^{serv}\) on the input rate \(\lambda\) for the \(MMAP\)s with different coefficients of correlation. It is expectable, that the value of \(N_{busy,1}\) increases with \(\lambda\) increasing. More interesting is that, for the same values of \(\lambda\), the mean number of servers servicing priority customers decreases with increasing correlation in the input flow. Such a relation between the values of \(N_{busy,1}\) for \(MMAP\)s with different correlations is explained by the mentioned above fact that, with less correlation, customers arrive more uniformly, which guarantees more uniform occupation of servers and an increase in the mean number of occupied servers. Figure 5 depicts the mean number of servers providing service to non-priority customers, \(N_{busy,2}\), as a function of the input rate \(\lambda\) and coefficients of correlation \(c_{1}\) and \(c_{2}\). Figure 4: Dependence of the mean number \(N_{busy,1}\) of servers providing service to priority customers on the input rate \(\lambda\) for the \(MMAP\)s with different coefficients of correlation. Figure 5: Dependence of the mean number \(N_{busy,2}\) of servers providing service to non-priority customers on the input rate \(\lambda\) for the \(MMAP\)s with different coefficients of correlation. It is seen from the figure, that the curves for different \(MMAP\)s first increase, and then decrease. Such a behavior of the curves can be explained by the mechanism of occupation of servers by non-priority customers: when the input rate is relatively small, an arriving non-priority customer often finds at least one free server and occupies it. This customer has a good chance of not being pushed out of service by a priority customer.