Optimal Release Time and Sensitivity Analysis Using a New NHPP Software Reliability Model with Probability of Fault Removal Subject to Operating Environments
Abstract
:1. Introduction
2. NHPP SRM
2.1. NHPP SRM
2.2. New NHPP SRM
- The fault detection rate function will be affected by the probability of fault removal on a failure.
3. Optimal Software Release
4. Numerical Examples
4.1. Criteria for Model Comparison
4.2. Results
4.2.1. Model Analysis
4.2.2. Optimal Software Release
4.2.3. Sensitivity Analysis
5. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
Acronym
HPP | homogeneous Poisson process |
NHPP | nonhomogeneous Poisson process |
SRM | software reliability model |
FDR | fault detection rate |
LSE | least squares estimate |
MSE | mean squared error |
RMSE | root mean squared error |
AIC | Akaike’s information criterion |
R2 | correlation index of the regression curve equation |
Adj R2 | adjected R2 |
SAE | sum of absolute error |
PRR | predictive ratio risk |
PP | predictive power |
Appendix A
Model | Time Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
GO | LCL | 2.9752 | 11.0372 | 19.8294 | 28.9373 | 38.2010 | 47.5388 | 56.9029 | 66.2622 | 75.5959 |
11.8375 | 23.5326 | 35.0871 | 46.5026 | 57.7808 | 68.9234 | 79.9320 | 90.8081 | 101.5535 | ||
UCL | 20.6998 | 36.0281 | 50.3449 | 64.0679 | 77.3607 | 90.3080 | 102.9611 | 115.3541 | 127.5111 | |
DS | LCL | −1.4475 | 2.3988 | 9.8762 | 19.7193 | 30.9757 | 42.9376 | 55.0882 | 67.0585 | 78.5932 |
3.0539 | 10.9049 | 21.9419 | 34.9465 | 49.0079 | 63.4565 | 77.8095 | 91.7285 | 104.9858 | ||
UCL | 7.5552 | 19.4109 | 34.0077 | 50.1736 | 67.0402 | 83.9755 | 100.5308 | 116.3985 | 131.3784 | |
IS | LCL | −1.3876 | 0.7345 | 4.9084 | 11.5269 | 20.9997 | 33.4459 | 48.4297 | 64.8712 | 81.2785 |
3.2709 | 8.0368 | 14.8268 | 24.1977 | 36.5783 | 52.0250 | 69.9771 | 89.1986 | 108.0540 | ||
UCL | 7.9295 | 15.3391 | 24.7452 | 36.8685 | 52.1569 | 70.6040 | 91.5245 | 113.5260 | 134.8295 | |
Y1 | LCL | 2.9706 | 11.0274 | 19.8152 | 28.9198 | 38.1810 | 47.5174 | 56.8810 | 66.2410 | 75.5762 |
11.8302 | 23.5194 | 35.0690 | 46.4810 | 57.7568 | 68.8981 | 79.9065 | 90.7836 | 101.5309 | ||
UCL | 20.6898 | 36.0113 | 50.3229 | 64.0422 | 77.3325 | 90.2787 | 102.9319 | 115.3262 | 127.4856 | |
Y2 | LCL | 1.7621 | 8.6986 | 16.7396 | 25.3788 | 34.3917 | 43.6577 | 53.1057 | 62.6911 | 72.3848 |
9.8437 | 20.3058 | 31.1057 | 42.0900 | 53.1750 | 64.3150 | 75.4850 | 86.6714 | 97.8668 | ||
UCL | 17.9252 | 31.9130 | 45.4717 | 58.8011 | 71.9582 | 84.9723 | 97.8644 | 110.6518 | 123.3489 | |
PNZ | LCL | −1.3876 | 0.7345 | 4.9084 | 11.5269 | 20.9997 | 33.4459 | 48.4297 | 64.8712 | 81.2785 |
3.2709 | 8.0368 | 14.8268 | 24.1977 | 36.5783 | 52.0250 | 69.9771 | 89.1986 | 108.0540 | ||
UCL | 7.9295 | 15.3391 | 24.7452 | 36.8685 | 52.1569 | 70.6040 | 91.5245 | 113.5260 | 134.8295 | |
PZ | LCL | −1.3854 | 0.7450 | 4.9321 | 11.5697 | 21.0687 | 33.5484 | 48.5716 | 65.0558 | 81.5054 |
3.2788 | 8.0561 | 14.8623 | 24.2557 | 36.6659 | 52.1496 | 70.1448 | 89.4123 | 108.3130 | ||
UCL | 7.9429 | 15.3671 | 24.7926 | 36.9416 | 52.2632 | 70.7509 | 91.7180 | 113.7689 | 135.1205 | |
DP1 | LCL | −1.4743 | −1.4746 | −0.0008 | 2.9470 | 7.3689 | 13.2649 | 20.6349 | 29.4790 | 39.7971 |
0.7370 | 2.9481 | 6.6333 | 11.7925 | 18.4257 | 26.5330 | 36.1144 | 47.1699 | 59.6994 | ||
UCL | 2.9484 | 7.3708 | 13.2673 | 20.6379 | 29.4825 | 39.8012 | 51.5939 | 64.8607 | 79.6016 | |
LCL | 26.5001 | 27.7130 | 29.7569 | 32.6558 | 36.4370 | 41.1292 | 46.7608 | 53.3593 | 60.9505 | |
DP2 | 43.4862 | 44.9904 | 47.5118 | 51.0621 | 55.6529 | 61.2957 | 68.0019 | 75.7827 | 84.6494 | |
UCL | 60.4722 | 62.2677 | 65.2666 | 69.4684 | 74.8688 | 81.4623 | 89.2430 | 98.2062 | 108.3484 | |
TC | LCL | −1.4553 | −1.1784 | 2.2130 | 9.2349 | 19.8279 | 33.4224 | 49.0474 | 65.5033 | 81.5698 |
0.7003 | 3.9242 | 10.5989 | 21.0540 | 35.0853 | 51.9963 | 70.7069 | 89.9303 | 108.3864 | ||
UCL | 2.8559 | 9.0268 | 18.9847 | 32.8732 | 50.3427 | 70.5702 | 92.3664 | 114.3573 | 135.2031 | |
3PD | LCL | −1.3728 | 0.7935 | 5.0173 | 11.6787 | 21.1710 | 33.5985 | 48.5203 | 64.8703 | 81.1846 |
3.3220 | 8.1446 | 14.9902 | 24.4031 | 36.7959 | 52.2106 | 70.0842 | 89.1976 | 107.9468 | ||
UCL | 8.0167 | 15.4957 | 24.9631 | 37.1276 | 52.4208 | 70.8228 | 91.6481 | 113.5249 | 134.7090 | |
NW | LCL | −1.5187 | 1.1251 | 6.4796 | 14.1086 | 23.7628 | 35.2697 | 48.4822 | 63.2088 | 79.0574 |
2.7625 | 8.7404 | 17.1454 | 27.6542 | 40.0675 | 54.2401 | 70.0392 | 87.2721 | 105.5167 | ||
UCL | 7.0437 | 16.3556 | 27.8111 | 41.1998 | 56.3722 | 73.2106 | 91.5961 | 111.3353 | 131.9759 | |
New | LCL | 2.5514 | 6.0537 | 10.3882 | 16.0978 | 23.7208 | 33.8252 | 46.8738 | 62.8212 | 80.4445 |
11.1541 | 16.5245 | 22.6460 | 30.2694 | 40.0147 | 52.4864 | 68.1359 | 86.8224 | 107.1017 | ||
UCL | 19.7568 | 26.9953 | 34.9038 | 44.4409 | 56.3087 | 71.1476 | 89.3979 | 110.8236 | 133.7590 | |
Model | Time Index | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
GO | LCL | 84.8889 | 94.1304 | 103.3121 | 112.4278 | 121.4725 | 130.4426 | 139.3350 | 148.1476 | 156.8786 |
112.1696 | 122.6580 | 133.0203 | 143.2580 | 153.3725 | 163.3654 | 173.2381 | 182.9920 | 192.6287 | ||
UCL | 139.4502 | 151.1856 | 162.7285 | 174.0882 | 185.2725 | 196.2882 | 207.1411 | 217.8365 | 228.3788 | |
DS | LCL | 89.5235 | 99.7463 | 109.2071 | 117.8870 | 125.7931 | 132.9502 | 139.3951 | 145.1722 | 150.3299 |
117.4374 | 129.0024 | 139.6461 | 149.3678 | 158.1903 | 166.1527 | 173.3047 | 179.7020 | 185.4031 | ||
UCL | 145.3513 | 158.2584 | 170.0852 | 180.8486 | 190.5874 | 199.3552 | 207.2143 | 214.2317 | 220.4763 | |
IS | LCL | 96.2253 | 108.7771 | 118.6274 | 125.9585 | 131.2036 | 134.8518 | 137.3398 | 139.0140 | 140.1303 |
125.0271 | 139.1636 | 150.1952 | 158.3745 | 164.2116 | 168.2647 | 171.0257 | 172.8822 | 174.1195 | ||
UCL | 153.8288 | 169.5500 | 181.7631 | 190.7905 | 197.2196 | 201.6776 | 204.7116 | 206.7504 | 208.1087 | |
Y1 | LCL | 84.8718 | 94.1167 | 103.3028 | 112.4238 | 121.4747 | 130.4518 | 139.3522 | 148.1736 | 156.9141 |
112.1500 | 122.6425 | 133.0098 | 143.2535 | 153.3750 | 163.3757 | 173.2572 | 183.0207 | 192.6678 | ||
UCL | 139.4283 | 151.1683 | 162.7169 | 174.0832 | 185.2752 | 196.2996 | 207.1621 | 217.8679 | 228.4216 | |
Y2 | LCL | 82.1664 | 92.0215 | 101.9394 | 111.9118 | 121.9320 | 131.9949 | 142.0961 | 152.2320 | 162.3994 |
109.0671 | 120.2701 | 131.4745 | 142.6797 | 153.8853 | 165.0912 | 176.2972 | 187.5033 | 198.7094 | ||
UCL | 135.9678 | 148.5186 | 161.0095 | 173.4476 | 185.8386 | 198.1875 | 210.4983 | 222.7746 | 235.0194 | |
PNZ | LCL | 96.2253 | 108.7771 | 118.6274 | 125.9585 | 131.2036 | 134.8518 | 137.3398 | 139.0140 | 140.1303 |
125.0271 | 139.1636 | 150.1952 | 158.3745 | 164.2116 | 168.2647 | 171.0257 | 172.8822 | 174.1195 | ||
UCL | 153.8288 | 169.5500 | 181.7631 | 190.7905 | 197.2196 | 201.6776 | 204.7116 | 206.7504 | 208.1087 | |
PZ | LCL | 96.4904 | 109.0742 | 118.9495 | 126.2992 | 131.5576 | 135.2151 | 137.7094 | 139.3877 | 140.5069 |
125.3267 | 139.4970 | 150.5551 | 158.7540 | 164.6051 | 168.6679 | 171.4356 | 173.2965 | 174.5367 | ||
UCL | 154.1629 | 169.9199 | 182.1608 | 191.2088 | 197.6527 | 202.1208 | 205.1618 | 207.2052 | 208.5666 | |
DP1 | LCL | 51.5893 | 64.8556 | 79.5959 | 95.8102 | 113.4987 | 132.6612 | 153.2977 | 175.4083 | 198.9930 |
73.7029 | 89.1805 | 106.1322 | 124.5579 | 144.4577 | 165.8316 | 188.6795 | 213.0014 | 238.7974 | ||
UCL | 95.8165 | 113.5055 | 132.6685 | 153.3056 | 175.4167 | 199.0019 | 224.0612 | 250.5945 | 278.6019 | |
LCL | 69.5581 | 79.2041 | 89.9088 | 101.6907 | 114.5672 | 128.5544 | 143.6675 | 159.9207 | 177.3277 | |
DP2 | 94.6130 | 105.6844 | 117.8746 | 131.1943 | 145.6542 | 161.2649 | 178.0369 | 195.9805 | 215.1061 | |
UCL | 119.6678 | 132.1647 | 145.8404 | 160.6978 | 176.7412 | 193.9754 | 212.4063 | 232.0403 | 252.8845 | |
TC | LCL | 96.2056 | 108.6911 | 118.6868 | 126.2064 | 131.5257 | 135.0653 | 137.2814 | 138.5872 | 139.3112 |
125.0048 | 139.0670 | 150.2617 | 158.6507 | 164.5696 | 168.5016 | 170.9609 | 172.4090 | 173.2117 | ||
UCL | 153.8040 | 169.4429 | 181.8365 | 191.0950 | 197.6135 | 201.9380 | 204.6404 | 206.2308 | 207.1121 | |
3PD | LCL | 96.0649 | 108.5914 | 118.4569 | 125.8343 | 131.1467 | 134.8761 | 137.4559 | 139.2308 | 140.4556 |
124.8457 | 138.9551 | 150.0047 | 158.2361 | 164.1483 | 168.2916 | 171.1544 | 173.1226 | 174.4800 | ||
UCL | 153.6266 | 169.3188 | 181.5526 | 190.6380 | 197.1499 | 201.7072 | 204.8530 | 207.0143 | 208.5043 | |
NW | LCL | 95.1400 | 109.8266 | 121.2536 | 128.6725 | 132.8455 | 135.0189 | 136.1257 | 136.6943 | 136.9930 |
123.8001 | 140.3413 | 153.1282 | 161.3963 | 166.0364 | 168.4501 | 169.6787 | 170.3096 | 170.6410 | ||
UCL | 152.4602 | 170.8561 | 185.0027 | 194.1201 | 199.2273 | 201.8814 | 203.2316 | 203.9248 | 204.2890 | |
New | LCL | 97.1777 | 110.5793 | 120.0313 | 126.3680 | 130.6527 | 133.6536 | 135.8425 | 137.4998 | 138.7947 |
126.1032 | 141.1857 | 151.7636 | 158.8307 | 163.5991 | 166.9341 | 169.3644 | 171.2031 | 172.6390 | ||
UCL | 155.0286 | 171.7921 | 183.4958 | 191.2934 | 196.5454 | 200.2146 | 202.8863 | 204.9064 | 206.4834 |
Model | Time Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
GO | LCL | 44.1844 | 81.6849 | 107.5241 | 125.0619 | 136.9017 | 144.8746 | 150.2361 | 153.8388 | 156.2583 |
64.9422 | 108.5177 | 137.7566 | 157.3755 | 170.5397 | 179.3727 | 185.2996 | 189.2764 | 191.9449 | ||
UCL | 85.6999 | 135.3506 | 167.9890 | 189.6892 | 204.1776 | 213.8708 | 220.3630 | 224.7141 | 227.6315 | |
DS | LCL | 25.7372 | 75.4099 | 111.9612 | 133.6497 | 145.3218 | 151.2622 | 154.1784 | 155.5743 | 156.2301 |
42.5368 | 101.3403 | 142.7351 | 166.9298 | 179.8675 | 186.4327 | 189.6511 | 191.1907 | 191.9138 | ||
UCL | 59.3364 | 127.2706 | 173.5090 | 200.2099 | 214.4131 | 221.6031 | 225.1239 | 226.8071 | 227.5975 | |
IS | LCL | 44.1752 | 81.6689 | 107.5036 | 125.0383 | 136.8760 | 144.8475 | 150.2081 | 153.8101 | 156.2292 |
64.9313 | 108.4996 | 137.7335 | 157.3492 | 170.5111 | 179.3427 | 185.2686 | 189.2448 | 191.9128 | ||
UCL | 85.6873 | 135.3302 | 167.9634 | 189.6601 | 204.1463 | 213.8379 | 220.3291 | 224.6795 | 227.5964 | |
Y1 | LCL | 47.0719 | 84.4861 | 108.9555 | 124.8918 | 135.3779 | 142.4181 | 147.2900 | 150.8015 | 153.4618 |
68.3705 | 111.7110 | 139.3638 | 157.1859 | 168.8487 | 176.6538 | 182.0440 | 185.9239 | 188.8605 | ||
UCL | 89.6692 | 138.9358 | 169.7721 | 189.4801 | 202.3195 | 210.8894 | 216.7980 | 221.0463 | 224.2593 | |
Y2 | LCL | 46.9038 | 84.3161 | 108.8575 | 124.8826 | 135.4480 | 142.5476 | 147.4564 | 150.9836 | 153.6403 |
68.1714 | 111.5173 | 139.2538 | 157.1757 | 168.9265 | 176.7971 | 182.2280 | 186.1250 | 189.0574 | ||
UCL | 89.4390 | 138.7185 | 169.6501 | 189.4688 | 202.4050 | 211.0466 | 216.9996 | 221.2664 | 224.4746 | |
PNZ | LCL | 46.8632 | 84.2786 | 108.8405 | 124.8881 | 135.4715 | 142.5829 | 147.4972 | 151.0244 | 153.6770 |
68.1233 | 111.4746 | 139.2347 | 157.1818 | 168.9526 | 176.8362 | 182.2731 | 186.1701 | 189.0980 | ||
UCL | 89.3834 | 138.6706 | 169.6289 | 189.4755 | 202.4337 | 211.0895 | 217.0490 | 221.3158 | 224.5189 | |
PZ | LCL | 44.1050 | 81.6039 | 107.4509 | 124.9995 | 136.8503 | 144.8330 | 150.2028 | 153.8121 | 156.2368 |
64.8476 | 108.4253 | 137.6743 | 157.3060 | 170.4827 | 179.3267 | 185.2628 | 189.2470 | 191.9212 | ||
UCL | 85.5903 | 135.2468 | 167.8978 | 189.6125 | 204.1150 | 213.8204 | 220.3227 | 224.6820 | 227.6056 | |
DP1 | LCL | −1.5930 | −1.0577 | 1.6061 | 6.3983 | 13.3188 | 22.3677 | 33.5451 | 46.8508 | 62.2849 |
1.0642 | 4.2568 | 9.5778 | 17.0271 | 26.6049 | 38.3111 | 52.1456 | 68.1086 | 86.1999 | ||
UCL | 3.7214 | 9.5712 | 17.5494 | 27.6560 | 39.8910 | 54.2544 | 70.7462 | 89.3664 | 110.1149 | |
LCL | 115.9952 | 116.6742 | 117.8113 | 119.4110 | 121.4782 | 124.0181 | 127.0358 | 130.5368 | 134.5268 | |
DP2 | 147.2522 | 148.0118 | 149.2832 | 151.0707 | 153.3789 | 156.2120 | 159.5743 | 163.4702 | 167.9039 | |
UCL | 178.5093 | 179.3493 | 180.7550 | 182.7305 | 185.2796 | 188.4060 | 192.1129 | 196.4036 | 201.2809 | |
TC | LCL | 55.3076 | 86.9606 | 107.6686 | 122.0984 | 132.5186 | 140.2258 | 146.0275 | 150.4553 | 153.8723 |
78.0664 | 114.5264 | 137.9188 | 154.0710 | 165.6732 | 174.2253 | 180.6481 | 185.5416 | 189.3135 | ||
UCL | 100.8252 | 142.0921 | 168.1690 | 186.0435 | 198.8277 | 208.2248 | 215.2686 | 220.6279 | 224.7546 | |
3PD | LCL | 52.2342 | 86.8857 | 108.5907 | 123.1337 | 133.3574 | 140.7969 | 146.3505 | 150.5778 | 153.8456 |
74.4613 | 114.4412 | 138.9543 | 155.2258 | 166.6051 | 174.8581 | 181.0052 | 185.6769 | 189.2840 | ||
UCL | 96.6883 | 141.9967 | 169.3178 | 187.3180 | 199.8527 | 208.9193 | 215.6600 | 220.7760 | 224.7224 | |
NW | LCL | 57.6766 | 83.8305 | 103.7251 | 120.1147 | 133.6045 | 143.9716 | 151.0377 | 155.2967 | 157.6696 |
80.8355 | 110.9641 | 133.4851 | 151.8567 | 166.8795 | 178.3735 | 186.1848 | 190.8846 | 193.5005 | ||
UCL | 103.9944 | 138.0978 | 163.2452 | 183.5986 | 200.1545 | 212.7754 | 221.3319 | 226.4725 | 229.3315 | |
New | LCL | 64.4950 | 79.9166 | 97.6041 | 117.6815 | 135.8098 | 146.3428 | 151.6415 | 154.6909 | 156.6830 |
88.7630 | 106.4987 | 126.5848 | 149.1380 | 169.3281 | 180.9967 | 186.8515 | 190.2164 | 192.4131 | ||
UCL | 113.0309 | 133.0808 | 155.5654 | 180.5946 | 202.8464 | 215.6506 | 222.0614 | 225.7420 | 228.1432 | |
Model | Time Index | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |
GO | LCL | 157.8827 | 158.9731 | 159.7050 | 160.1961 | 160.5257 | 160.7469 | 160.8953 | 160.9949 | |
193.7354 | 194.9368 | 195.7429 | 196.2839 | 196.6468 | 196.8903 | 197.0537 | 197.1634 | |||
UCL | 229.5881 | 230.9005 | 231.7809 | 232.3716 | 232.7679 | 233.0338 | 233.2122 | 233.3319 | ||
DS | LCL | 156.5338 | 156.6729 | 156.7360 | 156.7644 | 156.7771 | 156.7828 | 156.7853 | 156.7864 | |
192.2487 | 192.4020 | 192.4715 | 192.5028 | 192.5169 | 192.5231 | 192.5258 | 192.5271 | |||
UCL | 227.9635 | 228.1311 | 228.2071 | 228.2413 | 228.2566 | 228.2634 | 228.2664 | 228.2677 | ||
IS | LCL | 157.8533 | 158.9436 | 159.6753 | 160.1663 | 160.4959 | 160.7170 | 160.8654 | 160.9650 | |
193.7030 | 194.9042 | 195.7102 | 196.2510 | 196.6139 | 196.8574 | 197.0208 | 197.1304 | |||
UCL | 229.5527 | 230.8649 | 231.7452 | 232.3357 | 232.7320 | 232.9978 | 233.1762 | 233.2959 | ||
Y1 | LCL | 155.5912 | 157.3908 | 158.9870 | 160.4591 | 161.8567 | 163.2112 | 164.5421 | 165.8618 | |
191.2094 | 193.1933 | 194.9521 | 196.5734 | 198.1121 | 199.6027 | 201.0668 | 202.5181 | |||
UCL | 226.8275 | 228.9958 | 230.9172 | 232.6877 | 234.3675 | 235.9942 | 237.5916 | 239.1745 | ||
Y2 | LCL | 155.7490 | 157.5131 | 159.0605 | 160.4716 | 161.7971 | 163.0690 | 164.3073 | 165.5246 | |
191.3834 | 193.3281 | 195.0330 | 196.5872 | 198.0465 | 199.4463 | 200.8086 | 202.1473 | |||
UCL | 227.0178 | 229.1431 | 231.0055 | 232.7027 | 234.2959 | 235.8236 | 237.3099 | 238.7701 | ||
PNZ | LCL | 155.7784 | 157.5327 | 159.0686 | 160.4670 | 161.7790 | 163.0367 | 164.2604 | 165.4628 | |
191.4158 | 193.3497 | 195.0419 | 196.5821 | 198.0265 | 199.4107 | 200.7570 | 202.0794 | |||
UCL | 227.0532 | 229.1667 | 231.0153 | 232.6972 | 234.2741 | 235.7848 | 237.2536 | 238.6960 | ||
PZ | LCL | 157.8652 | 158.9586 | 159.6927 | 160.1855 | 160.5163 | 160.7384 | 160.8874 | 160.9874 | |
193.7161 | 194.9208 | 195.7294 | 196.2721 | 196.6364 | 196.8809 | 197.0450 | 197.1552 | |||
UCL | 229.5670 | 230.8830 | 231.7661 | 232.3588 | 232.7565 | 233.0235 | 233.2026 | 233.3229 | ||
DP1 | LCL | 79.8474 | 99.5383 | 121.3576 | 145.3053 | 171.3814 | 199.5859 | 229.9187 | 262.3800 | |
106.4196 | 128.7678 | 153.2443 | 179.8492 | 208.5825 | 239.4442 | 272.4343 | 307.5528 | |||
UCL | 132.9919 | 157.9972 | 185.1310 | 214.3931 | 245.7836 | 279.3026 | 314.9499 | 352.7256 | ||
LCL | 139.0116 | 143.9968 | 149.4885 | 155.4925 | 162.0147 | 169.0611 | 176.6375 | 184.7497 | ||
DP2 | 172.8795 | 178.4014 | 184.4737 | 191.1005 | 198.2860 | 206.0343 | 214.3493 | 223.2353 | ||
UCL | 206.7475 | 212.8060 | 219.4589 | 226.7086 | 234.5573 | 243.0075 | 252.0612 | 261.7209 | ||
TC | LCL | 156.5339 | 158.6235 | 160.2755 | 161.5893 | 162.6400 | 163.4841 | 164.1655 | 164.7175 | |
192.2487 | 194.5517 | 196.3712 | 197.8178 | 198.9741 | 199.9030 | 200.6526 | 201.2598 | |||
UCL | 227.9636 | 230.4798 | 232.4670 | 234.0462 | 235.3083 | 236.3219 | 237.1397 | 237.8021 | ||
3PD | LCL | 156.4035 | 158.4271 | 160.0431 | 161.3445 | 162.4010 | 163.2653 | 163.9776 | 164.5692 | |
192.1050 | 194.3353 | 196.1153 | 197.5483 | 198.7112 | 199.6622 | 200.4460 | 201.0966 | |||
UCL | 227.8065 | 230.2434 | 232.1875 | 233.7520 | 235.0213 | 236.0592 | 236.9143 | 237.6241 | ||
NW | LCL | 158.9588 | 159.6668 | 160.0664 | 160.2998 | 160.4407 | 160.5285 | 160.5848 | 160.6220 | |
194.9210 | 195.7008 | 196.1410 | 196.3980 | 196.5531 | 196.6498 | 196.7119 | 196.7528 | |||
UCL | 230.8833 | 231.7349 | 232.2156 | 232.4962 | 232.6656 | 232.7712 | 232.8389 | 232.8836 | ||
New | LCL | 158.0961 | 159.1551 | 159.9802 | 160.6421 | 161.1855 | 161.6398 | 162.0255 | 162.3571 | |
193.9706 | 195.1372 | 196.0460 | 196.7749 | 197.3732 | 197.8733 | 198.2978 | 198.6628 | |||
UCL | 229.8450 | 231.1194 | 232.1118 | 232.9078 | 233.5609 | 234.1069 | 234.5702 | 234.9686 |
Model | Time Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
GO | LCL | −1.6271 | −1.0439 | −0.0990 | 1.0319 | 2.2808 | 3.6126 | 5.0067 |
2.1481 | 4.2932 | 6.4354 | 8.5746 | 10.7108 | 12.8440 | 14.9743 | ||
UCL | 5.9234 | 9.6304 | 12.9698 | 16.1172 | 19.1407 | 22.0754 | 24.9418 | |
DS | LCL | −1.2338 | −1.6546 | −1.4302 | −0.7010 | 0.4162 | 1.8246 | 3.4442 |
0.4045 | 1.4969 | 3.1185 | 5.1372 | 7.4441 | 9.9496 | 12.5804 | ||
UCL | 2.0428 | 4.6484 | 7.6671 | 10.9754 | 14.4719 | 18.0745 | 21.7166 | |
IS | LCL | −1.5571 | −1.6348 | −1.3334 | −0.7248 | 0.1772 | 1.3684 | 2.8391 |
0.9391 | 2.0807 | 3.4533 | 5.0820 | 6.9847 | 9.1674 | 11.6194 | ||
UCL | 3.4352 | 5.7962 | 8.2399 | 10.8888 | 13.7923 | 16.9664 | 20.3997 | |
Y1 | LCL | −1.6486 | −1.1940 | −0.3781 | 0.6349 | 1.7818 | 3.0308 | 4.3631 |
1.9283 | 3.8795 | 5.8543 | 7.8533 | 9.8771 | 11.9263 | 14.0014 | ||
UCL | 5.5051 | 8.9530 | 12.0867 | 15.0718 | 17.9724 | 20.8217 | 23.6398 | |
Y2 | LCL | −1.6586 | −1.3833 | −0.7786 | 0.0033 | 0.9039 | 1.8930 | 2.9526 |
1.6343 | 3.2861 | 4.9553 | 6.6416 | 8.3448 | 10.0648 | 11.8014 | ||
UCL | 4.9272 | 7.9555 | 10.6892 | 13.2798 | 15.7857 | 18.2367 | 20.6502 | |
PNZ | LCL | −1.5571 | −1.6348 | −1.3332 | −0.7242 | 0.1783 | 1.3704 | 2.8425 |
0.9392 | 2.0809 | 3.4539 | 5.0833 | 6.9870 | 9.1710 | 11.6248 | ||
UCL | 3.4354 | 5.7967 | 8.2410 | 10.8908 | 13.7956 | 16.9715 | 20.4072 | |
PZ | LCL | −1.5571 | −1.6348 | −1.3334 | −0.7248 | 0.1772 | 1.3684 | 2.8391 |
0.9391 | 2.0807 | 3.4533 | 5.0820 | 6.9847 | 9.1674 | 11.6194 | ||
UCL | 3.4352 | 5.7962 | 8.2399 | 10.8888 | 13.7923 | 16.9664 | 20.3997 | |
DP1 | LCL | −0.7849 | −1.3204 | −1.6065 | −1.6432 | −1.4305 | −0.9684 | −0.2569 |
0.1247 | 0.4988 | 1.1223 | 1.9952 | 3.1175 | 4.4892 | 6.1103 | ||
UCL | 1.0343 | 2.3180 | 3.8511 | 5.6336 | 7.6655 | 9.9468 | 12.4775 | |
LCL | 0.8976 | 1.0355 | 1.2725 | 1.6168 | 2.0776 | 2.6654 | 3.3907 | |
DP2 | 8.3334 | 8.5809 | 9.0000 | 9.5960 | 10.3741 | 11.3392 | 12.4963 | |
UCL | 15.7692 | 16.1263 | 16.7275 | 17.5753 | 18.6705 | 20.0130 | 21.6018 | |
TC | LCL | −1.1910 | −1.6387 | −1.5204 | −0.9189 | 0.0932 | 1.4453 | 3.0679 |
0.3649 | 1.3143 | 2.7550 | 4.6141 | 6.8200 | 9.3009 | 11.9855 | ||
UCL | 1.9208 | 4.2672 | 7.0304 | 10.1471 | 13.5468 | 17.1565 | 20.9030 | |
3PD | LCL | −1.5571 | −1.6348 | −1.3332 | −0.7245 | 0.1776 | 1.3690 | 2.8398 |
0.9392 | 2.0810 | 3.4538 | 5.0827 | 6.9856 | 9.1684 | 11.6206 | ||
UCL | 3.4356 | 5.7968 | 8.2408 | 10.8899 | 13.7936 | 16.9679 | 20.4013 | |
NW | LCL | −1.5127 | −1.6433 | −1.3007 | −0.6009 | 0.3823 | 1.6042 | 3.0345 |
0.8205 | 1.9940 | 3.5580 | 5.3658 | 7.3796 | 9.5745 | 11.9322 | ||
UCL | 3.1537 | 5.6314 | 8.4167 | 11.3325 | 14.3770 | 17.5448 | 20.8300 | |
New | LCL | −1.6104 | −1.3350 | −0.9086 | −0.3265 | 0.4400 | 1.4274 | 2.6738 |
2.2732 | 3.4481 | 4.6398 | 5.9641 | 7.4891 | 9.2699 | 11.3527 | ||
UCL | 6.1567 | 8.2312 | 10.1882 | 12.2546 | 14.5381 | 17.1125 | 20.0317 | |
Model | Time Index | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
GO | LCL | 6.4495 | 7.9316 | 9.4462 | 10.9881 | 12.5533 | 14.1386 | 15.7414 |
17.1016 | 19.2259 | 21.3473 | 23.4658 | 25.5813 | 27.6939 | 29.8035 | ||
UCL | 27.7537 | 30.5203 | 33.2485 | 35.9435 | 38.6093 | 41.2492 | 43.8656 | |
DS | LCL | 5.2094 | 7.0666 | 8.9726 | 10.8928 | 12.7995 | 14.6715 | 16.4922 |
15.2774 | 17.9927 | 20.6888 | 23.3359 | 25.9113 | 28.3980 | 30.7837 | ||
UCL | 25.3453 | 28.9189 | 32.4049 | 35.7790 | 39.0231 | 42.1246 | 45.0752 | |
IS | LCL | 4.5662 | 6.5093 | 8.6112 | 10.8022 | 13.0068 | 15.1524 | 17.1769 |
14.3103 | 17.1884 | 20.1834 | 23.2124 | 26.1886 | 29.0312 | 31.6735 | ||
UCL | 24.0544 | 27.8675 | 31.7555 | 35.6225 | 39.3704 | 42.9099 | 46.1700 | |
Y1 | LCL | 5.7667 | 7.2337 | 8.7583 | 10.3364 | 11.9650 | 13.6419 | 15.3652 |
16.1032 | 18.2323 | 20.3893 | 22.5750 | 24.7900 | 27.0349 | 29.3106 | ||
UCL | 26.4397 | 29.2309 | 32.0204 | 34.8136 | 37.6149 | 40.4280 | 43.2559 | |
Y2 | LCL | 4.0712 | 5.2405 | 6.4546 | 7.7089 | 9.0000 | 10.3250 | 11.6816 |
13.5545 | 15.3237 | 17.1090 | 18.9101 | 20.7270 | 22.5594 | 24.4071 | ||
UCL | 23.0377 | 25.4069 | 27.7634 | 30.1113 | 32.4539 | 34.7937 | 37.1326 | |
PNZ | LCL | 4.5713 | 6.5167 | 8.6214 | 10.8157 | 13.0240 | 15.1738 | 17.2027 |
14.3180 | 17.1991 | 20.1976 | 23.2307 | 26.2116 | 29.0592 | 31.7069 | ||
UCL | 24.0648 | 27.8815 | 31.7739 | 35.6458 | 39.3991 | 42.9446 | 46.2111 | |
PZ | LCL | 4.5662 | 6.5093 | 8.6112 | 10.8022 | 13.0068 | 15.1524 | 17.1769 |
14.3103 | 17.1884 | 20.1834 | 23.2124 | 26.1886 | 29.0312 | 31.6735 | ||
UCL | 24.0543 | 27.8675 | 31.7555 | 35.6225 | 39.3704 | 42.9099 | 46.1700 | |
DP1 | LCL | 0.7040 | 1.9143 | 3.3740 | 5.0831 | 7.0416 | 9.2495 | 11.7068 |
7.9808 | 10.1007 | 12.4700 | 15.0887 | 17.9568 | 21.0743 | 24.4412 | ||
UCL | 15.2576 | 18.2871 | 21.5660 | 25.0943 | 28.8720 | 32.8991 | 37.1756 | |
LCL | 4.2639 | 5.2952 | 6.4939 | 7.8690 | 9.4287 | 11.1804 | 13.1313 | |
DP2 | 13.8500 | 15.4051 | 17.1661 | 19.1373 | 21.3231 | 23.7275 | 26.3548 | |
UCL | 23.4361 | 25.5151 | 27.8383 | 30.4056 | 33.2174 | 36.2746 | 39.5783 | |
TC | LCL | 4.8937 | 6.8584 | 8.9028 | 10.9735 | 13.0240 | 15.0150 | 16.9149 |
14.8047 | 17.6932 | 20.5914 | 23.4460 | 26.2115 | 28.8504 | 31.3334 | ||
UCL | 24.7156 | 28.5280 | 32.2799 | 35.9184 | 39.3990 | 42.6859 | 45.7519 | |
3PD | LCL | 4.5670 | 6.5101 | 8.6119 | 10.8028 | 13.0072 | 15.1525 | 17.1767 |
14.3115 | 17.1895 | 20.1844 | 23.2131 | 26.1890 | 29.0313 | 31.6732 | ||
UCL | 24.0560 | 27.8689 | 31.7568 | 35.6235 | 39.3709 | 42.9100 | 46.1697 | |
NW | LCL | 4.6512 | 6.4371 | 8.3780 | 10.4599 | 12.6649 | 14.9604 | 17.2800 |
14.4391 | 17.0837 | 19.8559 | 22.7443 | 25.7309 | 28.7786 | 31.8071 | ||
UCL | 24.2269 | 27.7302 | 31.3337 | 35.0287 | 38.7970 | 42.5968 | 46.3342 | |
New | LCL | 4.2106 | 6.0509 | 8.1732 | 10.5083 | 12.9377 | 15.3161 | 17.5104 |
13.7684 | 16.5204 | 19.5674 | 22.8105 | 26.0962 | 29.2462 | 32.1055 | ||
UCL | 23.3263 | 26.9899 | 30.9616 | 35.1128 | 39.2547 | 43.1762 | 46.7006 | |
Model | Time Index | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
GO | LCL | 17.3596 | 18.9914 | 20.6353 | 22.2899 | 23.9543 | 25.6274 | 27.3083 |
31.9102 | 34.0140 | 36.1149 | 38.2128 | 40.3078 | 42.4000 | 44.4892 | ||
UCL | 46.4609 | 49.0366 | 51.5945 | 54.1357 | 56.6614 | 59.1725 | 61.6700 | |
DS | LCL | 18.2495 | 19.9343 | 21.5408 | 23.0652 | 24.5055 | 25.8613 | 27.1334 |
33.0599 | 35.2212 | 37.2650 | 39.1905 | 40.9985 | 42.6914 | 44.2723 | ||
UCL | 47.8703 | 50.5081 | 52.9892 | 55.3157 | 57.4916 | 59.5216 | 61.4112 | |
IS | LCL | 19.0338 | 20.6944 | 22.1468 | 23.3929 | 24.4447 | 25.3206 | 26.0416 |
34.0684 | 36.1901 | 38.0319 | 39.6028 | 40.9225 | 42.0173 | 42.9159 | ||
UCL | 49.1031 | 51.6858 | 53.9170 | 55.8126 | 57.4002 | 58.7140 | 59.7903 | |
Y1 | LCL | 17.1339 | 18.9469 | 20.8036 | 22.7036 | 24.6466 | 26.6326 | 28.6614 |
31.6176 | 33.9569 | 36.3291 | 38.7349 | 41.1752 | 43.6507 | 46.1623 | ||
UCL | 46.1014 | 48.9669 | 51.8545 | 54.7662 | 57.7038 | 60.6689 | 63.6633 | |
Y2 | LCL | 13.0678 | 14.4820 | 15.9228 | 17.3890 | 18.8795 | 20.3933 | 21.9297 |
26.2700 | 28.1480 | 30.0408 | 31.9483 | 33.8703 | 35.8068 | 37.7574 | ||
UCL | 39.4723 | 41.8140 | 44.1588 | 46.5076 | 48.8612 | 51.2202 | 53.5851 | |
PNZ | LCL | 19.0643 | 20.7297 | 22.1869 | 23.4379 | 24.4946 | 25.3752 | 26.1009 |
34.1076 | 36.2350 | 38.0827 | 39.6594 | 40.9849 | 42.0855 | 42.9898 | ||
UCL | 49.1508 | 51.7404 | 53.9784 | 55.8809 | 57.4753 | 58.7957 | 59.8786 | |
PZ | LCL | 19.0338 | 20.6944 | 22.1468 | 23.3929 | 24.4447 | 25.3206 | 26.0416 |
34.0684 | 36.1901 | 38.0319 | 39.6028 | 40.9225 | 42.0173 | 42.9159 | ||
UCL | 49.1031 | 51.6858 | 53.9170 | 55.8126 | 57.4002 | 58.7140 | 59.7903 | |
DP1 | LCL | 14.4135 | 17.3696 | 20.5751 | 24.0301 | 27.7344 | 31.6881 | 35.8912 |
28.0575 | 31.9232 | 36.0384 | 40.4029 | 45.0168 | 49.8801 | 54.9928 | ||
UCL | 41.7015 | 46.4768 | 51.5016 | 56.7757 | 62.2992 | 68.0721 | 74.0944 | |
LCL | 15.2877 | 17.6556 | 20.2407 | 23.0483 | 26.0832 | 29.3501 | 32.8534 | |
DP2 | 29.2088 | 32.2933 | 35.6123 | 39.1692 | 42.9677 | 47.0112 | 51.3031 | |
UCL | 43.1299 | 46.9311 | 50.9838 | 55.2901 | 59.8522 | 64.6723 | 69.7527 | |
TC | LCL | 18.6995 | 20.3518 | 21.8610 | 23.2220 | 24.4349 | 25.5033 | 26.4341 |
33.6392 | 35.7538 | 37.6704 | 39.3879 | 40.9101 | 42.2452 | 43.4041 | ||
UCL | 48.5788 | 51.1559 | 53.4799 | 55.5537 | 57.3854 | 58.9871 | 60.3741 | |
3PD | LCL | 19.0334 | 20.6939 | 22.1463 | 23.3926 | 24.4448 | 25.3213 | 26.0433 |
34.0679 | 36.1895 | 38.0313 | 39.6023 | 40.9226 | 42.0182 | 42.9180 | ||
UCL | 49.1025 | 51.6850 | 53.9163 | 55.8121 | 57.4004 | 58.7151 | 59.7928 | |
NW | LCL | 19.4976 | 21.4257 | 22.8915 | 23.8548 | 24.4178 | 24.7248 | 24.8877 |
34.6629 | 37.1190 | 38.9717 | 40.1830 | 40.8888 | 41.2730 | 41.4767 | ||
UCL | 49.8281 | 52.8123 | 55.0519 | 56.5112 | 57.3598 | 57.8212 | 58.0657 | |
New | LCL | 19.4345 | 21.0601 | 22.4030 | 23.5015 | 24.4000 | 25.1390 | 25.7523 |
34.5821 | 36.6551 | 38.3556 | 39.7393 | 40.8664 | 41.7906 | 42.5556 | ||
UCL | 49.7296 | 52.2501 | 54.3082 | 55.9771 | 57.3329 | 58.4422 | 59.3590 |
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No. | Model | Model Type | ||
---|---|---|---|---|
1 | Goel–Okumoto [3] (GO) | Concave | ||
2 | Yamada et al. (Delayed S-shaped) [4] (DS) | S | ||
3 | Ohba (Inflection S-shaped) [5] (IS) | S | ||
4 | Yamada et al. (Imperfect Debugging 1) [25] (Y1) | Concave | ||
5 | Yamada et al. (Imperfect Debugging 2) [25] (Y2) | Concave | ||
6 | Pham et al. (Generalized Imperfect Debugging 1) [22] (PNZ) | Concave, S | ||
7 | Pham–Zhang (Generalized Imperfect Debugging 2) [26] (PZ) | + | Concave, S | |
8 | Pham (Dependent Parameter 1) [27] (DP1) | ) | ) )] | Convex |
9 | Pham (Dependent Parameter 2) [27] (DP2) | | Convex | |
10 | Chang et al. (Testing Coverage) [28] (TC) | Concave | ||
11 | Song et al. (Three parameter) [9] (3PD) | Concave, S | ||
12 | Song et al. (Weibull) [10] (NW) | S | ||
13 | New Proposed (New) | S |
No. | Criteria | Formula |
---|---|---|
1 | MSE | |
2 | RMSE | |
3 | AIC [29] | |
4 | [8] | |
5 | Adj [8] | |
6 | SAE [10] | |
7 | PRR [30] | |
8 | PP [30] |
Week Index | Dataset #1 | Dataset #2 | ||
---|---|---|---|---|
Failures | Cumulative Failures | Failures | Cumulative Failures | |
1 | 28 | 28 | 90 | 90 |
2 | 1 | 29 | 17 | 107 |
3 | 0 | 29 | 19 | 126 |
4 | 0 | 29 | 26 | 145 |
5 | 0 | 29 | 17 | 171 |
6 | 8 | 37 | 1 | 188 |
7 | 26 | 63 | 1 | 189 |
8 | 29 | 92 | 0 | 190 |
9 | 24 | 116 | 0 | 190 |
10 | 9 | 125 | 2 | 190 |
11 | 14 | 139 | 0 | 192 |
12 | 13 | 152 | 0 | 192 |
13 | 12 | 164 | 0 | 192 |
14 | 0 | 164 | 11 | 192 |
15 | 1 | 165 | 0 | 203 |
16 | 3 | 168 | 1 | 203 |
17 | 2 | 170 | 6 | 204 |
18 | 6 | 176 |
Week Index | Exposure Time (Cumulative System Test Hours) | Failures | Cumulative Failures |
---|---|---|---|
1 | 416 | 3 | 3 |
2 | 832 | 1 | 4 |
3 | 1248 | 0 | 4 |
4 | 1664 | 3 | 7 |
5 | 2080 | 2 | 9 |
6 | 2496 | 0 | 9 |
7 | 2912 | 1 | 10 |
8 | 3328 | 3 | 13 |
9 | 3744 | 4 | 17 |
10 | 4160 | 2 | 19 |
11 | 4576 | 4 | 23 |
12 | 4992 | 2 | 25 |
13 | 5408 | 5 | 30 |
14 | 5824 | 2 | 32 |
15 | 6240 | 4 | 36 |
16 | 6656 | 1 | 37 |
17 | 7072 | 2 | 39 |
18 | 7488 | 0 | 39 |
19 | 7904 | 0 | 39 |
20 | 8320 | 3 | 42 |
21 | 8736 | 1 | 43 |
Model | Dataset #1 | Dataset #2 | Dataset #3 |
---|---|---|---|
GO | = 984.237, = 0.0121 | = 197.387, = 0.399 | = 1546.477, = 0.00139 |
DS | = 226.111, = 0.1741 | = 192.528, = 0.882 | = 62.326, = 0.1185 |
IS | = 176.517, = 0.423, = 26.888 | = 197.354, = 0.399, = 0.000001 | = 46.539, = 0.241, = 12.233 |
Y1 | = 991.78, = 0.012, = 0.00 | = 182.550, = 0.467, = 0.007 | = 239.618, = 0.008, = 0.0197 |
Y2 | = 15.521, = 0.605, = 0.722 | = 182.934, = 0.464, = 0.0071 | = 174.77, = 0.011, = 0.0219 |
PNZ | = 176.517, = 0.423, = 0.000, = 26.888 | = 183.125, = 0.463, = 0.007, = 0.0001 | = 46.540, = 0.241 = 0.0001, = 12.233 |
PZ | = 0.310, = 0.423, = 0.000, = 26.888, = 176.940 | = 195.990, = 0.3987, = 1000.0, = 0.000, = 1.390 | = 0.610, = 0.241, = 1988.117, = 12.233, = 45.929 |
DP1 | = 0.000001, = 858.504 | = 0.000001, = 1031.599 | = 0.000002, = 249.70 |
DP2 | = 27641, = 0.006, = 0.159, = 43.0 | = 26,124.0, = 0.0044, = 0.000, = 147.0 | = 564.21, = 0.017 = 9.0 |
TC | = 0.05, = 2.500, = 101.0, = 14.00, = 174.0 | = 0.053., = 0.774, = 181.0, = 38.60, = 204.140 | = 1.863, = 9474.019, = 48.983 |
3PD | = 1.430, = 0.42, = 1042.4, = 0.09, = 178.9 | = 0.028, = 0.21, = 9.924, = 0.005, = 206.387 | = 999.493, = 0.17, = 46.843 |
NW | = 0.1363, = 10.245, = 0.1622, = 151.41, = 171.064 | = 0.18, = 5.714, = 0.08, = 3.77, = 196.852 | = 0.071, = 13.102, = 0.109, = 10.203, = 41.717 |
New | = 0.488, = 0.892, = 0.328, = 0.801, = 4644.6, = 0.942, = 184.23 | = 0.076, = 2.728, = 0.082, = 0.249, = 20,125.0, = 0.773, = 203.545 | = 0.93, = 0.78, = 0.44, = 1.95, = 400.001, = 0.590, = 50.001 |
Model | MSE | RMSE | AIC | R2 | Adj R2 | SAE | PRR | PP |
---|---|---|---|---|---|---|---|---|
GO | 299.3292 | 17.3011 | 261.4487 | 0.9247 | 0.9146 | 254.5092 | 2.7083 | 2.6703 |
DS | 202.8454 | 14.2424 | 286.6365 | 0.9490 | 0.9421 | 204.5240 | 70.0366 | 2.3678 |
IS | 116.2935 | 10.7839 | 237.2497 | 0.9726 | 0.9667 | 123.8812 | 65.0591 | 1.8223 |
Y1 | 319.2849 | 17.8685 | 263.4846 | 0.9247 | 0.9085 | 254.5864 | 2.7120 | 2.6668 |
Y2 | 326.5601 | 18.0710 | 271.4442 | 0.9230 | 0.9065 | 264.7073 | 4.2598 | 2.1294 |
PNZ | 124.6002 | 11.1624 | 239.2497 | 0.9726 | 0.9641 | 123.8812 | 65.0591 | 1.8223 |
PZ | 134.2981 | 11.5887 | 241.2417 | 0.9725 | 0.9611 | 124.7652 | 64.6969 | 1.8236 |
DP1 | 1482.3642 | 38.5015 | 451.7756 | 0.6270 | 0.5772 | 575.2253 | 1463.9339 | 4.2365 |
DP2 | 773.2907 | 27.8081 | 281.3004 | 0.8297 | 0.7773 | 403.4070 | 1.5855 | 3.2787 |
TC | 171.2268 | 13.0854 | 330.4311 | 0.9650 | 0.9504 | 136.5833 | 1563.7908 | 2.4065 |
3PD | 134.3453 | 11.5907 | 240.2708 | 0.9725 | 0.9611 | 124.2817 | 62.8017 | 1.8147 |
NW | 154.1849 | 12.4171 | 275.4208 | 0.9709 | 0.9588 | 128.3625 | 89.5199 | 1.8579 |
New | 93.2910 | 9.6587 | 182.2178 | 0.9839 | 0.9726 | 99.6157 | 3.1124 | 0.9340 |
Model | MSE | RMSE | AIC | R2 | Adj R2 | SAE | PRR | PP |
---|---|---|---|---|---|---|---|---|
GO | 80.6779 | 8.9821 | 184.3314 | 0.9388 | 0.9300 | 104.4025 | 0.1705 | 0.1013 |
DS | 232.6282 | 15.2522 | 331.8567 | 0.8234 | 0.7982 | 142.5442 | 1.2915 | 0.3330 |
IS | 86.4395 | 9.2973 | 186.3337 | 0.9388 | 0.9246 | 104.3703 | 0.1706 | 0.1013 |
Y1 | 78.8663 | 8.8807 | 157.4475 | 0.9441 | 0.9312 | 100.4119 | 0.1256 | 0.0860 |
Y2 | 78.8367 | 8.8790 | 157.8252 | 0.9442 | 0.9313 | 100.6173 | 0.1276 | 0.0866 |
PNZ | 84.9016 | 9.2142 | 159.8729 | 0.9442 | 0.9255 | 100.6068 | 0.1281 | 0.0867 |
PZ | 100.9894 | 10.0493 | 190.3321 | 0.9387 | 0.9108 | 104.3539 | 0.1719 | 0.1017 |
DP1 | 11,053.0563 | 105.1335 | 1208.2816 | −7.3888 | −8.5872 | 1516.4369 | 7828.3035 | 6.8736 |
DP2 | 769.2822 | 27.7359 | 480.3412 | 0.4940 | 0.3253 | 334.1283 | 0.4149 | 0.7120 |
TC | 72.2825 | 8.5019 | 158.9330 | 0.9561 | 0.9362 | 103.1963 | 0.0521 | 0.0479 |
3PD | 81.0903 | 9.0050 | 163.7968 | 0.9508 | 0.9284 | 106.3409 | 0.0729 | 0.0615 |
NW | 46.7395 | 6.8366 | 182.3811 | 0.9759 | 0.9650 | 65.5785 | 0.0293 | 0.0271 |
New | 25.2102 | 5.0210 | 131.1529 | 0.9872 | 0.9773 | 56.3823 | 0.0073 | 0.0072 |
Model | MSE | RMSE | AIC | R2 | Adj R2 | SAE | PRR | PP |
---|---|---|---|---|---|---|---|---|
GO | 6.7541 | 2.5989 | 78.3086 | 0.9714 | 0.9682 | 43.8536 | 0.6938 | 1.1445 |
DS | 3.2731 | 1.8092 | 81.0881 | 0.9862 | 0.9846 | 32.5598 | 44.3248 | 1.4297 |
IS | 1.8704 | 1.3676 | 76.9490 | 0.9925 | 0.9912 | 21.9615 | 5.9598 | 0.8969 |
Y1 | 6.2849 | 2.5070 | 81.0083 | 0.9748 | 0.9704 | 41.0689 | 0.6806 | 0.7548 |
Y2 | 31.0054 | 5.5683 | 81.1537 | 0.8757 | 0.8538 | 85.5914 | 1.5928 | 0.8203 |
PNZ | 1.9834 | 1.4083 | 78.9480 | 0.9925 | 0.9906 | 21.9572 | 5.9580 | 0.8970 |
PZ | 2.1042 | 1.4506 | 80.9490 | 0.9925 | 0.9900 | 21.9616 | 5.9599 | 0.8969 |
DP1 | 43.6887 | 6.6097 | 104.9786 | 0.8152 | 0.7946 | 121.7027 | 600.8316 | 4.5285 |
DP2 | 22.9775 | 4.7935 | 98.8587 | 0.9130 | 0.8913 | 79.7928 | 1.5190 | 6.5936 |
TC | 3.3002 | 1.8167 | 86.4351 | 0.9882 | 0.9843 | 28.8649 | 56.9751 | 1.5764 |
3PD | 2.1046 | 1.4507 | 80.9477 | 0.9925 | 0.9900 | 21.9661 | 5.9567 | 0.8967 |
NW | 2.0555 | 1.4337 | 86.1131 | 0.9927 | 0.9902 | 21.9748 | 8.2751 | 0.9430 |
New | 1.1626 | 1.0783 | 81.4086 | 0.9964 | 0.9944 | 15.7154 | 0.2447 | 0.1851 |
Operating Period | EC(T) | T* | Reliability |
---|---|---|---|
x = 9 | 19,024.95 | 88.0 | 0.8729 |
x = 12 | 19,098.35 | 96.1 | 0.8641 |
x = 15 (base) | 19,157.92 | 102.6 | 0.8564 |
x = 18 | 19,211.82 | 108.1 | 0.8496 |
x = 21 | 19,261.71 | 112.7 | 0.8431 |
Cost Coefficient | EC(T) | T* | Reliability |
---|---|---|---|
Base | 19,157.92 | 102.6 | 0.8564 |
C1 = 2.5 | 18,907.92 | 129.1 | 0.9074 |
C1 = 7.5 | 19,395.72 | 89.2 | 0.8135 |
C2 = 80 | 15,496.43 | 104.3 | 0.8608 |
C2 = 120 | 22,819.4 | 100.9 | 0.8518 |
C3 = 1500 | 19,078.56 | 92.6 | 0.8260 |
C3 = 2500 | 19,233.28 | 111.2 | 0.8767 |
C4 = 200 | 19,141.58 | 100.4 | 0.8504 |
C4 = 400 | 19,174.25 | 104.7 | 0.8618 |
Parameter | −30% | −20% | −10% | 0% | 10% | 20% | 30% |
---|---|---|---|---|---|---|---|
0.10039 | 0.06238 | 0.02924 | 0 | −0.02534 | −0.04873 | −0.07018 | |
0.04191 | 0.02437 | 0.01072 | 0 | −0.00877 | −0.01657 | −0.02242 | |
−0.00390 | −0.00195 | −0.00097 | 0 | 0.00097 | 0.00195 | 0.00292 | |
−0.09649 | −0.06140 | −0.02924 | 0 | 0.02729 | 0.05263 | 0.07602 | |
−0.09357 | −0.05945 | −0.02827 | 0 | 0.02632 | 0.05068 | 0.07407 | |
0.14327 | 0.08674 | 0.03996 | 0 | −0.03411 | −0.06530 | −0.09259 | |
−0.09747 | −0.06140 | −0.02924 | 0 | 0.02729 | 0.05263 | 0.07700 |
Parameter | −30% | −20% | −10% | 0% | 10% | 20% | 30% |
---|---|---|---|---|---|---|---|
−0.02531 | −0.01525 | −0.00698 | 0 | 0.00622 | 0.0115 | 0.01603 | |
−0.00002 | −0.00004 | −0.00005 | 0 | 0.00004 | −0.0001 | 0.00006 | |
0.00008 | 0.00020 | 0.00008 | 0 | −0.00004 | −0.0001 | −0.00004 | |
0.02151 | 0.01381 | 0.00673 | 0 | −0.00629 | −0.0123 | −0.01813 | |
0.02163 | 0.01389 | 0.00678 | 0 | −0.00633 | −0.01237 | −0.01794 | |
−0.02501 | −0.01529 | −0.00703 | 0 | 0.00626 | 0.01139 | 0.01608 | |
0.02147 | 0.01398 | 0.00682 | 0 | −0.00638 | −0.01247 | −0.01808 |
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Share and Cite
Song, K.Y.; Chang, I.H.; Pham, H. Optimal Release Time and Sensitivity Analysis Using a New NHPP Software Reliability Model with Probability of Fault Removal Subject to Operating Environments. Appl. Sci. 2018, 8, 714. https://doi.org/10.3390/app8050714
Song KY, Chang IH, Pham H. Optimal Release Time and Sensitivity Analysis Using a New NHPP Software Reliability Model with Probability of Fault Removal Subject to Operating Environments. Applied Sciences. 2018; 8(5):714. https://doi.org/10.3390/app8050714
Chicago/Turabian StyleSong, Kwang Yoon, In Hong Chang, and Hoang Pham. 2018. "Optimal Release Time and Sensitivity Analysis Using a New NHPP Software Reliability Model with Probability of Fault Removal Subject to Operating Environments" Applied Sciences 8, no. 5: 714. https://doi.org/10.3390/app8050714