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Maximum likelihood estimation and optimisation of parameters of software reliability models using evolutionary optimisation techniques Govindasamy P Research Scholar Department of Industrial Engineering College of Engineering, Guindy, Anna university, Chennai-600 025, Tamilnadu, India.
[email protected] Dillibabu R Associate Professor Department of Industrial Engineering College of Engineering, Guindy, Anna university, Chennai-600 025, Tamilnadu, India.
[email protected] Abstract — This article presents the method of parameter estimation of software reliability models using Maximum likelihood estimation. Goel-Okumoto model and delayed S- Shaped model were considered for this case study. After estimation, parameter optimization was done using three techniques namely Genetic Algorithm (GA), GA Multi Objective Optimization (GAMULOBJ), and Simulated Annealing (SA) algorithm. The objective of this paper is to propose suitable parameter estimation and optimization technique for software reliability models specifically the Non-Homogenous Poisson Process class. There were two unknown parameters in this study. Using real time failure data, the unknown parameters were estimated and optimized. A comparison is made between the considered techniques for selecting the best fit. A comparison criterion was also presented to ascertain the estimation accuracy of the employed optimization techniques namely GA, GAMULOBJ, and SA. Keywords—Parameter estimation; Optimisatio; Software reliability Models; Failures, Failure intensit; estimation error; Mean Square error; Mean Magnitude of relative error. I.
INTRODUCTION
Reliability is the need of the hour for all critical systems in this modern world [1]. Nowadays all the systems are driven by software and are called as embedded systems [2]. Quality is one of the important factors that affects the reputation and business of an organization [3]. There are six dimensions of quality, and reliability is one of the important dimensions. Quality is directly proportional to reliability [4]. Quality is fitness for use, and the degree of customer satisfaction [5]. Reliability is the degree to which a product or service performs its operation without failure for a stipulated period of time under predefined operating conditions [6]. So the reliability of a product must be ascertained before shipment of the end user.
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Software find applications in systems ranging from household devices like washing machines, micro wave oven, and induction stoves to critical systems like space missiles, aircrafts, defense, nuclear power plant, and medical devices[7]. For proper functioning of the systems, the reliability of the software must be to the maximum. The reliable performance of a system depends on the reliable software used for driving it. So there is a need to ascertain the reliability of software. Software reliability is defined as “the failure free operation of software for a given time period under pre-defined operating conditions” [8]. To ascertain the reliability of the software, software reliability models are often utilized. A software reliability model is a mathematical function that relates reliability and other metrics with time, software failure, error content, testing effort, and error propagation [9]. Around 300 models were developed by eminent researchers and software engineers from 1970s. But the model applicability is limited, and no model is universally suitable for all projects and situations. The non-homogenous Poisson process (NHPP) models are special class of software reliability models, and finds applications in many software projects [10]. Goel-Okumoto (G-O) model and Delayed S-Shaped model are prominent models in the NHPP category. G-O model was proposed by Goel and Okumoto in the year 1979 [11]. Yamada proposed delayed s-shaped model in the year 1984 [12]. Parameter estimation is one of the important tasks in software reliability modeling and estimation [13]. Probability plotting, and curve fitting are traditional parameter estimation techniques used before the evolution of computers. After computer evolution techniques like Maximum Likelihood Estimation (MLE) and Least Squares Estimation (LSE) were employed for estimation. Many heuristics and algorithms were developed by researchers to get the optimal parameter values. After estimation, optimization is used to arrive at the likelihood of the real value. Evolutionary optimization techniques like Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Artificial Bee Colony (ABC) optimization, and firefly search algorithm have given promising results [14]. The main aim of this paper is to propose suitable Parameter estimation and optimization technique for G-O model and delayed s-shaped model. The methodology is validated using software failure data collected from a real time project. The rest of the paper is organized as follows: Section II presents the related work on parameter estimation and optimization of software reliability models. Section III presents the methodology of our work. Section IV discusses the results and inferences. Finally section V concludes the paper and gives the limitations and future scope of our work.
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II.
RELATED WORK
A. G-O model and Delayed S-Shaped model The G-O model is a class of NHPP model, whose failure intensity function (FIF) is given by [11], 𝜆(𝑡) = 𝑎𝑏𝑒 −𝑏𝑡
(1)
Where, a is the expected number of software errors present b is the rate of fault detection t is the time of software operation The mean value function (MVF) is used to determine the cumulative number of failures at a time. The MVF is obtained by integrating the FIF with respect to the time interval (0,t). 𝑡
𝑚(𝑡) = ∫0 𝜆(𝑡)𝑑𝑡
(2)
The MVF of G-O model is given by, 𝑚(𝑡) = 𝑎(1 − 𝑒 −𝑏𝑡 )
(3)
The reliability function is given by, 𝑅(𝑡) = 𝑒 −𝑚(𝑡) 𝑅(𝑡) = 𝑒 −𝑎(1−𝑒
(4) −𝑏𝑡 )
(5)
The reliability at time t, after x hrs of operation is given by, 𝑅(𝑥|𝑡) = 𝑒 −(𝑚(𝑥+𝑡)−𝑚(𝑡)) 𝑅(𝑥|𝑡) = 𝑒
(6)
−(𝑎𝑒 −𝑏𝑡 (1−𝑒 −𝑏𝑥 ))
(7)
The delayed S-Shaped model has the failure intensity function given by [12], 𝜆(𝑡) = (1 − (1 + 𝑏𝑡))𝑎𝑏𝑒 −𝑏𝑡
(8)
The mean value function of the DS model is,
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𝑚(𝑡) = 𝑎(1 − (1 + 𝑏𝑡)𝑒 −𝑏𝑡 )
(9)
The reliability and the conditional reliability function are given as follows, 𝑅(𝑡) = 𝑒 −(𝑎(1−(1+𝑏𝑡))𝑒 𝑅(𝑥|𝑡) = 𝑒
−𝑏𝑡 )
(10)
−𝑎((1−𝑒 −𝑏𝑥 )𝑒 −𝑏𝑡 (1+𝑏𝑡)−𝑏𝑥𝑒 −𝑏(𝑥+𝑡) )
(11)
B. Parameter estimation of software reliability models Parameter estimation is the process of determining the nearby value of a real situation. In software reliability modeling, estimation of parameters is one of the major parts, and is sometimes difficult due to the non-linear property of the models. MLE and LSE are the two predominantly used techniques for estimation [15]. The two techniques provide closest value, but LSE does not hold good when the sample size is more. In this paper MLE is used for parameter estimation. In MLE, the unknown parameters are estimated by maximizing the likelihood function. For a model with probability function of f(x) is given by, 𝐿(𝑥|𝜃) = ∏𝑛𝑖=1 𝑓(𝑥𝑖|𝜃)
(12)
Where, n is the number of random variables, and θ is the unknown parameter. To minimize the computation, the likelihood function is converted into log likelihood function, by taking natural logarithm on both sides. The log likelihood function is given by ln[𝐿(𝑥|𝜃)] = ln[∏𝑛𝑖=1 𝑓(𝑥𝑖|𝜃)]
(13)
ln[𝐿(𝑥|𝜃)] = ∑𝑛𝑖=1 ln[𝑓(𝑥𝑖|𝜃)]
(14)
By equating the partial derivative of the log likelihood function to zero or by maximizing the function, we can estimate the unknown parameters. The Expectation maximization (EM) algorithm is the most commonly used method of MLE. In this the maximization of the likelihood function is done in an iterative manner which leads to the final solution. A modified form of EM algorithm called Expectation Conditional maximization (ECM) algorithm was proposed in [16]. In this instead of maximizing the likelihood function in an iterative manner, it is done using some conditions and constraints. C. Parameter optimization Techniques The estimated parameters by MLE and LSE sometimes have some limitations [17]. To overcome these limitations, GA was proposed to get the optimized value. Artificial Neural Network (ANN) was used to train the parameters and
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to get the optimal value. Particle swarm optimization (PSO) was used to arrive at the closest parameters. Reference [18] proposed ANN based parameter optimization for software reliability models and claims that their technique is 10 times faster than the PSO. ABC and ACO provided promising results. SA is a multi-objective probabilistic technique for getting the global optimum solution. It is a metaheuristic technique which provided nearest value. Genetic algorithm provides the optimal value by mutation of genes and off springs and is based on the Darwinian principle. The GAMULOBJ algorithm is based on GA and it gives the values by multi-objective optimization. III.
METHODOLOGY
Fig.1 shows the methodology of our proposed work. First the software reliability models (SRMs) to which the parameter estimation is to be is selected first. In our work, G-O model and Delayed S-Shaped model are selected for estimation. Then MLE is used for parameter estimation. The underlying assumption and steps are already discussed. The optimization is done using GA, GAMULOBJ, and SA. Then the estimation accuracy is validated using a comparison criterion. In our work, validation is done by comparing MSE, and MMRE.
Fig. 1 Methodology MSE is defined as “the mean value of the sum of the difference between the observed value and estimated value” [19]. MSE is given by,
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𝑀𝑆𝐸 = 𝑛 ∑𝑛𝑖=1(𝑂𝑖 − 𝐸𝑖)2
(15)
Where, ‘Oi’ is the ith observed value ‘Ei’ is the ith estimated value ‘n’ is the number of time interval or data MMRE is average of the difference between the observed value and estimated value with respect to the estimated value [19]. MMRE is defined in mathematical form as, 1 𝑛
𝑀𝑀𝑅𝐸 = ∑𝑛𝑖=1 |
𝑂𝑖−𝐸𝑖 | 𝐸𝑖
(16)
Oi, Ei, and n are same as that given for MSE. IV.
RESULTS AND DISCUSSIONS
The proposed methodology was validated using a case study. The failure data was collected from a project from a software development company in Chennai, Tamilnadu, India. The time interval is one week. Table I. shows the failure data of software. TABLE I. FAILURE DATA OF SOFTWARE Time interval 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Cumulative Testing
No. of failures
Cumulative No. of failures
80 29 26 60 15 11 19 16 8 22 10 11 17 4 15 3 10 14
80 109 135 195 210 221 240 256 264 286 296 307 324 328 343 346 356 370
hours 168 336 504 672 840 1008 1176 1284 1512 1680 1848 2016 2184 2352 2520 2688 2856 3024
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19 20 21 22 23 24
3192 3360 3528 3696 3864 4032
5 13 14 10 9 3
375 388 402 412 421 424
A. Parameter estimation using MLE The parameters of G-O model and delayed S-Shaped model were estimated using MLE. Optimization was done using GA, GA Multi objective optimization and SA algorithms. LSE and optimization was carried out using MATLAB 2014a. The unknown parameters are ‘a’, which is the expected total number of software faults, and ‘b’ which is the rate of fault detection. They are represented as â and b̂ respectively. The optimized parameter values after estimation is presented in table II. TABLE II. ESTIMATED PARAMETERS OF G-O MODEL AND DELAYED S-SHAPED MODEL Optimization Technique Software Reliability
GA
GAMULOBJ
SA
Models â
b̂
â
b̂
â
b̂
G-O Model
91.79
0.008
40.045
0.01
810.38
0.01
Delayed S-Shaped Model
100.964
0.01
97.957
0.01
667.282
0.01
B. Determination of best fit By using the failure intensity function of G-O model and Delayed S-shaped model, the failure intensity, number of failures, and the cumulative number of failures were calculated. Table III and IV shows the failure intensity, number of failures, and cumulative number of failures of G-O model and delayed S-Shaped model respectively. TABLE III. FAILURE RATE, NO. OF FAILURES, AND CUMULATIVE NO. OF FAILURES OF G-O MODEL Optimization techniques Time GA
Interval
1
2
GAMULOBJ
EFI
ENOF
ECNOF
EFI
ENOF
0.1913
32.148
32.14895
0.03385
5.6871
63
95021
021
2194
69
0.0499
16.768
48.91786
0.02861
9.6153
07
91673
694
7082
4
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ECNOF 5.687169
15.30251
SA EFI
ENOF
ECNOF
1.5103374
253.73668
253.73668
27
77
77
0.2814875
94.579830
348.31651
91
72
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5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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0.0130
6.5600
55.47787
0.02419
12.192
16
09728
667
156
55
0.0033
2.2811
57.75901
0.02045
13.742
95
37355
402
0428
69
0.0008
0.7436
58.50266
0.01728
14.521
85
51855
588
7848
79
0.0002
0.2327
58.73539
0.01461
14.731
31
33622
95
4349
26
6.02E-
0.0708
58.80621
0.01235
14.528
05
13186
268
4296
65
2.54E-
0.0325
58.83879
0.01108
14.238
05
86777
946
9557
99
0.0061
58.84499
0.00882
13.348
92629
209
8665
94
1.07E-
0.0017
58.84678
0.00746
12.538
06
94489
658
3346
42
2.79E-
0.0005
58.84730
0.00630
11.659
07
14804
138
9168
34
7.27E-
0.0001
58.84744
0.00533
10.752
08
46467
785
3479
29
1.89E-
4.1381
58.84748
0.00450
9.8469
08
7E-05
923
8677
51
4.94E-
1.1622
58.84750
0.00381
8.9644
09
6E-05
085
1428
78
1.29E-
3.2476
58.84750
0.00322
8.1194
09
8E-06
41
2005
52
3.36E-
9.0346
58.84750
0.00272
7.3213
10
2E-07
5
3734
98
8.77E-
2.5035
58.84750
0.00230
6.5759
11
E-07
525
2519
95
2.29E-
6.9132
58.84750
0.00194
5.8860
11
E-08
532
6443
45
5.96E-
1.9031
58.84750
0.00164
5.2522
12
3E-08
534
5433
23
4.1E-06
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27.49505
41.23774
55.75953
70.4908
85.01945
99.25844
112.6074
125.1458
136.8051
147.5574
157.4044
166.3689
174.4883
181.8097
188.3857
194.2718
199.524
0.0524619
26.440828
374.75734
62
66
71
0.0097775
6.5705098
381.32785
44
22
69
0.0018222
1.5307150
382.85857
8
5
2
0.0003396
0.3423425
383.20091
26
4
45
6.32974E-
0.0744376
383.27535
05
97
22
2.14955E-
0.0276002
383.30295
05
23
25
2.19865E-
0.0033243
383.30627
06
59
68
4.09771E-
0.0006884
383.30696
07
16
52
7.63707E-
0.0001411
383.30710
08
33
64
1.42335E-
2.86948E-
383.30713
08
05
51
2.65276E-
5.79362E-
383.30714
09
06
08
4.94405E-
1.16284E-
383.30714
10
06
2
9.21442E-
2.32203E-
383.30714
11
07
22
1.71733E-
4.61618E-
383.30714
11
08
23
3.20065E-
9.14106E-
383.30714
12
09
23
5.96518E-
1.80387E-
383.30714
13
09
23
1.11175E-
3.54872E-
383.30714
13
10
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1.55E-
5.2246
58.84750
0.00139
4.6736
12
E-09
535
0973
71
4.06E-
1.4307
58.84750
0.00117
4.1484
13
1E-09
535
5865
51
1.06E-
3.9089
58.84750
0.00099
3.6739
13
7E-10
535
4022
04
2.76E-
1.0658
58.84750
0.00084
3.2469
14
E-10
535
03
2
7.19E-
2.9004
58.84750
0.00071
2.8641
15
6E-11
535
0351
35
204.1977
208.3461
212.02
215.2669
218.1311
2.07202E-
6.96199E-
383.30714
14
11
23
3.86171E-
1.36241E-
383.30714
15
11
23
7.19722E-
2.66009E-
383.30714
16
12
23
1.34137E-
5.18307E-
383.30714
16
13
23
2.49997E-
1.00799E-
383.30714
17
13
23
TABLE IV. FAILURE RATE, NO. OF FAILURES, AND CUMULATIVE NO. OF FAILURES OF DELAYED SSHAPED MODEL Optimization techniques Time GA
Interval
GAMULOBJ
SA
EFI
ENOF
ECNOF
EFI
ENOF
ECNOF
EFI
ENOF
ECNOF
1
0.190052
31.92879
31.92879
0.052626
8.841104
8.841104
1.256076
211.0208
211.0208
2
0.035421
11.90139
43.83018
0.009808
3.295503
12.13661
0.2341
78.65758
289.6784
3
0.006602
3.327164
47.15735
0.001828
0.921294
13.0579
0.04363
21.98959
311.668
4
0.00123
0.826796
47.98414
0.000341
0.22894
13.28684
0.008132
5.464383
317.1324
5
0.000229
0.192617
48.17676
6.35E-05
0.053336
13.34018
0.001516
1.273024
318.4054
6
4.27E-05
0.043078
48.21984
1.18E-05
0.011928
13.35211
0.000282
0.28471
318.6901
7
7.96E-06
0.009367
48.2292
2.21E-06
0.002594
13.3547
5.26E-05
0.061906
318.752
8
2.7E-06
0.003473
48.23268
7.49E-07
0.000962
13.35566
1.79E-05
0.022954
318.775
9
2.77E-07
0.000418
48.2331
7.66E-08
0.000116
13.35578
1.83E-06
0.002765
318.7777
10
5.16E-08
8.66E-05
48.23318
1.43E-08
2.4E-05
13.3558
3.41E-07
0.000573
318.7783
11
9.61E-09
1.78E-05
48.2332
2.66E-09
4.92E-06
13.35581
6.35E-08
0.000117
318.7784
12
1.79E-09
3.61E-06
48.2332
4.96E-10
1E-06
13.35581
1.18E-08
2.39E-05
318.7785
13
3.34E-10
7.29E-07
48.2332
9.24E-11
2.02E-07
13.35581
2.21E-09
4.82E-06
318.7785
14
6.22E-11
1.46E-07
48.2332
1.72E-11
4.05E-08
13.35581
4.11E-10
9.67E-07
318.7785
15
1.16E-11
2.92E-08
48.2332
3.21E-12
8.09E-09
13.35581
7.66E-11
1.93E-07
318.7785
16
2.16E-12
5.81E-09
48.2332
5.98E-13
1.61E-09
13.35581
1.43E-11
3.84E-08
318.7785
17
4.03E-13
1.15E-09
48.2332
1.12E-13
3.19E-10
13.35581
2.66E-12
7.6E-09
318.7785
18
7.51E-14
2.27E-10
48.2332
2.08E-14
6.29E-11
13.35581
4.96E-13
1.5E-09
318.7785
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1.4E-14
4.47E-11
48.2332
3.87E-15
1.24E-11
13.35581
9.25E-14
2.95E-10
318.7785
20
2.61E-15
8.76E-12
48.2332
7.22E-16
2.43E-12
13.35581
1.72E-14
5.79E-11
318.7785
21
4.86E-16
1.71E-12
48.2332
1.35E-16
4.75E-13
13.35581
3.21E-15
1.13E-11
318.7785
22
9.06E-17
3.35E-13
48.2332
2.51E-17
9.27E-14
13.35581
5.99E-16
2.21E-12
318.7785
23
1.69E-17
6.52E-14
48.2332
4.67E-18
1.81E-14
13.35581
1.12E-16
4.31E-13
318.7785
24
3.15E-18
1.27E-14
48.2332
8.71E-19
3.51E-15
13.35581
2.08E-17
8.38E-14
318.7785
From the tabulated values, a graph was plotted for time versus expected failure intensity of G-O model and delayed S- shaped model for the estimated values of GA, GAMULOBJ, and SA. A figure 1 (a) and (b) shows the plot of time against failure intensity. Similarly plot was made for time against estimated number of failures, and time against estimated cumulative number of failures. Figure 2 (a) and (b) shows the plot of time versus expected number of failures for G-O model and delayed S- Shaped model. Figure 3 (a) and (b) shows the plot of time against the expected cumulative number of failures for G-O model and delayed S- Shaped model. In all the plots comparison was made between the observe values and optimal estimated values by GA, GA multi objective optimization, and SA.
Fig 2. Plot of time against expected failure intensity for (a) G-O model and (b) Delayed S-shaped model
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Fig 3. Plot of time against estimated number of failures for (a) G-O model and (b) Delayed S-Shaped model
Fig 4. Plot of time against estimated cumulative number of failures for (a) G-O model and (b) delayed S-Shaped model. From the figures, we can conclude that GA provides better fitting results with the observed values when compared to GAMULOBJ and SA.
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C. Validation of estimation accuracy The MSE and MMRE values are computed and is shown in Table V. TABLE V. MSE AND MMRE FOR ESTIMATED VALUES BY GA, GA MULTI OBJECTIVE OPTIMIZATION AND SA FOR G-O AND DELAYED S-SHAPED MODEL Software Reliability models (SRMs) Optimization
G-O model
Techniques
Delayed S-Shaped model
MSE
MMRE
MSE
MMRE
GA
356.04
9.426
406.6068
1.72
GAMULOBJ
368.57
6.53
544.6183
6.23
SA
1684.68
2.17
1071.818
2.61
From the table V, MSE and MMRE values is minimum for GA for both software reliability models. So we can say that GA provides a better optimization results when compared to GA multi objective optimization and Simulated Annealing algorithm. V.
CONCLUSION
In this paper, MLE was used to estimate the parameters of the two prominent Software Reliability models namely, G-O model and Delayed S- Shaped model. Genetic algorithm (GA), GA Multi Objective Optimization (GAMULOBJ), and Simulated Annealing (SA) are used for optimizing the estimated parameters. Real time data collected from a software project during the testing phase of software development was used for parameter estimation and validation. There are two unknown parameters. The expected failure intensity, Number of failures, and cumulative number of failures are then determined. Comparison was made between the observed values and estimated values, and this confirmed the superiority of GA in parameter optimization over GAMULOBJ and SA. MSE and MMRE values were also determined to confirm the best fit of GA in parameter optimization. VI.
LIMITATIONS AND FUTURE WORK
The optimization techniques used in this research work is three, and many techniques like Artificial Neural Network, Ant Colony Optimization, and Bee Colony Optimization have to be included for study. This is one of the limitations of this work. The parameter was estimated using MLE. This is the second limitation of this work. Since the use of a single estimation technique will lead to bias and homogeneity in the estimated values. In future, we will use MLE and LSE for parameter estimation, so that the homogeneity and bias in the estimated values can be avoided. Other optimization algorithms will be used so that the suitable technique with respect to different Software Reliability Models can be proposed.
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Acknowledgment This work was funded by Centre for Research, Anna University, Chennai-25., through ACRF. The process number is CFR/ACRF/2015/7 dated 21.01.2015. References [1] Rausand, Marvin, Reliability of safety-critical systems: theory and applications, John Wiley & Sons, 2014. [2] Koopman P, Reliability, safety, and security in everyday embedded systems, Lecture Notes in Computer Science., Sep 26 2007, 4746:1. [3] Besterfield DH, Quality control. Pearson Education India, 2004. [4] Mitra A, Fundamentals of quality control and improvement, John Wiley & Sons, April 2016. [5] Montgomery DC, Introduction to statistical quality control, John Wiley & Sons, December 2007. [6] Ebeling CE, An introduction to reliability and maintainability engineering, Tata McGraw-Hill Education, 2004. [7] Dunn WR, Designing safety-critical computer systems, Computer, November 2003, Vol.36, No.11, pp: 40-6. [8] Lyu, Michael R. "Handbook of software reliability engineering." 1996, pp. 3-25. [9] Ullah, N., Morisio, M. and Vetro, A., “Selecting the best reliability model to predict residual defects in open source software” Computer, Vol. 48, No.6, pp.50-58. [10] Huang, C.Y., Lyu, M.R. and Kuo, S.Y., “A unified scheme of some non-homogenous Poisson process models for software reliability estimation”, IEEE Transactions on Software Engineering, Vol. 29, No. 3, pp.261-269. [11] Goel, A.L. and Okumoto, K., “Time-dependent error-detection rate model for software reliability and other performance measures”, IEEE transactions on Reliability, Vol. 28, No. 3, pp.206-211. [12] Yamada, S., Ohba, M. and Osaki, S., “S-shaped software reliability growth models and their applications”, IEEE Transactions on Reliability, 1984, Vol. 33, No.4, pp.289-292. [13] Meyfroyt, P. H. A, “Parameter estimation for software reliability models”, Diss. Ph. D. Thesis, Universidad Carlos III de Madrid, Madrid, 2012. [14] Gokhale, S.S., “Architecture-based software reliability analysis: Overview and limitations”, IEEE Transactions on dependable and secure computing, Vol. 4, No.1, pp.32-40. [15] Park J, Kim HJ, Shin JH, Baik J, “An embedded software reliability model with consideration of hardware related software failures”, InSoftware Security and Reliability (SERE), IEEE Sixth International Conference, June 2012, pp. 207-214. [16] Zeephongsekul P, Jayasinghe CL, Fiondella L, Nagaraju V, “Maximum-likelihood estimation of parameters of NHPP software reliability models using expectation conditional maximization algorithm”, IEEE Transactions on Reliability, September 2016, Vol. 65, No. 3, pp. 1571-83.
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[17] Nagaraju V, Fiondella L, Zeephongsekul P, Jayasinghe CL, Wandji T, “Performance Optimized Expectation Conditional Maximization Algorithms for Nonhomogeneous Poisson Process Software Reliability Models”, IEEE Transactions on Reliability, September 2017, Vol. 66, No. 3, pp.722-34. [18] Zheng C, Liu X, Huang S, Yao Y, “A parameter estimation method for software reliability models. Procedia engineering”, January 2011, Vol.1, No.15, pp.3477-3481. [19] Li Q, Pham H, “ A testing-coverage software reliability model considering fault removal efficiency and error generation”, PloS one, July 2017, Vol.12, No.7, e0181524.
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