## 5 Numerical Example  
In this numerical example, we investigate the impact of the parameters \(\gamma_{r},\;r=1,2\), which define the rates of type-\(r\) customers' transition from the corresponding storage to the infinite buffer, on the main performance measures of the system. The main goal of the numerical example is to show how to optimize the access of priority and non-priority customers via appropriately choosing the parameters \(\gamma_{r},\;r=1,2\). Let us assume that the arrival flow of customers is modeled by the \(MMAP\) arrival process defined by the following matrices:  
\[D_{0}=\left( \begin{array}{cc}-6.759&0\\ 0&-0.21941\end{array} \right),\,D_{1}=\left( \begin{array}{cc}2.238&0.015\\ 0.04072&0.03242\end{array} \right),\,D_{2}=\left( \begin{array}{cc}4.476&0.03 \\ 0.08144&0.06483\end{array} \right).\]  
The total rate of customers' (priority and non-priority) arrival to the system is \(\lambda=4.99852\). The coefficient of correlation of successive inter-arrival times in this arrival process is 0.2, and the squared coefficient of variation is 12.3467. The average intensity of priority customers' arrival is \(\lambda_{1}=1.66617\), and the average intensity of non-priority customers arrival is \(\lambda_{2}=3.33235\). We assume that the capacity of Storage 1 is \(K=5\) and the capacity of Storage 2 is \(R=7\). The intensities of impatience in Storages 1 and 2 are equal to \(\alpha_{1}=0.05\) and \(\alpha_{2}=0.04\), and the intensity \(\varphi\) of impatience in the buffer is equal to 0.06. The \(PH\) service process is defined by the vector \(\boldsymbol{\beta}=(0.1,0.9)\) and the matrix \(S=\left( \begin{array}{cc}-5&0.2\\ 0.5&-8\end{array} \right).\) The mean service time is \(b_{1}=0.144612\). Let us vary the intensity \(\gamma_{1}\) over the interval [0,5] with a step of 0.1. The intensity \(\gamma_{2}\) is varied over the interval \([0,\gamma_{1}]\) also with the step of 0.1. For the computations, we use a computer with an Intel Core i7-8700 CPU and 16 GB RAM and Mathematica 11. In this numerical example, the number of different pairs \(\gamma_{1},\,\gamma_{2}\) is equal to 1326. The total computation time (for all different pairs \(\gamma_{1},\,\gamma_{2}\)) is about 20 min, and the computation time for the fixed values of \(\gamma_{1},\,\gamma_{2}\) is less than 1 s.  
The dependencies of the loss probability \(P_{1}^{ent-loss}\) of an arbitrary priority customer upon arrival due to Storage 1 overflow and the loss probability \(P_{2}^{ent-loss}\) of an arbitrary non-priority customer upon arrival due to Storage 2 overflow on different values of \(\gamma_{1}\) and \(\gamma_{2}\) are illustrated in Figures 2 and 3. The dependencies of the loss probability \(P_{1}^{imp-loss}\) of an arbitrary priority customer due to impatience in Storage 1 and the loss probability \(P_{2}^{imp-loss}\) of an arbitrary non-priority customer due to impatience in Storage 2 on different values of \(\gamma_{1}\) and \(\gamma_{2}\) are illustrated in Figures 4 and 5. As is seen from Figure 2, the loss probability \(P_{1}^{ent-loss}\) of a priority customer due to Storage 1 overflow grows when the intensity \(\gamma_{2}\) increases. This is easily explained by the fact that with the increase of \(\gamma_{2}\), the loss probabilities of Type-2 customers due to buffer overflow and due to impatience essentially decrease; see Figures 3 and 5. Thus, more Type-2 customers arrive at the infinite buffer, and the server becomes idle less often, while Type-1 customers are rarely chosen from Storage 1 for service without visiting the infinite buffer. Therefore, the number of Type-1 customers in the buffer grows, which causes the increase of the loss probability \(P_{1}^{ent-loss}\). For the same reasons, the growth of the probability \(P_{1}^{imp-loss}\) with the increase of the intensity \(\gamma_{2}\) (see Figure 4) can be explained.