## 5 Numerical Example
The dependencies of the probability \(P_{1}^{loss}\) of an arbitrary priority customer loss from Storage 1 and the probability \(P_{2}^{loss}\) of an arbitrary non-priority customer loss from Storage 2 on different values of \(\gamma_{1}\) and \(\gamma_{2}\) are illustrated in Figures 6 and 7. Figure 4: Dependence of the probability \(P_{1}^{imp-loss}\) that an arbitrary customer will be lost due to impatience in Storage 1 on \(\gamma_{1}\) and \(\gamma_{2}\). Figure 5: Dependence of the probability \(P_{2}^{imp-loss}\) that an arbitrary customer will be lost due to impatience in Storage 2 on \(\gamma_{1}\) and \(\gamma_{2}\). It is seen from Figure 6 that the minimal value of the loss probability of an arbitrary priority customer from Storage 1 is achieved when \(\gamma_{2}=0\). Note, that if \(\gamma_{2}=0\), then Type-2 customers cannot transit to the infinite buffer. Thus, a Type-2 customer can be chosen for service only in the case of the absence of Type-1 customers in the storage and system. Thus, if \(\gamma_{2}=0\), the system under study behaves as some kind of system with non-preemptive priority of Type-1 customers over Type-2 customers. If we fix \(\gamma_{1}=\gamma_{2}=0\), we obtain the system with two finite buffers and non-preemptive priority. If we choose huge \(\gamma_{1}\) and \(\gamma_{2}=0\), we obtain the system with the finite buffer for non-priority customers, the infinite buffer for priority customers, and non-preemptive priority. However, when \(\gamma_{2}\) is equal to zero, the loss probability of Type-2 customers is very high; see Figure 7.  
The dependencies of the loss probability \(P^{imp-loss}\) of an arbitrary customer due to impatience from the infinite buffer and the probability \(P_{loss}\) of an arbitrary customer loss on different values of \(\gamma_{1}\) and \(\gamma_{2}\) are illustrated in Figures 8 and 9. Figure 8: Dependence of the probability \(P^{imp-loss}\) that an arbitrary customer will be lost due to impatience from the infinite buffer on \(\gamma_{1}\) and \(\gamma_{2}\). Figure 7: Dependence of the probability \(P^{loss}_{2}\) of an arbitrary non-priority customer loss from Storage 2 on \(\gamma_{1}\) and \(\gamma_{2}\). As is seen from Figure 8, the probability that an arbitrary customer will be lost due to impatience from the infinite buffer grows with the increase of both intensities \(\gamma_{1}\) and \(\gamma_{2}\). This can be explained by the fact that when \(\gamma_{1}\) and \(\gamma_{2}\) grow, the number of customers in the infinite buffer grows, which causes the increase of the loss probability \(P^{imp-loss}\). The minimal value of the loss probability of an arbitrary customer \(P_{loss}\) is achieved for \(\gamma_{1}=2.8\) and \(\gamma_{2}=2.4\) and is equal to 0.04824. Minimization of the loss probability \(P_{loss}\) can be considered as a problem of the optimization of the system operation. However, in the system under study, the importance (value) of Type-1 and Type-2 customers can be different. The loss of a priority customer can be more essential than the loss of a non-priority customer. Let us assume that the quality of system operation is described by the following economic criterion:  
\[EM(\gamma_{1},\gamma_{2})=a\lambda_{1}P_{1}^{loss}+b\lambda_{2}P_{2}^{loss}+c \lambda_{in}P^{imp-loss}\]  
where \(a\) is a charge paid by the system for the loss of a priority customer from Storage 1, \(b\) is a charge paid for the loss of a non-priority customer from Storage 2, and \(c\) is a charge paid for the loss of a customer from the infinite buffer. The economic criterion \(EM\) describes the average losses of the system per unit of time. Thus, to optimize the system operation, we have to determine the values of the intensities \(\gamma_{1}\) and \(\gamma_{2}\) for which the economical criterion \(EM\) admits the minimal value. We fix the following cost coefficients in the cost criterion: \(a=10\), \(b=6\), \(c=10\).