Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning
Abstract
:1. Introduction
2. Software Reliability Model
2.1. Generalized Software Relability Model
2.2. Software Reliablity Model Using Deep Learning
2.2.1. Deep Neural Network
2.2.2. Recurrent Neural Network
3. Numerical Example
3.1. Data Information
3.2. Criteria
3.3. Results
3.3.1. Results of Dataset 1
3.3.2. Results of Dataset 2
3.4. Confidence Interval
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Index | Failures | Cumulative Failures | Index | Failures | Cumulative Failures | Index | Failures | Cumulative Failures |
---|---|---|---|---|---|---|---|---|
1 | 3 | 3 | 18 | 4 | 19 | 35 | 0 | 49 |
2 | 0 | 3 | 19 | 1 | 20 | 36 | 1 | 50 |
3 | 1 | 4 | 20 | 4 | 24 | 37 | 2 | 52 |
4 | 0 | 4 | 21 | 0 | 24 | 38 | 3 | 55 |
5 | 0 | 4 | 22 | 0 | 24 | 39 | 1 | 56 |
6 | 0 | 4 | 23 | 0 | 24 | 40 | 2 | 58 |
7 | 0 | 4 | 24 | 2 | 26 | 41 | 0 | 58 |
8 | 0 | 4 | 25 | 0 | 26 | 42 | 0 | 58 |
9 | 0 | 4 | 26 | 1 | 27 | 43 | 1 | 59 |
10 | 2 | 6 | 27 | 1 | 28 | 44 | 4 | 63 |
11 | 2 | 8 | 28 | 5 | 33 | 45 | 2 | 65 |
12 | 3 | 11 | 29 | 2 | 35 | 46 | 1 | 66 |
13 | 0 | 11 | 30 | 2 | 37 | 47 | 2 | 68 |
14 | 2 | 13 | 31 | 2 | 39 | 48 | 3 | 71 |
15 | 0 | 13 | 32 | 2 | 41 | 49 | 3 | 74 |
16 | 2 | 15 | 33 | 4 | 45 | 50 | 2 | 76 |
17 | 0 | 15 | 34 | 4 | 49 |
Index | Failures | Cumulative Failures | Index | Failures | Cumulative Failures | Index | Failures | Cumulative Failures | Index | Failures | Cumulative Failures |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 34 | 0 | 25 | 67 | 0 | 85 | 100 | 3 | 197 |
2 | 0 | 1 | 35 | 1 | 26 | 68 | 1 | 86 | 101 | 6 | 203 |
3 | 1 | 2 | 36 | 0 | 26 | 69 | 2 | 88 | 102 | 3 | 206 |
4 | 0 | 2 | 37 | 0 | 26 | 70 | 3 | 91 | 103 | 6 | 212 |
5 | 2 | 4 | 38 | 1 | 27 | 71 | 1 | 92 | 104 | 9 | 221 |
6 | 0 | 4 | 39 | 0 | 27 | 72 | 3 | 95 | 105 | 6 | 227 |
7 | 0 | 4 | 40 | 3 | 30 | 73 | 6 | 101 | 106 | 3 | 230 |
8 | 1 | 5 | 41 | 3 | 33 | 74 | 9 | 110 | 107 | 12 | 242 |
9 | 0 | 5 | 42 | 1 | 34 | 75 | 2 | 112 | 108 | 6 | 248 |
10 | 0 | 5 | 43 | 2 | 36 | 76 | 6 | 118 | 109 | 13 | 261 |
11 | 1 | 6 | 44 | 3 | 39 | 77 | 3 | 121 | 110 | 8 | 269 |
12 | 0 | 6 | 45 | 0 | 39 | 78 | 4 | 125 | 111 | 10 | 279 |
13 | 0 | 6 | 46 | 6 | 45 | 79 | 4 | 129 | 112 | 5 | 284 |
14 | 2 | 8 | 47 | 5 | 50 | 80 | 4 | 133 | 113 | 3 | 287 |
15 | 0 | 8 | 48 | 4 | 54 | 81 | 0 | 133 | 114 | 11 | 298 |
16 | 1 | 9 | 49 | 2 | 56 | 82 | 3 | 136 | 115 | 15 | 313 |
17 | 0 | 9 | 50 | 1 | 57 | 83 | 1 | 137 | 116 | 12 | 325 |
18 | 2 | 11 | 51 | 0 | 57 | 84 | 4 | 141 | 117 | 10 | 335 |
19 | 1 | 12 | 52 | 3 | 60 | 85 | 1 | 142 | 118 | 15 | 350 |
20 | 0 | 12 | 53 | 1 | 61 | 86 | 0 | 142 | 119 | 16 | 366 |
21 | 1 | 13 | 54 | 1 | 62 | 87 | 3 | 145 | 120 | 12 | 378 |
22 | 2 | 15 | 55 | 3 | 65 | 88 | 4 | 149 | 121 | 8 | 386 |
23 | 2 | 17 | 56 | 1 | 66 | 89 | 3 | 152 | 122 | 12 | 398 |
24 | 0 | 17 | 57 | 1 | 67 | 90 | 6 | 158 | 123 | 8 | 406 |
25 | 0 | 17 | 58 | 2 | 69 | 91 | 7 | 165 | 124 | 6 | 412 |
26 | 0 | 17 | 59 | 3 | 72 | 92 | 6 | 171 | 125 | 10 | 422 |
27 | 0 | 17 | 60 | 3 | 75 | 93 | 2 | 173 | 126 | 10 | 432 |
28 | 1 | 18 | 61 | 1 | 76 | 94 | 4 | 177 | 127 | 17 | 449 |
29 | 1 | 19 | 62 | 2 | 78 | 95 | 5 | 182 | 128 | 6 | 455 |
30 | 1 | 20 | 63 | 0 | 78 | 96 | 1 | 183 | 129 | 21 | 476 |
31 | 3 | 23 | 64 | 3 | 81 | 97 | 3 | 186 | 130 | 26 | 502 |
32 | 2 | 25 | 65 | 1 | 82 | 98 | 6 | 192 | |||
33 | 0 | 25 | 66 | 3 | 85 | 99 | 2 | 194 |
Time | Real | GO | DS | YID | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
40 | 58 | 49.98 | 63.84 | 36.13 | 57.58 | 72.46 | 42.71 | 59.04 | 74.10 | 43.98 |
41 | 58 | 51.22 | 65.24 | 37.19 | 59.51 | 74.63 | 44.39 | 61.44 | 76.80 | 46.07 |
42 | 58 | 52.45 | 66.64 | 38.26 | 61.44 | 76.80 | 46.08 | 63.88 | 79.54 | 48.21 |
43 | 59 | 53.68 | 68.04 | 39.32 | 63.36 | 78.96 | 47.76 | 66.36 | 82.33 | 50.39 |
44 | 63 | 54.91 | 69.44 | 40.39 | 65.27 | 81.10 | 49.43 | 68.88 | 85.15 | 52.62 |
45 | 65 | 56.14 | 70.83 | 41.46 | 67.17 | 83.23 | 51.11 | 71.45 | 88.01 | 54.88 |
46 | 66 | 57.37 | 72.22 | 42.53 | 69.06 | 85.35 | 52.77 | 74.05 | 90.92 | 57.19 |
47 | 68 | 58.60 | 73.60 | 43.60 | 70.94 | 87.45 | 54.43 | 76.70 | 93.86 | 59.53 |
48 | 71 | 59.83 | 74.99 | 44.67 | 72.81 | 89.54 | 56.09 | 79.38 | 96.85 | 61.92 |
49 | 74 | 61.05 | 76.37 | 45.74 | 74.67 | 91.61 | 57.73 | 82.11 | 99.87 | 64.35 |
Time | Real | PNZ | PZ | TP | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
40 | 58 | 163.96 | 189.05 | 138.86 | 28.51 | 38.97 | 18.04 | 58.28 | 73.24 | 43.31 |
41 | 58 | 168.86 | 194.33 | 143.39 | 29.87 | 40.58 | 19.16 | 60.47 | 75.71 | 45.23 |
42 | 58 | 173.80 | 199.64 | 147.96 | 31.28 | 42.24 | 20.31 | 62.69 | 78.21 | 47.17 |
43 | 59 | 178.77 | 204.97 | 152.56 | 32.73 | 43.94 | 21.52 | 64.93 | 80.72 | 49.13 |
44 | 63 | 183.77 | 210.34 | 157.20 | 34.24 | 45.70 | 22.77 | 67.19 | 83.26 | 51.13 |
45 | 65 | 188.81 | 215.74 | 161.88 | 35.79 | 47.52 | 24.06 | 69.48 | 85.81 | 53.14 |
46 | 66 | 193.88 | 221.17 | 166.59 | 37.40 | 49.38 | 25.41 | 71.78 | 88.39 | 55.18 |
47 | 68 | 198.98 | 226.63 | 171.33 | 39.05 | 51.30 | 26.80 | 74.11 | 90.99 | 57.24 |
48 | 71 | 204.12 | 232.12 | 176.11 | 40.76 | 53.28 | 28.25 | 76.46 | 93.60 | 59.32 |
49 | 74 | 209.28 | 237.64 | 180.93 | 42.53 | 55.31 | 29.75 | 78.83 | 96.24 | 61.43 |
Time | Real | TC | Vtub | DPF | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
40 | 58 | 57.84 | 72.75 | 42.94 | 29.64 | 40.31 | 18.97 | 58.94 | 73.98 | 43.89 |
41 | 58 | 59.53 | 74.65 | 44.40 | 31.00 | 41.91 | 20.08 | 61.33 | 76.68 | 45.98 |
42 | 58 | 61.13 | 76.45 | 45.80 | 32.38 | 43.54 | 21.23 | 63.78 | 79.43 | 48.13 |
43 | 59 | 62.64 | 78.15 | 47.13 | 33.80 | 45.19 | 22.40 | 66.28 | 82.23 | 50.32 |
44 | 63 | 64.07 | 79.76 | 48.38 | 35.23 | 46.87 | 23.60 | 68.82 | 85.08 | 52.56 |
45 | 65 | 65.42 | 81.27 | 49.57 | 36.69 | 48.57 | 24.82 | 71.41 | 87.97 | 54.85 |
46 | 66 | 66.68 | 82.68 | 50.67 | 38.18 | 50.29 | 26.07 | 74.05 | 90.92 | 57.19 |
47 | 68 | 67.86 | 84.00 | 51.71 | 39.68 | 52.02 | 27.33 | 76.75 | 93.92 | 59.57 |
48 | 71 | 68.95 | 85.23 | 52.68 | 41.20 | 53.78 | 28.62 | 79.49 | 96.96 | 62.01 |
49 | 74 | 69.97 | 86.36 | 53.57 | 42.74 | 55.55 | 29.92 | 82.28 | 100.06 | 64.50 |
Time | Real | UDPF | DNN | RNN | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
40 | 58 | 62.68 | 78.19 | 47.16 | 60.01 | 75.20 | 44.83 | 59.46 | 74.57 | 44.34 |
41 | 58 | 64.97 | 80.77 | 49.18 | 59.02 | 74.07 | 43.96 | 59.24 | 74.32 | 44.15 |
42 | 58 | 67.29 | 83.37 | 51.21 | 59.02 | 74.07 | 43.96 | 59.19 | 74.27 | 44.11 |
43 | 59 | 69.62 | 85.97 | 53.26 | 61.01 | 76.32 | 45.70 | 60.24 | 75.45 | 45.03 |
44 | 63 | 71.96 | 88.58 | 55.33 | 64.39 | 80.12 | 48.66 | 64.71 | 80.48 | 48.95 |
45 | 65 | 74.31 | 91.20 | 57.41 | 67.01 | 83.06 | 50.97 | 66.64 | 82.64 | 50.64 |
46 | 66 | 76.67 | 93.83 | 59.50 | 68.01 | 84.18 | 51.85 | 67.41 | 83.50 | 51.32 |
47 | 68 | 79.03 | 96.46 | 61.61 | 70.01 | 86.41 | 53.61 | 69.44 | 85.77 | 53.11 |
48 | 71 | 81.41 | 99.10 | 63.73 | 71.70 | 88.30 | 55.11 | 72.65 | 89.35 | 55.94 |
49 | 74 | 83.79 | 101.73 | 65.85 | 74.70 | 91.65 | 57.76 | 75.82 | 92.88 | 58.75 |
Time | Real | LSTM | GRU | |||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | |||||
40 | 58 | 59.51 | 74.63 | 44.39 | 59.72 | 74.86 | 44.57 | |||
41 | 58 | 59.42 | 74.53 | 44.31 | 59.27 | 74.36 | 44.18 | |||
42 | 58 | 59.36 | 74.46 | 44.26 | 59.13 | 74.20 | 44.06 | |||
43 | 59 | 60.33 | 75.55 | 45.10 | 60.26 | 75.47 | 45.04 | |||
44 | 63 | 64.53 | 80.27 | 48.78 | 64.78 | 80.56 | 49.00 | |||
45 | 65 | 66.53 | 82.52 | 50.55 | 66.78 | 82.80 | 50.77 | |||
46 | 66 | 67.46 | 83.55 | 51.36 | 67.54 | 83.65 | 51.43 | |||
47 | 68 | 69.45 | 85.78 | 53.11 | 69.70 | 86.06 | 53.34 | |||
48 | 71 | 72.54 | 89.24 | 55.85 | 72.61 | 89.31 | 55.91 | |||
49 | 74 | 75.55 | 92.58 | 58.51 | 75.50 | 92.53 | 58.47 |
Time | Real | GO | DS | YID | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
45 | 65 | 58.35 | 73.32 | 43.38 | 65.02 | 80.82 | 49.21 | 66.13 | 82.07 | 50.19 |
46 | 66 | 59.64 | 74.78 | 44.50 | 66.73 | 82.74 | 50.72 | 68.13 | 84.31 | 51.96 |
47 | 68 | 60.93 | 76.23 | 45.63 | 68.42 | 84.63 | 52.21 | 70.15 | 86.56 | 53.73 |
48 | 71 | 62.22 | 77.68 | 46.76 | 70.10 | 86.51 | 53.69 | 72.17 | 88.82 | 55.52 |
49 | 74 | 63.51 | 79.14 | 47.89 | 71.76 | 88.36 | 55.16 | 74.19 | 91.08 | 57.31 |
Time | Real | PNZ | PZ | TP | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
45 | 65 | 68.29 | 84.49 | 52.10 | 89.44 | 107.98 | 70.90 | 35.16 | 46.78 | 23.54 |
46 | 66 | 69.86 | 86.25 | 53.48 | 90.03 | 108.62 | 71.43 | 36.28 | 48.09 | 24.48 |
47 | 68 | 71.38 | 87.94 | 54.82 | 90.57 | 109.23 | 71.92 | 37.40 | 49.39 | 25.41 |
48 | 71 | 72.85 | 89.58 | 56.12 | 91.08 | 109.79 | 72.38 | 38.51 | 50.67 | 26.35 |
49 | 74 | 74.27 | 91.16 | 57.38 | 91.55 | 110.31 | 72.80 | 39.61 | 51.95 | 27.28 |
Time | Real | TC | Vtub | DPF | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
45 | 65 | 65.76 | 81.65 | 49.86 | 65.89 | 81.80 | 49.98 | 63.71 | 79.35 | 48.06 |
46 | 66 | 67.66 | 83.78 | 51.53 | 67.78 | 83.91 | 51.64 | 64.82 | 80.60 | 49.04 |
47 | 68 | 69.56 | 85.90 | 53.21 | 69.66 | 86.02 | 53.30 | 65.85 | 81.75 | 49.94 |
48 | 71 | 71.45 | 88.02 | 54.89 | 71.53 | 88.11 | 54.95 | 66.80 | 82.82 | 50.78 |
49 | 74 | 73.35 | 90.13 | 56.56 | 73.39 | 90.19 | 56.60 | 67.68 | 83.81 | 51.56 |
Time | Real | UDPF | DNN | RNN | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
45 | 65 | 83.99 | 101.95 | 66.03 | 66.72 | 82.73 | 50.71 | 66.52 | 82.50 | 50.53 |
46 | 66 | 85.64 | 103.78 | 67.51 | 68.57 | 84.80 | 52.34 | 67.43 | 83.52 | 51.33 |
47 | 68 | 87.28 | 105.59 | 68.97 | 69.72 | 86.08 | 53.35 | 69.45 | 85.78 | 53.11 |
48 | 71 | 88.89 | 107.36 | 70.41 | 71.70 | 88.30 | 55.11 | 72.54 | 89.23 | 55.85 |
49 | 74 | 90.47 | 109.11 | 71.83 | 74.70 | 91.64 | 57.76 | 75.61 | 92.65 | 58.57 |
Time | Real | LSTM | GRU | |||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | |||||
45 | 65 | 66.55 | 82.54 | 50.56 | 66.52 | 82.51 | 50.54 | |||
46 | 66 | 67.46 | 83.56 | 51.37 | 67.45 | 83.55 | 51.36 | |||
47 | 68 | 69.50 | 85.84 | 53.16 | 69.50 | 85.84 | 53.16 | |||
48 | 71 | 72.57 | 89.27 | 55.87 | 72.61 | 89.31 | 55.91 | |||
49 | 74 | 75.58 | 92.62 | 58.54 | 75.69 | 92.74 | 58.64 |
Time | Real | GO | DS | YID | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
104 | 221 | 162.41 | 187.39 | 137.43 | 210.86 | 239.32 | 182.40 | 212.33 | 240.89 | 183.77 |
105 | 227 | 163.96 | 189.06 | 138.87 | 214.68 | 243.40 | 185.96 | 216.29 | 245.12 | 187.47 |
106 | 230 | 165.52 | 190.73 | 140.30 | 218.53 | 247.50 | 189.55 | 220.29 | 249.38 | 191.20 |
107 | 242 | 167.07 | 192.40 | 141.74 | 222.40 | 251.63 | 193.17 | 224.32 | 253.67 | 194.96 |
108 | 248 | 168.62 | 194.07 | 143.17 | 226.31 | 255.79 | 196.82 | 228.38 | 258.00 | 198.76 |
109 | 261 | 170.18 | 195.74 | 144.61 | 230.24 | 259.98 | 200.50 | 232.48 | 262.36 | 202.60 |
110 | 269 | 171.73 | 197.41 | 146.04 | 234.21 | 264.20 | 204.21 | 236.61 | 266.76 | 206.47 |
111 | 279 | 173.28 | 199.08 | 147.48 | 238.20 | 268.45 | 207.95 | 240.78 | 271.20 | 210.37 |
112 | 284 | 174.83 | 200.75 | 148.92 | 242.22 | 272.73 | 211.72 | 244.99 | 275.67 | 214.31 |
113 | 287 | 176.38 | 202.41 | 150.35 | 246.27 | 277.03 | 215.52 | 249.23 | 280.17 | 218.28 |
114 | 298 | 177.94 | 204.08 | 151.79 | 250.35 | 281.37 | 219.34 | 253.50 | 284.71 | 222.29 |
115 | 313 | 179.49 | 205.75 | 153.23 | 254.46 | 285.73 | 223.20 | 257.81 | 289.28 | 226.34 |
116 | 325 | 181.04 | 207.41 | 154.67 | 258.60 | 290.12 | 227.08 | 262.15 | 293.88 | 230.41 |
117 | 335 | 182.59 | 209.08 | 156.11 | 262.76 | 294.53 | 230.99 | 266.53 | 298.52 | 234.53 |
118 | 350 | 184.14 | 210.74 | 157.54 | 266.96 | 298.98 | 234.93 | 270.94 | 303.20 | 238.68 |
119 | 366 | 185.69 | 212.40 | 158.98 | 271.18 | 303.45 | 238.90 | 275.38 | 307.91 | 242.86 |
120 | 378 | 187.24 | 214.06 | 160.42 | 275.42 | 307.95 | 242.90 | 279.86 | 312.65 | 247.07 |
121 | 386 | 188.79 | 215.73 | 161.86 | 279.70 | 312.48 | 246.92 | 284.38 | 317.43 | 251.33 |
122 | 398 | 190.35 | 217.39 | 163.30 | 284.00 | 317.03 | 250.97 | 288.93 | 322.24 | 255.61 |
123 | 406 | 191.90 | 219.05 | 164.74 | 288.34 | 321.62 | 255.05 | 293.51 | 327.09 | 259.93 |
124 | 412 | 193.45 | 220.71 | 166.19 | 292.69 | 326.23 | 259.16 | 298.13 | 331.97 | 264.28 |
125 | 422 | 195.00 | 222.37 | 167.63 | 297.08 | 330.86 | 263.30 | 302.78 | 336.88 | 268.67 |
126 | 432 | 196.55 | 224.02 | 169.07 | 301.49 | 335.53 | 267.46 | 307.46 | 341.83 | 273.10 |
127 | 449 | 198.10 | 225.68 | 170.51 | 305.93 | 340.22 | 271.65 | 312.18 | 346.81 | 277.55 |
128 | 455 | 199.65 | 227.34 | 171.95 | 310.40 | 344.93 | 275.87 | 316.94 | 351.83 | 282.04 |
129 | 476 | 201.19 | 229.00 | 173.39 | 314.90 | 349.68 | 280.11 | 321.73 | 356.88 | 286.57 |
Time | Real | PNZ | PZ | TP | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
104 | 221 | 487.7756 | 531.063 | 444.488 | 215.524 | 244.299 | 186.750 | 54.79 | 69.30 | 40.28 |
105 | 227 | 495.2823 | 538.902 | 451.663 | 219.663 | 248.712 | 190.614 | 55.61 | 70.23 | 40.99 |
106 | 230 | 502.8471 | 546.799 | 458.896 | 223.824 | 253.147 | 194.501 | 56.43 | 71.16 | 41.71 |
107 | 242 | 510.4698 | 554.753 | 466.186 | 228.005 | 257.600 | 198.409 | 57.26 | 72.10 | 42.43 |
108 | 248 | 518.1506 | 562.766 | 473.535 | 232.205 | 262.072 | 202.338 | 58.10 | 73.04 | 43.16 |
109 | 261 | 525.8893 | 570.837 | 480.942 | 236.421 | 266.558 | 206.284 | 58.94 | 73.99 | 43.90 |
110 | 269 | 533.6859 | 578.965 | 488.407 | 240.654 | 271.059 | 210.248 | 59.79 | 74.95 | 44.64 |
111 | 279 | 541.5405 | 587.152 | 495.929 | 244.900 | 275.572 | 214.227 | 60.65 | 75.91 | 45.38 |
112 | 284 | 549.4528 | 595.396 | 503.510 | 249.157 | 280.095 | 218.219 | 61.51 | 76.88 | 46.14 |
113 | 287 | 557.423 | 603.698 | 511.148 | 253.425 | 284.627 | 222.223 | 62.38 | 77.86 | 46.90 |
114 | 298 | 565.4509 | 612.058 | 518.844 | 257.701 | 289.165 | 226.237 | 63.25 | 78.84 | 47.67 |
115 | 313 | 573.5364 | 620.476 | 526.597 | 261.983 | 293.708 | 230.259 | 64.14 | 79.83 | 48.44 |
116 | 325 | 581.6795 | 628.951 | 534.408 | 266.270 | 298.253 | 234.288 | 65.03 | 80.83 | 49.22 |
117 | 335 | 589.88 | 637.483 | 542.277 | 270.560 | 302.800 | 238.321 | 65.92 | 81.84 | 50.01 |
118 | 350 | 598.138 | 646.073 | 550.203 | 274.851 | 307.345 | 242.356 | 66.82 | 82.85 | 50.80 |
119 | 366 | 606.4533 | 654.721 | 558.186 | 279.140 | 311.887 | 246.393 | 67.73 | 83.87 | 51.60 |
120 | 378 | 614.8259 | 663.425 | 566.226 | 283.427 | 316.424 | 250.430 | 68.65 | 84.89 | 52.41 |
121 | 386 | 623.2555 | 672.187 | 574.324 | 287.709 | 320.954 | 254.463 | 69.57 | 85.92 | 53.23 |
122 | 398 | 631.7421 | 681.006 | 582.479 | 291.984 | 325.476 | 258.493 | 70.51 | 86.96 | 54.05 |
123 | 406 | 640.2856 | 689.881 | 590.690 | 296.251 | 329.987 | 262.516 | 71.44 | 88.01 | 54.88 |
124 | 412 | 648.8858 | 698.813 | 598.958 | 300.508 | 334.485 | 266.531 | 72.39 | 89.06 | 55.71 |
125 | 422 | 657.5427 | 707.802 | 607.283 | 304.753 | 338.969 | 270.537 | 73.34 | 90.13 | 56.56 |
126 | 432 | 666.256 | 716.847 | 615.665 | 308.985 | 343.437 | 274.532 | 74.30 | 91.20 | 57.41 |
127 | 449 | 675.0257 | 725.949 | 624.102 | 313.200 | 347.888 | 278.513 | 75.27 | 92.27 | 58.26 |
128 | 455 | 683.8516 | 735.107 | 632.596 | 317.399 | 352.318 | 282.480 | 76.24 | 93.36 | 59.13 |
129 | 476 | 692.7334 | 744.320 | 641.147 | 321.579 | 356.727 | 286.431 | 77.23 | 94.45 | 60.00 |
Time | Real | TC | Vtub | DPF | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
104 | 221 | 212.53 | 241.10 | 183.96 | 215.13 | 243.88 | 186.38 | 210.95 | 239.42 | 182.48 |
105 | 227 | 216.44 | 245.27 | 187.60 | 219.32 | 248.34 | 190.29 | 214.13 | 242.82 | 185.45 |
106 | 230 | 220.38 | 249.48 | 191.28 | 223.54 | 252.85 | 194.24 | 217.27 | 246.16 | 188.38 |
107 | 242 | 224.35 | 253.71 | 194.99 | 227.80 | 257.39 | 198.22 | 220.36 | 249.46 | 191.27 |
108 | 248 | 228.35 | 257.97 | 198.73 | 232.10 | 261.97 | 202.24 | 223.41 | 252.70 | 194.11 |
109 | 261 | 232.38 | 262.26 | 202.50 | 236.44 | 266.58 | 206.30 | 226.40 | 255.89 | 196.91 |
110 | 269 | 236.44 | 266.58 | 206.30 | 240.82 | 271.23 | 210.40 | 229.34 | 259.02 | 199.66 |
111 | 279 | 240.53 | 270.93 | 210.13 | 245.23 | 275.92 | 214.53 | 232.23 | 262.09 | 202.36 |
112 | 284 | 244.65 | 275.31 | 213.99 | 249.67 | 280.64 | 218.70 | 235.06 | 265.11 | 205.01 |
113 | 287 | 248.80 | 279.71 | 217.88 | 254.16 | 285.40 | 222.91 | 237.83 | 268.06 | 207.61 |
114 | 298 | 252.97 | 284.15 | 221.80 | 258.67 | 290.20 | 227.15 | 240.55 | 270.95 | 210.15 |
115 | 313 | 257.18 | 288.61 | 225.75 | 263.23 | 295.03 | 231.43 | 243.21 | 273.78 | 212.65 |
116 | 325 | 261.41 | 293.10 | 229.72 | 267.81 | 299.89 | 235.74 | 245.81 | 276.54 | 215.09 |
117 | 335 | 265.68 | 297.62 | 233.73 | 272.44 | 304.79 | 240.09 | 248.36 | 279.25 | 217.47 |
118 | 350 | 269.97 | 302.17 | 237.76 | 277.09 | 309.72 | 244.47 | 250.84 | 281.89 | 219.80 |
119 | 366 | 274.29 | 306.75 | 241.82 | 281.78 | 314.68 | 248.88 | 253.27 | 284.46 | 222.08 |
120 | 378 | 278.63 | 311.35 | 245.91 | 286.51 | 319.68 | 253.33 | 255.63 | 286.97 | 224.30 |
121 | 386 | 283.01 | 315.98 | 250.03 | 291.26 | 324.71 | 257.81 | 257.94 | 289.42 | 226.46 |
122 | 398 | 287.41 | 320.64 | 254.18 | 296.05 | 329.78 | 262.33 | 260.19 | 291.80 | 228.57 |
123 | 406 | 291.84 | 325.32 | 258.35 | 300.87 | 334.87 | 266.87 | 262.37 | 294.12 | 230.63 |
124 | 412 | 296.29 | 330.03 | 262.55 | 305.72 | 340.00 | 271.45 | 264.50 | 296.38 | 232.63 |
125 | 422 | 300.77 | 334.77 | 266.78 | 310.61 | 345.15 | 276.07 | 266.57 | 298.57 | 234.57 |
126 | 432 | 305.28 | 339.53 | 271.04 | 315.52 | 350.34 | 280.71 | 268.58 | 300.70 | 236.46 |
127 | 449 | 309.82 | 344.32 | 275.32 | 320.47 | 355.56 | 285.38 | 270.54 | 302.77 | 238.30 |
128 | 455 | 314.38 | 349.13 | 279.63 | 325.44 | 360.80 | 290.09 | 272.43 | 304.78 | 240.08 |
129 | 476 | 318.97 | 353.97 | 283.96 | 330.45 | 366.08 | 294.82 | 274.27 | 306.73 | 241.81 |
Time | Real | UDPF | DNN | RNN | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
104 | 221 | 207.30 | 235.52 | 179.08 | 224.90 | 254.29 | 195.51 | 223.38 | 252.67 | 194.09 |
105 | 227 | 210.58 | 239.02 | 182.13 | 231.12 | 260.92 | 201.32 | 229.02 | 258.68 | 199.36 |
106 | 230 | 213.86 | 242.52 | 185.20 | 232.86 | 262.76 | 202.95 | 233.04 | 262.96 | 203.12 |
107 | 242 | 217.14 | 246.02 | 188.26 | 245.66 | 276.38 | 214.94 | 243.36 | 273.94 | 212.78 |
108 | 248 | 220.43 | 249.53 | 191.33 | 252.12 | 283.24 | 221.00 | 249.67 | 280.64 | 218.70 |
109 | 261 | 223.72 | 253.03 | 194.40 | 264.58 | 296.46 | 232.70 | 262.29 | 294.03 | 230.54 |
110 | 269 | 227.01 | 256.54 | 197.48 | 272.98 | 305.36 | 240.59 | 270.43 | 302.66 | 238.20 |
111 | 279 | 230.30 | 260.05 | 200.56 | 282.82 | 315.78 | 249.86 | 280.34 | 313.16 | 247.52 |
112 | 284 | 233.60 | 263.55 | 203.64 | 288.04 | 321.30 | 254.77 | 285.86 | 319.00 | 252.72 |
113 | 287 | 236.89 | 267.06 | 206.73 | 289.86 | 323.22 | 256.49 | 289.90 | 323.27 | 256.53 |
114 | 298 | 240.19 | 270.56 | 209.81 | 301.74 | 335.79 | 267.69 | 299.40 | 333.31 | 265.48 |
115 | 313 | 243.48 | 274.07 | 212.90 | 316.42 | 351.28 | 281.55 | 314.27 | 349.02 | 279.52 |
116 | 325 | 246.78 | 277.57 | 215.99 | 328.66 | 364.19 | 293.13 | 326.29 | 361.70 | 290.89 |
117 | 335 | 250.08 | 281.07 | 219.08 | 338.82 | 374.90 | 302.74 | 336.33 | 372.28 | 300.39 |
118 | 350 | 253.37 | 284.57 | 222.17 | 353.42 | 390.27 | 316.57 | 351.27 | 388.01 | 314.54 |
119 | 366 | 256.67 | 288.07 | 225.27 | 369.34 | 407.00 | 331.67 | 367.27 | 404.83 | 329.71 |
120 | 378 | 259.96 | 291.56 | 228.36 | 381.66 | 419.95 | 343.37 | 379.29 | 417.46 | 341.12 |
121 | 386 | 263.25 | 295.06 | 231.45 | 389.98 | 428.68 | 351.27 | 387.43 | 426.01 | 348.85 |
122 | 398 | 266.54 | 298.54 | 234.55 | 401.66 | 440.94 | 362.38 | 399.30 | 438.46 | 360.13 |
123 | 406 | 269.83 | 302.03 | 237.64 | 409.98 | 449.66 | 370.29 | 407.43 | 446.99 | 367.87 |
124 | 412 | 273.12 | 305.51 | 240.73 | 416.12 | 456.10 | 376.14 | 413.69 | 453.55 | 373.82 |
125 | 422 | 276.41 | 308.99 | 243.82 | 425.82 | 466.27 | 385.37 | 423.36 | 463.68 | 383.03 |
126 | 432 | 279.69 | 312.47 | 246.91 | 435.82 | 476.74 | 394.90 | 433.34 | 474.14 | 392.54 |
127 | 449 | 282.97 | 315.94 | 250.00 | 452.26 | 493.94 | 410.57 | 450.27 | 491.86 | 408.68 |
128 | 455 | 286.24 | 319.40 | 253.08 | 459.12 | 501.12 | 417.12 | 456.64 | 498.53 | 414.76 |
129 | 476 | 289.52 | 322.86 | 256.17 | 478.93 | 521.83 | 436.04 | 477.26 | 520.08 | 434.44 |
Time | Real | LSTM | GRU | |||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | |||||
104 | 221 | 224.52 | 253.89 | 195.15 | 224.46 | 253.82 | 195.09 | |||
105 | 227 | 230.52 | 260.27 | 200.76 | 230.47 | 260.23 | 200.72 | |||
106 | 230 | 233.63 | 263.59 | 203.67 | 233.09 | 263.02 | 203.17 | |||
107 | 242 | 245.44 | 276.14 | 214.73 | 245.49 | 276.20 | 214.78 | |||
108 | 248 | 251.49 | 282.58 | 220.41 | 251.49 | 282.58 | 220.41 | |||
109 | 261 | 264.28 | 296.14 | 232.42 | 264.73 | 296.62 | 232.84 | |||
110 | 269 | 272.32 | 304.67 | 239.98 | 272.77 | 305.14 | 240.40 | |||
111 | 279 | 282.19 | 315.11 | 249.26 | 282.79 | 315.75 | 249.83 | |||
112 | 284 | 287.44 | 320.67 | 254.21 | 287.72 | 320.96 | 254.47 | |||
113 | 287 | 290.57 | 323.98 | 257.16 | 290.25 | 323.64 | 256.86 | |||
114 | 298 | 301.38 | 335.41 | 267.36 | 301.60 | 335.64 | 267.56 | |||
115 | 313 | 316.31 | 351.17 | 281.46 | 316.73 | 351.62 | 281.85 | |||
116 | 325 | 328.25 | 363.76 | 292.74 | 328.79 | 364.33 | 293.25 | |||
117 | 335 | 338.21 | 374.25 | 302.16 | 338.78 | 374.85 | 302.70 | |||
118 | 350 | 353.18 | 390.02 | 316.35 | 353.75 | 390.62 | 316.89 | |||
119 | 366 | 369.30 | 406.97 | 331.64 | 369.74 | 407.43 | 332.05 | |||
120 | 378 | 381.25 | 419.52 | 342.98 | 381.69 | 419.98 | 343.40 | |||
121 | 386 | 389.30 | 427.97 | 350.63 | 389.72 | 428.41 | 351.02 | |||
122 | 398 | 401.13 | 440.39 | 361.88 | 401.67 | 440.96 | 362.39 | |||
123 | 406 | 409.25 | 448.90 | 369.60 | 409.70 | 449.38 | 370.03 | |||
124 | 412 | 415.37 | 455.31 | 375.42 | 415.78 | 455.75 | 375.82 | |||
125 | 422 | 425.18 | 465.59 | 384.76 | 425.76 | 466.20 | 385.32 | |||
126 | 432 | 435.14 | 476.03 | 394.26 | 435.72 | 476.64 | 394.81 | |||
127 | 449 | 452.25 | 493.93 | 410.56 | 452.72 | 494.43 | 411.02 | |||
128 | 455 | 458.45 | 500.42 | 416.49 | 458.79 | 500.77 | 416.81 | |||
129 | 476 | 479.47 | 522.38 | 436.55 | 479.79 | 522.72 | 436.86 |
Time | Real | GO | DS | YID | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
117 | 335 | 215.85 | 244.65 | 187.06 | 284.80 | 317.88 | 251.73 | 285.07 | 318.16 | 251.98 |
118 | 350 | 217.69 | 246.61 | 188.77 | 289.52 | 322.87 | 256.17 | 289.81 | 323.18 | 256.45 |
119 | 366 | 219.53 | 248.57 | 190.49 | 294.27 | 327.89 | 260.65 | 294.59 | 328.24 | 260.95 |
120 | 378 | 221.37 | 250.53 | 192.21 | 299.06 | 332.95 | 265.16 | 299.41 | 333.33 | 265.50 |
121 | 386 | 223.21 | 252.49 | 193.93 | 303.88 | 338.05 | 269.71 | 304.26 | 338.45 | 270.08 |
122 | 398 | 225.05 | 254.45 | 195.65 | 308.74 | 343.18 | 274.30 | 309.16 | 343.62 | 274.69 |
123 | 406 | 226.89 | 256.41 | 197.37 | 313.63 | 348.34 | 278.92 | 314.08 | 348.82 | 279.35 |
124 | 412 | 228.73 | 258.37 | 199.08 | 318.56 | 353.54 | 283.58 | 319.05 | 354.05 | 284.04 |
125 | 422 | 230.57 | 260.33 | 200.80 | 323.52 | 358.78 | 288.27 | 324.05 | 359.33 | 288.76 |
126 | 432 | 232.40 | 262.28 | 202.53 | 328.52 | 364.05 | 293.00 | 329.08 | 364.64 | 293.53 |
127 | 449 | 234.24 | 264.24 | 204.25 | 333.56 | 369.35 | 297.76 | 334.16 | 369.98 | 298.33 |
128 | 455 | 236.08 | 266.20 | 205.97 | 338.63 | 374.69 | 302.56 | 339.27 | 375.37 | 303.16 |
129 | 476 | 237.92 | 268.15 | 207.69 | 343.73 | 380.07 | 307.39 | 344.41 | 380.79 | 308.04 |
Time | Real | PNZ | PZ | TP | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
117 | 335 | 637.8859 | 687.388 | 588.383 | 312.9409 | 347.614 | 178.36 | 74.02 | 90.88 | 57.15 |
118 | 350 | 648.5408 | 698.455 | 598.627 | 320.3052 | 355.383 | 179.16 | 74.99 | 91.96 | 58.01 |
119 | 366 | 659.313 | 709.640 | 608.986 | 327.825 | 363.313 | 179.95 | 75.97 | 93.05 | 58.88 |
120 | 378 | 670.2031 | 720.944 | 619.462 | 335.505 | 371.406 | 180.74 | 76.95 | 94.14 | 59.76 |
121 | 386 | 681.2118 | 732.368 | 630.056 | 343.347 | 379.665 | 181.51 | 77.94 | 95.24 | 60.64 |
122 | 398 | 692.3398 | 743.912 | 640.768 | 351.355 | 388.094 | 182.28 | 78.94 | 96.35 | 61.52 |
123 | 406 | 703.5877 | 755.577 | 651.598 | 359.532 | 396.697 | 183.04 | 79.94 | 97.47 | 62.42 |
124 | 412 | 714.956 | 767.364 | 662.548 | 367.883 | 405.476 | 183.79 | 80.95 | 98.59 | 63.32 |
125 | 422 | 726.4455 | 779.273 | 673.618 | 376.411 | 414.437 | 184.54 | 81.97 | 99.72 | 64.23 |
126 | 432 | 738.0568 | 791.305 | 684.809 | 385.119 | 423.582 | 185.27 | 83.00 | 100.85 | 65.14 |
127 | 449 | 749.7904 | 803.460 | 696.121 | 394.011 | 432.916 | 186.00 | 84.03 | 101.99 | 66.06 |
128 | 455 | 761.647 | 815.739 | 707.555 | 403.091 | 442.443 | 186.72 | 85.07 | 103.14 | 66.99 |
129 | 476 | 773.6271 | 828.143 | 719.111 | 412.364 | 452.165 | 187.43 | 86.11 | 104.30 | 67.92 |
Time | Real | TC | Vtub | DPF | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
117 | 335 | 300.84 | 334.83 | 266.84 | 315.55 | 350.37 | 280.74 | 314.95 | 349.74 | 280.17 |
118 | 350 | 306.57 | 340.89 | 272.26 | 322.66 | 357.87 | 287.45 | 322.44 | 357.63 | 287.24 |
119 | 366 | 312.37 | 347.01 | 277.73 | 329.85 | 365.45 | 294.25 | 330.07 | 365.68 | 294.46 |
120 | 378 | 318.23 | 353.20 | 283.27 | 337.12 | 373.10 | 301.13 | 337.86 | 373.89 | 301.83 |
121 | 386 | 324.15 | 359.44 | 288.86 | 344.46 | 380.84 | 308.09 | 345.80 | 382.25 | 309.35 |
122 | 398 | 330.13 | 365.74 | 294.51 | 351.88 | 388.65 | 315.12 | 353.90 | 390.77 | 317.03 |
123 | 406 | 336.16 | 372.10 | 300.23 | 359.37 | 396.53 | 322.22 | 362.16 | 399.46 | 324.86 |
124 | 412 | 342.26 | 378.52 | 306.00 | 366.93 | 404.48 | 329.39 | 370.58 | 408.31 | 332.85 |
125 | 422 | 348.42 | 385.00 | 311.83 | 374.56 | 412.49 | 336.63 | 379.17 | 417.33 | 341.00 |
126 | 432 | 354.63 | 391.54 | 317.72 | 382.25 | 420.57 | 343.93 | 387.92 | 426.52 | 349.31 |
127 | 449 | 360.91 | 398.14 | 323.67 | 389.99 | 428.70 | 351.29 | 396.83 | 435.88 | 357.79 |
128 | 455 | 367.25 | 404.81 | 329.69 | 397.79 | 436.89 | 358.70 | 405.92 | 445.41 | 366.43 |
129 | 476 | 373.64 | 411.53 | 335.76 | 405.65 | 445.12 | 366.17 | 415.18 | 455.12 | 375.25 |
Time | Real | UDPF | DNN | RNN | ||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | Prediction | Upper | Lower | ||
117 | 335 | 313.06 | 347.74 | 278.38 | 342.30 | 378.56 | 306.04 | 344.31 | 380.68 | 307.94 |
118 | 350 | 319.88 | 354.94 | 284.83 | 357.85 | 394.93 | 320.77 | 359.54 | 396.70 | 322.38 |
119 | 366 | 326.81 | 362.24 | 291.37 | 373.93 | 411.83 | 336.03 | 375.59 | 413.58 | 337.61 |
120 | 378 | 333.83 | 369.64 | 298.02 | 385.60 | 424.09 | 347.11 | 387.48 | 426.06 | 348.90 |
121 | 386 | 340.96 | 377.15 | 304.77 | 392.86 | 431.70 | 354.01 | 395.19 | 434.15 | 356.22 |
122 | 398 | 348.19 | 384.77 | 311.62 | 405.60 | 445.08 | 366.13 | 407.38 | 446.94 | 367.82 |
123 | 406 | 355.53 | 392.49 | 318.57 | 412.86 | 452.68 | 373.03 | 415.14 | 455.07 | 375.20 |
124 | 412 | 362.97 | 400.31 | 325.63 | 417.20 | 457.23 | 377.16 | 420.72 | 460.92 | 380.52 |
125 | 422 | 370.52 | 408.25 | 332.79 | 429.30 | 469.91 | 388.69 | 431.01 | 471.71 | 390.32 |
126 | 432 | 378.17 | 416.29 | 340.06 | 439.30 | 480.38 | 398.22 | 441.16 | 482.33 | 400.00 |
127 | 449 | 385.93 | 424.44 | 347.43 | 457.00 | 498.90 | 415.10 | 458.57 | 500.55 | 416.60 |
128 | 455 | 393.80 | 432.70 | 354.91 | 460.20 | 502.24 | 418.15 | 464.02 | 506.24 | 421.80 |
129 | 476 | 401.78 | 441.06 | 362.49 | 484.30 | 527.43 | 441.17 | 485.63 | 528.82 | 442.44 |
Time | Real | LSTM | GRU | |||||||
Prediction | Upper | Lower | Prediction | Upper | Lower | |||||
117 | 335 | 344.69 | 381.08 | 308.30 | 341.00 | 377.20 | 304.81 | |||
118 | 350 | 359.89 | 397.08 | 322.71 | 356.47 | 393.48 | 319.47 | |||
119 | 366 | 376.04 | 414.05 | 338.04 | 372.84 | 410.68 | 334.99 | |||
120 | 378 | 388.18 | 426.80 | 349.56 | 384.75 | 423.19 | 346.30 | |||
121 | 386 | 396.10 | 435.10 | 357.09 | 392.09 | 430.91 | 353.28 | |||
122 | 398 | 408.02 | 447.61 | 368.43 | 404.27 | 443.68 | 364.86 | |||
123 | 406 | 415.90 | 455.87 | 375.93 | 411.73 | 451.50 | 371.95 | |||
124 | 412 | 421.43 | 461.67 | 381.20 | 416.92 | 456.94 | 376.90 | |||
125 | 422 | 431.44 | 472.15 | 390.73 | 427.32 | 467.83 | 386.80 | |||
126 | 432 | 441.49 | 482.67 | 400.31 | 437.55 | 478.54 | 396.55 | |||
127 | 449 | 458.83 | 500.81 | 416.85 | 455.32 | 497.14 | 413.50 | |||
128 | 455 | 464.67 | 506.92 | 422.42 | 460.27 | 502.32 | 418.22 | |||
129 | 476 | 485.82 | 529.02 | 442.62 | 482.13 | 525.17 | 439.09 |
References
- Jelinski, Z.; Moranda, P. Software reliability research. In Statistical Computer Performance Evaluation; Elsevier: Amsterdam, The Netherlands, 1972; pp. 465–484. [Google Scholar]
- Goel, A.L.; Okumoto, K. Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. 1979, 28, 206–211. [Google Scholar] [CrossRef]
- Yamada, S.; Ohba, M.; Osaki, S. S-shaped reliability growth modeling for software error detection. IEEE Trans. Reliab. 1983, 32, 475–484. [Google Scholar] [CrossRef]
- Ohba, M. Inflection S-shaped software reliability growth model. In Stochastic Models in Reliability Theory; Osaki, S., Hatoyama, Y., Eds.; Springer: Berlin, Germany, 1984; pp. 144–162. [Google Scholar]
- Zhang, X.M.; Teng, X.L.; Pham, H. Considering fault removal efficiency in software reliability assessment. IEEE Trans. Syst. Man. Cybern. Part Syst. Hum. 2003, 33, 114–120. [Google Scholar] [CrossRef]
- Yamada, S.; Tokuno, K.; Osaki, S. Imperfect debugging models with fault introduction rate for software reliability assessment. Int. J. Syst. Sci. 1992, 23, 2241–2252. [Google Scholar] [CrossRef]
- Pham, H.; Zhang, X. An NHPP software reliability models and its comparison. Int. J. Reliab. Qual. Saf. Eng. 1997, 4, 269–282. [Google Scholar] [CrossRef]
- Pham, H.; Nordmann, L.; Zhang, X. A general imperfect software debugging model with S-shaped fault detection rate. IEEE Trans. Reliab. 1999, 48, 169–175. [Google Scholar] [CrossRef]
- Teng, X.; Pham, H. A new methodology for predicting software reliability in the random field environments. IEEE Trans. Reliab. 2006, 55, 458–468. [Google Scholar] [CrossRef]
- Kapur, P.K.; Pham, H.; Anand, S.; Yadav, K. A unified approach for developing software reliability growth models in the presence of imperfect debugging and error generation. IEEE Trans. Reliab. 2011, 60, 331–340. [Google Scholar] [CrossRef]
- Roy, P.; Mahapatra, G.S.; Dey, K.N. An NHPP software reliability growth model with imperfect debugging and error generation. Int. J. Reliab. Qual. Saf. Eng. 2014, 21, 1450008. [Google Scholar] [CrossRef]
- Chang, I.H.; Pham, H.; Lee, S.W.; Song, K.Y. A testing-coverage software reliability model with the uncertainty of operation environments. Int. J. Syst. Sci. Oper. Logist. 2014, 1, 220–227. [Google Scholar] [CrossRef]
- Pham, H. A new software reliability model with Vtub-Shaped fault detection rate and the uncertainty of operating environments. Optimization 2014, 63, 1481–1490. [Google Scholar] [CrossRef]
- Pham, L.; Pham, H. Software reliability models with time-dependent hazard function based on bayesian approach. IEEE Trans. Syst. Man, Cybern.-Part A Syst. Humans 2000, 30, 25–35. [Google Scholar] [CrossRef]
- Kim, Y.S.; Song, K.Y.; Pham, H.; Chang, I.H. A software reliability model with dependent failure and optimal release time. Symmetry 2022, 14, 343. [Google Scholar] [CrossRef]
- Lee, D.H.; Chang, I.H.; Pham, H. Software reliability growth model with dependent failures and uncertain operating environments. Appl. Sci. 2022, 12, 12383. [Google Scholar] [CrossRef]
- Kim, Y.S.; Chang, I.H.; Lee, D.H. Non-parametric software reliability model using deep neural network and NHPP software reliability growth model comparison. J. Korean Data Anal. Soc. 2020, 22, 2371–2382. [Google Scholar] [CrossRef]
- Miyamoto, S.; Tamura, Y.; Yamada, S. Reliability assessment tool based on deep learning and data preprocessing for OSS. Am. J. Oper. Res. 2022, 12, 111–125. [Google Scholar] [CrossRef]
- Wang, J.; Zhang, C. Software reliability prediction using a deep learning model based on the RNN encoder–decoder. Reliab. Eng. Syst. Saf. 2018, 170, 73–82. [Google Scholar] [CrossRef]
- Oveisi, S.; Moeini, A.; Mirzaei, S. LSTM encoder-decoder dropout model in software reliability prediction. Int. J. Reliab. Risk Saf. Theory Appl. 2021, 4, 1–12. [Google Scholar] [CrossRef]
- Raamesh, L.; Jothi, S.; Radhika, S. Enhancing software reliability and fault detection using hybrid brainstorm optimization-based LSTM model. IETE J. Res. 2022, 1–15. [Google Scholar] [CrossRef]
- Karunanithi, N.; Whitley, D.; Malaiya, Y.K. Using neural networks in reliability prediction. IEEE Softw. 1992, 9, 53–59. [Google Scholar] [CrossRef]
- Singh, Y.; Kumar, P. Prediction of software reliability using feed forward neural networks. In Proceedings of the 2010 International Conference on Computational Intelligence and Software Engineering, Wuhan, China, 10–12 December 2010; pp. 1–5. [Google Scholar]
- Graves, A. Generating sequences with recurrent neural networks. arXiv 2013, arXiv:1308.0850. [Google Scholar]
- Bengio, Y.; Simard, P.; Frasconi, P. Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Netw. 1994, 5, 157–166. [Google Scholar] [CrossRef] [PubMed]
- Jozefowicz, R.; Zaremba, W.; Sutskever, I. An empirical exploration of recurrent network architectures In Proceedings of the 32nd International Conference on Machine Learning (ICML-15), Lille, France, 6–11 July 2015.
- Gers, F.A.; Schmidhuber, J.; Cummins, F. Learning to forget: Continual prediction with LSTM. Neural Comput. 2000, 12, 2451–2471. [Google Scholar] [CrossRef]
- Inoue, S.; Yamada, S. Discrete software reliability assessment with discretized NHPP models. Comput. Math. Appl. 2006, 51, 161–170. [Google Scholar] [CrossRef]
- Askari, R.; Sebt, M.V.; Amjadian, A. A multi-product EPQ model for defective production and inspection with single machine, and operational constraints: Stochastic programming approach. Commun. Comput. Infor. Sci. 2021, 1458, 161–193. [Google Scholar]
- Jeske, D.R.; Zhang, X. Some successful approaches to software reliability modeling in industry. J. Syst. Softw. 2005, 74, 85–99. [Google Scholar] [CrossRef]
- Iqbal, J. Software reliability growth models: A comparison of linear and exponential fault content functions for study of imperfect debugging situations. Cogent Eng. 2017, 4, 1286739. [Google Scholar] [CrossRef]
- Pillai, K.; Sukumaran Nair, V.S. A model for software development effort and cost estimation. IEEE Trans. Softw. Eng. 1997, 23, 485–497. [Google Scholar] [CrossRef]
- Souza, R.L.C.; Ghasemi, A.; Saif, A.; Gharaei, A. Robust job-shop scheduling under deterministic and stochastic unavailability constraints due to preventive and corrective maintenance. Comput. Ind. Eng. 2022, 168, 108130. [Google Scholar] [CrossRef]
- Li, Q.; Pham, H. NHPP software reliability model considering the uncertainty of operating environments with imperfect debugging and testing coverage. Appl. Math. Model. 2017, 51, 68–85. [Google Scholar] [CrossRef]
- Pham, H. System Software Reliability; Springer: London, UK, 2006. [Google Scholar]
No. | Model | Mean Value Function | Note |
---|---|---|---|
1 | Goel-Okumoto (GO) [2] | Concave | |
2 | Yamada et al. (DS) [3] | S-shape | |
3 | Yamada et al. (YID) [6] | Concave | |
4 | Pham-Zhang (PZ) [7] | Both | |
5 | Pham et al. (PNZ) [8] | Both | |
6 | Teng-Pham (TP) [9] | S-shape | |
7 | Chang et al. (TC) [12] | Both | |
8 | Pham (Vtub) [13] | S-shape | |
9 | Kim et al. (DPF) [15] | S-shape, Dependent | |
10 | Lee et al. (UDPF) [16] | S-shape Dependent |
No. | Model | 80% | 90% |
---|---|---|---|
1 | GO | ||
2 | DS | ||
3 | YID | , | 1, |
4 | PZ | , | , |
5 | PNZ | , | , |
6 | TP | , , | , |
7 | TC | , , | , |
8 | Vtub | , | , |
9 | DPF | , , | , , |
10 | UDPF | , , | . , |
11 | DNN | α = 0.005, hidden layers = 3, optimizer = Adam, epoch = 200 | |
12 | RNN | α = 0.0001, hidden layers = 2, optimizer = Adam, epoch = 200 | |
13 | LSTM | α = 0.0001, hidden layers = 2, optimizer = Adam, epoch = 200 | |
14 | GRU | α = 0.0001, hidden layers = 2, optimizer = Adam, epoch = 200 |
No. | Model | MSE | MAE | PRR | PP | R2 | PRV | RMSPE | MEOP | TS | PC | preSSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | GO | 21.1902 | 4.0601 | 4.7030 | 11.9204 | 0.9254 | 4.2944 | 4.6544 | 3.9586 | 16.0037 | 60.5184 | 768.422 |
2 | DS | 3.9990 | 1.6435 | 1853.84 | 3.9028 | 0.9859 | 1.9503 | 2.0240 | 1.6024 | 6.9523 | 28.0021 | 64.894 |
3 | YID | 2.7404 | 1.2988 | 26.5655 | 2.4337 | 0.9904 | 1.6685 | 1.6768 | 1.2663 | 5.7552 | 20.6324 | 454.284 |
4 | PZ | 2.8043 | 1.2790 | 15.2300 | 2.6449 | 0.9901 | 1.6951 | 1.6965 | 1.2471 | 5.8219 | 21.0819 | 8306.771 |
5 | PNZ | 2.7172 | 1.2864 | 22.5012 | 2.4253 | 0.9904 | 1.6634 | 1.6698 | 1.2543 | 5.7308 | 20.4666 | 150,716.8 |
6 | TP | 2.7168 | 1.2888 | 18.3855 | 2.3926 | 0.9904 | 1.6648 | 1.6697 | 1.2566 | 5.7303 | 20.4634 | 7839.272 |
7 | TC | 3.1185 | 1.4057 | 221.7352 | 3.0482 | 0.9890 | 1.7648 | 1.7884 | 1.3705 | 6.1394 | 23.1529 | 224.970 |
8 | Vtub | 2.8788 | 1.3403 | 108.599 | 2.8416 | 0.9899 | 1.6990 | 1.7184 | 1.3068 | 5.8987 | 21.5931 | 455.315 |
9 | DPF | 3.1625 | 1.3642 | 0.9386 | 1.9608 | 0.9889 | 1.7976 | 1.8015 | 1.3301 | 6.1826 | 23.4260 | 47.645 |
10 | UDPF | 2.5427 | 1.2164 | 0.8600 | 1.2500 | 0.9911 | 1.6154 | 1.6154 | 1.1860 | 5.5437 | 19.1722 | 876.104 |
11 | DNN | 2.4373 | 1.4791 | 0.7005 | 1.1789 | 0.9914 | 0.5061 | 1.5633 | 1.4421 | 5.4275 | 18.3463 | 25.294 |
12 | RNN | 2.1147 | 1.4127 | 0.7380 | 1.2844 | 0.9926 | 0.3495 | 1.4553 | 1.3774 | 5.0557 | 15.5784 | 22.274 |
13 | LSTM | 2.0069 | 1.4104 | 0.8014 | 1.4432 | 0.9929 | 0.1345 | 1.4168 | 1.3751 | 4.9250 | 14.5574 | 21.570 |
14 | GRU | 2.0159 | 1.3983 | 0.6706 | 1.1032 | 0.9929 | 0.2493 | 1.4204 | 1.3634 | 4.9362 | 14.6454 | 23.872 |
No. | Model | MSE | MAE | PRR | PP | R2 | PRV | RMSPE | MEOP | TS | PC | preSSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | GO | 22.6581 | 4.3155 | 4.7797 | 13.4021 | 0.9407 | 4.4490 | 4.8071 | 4.2196 | 14.1477 | 69.6287 | 321.655 |
2 | DS | 3.9569 | 1.6600 | 1636.27 | 3.7881 | 0.9896 | 1.9677 | 2.0112 | 1.6231 | 5.9122 | 31.2376 | 6.537 |
3 | YID | 3.5588 | 1.5488 | 58.9754 | 2.7164 | 0.9907 | 1.9001 | 1.9081 | 1.5144 | 5.6069 | 28.9044 | 11.837 |
4 | PZ | 3.1010 | 1.4570 | 31.2349 | 2.4008 | 0.9919 | 1.7750 | 1.7812 | 1.4246 | 5.2339 | 25.8753 | 2395.537 |
5 | PNZ | 3.1296 | 1.4638 | 27.1085 | 2.3403 | 0.9918 | 1.7798 | 1.7893 | 1.4313 | 5.2580 | 26.0772 | 40.741 |
6 | TP | 3.2108 | 1.4898 | 26.6591 | 2.3619 | 0.9916 | 1.8069 | 1.8125 | 1.4567 | 5.3257 | 26.6407 | 4948.058 |
7 | TC | 3.6277 | 1.5687 | 262.6646 | 3.1326 | 0.9905 | 1.9117 | 1.9263 | 1.5338 | 5.6609 | 29.3263 | 6.367 |
8 | Vtub | 3.4308 | 1.5331 | 82.7369 | 2.6383 | 0.9910 | 1.8662 | 1.8735 | 1.4990 | 5.5051 | 28.0984 | 7.340 |
9 | DPF | 3.0682 | 1.3714 | 0.8756 | 1.7908 | 0.9920 | 1.7676 | 1.7718 | 1.3409 | 5.2061 | 25.6412 | 65.275 |
10 | UDPF | 2.9984 | 1.3889 | 0.6386 | 0.9113 | 0.9922 | 1.7515 | 1.7516 | 1.3581 | 5.1465 | 25.1349 | 1709.197 |
11 | DNN | 2.5868 | 1.5030 | 0.7327 | 1.3150 | 0.9932 | 0.5792 | 1.6107 | 1.4696 | 4.7803 | 21.8866 | 13.471 |
12 | RNN | 2.0731 | 1.4332 | 0.8317 | 1.5417 | 0.9946 | 0.1402 | 1.4400 | 1.4013 | 4.2794 | 17.0167 | 11.397 |
13 | LSTM | 2.0257 | 1.4179 | 0.7584 | 1.3192 | 0.9947 | 0.1251 | 1.4234 | 1.3864 | 4.2302 | 16.5077 | 11.755 |
14 | GRU | 1.9961 | 1.4061 | 0.7593 | 1.331 | 0.9948 | 0.1392 | 1.413 | 1.3749 | 4.1991 | 16.1831 | 12.103 |
No. | Model | 80% | 90% |
---|---|---|---|
1 | GO | ||
2 | DS | ||
3 | YID | , | , |
4 | PZ | , , | , |
5 | PNZ | , , | , |
6 | TP | , , | , |
7 | TC | ,, , , | , , |
8 | Vtub | , , | , , |
9 | DPF | , , | , , |
10 | UDPF | , , | , , |
11 | DNN | α =0.000001, hidden layers = 3, optimizer = Adam, epoch = 200 | |
12 | RNN | α = 0.00001, hidden layers = 2, optimizer = Adam, epoch = 200 | |
13 | LSTM | α = 0.00001, hidden layers = 2, optimizer = Adam, epoch = 200 | |
14 | GRU | α = 0.00001, hidden layers = 2, optimizer = Adam, epoch = 200 |
No. | Model | MSE | MAE | PRR | PP | R2 | PRV | RMSPE | MEOP | TS | PC | preSSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | GO | 514.6996 | 20.0326 | 17.3582 | 100.887 | 0.8664 | 21.0794 | 22.7819 | 19.8400 | 23.7311 | 322.5348 | 735,386.2 |
2 | DS | 13.7076 | 3.0740 | 2256.700 | 8.8015 | 0.9964 | 3.6874 | 3.7202 | 3.0445 | 3.8728 | 135.8146 | 202,541.7 |
3 | YID | 12.1406 | 2.8671 | 215.3384 | 7.1101 | 0.9968 | 3.4643 | 3.5010 | 2.8395 | 3.6447 | 129.5631 | 184,160.8 |
4 | PZ | 10.0437 | 2.5378 | 3299.707 | 9.6331 | 0.9974 | 3.1828 | 3.1847 | 2.5134 | 3.3150 | 119.7981 | 174,920.2 |
5 | PNZ | 9.6155 | 2.4483 | 7.2913 | 1.9345 | 0.9975 | 3.1155 | 3.1160 | 2.4248 | 3.2436 | 117.5539 | 1,643,848 |
6 | TP | 11.0679 | 2.6448 | 5.4184 | 1.7716 | 0.9971 | 3.3366 | 3.3430 | 2.6194 | 3.4800 | 124.7987 | 2,037,721 |
7 | TC | 13.4688 | 3.0404 | 2453.543 | 9.2252 | 0.9965 | 3.6077 | 3.6872 | 3.0112 | 3.8389 | 134.9098 | 189,842.4 |
8 | Vtub | 10.7640 | 2.7480 | 5303.130 | 7.7007 | 0.9972 | 3.2707 | 3.2966 | 2.7216 | 3.4318 | 123.3649 | 159,551.2 |
9 | DPF | 16.6392 | 3.3789 | 5.4085 | 88.826 | 0.9957 | 4.0398 | 4.0985 | 3.3464 | 4.2668 | 145.7959 | 305,246.7 |
10 | UDPF | 22.5801 | 3.8694 | 99.4808 | 9.7126 | 0.9941 | 4.6880 | 4.7742 | 3.8322 | 4.9705 | 161.5192 | 274,796.2 |
11 | DNN | 5.8181 | 2.1792 | 1.5342 | 4.0310 | 0.9985 | 1.0391 | 2.4143 | 2.1583 | 2.5231 | 91.6808 | 355.63 |
12 | RNN | 4.8223 | 2.0739 | 2.1101 | 9.9452 | 0.9987 | 0.7255 | 2.1971 | 2.0539 | 2.2970 | 82.0123 | 71.06 |
13 | LSTM | 4.6712 | 2.0805 | 2.3802 | 16.1958 | 0.9988 | 0.5882 | 2.1621 | 2.0605 | 2.2608 | 80.3730 | 290.41 |
14 | GRU | 4.5494 | 2.0688 | 2.1001 | 9.9841 | 0.9988 | 0.5218 | 2.1336 | 2.0489 | 2.2311 | 79.0128 | 347.92 |
No. | Model | MSE | MAE | PRR | PP | R2 | PRV | RMSPE | MEOP | TS | PC | preSSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | GO | 1278.733 | 30.0077 | 23.6240 | 173.2001 | 0.8256 | 33.3020 | 35.8928 | 29.7512 | 28.0264 | 415.9016 | 426,981.8 |
2 | DS | 98.7794 | 7.0277 | 2204.006 | 8.9742 | 0.9865 | 9.8441 | 9.9807 | 6.9676 | 7.7895 | 267.3789 | 114,300.6 |
3 | YID | 97.1781 | 6.9453 | 840.7242 | 8.4733 | 0.9867 | 9.7596 | 9.8995 | 6.8859 | 7.7261 | 266.4310 | 113,140.1 |
4 | PZ | 27.9916 | 4.2381 | 677.6398 | 67.5837 | 0.9962 | 5.3037 | 5.3136 | 4.2019 | 4.1466 | 194.2418 | 26,776.2 |
5 | PNZ | 39.4936 | 4.7960 | 18.9956 | 3.8443 | 0.9946 | 6.2657 | 6.3113 | 4.7550 | 4.9254 | 214.2075 | 1,165,855 |
6 | TP | 13.6426 | 2.9286 | 4.0108 | 1.6448 | 0.9981 | 3.6944 | 3.7095 | 2.9035 | 2.8948 | 152.5569 | 1,390,597 |
7 | TC | 64.3763 | 6.6342 | 19,228.33 | 14.4663 | 0.9912 | 7.8678 | 8.0567 | 6.5775 | 6.2884 | 242.5466 | 65,012.80 |
8 | Vtub | 26.1124 | 3.8598 | 3.7169 | 1.4301 | 0.9964 | 5.1293 | 5.1322 | 3.8268 | 4.0050 | 190.2112 | 23,868.05 |
9 | DPF | 50.3622 | 6.2521 | 9.2044 | 251.4521 | 0.9931 | 7.0064 | 7.1264 | 6.1987 | 5.5620 | 228.3074 | 28,590.91 |
10 | UDPF | 36.7890 | 4.7296 | 4.2437 | 47.5652 | 0.9950 | 6.0896 | 6.0917 | 4.6891 | 4.7537 | 210.0929 | 33,114.75 |
11 | DNN | 11.4930 | 2.8455 | 1.6236 | 4.2855 | 0.9984 | 1.8509 | 3.3945 | 2.8211 | 2.6570 | 142.612 | 680.76 |
12 | RNN | 10.3925 | 2.8168 | 2.9032 | 22.1128 | 0.9986 | 1.5746 | 3.2271 | 2.7928 | 2.5266 | 136.7744 | 1122.65 |
13 | LSTM | 11.7785 | 2.8708 | 2.9508 | 40.5778 | 0.9984 | 1.8888 | 3.4365 | 2.8463 | 2.6898 | 144.0352 | 1251.45 |
14 | GRU | 8.8655 | 2.8335 | 2.8749 | 19.0924 | 0.9988 | 0.9189 | 2.9787 | 2.8092 | 2.3336 | 127.5573 | 468.03 |
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Kim, Y.S.; Song, K.Y.; Chang, I.H. Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning. Appl. Sci. 2023, 13, 6730. https://doi.org/10.3390/app13116730
Kim YS, Song KY, Chang IH. Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning. Applied Sciences. 2023; 13(11):6730. https://doi.org/10.3390/app13116730
Chicago/Turabian StyleKim, Youn Su, Kwang Yoon Song, and In Hong Chang. 2023. "Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning" Applied Sciences 13, no. 11: 6730. https://doi.org/10.3390/app13116730
APA StyleKim, Y. S., Song, K. Y., & Chang, I. H. (2023). Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning. Applied Sciences, 13(11), 6730. https://doi.org/10.3390/app13116730