1 Introduction
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N(t) The total number of software failures by time t based on NHPP.
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m(t) The expected number of software failures by time t, i.e., \(m( t)=E[N(t)]\).
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a(t) Fault content function.
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L The maximum number of faults software is able to contain.
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b(t) Software fault detection rate per fault per unit of time.
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c(t) Non-removed error rate per unit of time.
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\(\lambda (t)\) Failure intensity function.
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\(R(x\vert t)\) Software reliability function by time x given a mission time t.
2 Software reliability modeling
Model name | Model type | MVF (m(t)) |
---|---|---|
Goel–Okumoto (G–O) model [4] | Concave |
\(m( t)=a( {1-e^{bt}})\)
|
Delayed S-shaped model [6] | S-shaped |
\(m( t)=a(1-(1+bt)e^{-bt})\)
|
Inflection S-shaped model [7] | S-shaped |
\(m( t)=\frac{a(1-e^{-bt})}{1+\beta e^{-bt}}\)
|
Yamada imperfect debugging model [29] | Concave |
\(m( t)=a\left[ {1-e^{-bt}} \right] \left[ {1-\frac{\alpha }{b}} \right] +\alpha at\)
|
PNZ model [8] | S-shaped and concave |
\(m( t)=\frac{a[(1-e^{-bt)}( {1-\frac{\alpha }{b}})+\alpha t]}{1+\beta e^{-bt}}\)
|
Pham-Zhang model [30] | S-shaped and concave |
\(m( t)=\frac{1}{1+\beta e^{-bt}}[( {c+a})( {1-e^{-bt}})-\frac{ab}{b-\alpha }(e^{-\alpha t}-e^{-bt})]\)
|
Dependent-parameter model [31] | S-shaped and concave |
\(m( t)=\alpha ( {1+\gamma t})(\gamma t+e^{-\gamma t}-1)\)
|
S-shaped and concave |
\(m( t)=m_0 ( {\frac{\gamma t+1}{rt_0 +1}})e^{-\gamma ( {t-t_0 })}+\alpha ( {\gamma t+1})[\gamma t-1+(1-\gamma t_0 )e^{-\gamma (t-t_0 )}]\)
| |
Loglog fault-detection rate model [18] | Concave |
\(m( t)=N(1-e^{-(a^{t^b}-1)})\)
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Proposed model | S-shaped and concave |
\(m( t)=\frac{\beta +e^{bt}}{\frac{b}{L(b-c)}\left[ {e^{bt}-e^{ct}} \right] +\frac{1+\beta }{m_0 }e^{ct}}\)
|
3 Parameter estimation and goodness-of-fit criteria
4 Model evaluation and comparison
4.1 Software failure data description
Week index | Exposure time (cumulative system test hours) | Fault | Cumulative fault |
---|---|---|---|
1 | 356 | 1 | 1 |
2 | 712 | 0 | 1 |
3 | 1068 | 1 | 2 |
4 | 1424 | 1 | 3 |
5 | 1780 | 2 | 5 |
6 | 2136 | 0 | 5 |
7 | 2492 | 0 | 5 |
8 | 2848 | 3 | 8 |
9 | 3204 | 1 | 9 |
10 | 3560 | 2 | 11 |
11 | 3916 | 2 | 13 |
12 | 4272 | 2 | 15 |
13 | 4628 | 4 | 19 |
14 | 4984 | 0 | 19 |
15 | 5340 | 3 | 22 |
16 | 5696 | 0 | 22 |
17 | 6052 | 1 | 23 |
18 | 6408 | 1 | 24 |
19 | 6764 | 0 | 24 |
20 | 7120 | 0 | 24 |
21 | 7476 | 2 | 26 |
Week index | Exposure time (cumulative system test hours) | Fault | Cumulative fault |
---|---|---|---|
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 |
4.2 Model comparison
Model name | MSE | PRR | PP | AIC | Parameter estimate |
---|---|---|---|---|---|
Goel–Okumoto (G–O) | 5.944 | 1.818 | 8.165 | 66.211 |
\( \hat{a}=62.0395\)
\( \hat{b}=0.0243\)
|
Delayed S-shaped | 1.609 | 14.546 | 0.981 | 64.230 |
\( \hat{a}=44.221\)
\( \hat{b} =0.1007\)
|
Inflection S-shaped | 0.709 | 1.714 | 0.512 | 63.938 |
\( \hat{a} =27.247\)
\( \hat{b} =0.269\)
\( \hat{\beta } =17.255\)
|
Yamada imperfect debugging | 2.602 | 0.840 | 0.757 | 66.710 |
\( \hat{a} =1.8643\)
\( \hat{b} =0.25\)
\( \hat{\alpha } =0.8418\)
|
PNZ Model | 2.479 | 2.954 | 0.690 | 68.611 |
\( \hat{a} =1.5556\)
\( \hat{b} =0.3239\)
\( \hat{\alpha } =0.9689\)
\( \hat{\beta } =0.9999\)
|
Pham-Zhang model | 3.429 | 1.982 | 1.187 | 70.617 |
\( \hat{a} =13.394\)
\( \hat{b} =0.2671\)
\( \hat{\alpha } =0.5113\)
\( \hat{\beta } =9.0131\)
\( \hat{c} =12.0336\)
|
Dependent-parameter model | 15.741 | 287.191 | 3.768 | 77.541 |
\( \hat{\alpha } =0.0872\)
\( \hat{\gamma } =0.9523\)
|
Dependent-parameter model with \(m_0 \ne 0\), \(t_0 \ne 0\)
| 13.477 | 2.136 | 1.189 | 77.621 |
\( \hat{\alpha } =6206\)
\( \hat{\gamma } =0.0048\)
\(t_0 =1\)
\(m_0 =1\)
|
Loglog fault-detection rate model | 71.241 | 11.736 | 15.475 | 93.592 |
\( \hat{N} =15.403\)
\( \hat{a} =1.181\)
\( \hat{b} =0.567\)
|
Proposed model | 0.630 | 0.408 | 0.526 | 65.777 |
\( \hat{m}_0 =1\)
\( \hat{L}=49.7429\)
\( \hat{\beta } =0.2925\)
\( \hat{b} =0.6151\)
\( \hat{c} =0.292\)
|
Model name | MSE | PRR | PP | AIC | Parameter estimate |
---|---|---|---|---|---|
Goel–Okumoto (G–O) | 6.607 | 0.687 | 1.099 | 74.752 |
\(\hat{a}=98295\)
\(\hat{b}=5.2E-8\)
|
Delayed S-shaped | 3.273 | 44.267 | 1.429 | 77.502 |
\( \hat{a}=62.3\)
\(\hat{b}=2.85E-4\)
|
Inflection S-shaped | 1.871 | 5.938 | 0.895 | 73.359 |
\( \hat{a} =46.6\)
\( \hat{b}=5.78E-4\)
\( \hat{\beta } =12.2\)
|
Yamada imperfect debugging | 4.982 | 4.296 | 0.809 | 78.054 |
\(\hat{a}=1.5\)
\(\hat{b}=1.1E-3\)
\(\hat{\alpha }=3.8E-3\)
|
PNZ Model | 1.994 | 6.834 | 0.957 | 75.501 |
\( \hat{a}=45.99\)
\( \hat{b} =6.0E-4\)
\( \hat{\alpha } =0\)
\( \hat{\beta } =13.24\)
|
Pham-Zhang model | 2.119 | 6.762 | 0.952 | 77.502 |
\( \hat{a} =0.06\)
\( \hat{b} =6.0E-4\)
\( \hat{\alpha } =1.0E-4\)
\( \hat{\beta }=13.2\)
\( \hat{c} =45.9\)
|
Dependent-parameter model | 43.689 | 601.336 | 4.530 | 101.386 |
\( \hat{\alpha } =3.0E-6\)
\( \hat{\gamma } =0.49\)
|
Dependent-parameter model with \(m_0 \ne 0\), \(t_0 \ne 0\)
| 35.398 | 2.250 | 1.167 | 87.667 |
\( \hat{\alpha } =890996\)
\( \hat{\gamma } =1.2E-6\)
\(t_0 =832\)
\(m_0 =4\)
|
Loglog fault-detection rate model | 219.687 | 13.655 | 4.383 | 114.807 |
\( \hat{N} =231.92\)
\( \hat{a} =1.019\)
\( \hat{b}=0.489\)
|
Proposed model | 1.058 | 0.163 | 0.144 | 68.316 |
\( \hat{m}_0 =3\)
\( \hat{L} =59.997\)
\( \hat{\beta }=0.843\)
\( \hat{b}=0.409\)
\( \hat{c}=0.108\)
|
Testing time (weeks) | CPU hours | Defects found | Predicted total defects by G-O | Predicted total defects by Zhang–Teng–Pham model | Predicted total defects by proposed model |
---|---|---|---|---|---|
1 | 519 | 16 | – | – | – |
2 | 968 | 24 | – | – | – |
3 | 1430 | 27 | – | – | – |
4 | 1893 | 33 | – | – | – |
5 | 2490 | 41 | – | – | – |
6 | 3058 | 49 | – | – | – |
7 | 3625 | 54 | – | – | – |
8 | 4422 | 58 | – | – | – |
9 | 5218 | 69 | – | – | – |
10 | 5823 | 75 | 98 | 74.7 | 75.5 |
11 | 6539 | 81 | 107 | 80.1 | 80.8 |
12 | 7083 | 86 | 116 | 85.2 | 85.1 |
13 | 7487 | 90 | 123 | 90.1 | 88.5 |
14 | 7846 | 93 | 129 | 94.6 | 91.2 |
15 | 8205 | 96 | 129 | 98.9 | 93.2 |
16 | 8564 | 98 | 134 | 102.9 | 94.7 |
17 | 8923 | 99 | 139 | 106.8 | 95.8 |
18 | 9282 | 100 | 138 | 110.4 | 96.6 |
19 | 9641 | 100 | 135 | 111.9 | 97.2 |
20 | 10,000 | 100 | 133 | 112.2 | 97.6 |
Predicted MSE | 1359.222 | 82.66 | 10.120 | ||
Predicted AIC | 149.60 | 186.468 | 169.667 | ||
Predicted PRR | 0.756 | 0.041 | 0.007 | ||
Predicted PP | 1.395 | 0.050 | 0.006 |