## 4 Performance Measures of the System  
Having computed the vectors of the stationary probabilities \(\pi_{i},\;i\geq 0,\) it is possible to compute a variety of the performance measures of the system.  
The average number of customers in the system is computed by:  
\[L=\sum_{i=1}^{\infty}\sum_{k=0}^{K}\sum_{r=0}^{R}(i+k+r)\pi(i,k,r)\mathbf{e}.\]  
The average number of customers in the buffer is computed by:  
\[N^{buf}=\sum_{i=2}^{\infty}(i-1)\pi_{i}\mathbf{e}.\]  
The average number of customers in Storage 1 is computed by:  
\[N_{1}^{stor}=\sum_{i=1}^{\infty}\sum_{k=1}^{K}k\pi(i,k)\mathbf{e}.\]  
The average number of customers in Storage 2 is computed by:  
\[N_{2}^{stor}=\sum_{i=1}^{\infty}\sum_{k=0}^{K}\sum_{r=1}^{R}r\pi(i,k,r) \mathbf{e}.\]  
The loss probability of an arbitrary priority customer upon arrival due to Storage 1 overflow is computed by:  
\[P_{1}^{ent-loss}=\frac{1}{\lambda_{1}}\sum_{r=0}^{R}\sum_{i=1}^{\infty}\pi(i, K,r)(D_{1}\otimes I_{M})\mathbf{e}.\]  
The loss probability of an arbitrary non-priority customer upon arrival due to Storage 2 overflow is computed by:  
\[P_{2}^{ent-loss}=\frac{1}{\lambda_{2}}\sum_{k=0}^{K}\sum_{i=1}^{\infty}\pi(i, k,R)(D_{2}\otimes I_{M})\mathbf{e}.\]The loss probability of an arbitrary priority customer due to impatience in Storage 1 is computed by:  
\[P_{1}^{imp-loss}=\frac{1}{\lambda_{1}}\sum_{i=1}^{\infty}\sum_{k=1}^{K}k\alpha_{ 1}\boldsymbol{\pi}(i,k)\mathbf{e}=\frac{\alpha_{1}}{\lambda_{1}}N_{1}^{stor}.\]  
The loss probability of an arbitrary non-priority customer due to impatience in Storage 2 is computed by:  
\[P_{2}^{imp-loss}=\frac{1}{\lambda_{2}}\sum_{i=1}^{\infty}\sum_{k=0}^{K}\sum_{ r=1}^{R}r\alpha_{2}\boldsymbol{\pi}(i,k,r)\mathbf{e}=\frac{\alpha_{2}}{ \lambda_{2}}N_{2}^{stor}.\]  
The intensity of the output flow of successfully served customers is computed by:  
\[\lambda_{out}=\sum_{i=1}^{\infty}\boldsymbol{\pi}_{i}(\mathbf{e}_{(K+1)(R+1)W} \otimes\mathbf{S}_{0}).\]  
The probability of an arbitrary customer loss is computed by:  
\[P_{loss}=1-\frac{\lambda_{out}}{\lambda}. \tag{7}\]  
The probability of an arbitrary priority customer loss from Storage 1 is computed by:  
\[P_{1}^{loss}=P_{1}^{ent-loss}+P_{1}^{imp-loss}.\]  
The probability of an arbitrary non-priority customer loss from Storage 2 is computed by:  
\[P_{2}^{loss}=P_{2}^{ent-loss}+P_{2}^{imp-loss}.\]  
The intensity of the input flow of customers into the system (to the buffer or directly to the server) is computed by:  
\[\lambda_{in}=(1-P_{1}^{loss})\lambda_{1}+(1-P_{2}^{loss})\lambda_{2}.\]  
The loss probability of an arbitrary customer due to impatience in the system is computed by:  
\[P^{imp-loss}=\frac{1}{\lambda_{in}}\sum_{i=2}^{\infty}(i-1)\varphi\boldsymbol {\pi}_{i}\mathbf{e}.\]  
**Remark 4**.: _The alternative to Formula (7) for the computation of the probability \(P_{loss}\) of an arbitrary customer loss is:_  
\[P_{loss}=\frac{\lambda_{1}P_{1}^{loss}+\lambda_{2}P_{2}^{loss}+\lambda_{in} P^{imp-loss}}{\lambda}.\]  
_The existence of two different formulas for the probability \(P_{loss}\) can be helpful for the verification of the results of calculation of the stationary distribution of the Markov chain \(\xi_{t}\)._