Journal of Systems Engineering and Electronics ›› 2018, Vol. 29 ›› Issue (5): 1089-1100.doi: 10.21629/JSEE.2018.05.19
• Reliability • Previous Articles Next Articles
Xiaolin SHI1,*(
), Pu LU2(), Yimin SHI2()
Received:2017-06-17
Online:2018-10-26
Published:2018-11-14
Contact:
Xiaolin SHI
E-mail:linda20016@163.com;1464293801@163.com;ymshi@nwpu.edu.cn
About author:SHI Xiaolin was born in 1980. She received her B.S., M.S., and Ph.D. degrees in electrical engineering from Northwestern Polytechnical University. She is currently working as an associate professor of School of Electronics Engineering at Xi’an University of Posts & Telecommunications. Her research interests include system reliability analysis and statistical inference for masked data. E-mail: Supported by:Xiaolin SHI, Pu LU, Yimin SHI. Inference and optimal design on step-stress partially accelerated life test for hybrid system with masked data[J]. Journal of Systems Engineering and Electronics, 2018, 29(5): 1089-1100.
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Fig 1
Series-parallel system"
Fig 2
Parallel-series system"
Table 1
A-MLE of parameter and reliability index and MSE for different $\mathit{\boldsymbol{(n, R)}}$ and CS $\mathit{\boldsymbol{(p = 0.3)}}$ "
| n | CS | $T_0 = 0.6, p = 0.3$ | |||||
| $\widehat {\alpha }$ (MSE) | $\widehat {\beta }$ (MSE) | $\widehat {\lambda }$ (MSE) | $\widehat {b}$ (MSE) | $\widehat {R}_S $ (MSE) | $\widehat {h}_S $ (MSE) | ||
| (40, 20) | 1 | 0.342 3 | 0.733 7 | 0.480 5 | 2.357 2 | 0.838 0 | 0.599 3 |
| (0.044 8) | (0.080 1) | (0.569 3) | (1.972 7) | (0.007 2) | (0.059 2) | ||
| 2 | 0.259 7 | 0.691 5 | 0.397 6 | 1.782 3 | 0.909 4 | 0.457 2 | |
| (0.035 1) | (0.219 0) | (1.904 3) | (1.752 6) | (0.003 4) | (0.067 2) | ||
| 3 | 0.268 8 | 0.718 8 | 0.574 5 | 2.202 1 | 0.825 8 | 0.403 9 | |
| (0.028 9) | (0.086 8) | (1.326 3) | (1.802 2) | (0.002 6) | (0.041 1) | ||
| (40, 25) | 1 | 0.312 1 | 0.835 9 | 0.485 4 | 1.977 0 | 0.883 9 | 0.490 8 |
| (0.019 4) | (0.070 1) | (1.021 4) | (1.650 6) | (0.006 8) | (0.023 7) | ||
| 2 | 0.264 5 | 0.698 3 | 0.414 3 | 1.839 8 | 0.900 9 | 0.487 4 | |
| (0.023 9) | (0.195 1) | (1.846 0) | (1.500 2) | (0.003 1) | (0.066 9) | ||
| 3 | 0.288 2 | 0.720 7 | 0.557 4 | 2.211 8 | 0.824 0 | 0.424 9 | |
| (0.023 1) | (0.076 9) | (1.087 4) | (1.394 9) | (0.002 6) | (0.025 0) | ||
| (50, 35) | 1 | 0.296 1 | 0.837 1 | 0.456 2 | 2.001 9 | 0.881 8 | 0.450 4 |
| (0.012 6) | (0.028 6) | (0.516 5) | (1.411 1) | (0.002 6) | (0.017 9) | ||
| 2 | 0.261 9 | 0.754 7 | 0.511 6 | 1.861 9 | 0.895 8 | 0.484 5 | |
| (0.014 2) | (0.049 3) | (1.128 4) | (1.211 6) | (0.001 5) | (0.052 3) | ||
| 3 | 0.285 2 | 0.723 1 | 0.557 6 | 1.908 1 | 0.855 1 | 0.418 6 | |
| (0.014 6) | (0.030 5) | (0.572 0) | (1.303 4) | (0.002 3) | (0.023 5) | ||
Table 2
A-MLE of parameter and reliability index and MSE for different $\mathit{\boldsymbol{(n, R)}}$ and CS $\mathit{\boldsymbol{(p = 0.8)}}$ "
| n | CS | $T_0 = 0.6, p = 0.3$ | |||||
| $\widehat {\alpha }$ (MSE) | $\widehat {\beta }$ (MSE) | $\widehat {\lambda }$ (MSE) | $\widehat {b}$ (MSE) | $\widehat {R}_S $ (MSE) | $\widehat {h}_S $ (MSE) | ||
| (40, 20) | 1 | 0.444 6 | 0.875 2 | 0.450 1 | 2.162 1 | 0.892 4 | 0.394 0 |
| (0.082 0) | (0.084 1) | (1.043 2) | (2.571 3) | (0.008 1) | (0.033 2) | ||
| 2 | 0.239 7 | 0.745 6 | 0.572 0 | 1.849 7 | 0.912 7 | 0.308 7 | |
| (0.038 2) | (0.241 0) | (1.921 6) | (1.827 6) | (0.003 8) | (0.066 7) | ||
| 3 | 0.269 8 | 0.759 6 | 0.588 7 | 2.227 5 | 0.845 6 | 0.396 1 | |
| (0.032 4) | (0.087 1) | (1.674 6) | (1.857 8) | (0.002 8) | (0.051 4) | ||
| (40, 25) | 1 | 0.283 7 | 0.848 1 | 0.512 1 | 2.039 5 | 0.895 0 | 0.400 4 |
| (0.029 0) | (0.072 2) | (1.019 2) | (2.145 4) | (0.001 7) | (0.030 9) | ||
| 2 | 0.248 0 | 0.753 3 | 0.500 3 | 1.883 9 | 0.909 6 | 0.323 0 | |
| (0.024 2) | (0.224 5) | (1.903 9) | (1.792 1) | (0.003 4) | (0.051 1) | ||
| 3 | 0.226 0 | 0.755 9 | 0.541 1 | 2.076 7 | 0.869 3 | 0.414 6 | |
| (0.030 8) | (0.082 0) | (1.601 8) | (2.542 2) | (0.002 7) | (0.045 1) | ||
| (50, 35) | 1 | 0.266 5 | 0.856 8 | 0.454 1 | 1.980 0 | 0.895 0 | 0.414 2 |
| (0.012 7) | (0.030 1) | (0.589 4) | (1.515 5) | (0.003 6) | (0.023 2) | ||
| 2 | 0.241 5 | 0.832 9 | 0.542 6 | 1.970 6 | 0.901 6 | 0.369 8 | |
| (0.022 4) | (0.051 2) | (1.280 9) | (1.296 2) | (0.001 9) | (0.028 9) | ||
| 3 | 0.263 9 | 0.762 0 | 0.611 2 | 0.902 5 | 0.866 7 | 0.409 5 | |
| (0.015 7) | (0.037 5) | (0.785 3) | (2.298 6) | (0.003 4) | (0.038 6) | ||
Table 3
Average value of 95% CIs of parameters and corresponding CIL, CP $\mathit{\boldsymbol{(T_0 = 1.2, (n, R) = (50, 5))}}$ "
| MP | CS | Parameter | ACIs\CIL CP | Boot-p CIs \ CIL CP | Boot-t CIs \ CIL CP |
| p=0.3 | 1 | α | (0.112 1, 0.556 3)\0.444 1 95.0% | (0.111 0, 0.518 8)\0.407 7 95.6% | (0.191 4, 0.456 4)\0.264 9 95.4% |
| β | (0.250 9, 1.151 8)\0.900 8 95.3% | (0.610 4, 0.975 3)\0.364 9 95.7% | (0.760 9, 1.080 2)\0.319 2 95.4% | ||
| λ | ( – 0.282 5, 1.072 6)\1.355 1 95.0% | (0.145 9, 0.864 5)\0.718 6 96.4% | ( – 0.037 1, 0.819 1)\0.856 2 96.1% | ||
| b | ( – 0.308 7, 2.432 3)\2.741 8 95.0% | (1.183 1, 2.619 6)\1.436 5 95.6% | (1.371 0, 2.472 3)\1.101 3 95% | ||
| 2 | α | (0.051 5, 0.525 9)\0.474 3 94.9% | (0.106 4, 0.548 9)\0.442 5 95.0% | (0.254 5, 0.408 4)\0.153 8 95.0% | |
| β | (0.432 2, 0.926 2)\0.494 0 94.7% | (0.564 2, 0.945 6)\0.381 4 95.5% | (0.695 2, 0.962 0)\0.266 8 95.3% | ||
| λ | ( – 0.044 6, 1.201 3)\1.245 9 94.8% | (0.114 1, 0.854 9)\0.740 8 95.4% | (0.250 2, 0.830 7)\0.580 5 95.0% | ||
| b | (0.341 9, 2.793 7)\2.451 7 94.7% | (1.788 9, 2.475 3)\0.686 3 95.5% | (1.757 8, 2.399 2)\0.641 4 95.3% | ||
| 3 | α | (0.222 6, 0.570 4)\0.347 7 94.5% | (0.125 6, 0.547 8)\0.422 2 95.0% | (0.282 5, 0.416 4)\0.133 9 94.9% | |
| β | ( – 0.286 8, 1.722 1)\2.009 1 94.5% | (0.611 8, 0.985 9)\0.374 1 95.2% | (0.511 3, 0.891 6)\0.380 2 95.1% | ||
| λ | ( – 0.684 6, 1.798 4)\2.483 0 94.3% | (0.270 4, 0.849 1)\0.578 6 95.2% | (0.382 8, 0.636 5)\0.253 6 95.1% | ||
| b | (1.225 9, 2.720 5)\1.494 6 94.0% | (1.458 6, 2.275 7)\0.817 1 95.5% | (1.602 1, 2.349 7)\0.747 6 95.2% | ||
| p=0.8 | 1 | α | ( – 0.006 5, 0.857 7)\0.864 2 93.8% | (0.143 9, 0.639 9)\0.495 9 94.1% | (0.210 1, 0.357 1)\0.146 9 93.9% |
| β | (0.072 2, 2.713 6)\2.641 3 94.2% | (0.552 3, 1.065 1)\0.512 7 94.3% | (0.600 4, 0.940 6)\0.340 2 93.7% | ||
| λ | ( – 0.310 6, 1.362 4)\1.673 0 93.6% | ( – 0.009 9, 1.648 7)\1.658 6 96.1 % | ( – 0.468 3, 0.647 9)\1.116 3 94.1% | ||
| b | (0.302 1, 3.282 3)\2.980 2 92.2% | (1.380 4, 3.029 4)\1.648 9 92.7% | (1.561 5, 3.007 3)\1.445 8 92.3% | ||
| 2 | α | ( – 0.000 7, 0.566 1)\0.566 8 92.2% | (0.072 4, 0.606 1)\0.533 7 93.4% | (0.175 4, 0.364 7)\0.189 2 93.2% | |
| β | ( – 0.594 8, 1.400 8)\1.995 6 91.8% | (0.712 7, 1.777 3)\1.064 6 94.7% | (0.710 2, 1.015 9)\0.305 7 94.6% | ||
| λ | ( – 0.085 7, 1.765 1)\1.850 8 92.7% | (0.075 9, 1.588 5)\1.512 6 93.9% | (0.173 3, 0.779 1)\0.605 8 92.8% | ||
| b | ( – 0.301 2, 2.442 9)\2.744 1 92.6% | (1.092 5, 2.468 2)\1.375 7 92.8% | (1.616 3, 3.088 5)\1.472 2 92.6% | ||
| 3 | α | (0.222 9, 0.594 9)\0.372 0 93.3% | (0.119 8, 0.601 1)\0.481 2 93.9% | (0.209 4, 0.454 8)\0.245 3 93.4% | |
| β | ( – 0.582 7, 1.478 9)\2.061 6 93.1% | (0.388 1, 0.925 9)\0.537 8 94.7% | (0.505 6, 0.988 4)\0.482 9 94.1% | ||
| λ | ( – 0.585 6, 2.292 3)\2.877 9 93.7% | (0.127 3, 1.153 6)\1.026 3 94.6% | (0.301 1, 0.908 0)\0.606 8 94.3% | ||
| b | (0.937 1, 2.871 0)\1.933 9 93.1% | (1.208 2, 2.550 0)\1.341 7 94.2% | (1.158 7, 2.405 6)\1.246 8 94.0% |
Table 4
Optimal time of changing stress levels under the two optimalcriteria"
| CS | $p = 0.3$ | $p = 0.8$ | |||||
| $\tau _D^\ast $ | $\tau _A^\ast $ | $\tau _D^\ast $ | $\tau _A^\ast $ | ||||
| T0 = 0.6 | (40, 20) | 1 | 0.315 1 | 0.300 6 | 0.382 5 | 0.306 1 | |
| 2 | 0.373 3 | 0.293 1 | 0.340 1 | 0.362 0 | |||
| 3 | 0.385 5 | 0.308 6 | 0.381 8 | 0.299 3 | |||
| (40, 25) | 1 | 0.349 4 | 0.302 9 | 0.393 9 | 0.308 1 | ||
| 2 | 0.374 2 | 0.313 6 | 0.369 8 | 0.363 5 | |||
| 3 | 0.393 8 | 0.312 7 | 0.382 7 | 0.300 8 | |||
| (50, 35) | 1 | 0.388 2 | 0.311 6 | 0.411 5 | 0.318 3 | ||
| 2 | 0.392 3 | 0.357 4 | 0.377 6 | 0.370 3 | |||
| 3 | 0.391 3 | 0.317 0 | 0.383 9 | 0.304 9 | |||
| T0 = 1.2 | (40, 20) | 1 | 0.305 9 | 0.355 0 | 0.337 7 | 0.342 6 | |
| 2 | 0.325 8 | 0.342 3 | 0.338 1 | 0.375 4 | |||
| 3 | 0.306 3 | 0.342 5 | 0.353 9 | 0.330 2 | |||
| (40, 25) | 1 | 0.316 3 | 0.356 5 | 0.339 8 | 0.345 9 | ||
| 2 | 0.366 9 | 0.345 8 | 0.350 9 | 0.376 1 | |||
| 3 | 0.311 0 | 0.344 8 | 0.372 8 | 0.332 7 | |||
| (50, 35) | 1 | 0.302 9 | 0.362 6 | 0.344 6 | 0.351 8 | ||
| 2 | 0.375 1 | 0.358 3 | 0.365 3 | 0.377 1 | |||
| 3 | 0.334 8 | 0.347 4 | 0.379 7 | 0.332 5 | |||
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