# Stochastic Analysis of a Priority Standby System under Preventive Maintenance
## 8 Numerical Illustration  
In order to demonstrate the usefulness of the proposed system, we investigate the effect of PM and the other parameters on the reliability measures through the following numerical illustrations below. By setting \(\alpha=100,k_{1}=30,k_{2}=10,k_{3}=20,k_{4}=10\) and  
\[F_{\ell}(t)= 1-(1+\lambda_{\ell}t)e^{-\lambda_{\ell}t},\qquad\qquad\lambda_{\ell}>0, \qquad\ell=1,2,3,4,\] \[G(t)= 1-e^{-vt},\] \[B_{\ell}(t)= 1-e^{-\alpha_{\ell}t},\ \ell=1,2.\]  
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\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \multicolumn{6}{c}{**AV (t0)**} & \multicolumn{6}{c}{**Cost (\(\mathbf{t_{0}}\))**} \\ \hline  & & & & \multicolumn{2}{c}{\(\mathbf{p=0.3}\)} & \multicolumn{2}{c}{\(\mathbf{p=0.8}\)} & \multicolumn{2}{c}{\(\mathbf{p=0.3}\)} & \multicolumn{2}{c}{\(\mathbf{p=0.8}\)} \\ \hline  & & & & **OT** & **OV** & **OT** & **OV** & **OT** & **OV** & **OT** & **OV** \\ \hline
1.5 & 2.5 & 0.01 & 0.001 & 7.0 & 0.9641 & 7.0 & 0.9488 & 8.2 & 95.47 & 9.3 & 93.42 \\  & & & 0.01 & 6.2 & 0.9627 & 6.4 & 0.9475 & 7.0 & 95.18 & 7.8 & 93.09 \\  & & & 0.05 & 3.8 & 0.9435 & 4.1 & 0.9287 & 3.9 & 92.16 & 4.4 & 89.84 \\  & & 0.05 & 0.001 & 4.0 & 0.9542 & 4.2 & 0.9395 & 4.2 & 93.67 & 4.6 & 91.37 \\  & & & 0.01 & 3.9 & 0.9537 & 4.1 & 0.9390 & 4.1 & 93.57 & 4.5 & 91.27 \\  & & & 0.05 & 3.3 & 0.9413 & 3.5 & 0.9267 & 3.4 & 91.67 & 3.7 & 89.26 \\  & & 0.10 & 0.001 & 2.9 & 0.9486 & 3.0 & 0.9301 & 3.0 & 92.06 & 3.3 & 89.59 \\  & & & 0.01 & 2.9 & 0.9443 & 3.0 & 0.9298 & 3.0 & 92.00 & 3.3 & 89.53 \\  & & & 0.05 & 2.5 & 0.9363 & 2.7 & 0.9218 & 2.7 & 90.76 & 3.0 & 88.23 \\ \hline
3.0 & 0.01 & 0.001 & 7.2 & 0.9684 & 7.1 & 0.9500 & 8.1 & 95.97 & 9.3 & 93.57 \\  & & & 0.01 & 6.2 & 0.9671 & 6.4 & 0.9487 & 6.9 & 95.68 & 7.8 & 93.22 \\  & & & 0.05 & 3.8 & 0.9437 & 4.1 & 0.9298 & 4.0 & 92.67 & 4.3 & 89.97 \\  & & 0.05 & 0.001 & 3.9 & 0.9585 & 4.2 & 0.9406 & 4.1 & 94.17 & 4.6 & 91.50 \\  & & & 0.01 & 3.8 & 0.9580 & 4.2 & 0.9401 & 4.1 & 94.07 & 4.5 & 91.39 \\  & & & 0.05 & 3.2 & 0.9455 & 3.4 & 0.9278 & 3.3 & 92.18 & 3.7 & 89.38 \\  & & 0.10 & 0.001 & 2.8 & 0.9487 & 3.0 & 0.9311 & 2.9 & 92.57 & 3.3 & 89.71 \\  & & & 0.01 & 2.7 & 0.9483 & 3.0 & 0.9308 & 2.9 & 92.52 & 3.4 & 89.63 \\  & & & 0.05 & 2.5 & 0.9406 & 2.7 & 0.9228 & 2.6 & 91.27 & 3.0 & 88.34 \\ \hline
2.0 & 2.5 & 0.01 & 0.001 & 7.3 & 0.9702 & 7.1 & 0.9649 & 7.9 & 96.21 & 8.7 & 95.33 \\  & & & 0.01 & 7.1 & 0.9703 & 6.1 & 0.9636 & 7.9 & 96.21 & 7.4 & 95.00 \\  & & & 0.05 & 3.8 & 0.9488 & 3.9 & 0.9444 & 3.9 & 92.98 & 4.2 & 91.84 \\  & & 0.05 & 0.001 & 3.9 & 0.9608 & 4.0 & 0.9554 & 4.1 & 94.51 & 4.3 & 93.39 \\  & & & 0.01 & 3.9 & 0.9602 & 3.9 & 0.9549 & 4.0 & 94.42 & 4.2 & 93.29 \\  & & & 0.05 & 3.2 & 0.9479 & 3.2 & 0.9426 & 3.3 & 92.55 & 3.4 & 92.32 \\  & & 0.10 & 0.001 & 2.7 & 0.9518 & 2.9 & 0.9461 & 2.9 & 93.03 & 3.1 & 91.73 \\  & & & 0.01 & 2.7 & 0.9515 & 2.8 & 0.9460 & 2.9 & 92.97 & 3.0 & 91.67 \\  & & & 0.05 & 2.5 & 0.9435 & 2.6 & 0.9380 & 2.6 & 91.73 & 2.7 & 90.37 \\ \hline
3.0 & 0.01 & 0.001 & 6.8 & 0.9749 & 7.0 & 0.9662 & 7.8 & 96.72 & 7.8 & 95.45 \\  & & & 0.01 & 6.8 & 0.9734 & 6.3 & 0.9648 & 6.7 & 96.44 & 7.3 & 95.15 \\  & & & 0.05 & 3.7 & 0.9543 & 3.9 & 0.9456 & 3.8 & 93.52 & 4.1 & 91.98 \\  & & 0.05 & 0.001 & 3.8 & 0.9652 & 4.0 & 0.9566 & 4.0 & 95.04 & 4.3 & 93.53 \\  & & & 0.01 & 3.7 & 0.9648 & 4.0 & 0.9536 & 3.9 & 94.95 & 4.2 & 93.43 \\  & & & 0.05 & 3.0 & 0.9525 & 3.2 & 0.3938 & 3.2 & 93.1 & 3.4 & 91.46 \\  & & 0.10 & 0.001 & 2.7 & 0.9564 & 2.8 & 0.9475 & 3.0 & 93.54 & 3.0 & 91.87 \\  & & & 0.01 & 2.7 & 0.9561 & 2.8 & 0.9474 & 2.8 & 93.53 & 3.0 & 91.81 \\  & & & 0.05 & 2.4 & 0.9481 & 2.5 & 0.9392 & 2.5 & 92.30 & 2.7 & 90.52 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The optimal time (OT) and optimal value (OV) of the availability and cost function of the system in the steady state when \(\
u=0.5,\delta=1,\lambda_{3}=0.3,\lambda_{4}=0.5\).