Research Article Software Reliability Growth Model with...

Research ArticleSoftware Reliability Growth Model with Partial DifferentialEquation for Various Debugging Processes

Jiajun Xu and Shuzhen Yao

School of Computer Science and Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Jiajun Xu xujiajunbuaaeducn

Received 19 September 2015 Revised 8 December 2015 Accepted 20 December 2015

Academic Editor Jean-Christophe Ponsart

Copyright copy 2016 J Xu and S Yao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Most Software Reliability Growth Models (SRGMs) based on the Nonhomogeneous Poisson Process (NHPP) generally assumeperfect or imperfect debugging However environmental factors introduce great uncertainty for SRGMs in the development andtesting phaseWepropose a novelNHPPmodel based on partial differential equation (PDE) to quantify the uncertainties associatedwith perfect or imperfect debugging process We represent the environmental uncertainties collectively as a noise of arbitrarycorrelation Under the new stochastic framework one could compute the full statistical information of the debugging process forexample its probabilistic density function (PDF) Through a number of comparisons with historical data and existing methodssuch as the classic NHPP model the proposed model exhibits a closer fitting to observation In addition to conventional focuson the mean value of fault detection the newly derived full statistical information could further help software developers makedecisions on system maintenance and risk assessment

1 Introduction

Software reliability defined as the probability of failure-freeoperation under certain conditions and by specific time [1]is one of the significant attributes of software systems devel-opment life cycle As the software systems mature with evergrowing complexity evaluation prediction and improve-ment of their reliability become a crucial and daunting taskfor developers in both the development and testing phasesNumerous Software Reliability Growth Models (SRGMs)have been developed [2ndash5] which generally agree that thefault debugging process is a Nonhomogeneous Poisson Pro-cess (NHPP) under various diverging assumptions perfectand imperfect debugging phenomenon [6 7]

A common assumption is that removing detected faultswill not introduce new faults usually called perfect debug-ging phenomenon Based on earlier works of Jelinski andMoranda [8] Goel andOkumoto [9] developed the exponen-tial Software Reliability Growth Models (G-O model) with aconstant fault detection rate In latter models the smoothlychangeable fault detection rates are incorporated Yamada etal [10] found the fault detection rate fit to a S-shape growth

curve and proposed the delay S-shaped SRGM Employingthe learning phenomenon in software fault detection processOhba [11] later developed an alternative inflection S-shapedSRGM Further work was conducted by Yamada and others[12] by incorporating the relationship between thework effortand the number of faults detected into themodel Huang et al[13] stated that fault detection and repair process are differentin software development and operation They proposed aunified theory to merge multiple moves into a NHPP-basedSRGM and found a regularly changeable fault detection ratein the detection process Later on Huang and Lyu [6] alsousedmultiple change points to deal with imperfect debuggingproblemsThe common assumption is that the removal of thedetected faults will not introduce new faults This scenariois known as perfect debugging However given the com-plexity of the software debugging process debugging activitymay be imperfect in practical software development environ-ment and thus those perfect models may oversimplify theunderlying dynamics

To solve such problems imperfect debugging processesare incorporated in lattermodelsWhenever the software sys-tem is developed new errors can be introduced into the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2476584 13 pageshttpdxdoiorg10115520162476584

2 Mathematical Problems in Engineering

software during developmenttesting phase and faults maynot be corrected perfectly This phenomenon is known asimperfect debugging According to the above two phe-nomenons SRGMs can be divided into two categoriesperfect and imperfect debugging model Kapur and Garg[14] discussed their fault removal phenomenon that as theteam gainsmore experience theymay remove detected faultswithout causing new ones However the testing team maynot correct a fault perfectly and the original fault may remainor generate new faults leading to a phenomenon known asimperfect debugging It was Goel [15] who first introducedthe concept of imperfect debugging Latter Ohba and Chou[16] developed an error generation model based on the G-Omodel named as an imperfect debugging model Pham et al[17] proposed a general imperfect software debugging model(P-N-Z model) Pham [18] also developed an SRGM formultiple failure types incorporating error generation Zhanget al [19] proposed a testing efficiency model that includesboth imperfect debugging and error generation Kapur et al[20] proposed a flexible SRGM with imperfect debuggingand error generation using a logistic function for the faultdetection rate which reflects the efficiency of the testingand removal team Employing the learning phenomenon insoftware fault detection process Yamada et al [21] later devel-oped an imperfect debugging model In imperfect softwaredebugging models the fault introduction rate per fault isgenerally assumed to be constant or decreasing changes overtime [22] Recently Wang et al [22] also proposed a modelto represent the imperfect debugging process with a log-logistic distribution To sum up most models presume adeterministic relationship between the cumulative numberof detected faults and the time span of the software faultdebugging process

However software debugging is a stochastic and uncer-tain process which can be effected by many environmentalfactors such as the distribution of resources strategy andrunning environment [23] In particular in the uncertainnetwork environment the aforementioned assumptions ofdeterministic model would become problematic The envi-ronmental noise introduces great uncertainties that signifi-cantly affects traditional debugging process To address suchchallenges new stochastic models [24ndash27] were proposedwhich treat the debugging process as perfect and stochas-tic they assume each failure occurrence is independentand follows identical random distribution By employingthe white noise that is temporally uncorrelated randomvariables for the environmental factors collectively theyused a flexible stochastic differential equation to model theirregular changes Comparing to conventional models suchwhite-noise-based approach assumes the perfect debuggingand no doubt provides closer approximation to uncertainfluctuations in reality but with great mathematical simplicityDebugging activity is usually imperfect in piratical softwaredevelopment and recent data [28 29] show that the faultdetection is highly susceptible to noise and is generally cor-related thus earlier assumptions become problematic Thusbecause of its mathematical simplicity it may also consider-ably underestimate the imperfect debugging process and thetemporal correlation in a dynamic environment

In this study we propose an alternative stochastic frame-work based on partial differential equation (PDE) to describethe perfectimperfect debugging processes in which theenvironmental noise is of arbitrary correlation Details of themodel is provided in Section 2 with an equation for the prob-abilistic density function (PDF) of system states In Section 3we validate our approach with both historical failure dataand results from conventional methods main features of themodel including the temporal evolutions of its full statisti-cal information are later addressed in Section 4 The finalconclusions are given in Section 5

2 Problem Formulation

21Model Formulation Widely used inmany practical appli-cations NHPP model assumes the following

(1) the fault debugging process is modeled as a stochasticprocess within a continuous state space

(2) the system is subject to failures at random timescaused by the remaining errors in the system

(3) the mean number of detected faults in a time intervalis proportional to the mean number of remainingfaults

(4) when a fault is detected it may be removed with theprobability 119901

(5) when a fault is removed new faults may be generatedwith a constant probability 119902

In order to describe the stochastic and uncertain behaviorof the fault debugging process in a continuous state space thetemporal evolution of119872(119905) is routinely described as [12]

d119872d119905

= 119903 (119905) (119886 + 119902119872 minus 119901119872) (1)

where 119872(119905) is the expected number of faults detected in thetime interval (0 119905] 119886 denotes the number of total faults and119903(119905) is a nonnegative function representing the fault detectionrate per remaining fault at testing time 119905 We further notethat 119902(119905)119872 is the number of newly introduced bugs while119901(119905)119872 means the number of successful removal togetherthey represent a (perfectimperfect) debugging process inwhich probabilities 119902 and 119901 are assigned [30ndash32] Without aloss of generality we denote 119896(119905) = 119901(119905) minus 119902(119905) to facilitatesubsequent presentation and 119896 = 1 indicates a perfectdebugging process

Since 119903(119905) is subject to a number of random environmen-tal effects [23] we follow the previous studies [24 25 27] andrepresent 119903(119905) as a sum of its mean value ⟨119903(119905)⟩ and a zero-mean (random) fluctuation 119903

1015840(119905)

119903 (119905) = ⟨119903 (119905)⟩ + 1199031015840

(119905) ⟨1199031015840

(119905)⟩ = 0 (2)

By substituting (2) into (1) we obtain a stochastic differ-ential equation

d119872d119905

= [⟨119903 (119905)⟩ + 1199031015840

(119905)] [119886 minus 119896 (119905)119872] (3)

Mathematical Problems in Engineering 3

In contrast to earlier works of [24 25 27] which approx-imate the fluctuations term 119903

1015840(119905) as a white noise we relax

such assumption and denote the noisersquos two-point covariancemore broadly as

⟨1199031015840

(119905) 1199031015840

(119904)⟩ = 1205902

119903120588 (

119905 minus 119904

120591) (4)

where 1205902119903represents the strength of the noise 120588 denotes its

correlation function 119905 and 119904 indicate two temporal pointsand 120591 is the noise correlation time

22 PDF Method Our goal is to derive an equation for theprobabilistic density function of 119872 that is 119891

119872(119898 119905) To be

specific we adopt the PDF method originally developed inthe context of turbulent flow [33] and use the Dirac deltafunction 120575(sdot) to define a ldquorawrdquo PDF of system states119872 at time119905

Π (119898119872 119905) = 120575 [119872 (119905) minus 119898] (5)

whose ensemble average over random realisations of119872 yields119891119872(119898 119905)

⟨Π⟩ equiv int120575 (1198721015840minus 119898)119891

119872(1198721015840 119905) d1198721015840 = 119891

119872(119898 119905) (6)

Following recent study on PDFmethod [34] we multiplythe original equation (3) with 120597Π120597119898 and apply the prop-erties of the Dirac delta function 120575(119898 minus 119872)119892(119872) = 120575(119898 minus

119872)119892(119898) This would yield the stochastic evolution equationfor Π in the probability space (119898 isin Ω)

120597Π

120597119905= minus

120597

120597119898[⟨119903 (119905)⟩ + 119903

1015840

(119905)] [119886 minus 119896 (119905)119898]Π (7)

Now we take the ensemble average of (7) and apply Def-inition (6) by employing a closure approximation variouslyknown as the large-eddy diffusivity (LED) closure [35] or theweak approximation [36] an equation of 119891

119872can be obtained

120597119891119872

120597119905= minus

120597

120597119898(V119891119872) +

120597

120597119898(D

120597119891119872

120597119898) (8a)

where the eddy diffusivityD and effective velocityV are

D (119898 119905)

asymp int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩ [119886 minus 119896 (119905)119898] [119886 minus 119896 (119904) 119910]

sdotG119889d119910 d119904

(8b)

V (119898 119905) asymp [119886 minus 119896 (119905)119898] ⟨119903 (119905)⟩ + int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩

sdot [119886 minus 119896 (119905)119898] 119896 (119904)G119889d119910 d119904

(8c)

And Greenrsquos functionG119889(119910 119904 119898 119905) satisfies the deterministic

partial differential equation

120597G119889

120597119904+ [119886 minus 119896 (119904) 119910] ⟨119903 (119904)⟩

120597G119889

120597119910

= minus120575 (119910 minus 119898) 120575 (119904 minus 119905)

(8d)

subject to the homogeneous initial (and boundary) con-ditions corresponding to their (possibly inhomogeneous)counterparts for the raw PDF (7)

3 Model Validation

In this section we validate our PDE model (8a) (8b) (8c)and (8d) by computing the mean fault detection ⟨119872⟩ andcompare it with historical data and two other methodsnamely the classic deterministic SRGMs (generalised NHPPmodel) and the popular stochastic SRGMs (white-noisemodel) As shown in Table 1 four sets of data are used thefirst one (DS1) is from WebERP system [37] the second(DS2) originated from the open source project managementsoftware [38] and the third (DS3) and fourth (DS4) arefrom the first and fourth product releases of the softwareproducts at Tandem Computers [32] Among those four setsof data the two UDPTCP-based systems (DS1 and DS2) aremore likely to be affected by environmental factors whereasthe TCP-based software development at Tandem ComputersInc takes place in a more enclosedstable environmentUDP is a connectionless-oriented transport protocol and willintroduce larger correlation strength for system based on itwhile TCP is connection-oriented and will generate smallercorrelation strengthWe encapsulate such difference in termsof noise variance for example larger 1205902

119903for UDPTCP-based

systems (DS1 and DS2) Moreover DS1 DS2 and DS3 arefrom the imperfect debugging process while DS4 is from theperfect debugging process we demonstrate this diverse withvarious configurations of the subsequent presentation 119896 Thetesting time of DS1 DS2 DS3 and DS4 is 60 (months) 29(weeks) 20 (weeks) and 19 (weeks) respectively For detailedinformation please refer to the data values tabulated in theAppendix

It is also noted here that we consider a special casethat is constant 119896(119905) = 119896 by adopting earlier works[30 39 40] A stationary Gaussian process is prescribed tothe fluctuation term 1199031015840 sim N(0 120590

2

119903) with an exponential

correlation 120588(119905 119904 120591) = exp(minus|119905 minus 119904|120591)

31 Model Solution for Constant 119896 For model validation weadopt the popular generalized Goel-Okumoto model (G-O)as this enables us to derive an analytical solution of (3) usingseparation of variables The expression for the mean faultdetection rate per remaining fault using the generalised G-Omodel is given by [41]

⟨119903 (119905)⟩ = 119887119905119889 (9)

where 119887 isin [0 1] is a constant and 119889 is the shape factorFollowing the PDF method we can obtain an analytical

solution of 119891119872

119891119872(119898 119905) =

exp [minus ( + )2

21205902

119868]

120590119868(119886 minus 119896119898)radic2120587

119898 lt 119886

0 119898 ge 119886

(10a)


Table 1 Data sets for model validation

Data set Time unit Detected failure Description

DS1 [37 38] 60 months 146 Web-based integrated accounting ERP system (WebERP) from August 2003 to July 2008 whichuses HTTP on top of UDPTCP

DS2 [37 38] 49 weeks 94 Open source project management software (OpenProj) from August 2007 to July 2008 usedwith Remote Display Protocol (RDP) over UDPTCP

DS3 [32] 20 weeks 100 Release 1 of software testing data collected from Tandem Computers which uses a simpleSession Control Protocol (SCP) on top of TCP

DS4 [32] 19 weeks 42 Release 4 of software testing data collected from Tandem Computers applying (SCP) on top ofTCP

Table 2 Approximation schemes [51] of PDF for the random integral 1198681199031015840 = int119905

01199031015840(1199051015840)d1199051015840 = sum

119873

119894=1int119894Δ

(119894minus1)Δ1199031015840(1199051015840)d119905 where Δ = 119905119873

Cases 119905 ≪ 120591 119905 ≫ 120591 Others

Variance 1205902119868

1205902

1199031199052

21205902

119903119905 2119873120590

2

119903[int

Δ

0

(Δ minus 1199051015840

) 120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

int

Δ

0

int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus 11990510158401015840

) d1199051015840d11990510158401015840]

d120590119868

d119905(119905 119879) 120590

119903

1

radic2(119879 + 119905)

1

120590119868

int

Δ

0

120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

[int

Δ

0

119894120588 (119894Δ minus 11990510158401015840

) d11990510158401015840 minus int

Δ

0

(119894 minus 1) 120588 [(119894 minus 1) Δ minus 11990510158401015840

] d11990510158401015840 + int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus Δ) d1199051015840]

where

=ln (1 minus 119896119898119886)

119896

=119887119905119889+1

119889 + 1

(10b)

and 1205902

119868is the variance of the random integral 119868

1199031015840 = int119905

01199031015840(119904)d119904

whose PDF follows a Gaussian distribution 1198911198681199031015840sim N(0 120590

2

119868)

as outlined in Table 2The cumulative density function (CDF) 119865

119872(119898 119905) can be

derived as 119865119872(119898 119905) = int119891

119872(1198981015840 119905)d1198981015840

119865119872(119898 119905) =

1

2[1 minus erf ( +

120590119868radic2

)] 119898 lt 119886

1 119898 ge 119886

(11)

and erf(lowast) isin [minus1 1] represents the error functionThe mean fault detection ⟨119872⟩ is then⟨119872⟩

=119886

119896int

infin

minusinfin

1 minus exp[minus119896(1198681199031015840 +

119887119905119889+1

119889 + 1)]119891

1198681199031015840d1198681199031015840

(12)

It is noted here that the fault detection rate model(12) describes an imperfect debugging process for constant119896 In many circumstances time delay failed removal andnew faults would lead to the problem that 119896(119905) is a time-dependent variable [6 7 13 42] so that we are not ableto get an analytical solution ⟨119872⟩ and can only obtain anumerical solution In the current study our focus lies onthe uncertainty quantification framework of environmentalfactors in imperfect debugging process without a loss ofgenerality subsequent approach could also be extended to theimperfect debugging process with time-dependent 119896(119905)

32 Performance Criteria To evaluate the performance ofour PDE model we employ four comparison criteria forexample the predicted relative error (PRE) mean squareerror (MSE) Coefficient of Determination (1198772) and rootmean square prediction error (RMSPE)

321 Predicted Relative Error (PRE) The relative differencebetween observed and estimated number of faults at time 119905known as predictive validity [7 43] is as follows

PRE (119905) = 119872 (119905) minus 119873 (119905)

119873 (119905) (13)

where 119873 is the fault number from observed data at time 119905Hence for a better fit to the data we seek PRE(119905) rarr 0 [44]

322 Mean Square Error (MSE) Consider

MSE (119905) =sum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

119873119900

(14)

where 119872(119905119894) and 119873(119905

119894) denote the estimated and observed

number of faults detected at time 119905119894 (119894 = 1 2 119870)

respectively119870 is the total number of observation data pointsA lower value of MSE(119905) represents better model prediction[41 45]

323 Coefficient of Determination (1198772) We define this coef-ficient as the ratio of the sum of squares resulting from thetrend model to that from constant model

1198772

(119905) = 1 minussum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

sum119870

119894=1[119873 (119905119894) minus 119873]

2 (15)

where119873 is the average of detected faults119873 = sum119870

119894=1119873(119905119894)119870

1198772

isin [0 1] measures the percentage of the total variationabout the mean accounted for the fitted curve The lager 1198772is the better the model explains the variation in the data


324 Root Mean Square Prediction Error (RMSPE) Theaverage of prediction errors (PEs) is known as Bias [46]The standard deviation of predicted relative variation (PRV)is known as predicted relative variation [46] The rootmean square prediction error (RMSPE) is a measure of thecloseness with which the model predicts the observation andis a measure of closeness with which a model predicts theobservation [46]

RMSPE (119905) = radicBias2 + PRV2 (16)

Given

PE (119905119894) = 119872(119905

119894) minus 119873 (119905

119894) (17a)

Bias =119870

sum

119894=1

[PE (119905119894)

119870] (17b)

PRV (119905) = radicsum119870

119894=1(PE119894minus Bias)2

119870 minus 1 (17c)

The lower the values of Bias PRV and RMSPE the better thegoodness of fit

33 Model Parameter Estimation There are six input param-eters for our PDEmethod the total number of initial faults 119886the fault detection rate 119887 the shape factor 119889 the subsequentpresentation 119896 the noise strength 120590

2

119903 and correlation time 120591

Given a data set we used SPSS to estimate the parameters 119886119887 and 119889 and the methods for estimating three parametersinclude method of moments least square method maxi-mum likelihood and Bayesian methods [47ndash49] For thesubsequent presentation 119896 we use the empirical value of[30 39 40] About the noise parameters we adopt earlierworks on white noise [24] for the strength of our correlatednoise 120590

2

119903 which yields the optimal solution as discussed

in the sensitivity analysis Section 35 Regarding the noisecorrelation time 120591 we take the fact that a software failure istypically repaired within days [50] and hence set 120591 = 01

(month) or (week) for later analysis A stationary Gaussianprocess is prescribed to the fluctuation term 1199031015840 sim N(0 120590

2

119903)

with an exponential correlation 120588(119905 119904 120591) = exp(minus|119905 minus 119904|120591)

34 Validation Results In this section we validate our PDEmodel (12) by computing the mean fault detection ⟨119872⟩

and compare it with historical data and two other methodsnamely the classic deterministic SRGMs (generalised NHPPmodel) and the popular stochastic SRGMs (white-noisemodel) Figure 1 shows expected number of faults ⟨119872⟩ fromour PDE model the G-O model and the white-noise modelagainst four data sets over the testing time respectivelyParameters of all three models are adjusted for their bestfit As demonstrated in Figure 1 though our PDE modeloverestimates the number of faults for DS1 and leads tounderestimation for DS2 DS3 and DS4 overall it providesthe overall best match to the four data sets compared withthe G-O model and the white-noise model

Figure 2 shows the PRE results for our PDE model G-Omodel and white-noise model over the testing time Similar

trends are exhibited for all three models large estimationerror (deviation from zero) at the beginning but gradualconvergence to observation at later time This is expectedsince few data can be utilised at earlier time to gauge themodelsrsquo parameters as time elapses and more data becomeavailable the modelsrsquo accuracy improve and hence their PREapproaches zero Overall the PDE model yields the smallestPRE

We tabulated the aforementioned evaluation results ofour PDEmodel G-Omodel and white-noise model for DS1DS2 DS3 and DS4 in Table 3 Despite a slightly higher valueof PRV for DS1 the PDE model presents best goodness-to-fitwith all four data sets in all performance indicators

35 Sensitivity Analysis Having validated our PDE modelwith four observation real data sets we now conduct thesensitivity analysis to study the impact of correlation time 120591strength 120590

2

119903 and function 120588(119905 119904 120591) on our fault prediction

We first investigate the impact of the correlation time 120591Figure 3 andTable 4 present the PREMSE1198772 Bias PRV andRMSPE results on DS2 for our PDE model with parameters119886 = 156 119887 = 05446 119889 = 0026 1205902

119903= 00002228 and 119896 =

05092 G-O model with parameters 119886 = 94 and 119887 = 0146and white-noise model with parameters 119886 = 97 119887 = 016246119889 = 0026 1205902

119903= 00002228 and 119896 = 05092 It demonstrates

that the PDE model fits best the real data and the repair timeis 120591 = 01 week Hence a correlation time 120591 = 01 (month orweek) is a good approximation of fault repair time in reality

Next we study the impact of correlation strength Figure 4and Table 5 show the results with DS2 for our PDE modelwith parameters 119886 = 156 119887 = 05446 119889 = 0026 and 120591 = 01G-Omodel with parameters 119886 = 94 and 119887 = 0146 andwhite-noise model with parameters 119886 = 97 119887 = 016246 119889 = 0026and 119896 = 05092 It demonstrates that 1205902

119903has a large effect on

the white-noise model while its influence on the PDE modelis very limited

Lastly we conduct the sensitivity analysis of the correla-tion function Five types of correlation function exponentialbesselj normal cosine and cubic are studied Figure 5 andTable 6 show the results on DS2 for our PDE model withparameters 119886 = 156 119887 = 05446 119889 = 0026 1205902

119903= 00002228

119896 = 05092 and 120591 = 01 G-O model with parameters 119886 = 94

and 119887 = 0146 and white-noise model with parameters 119886 =

97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903= 00002228

It is clear that the correlation type significantly affect ourmodel estimation It also demonstrates that the PDE modelwith exponential correlation function fits best

4 Reliability Assessment

Having validated our model with historical data wenow apply our framework first to study the detectionprocess behaviour through its full statistical informationin Section 41 and later to assess software reliability inSection 42

41 Temporal Evolution of PDF In addition to conventionalfocus on the mean value of fault detection ⟨119872⟩ we now


0 10 20 30 40 50 60

PDE modelG-O model

White-noise modelDS1

0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)

Figure 1 Expected number of faults ⟨119872⟩ computed from the PDEmodel G-Omodel and white-noise model are compared with four datasets (a) DS1 (b) DS2 (c) DS3 and (d) DS4

include full statistical information that is the probabilisticdensity function (PDF) and the cumulative density function(CDF) to study system behaviour

Three snapshots of the PDF 119891119872(119898 119905) and CDF 119865

119872(119898 119905)

at time 119905 = 2 10 and 50 weeks are plotted in Figure 6respectively using 119886 = 156 119887 = 0005446 119889 = 00261205902

119903= 00002228 119896 = 05092 and 120591 = 01 The uncertainty

of our prediction encapsulated in the width of the PDF andthe slope of the CDF initially rises from 1 to 10 weeks itlater falls at 50 weeks as most of the faults are detectedHere we propose employing the CDF as a criteria to ensurecertain levels of reliability For example in Figure 6 for a

90 software reliability in 2 10 and 50 weeks the minimumnumber of faults detected must be 3 10 and 44 respectively

42 Software Reliability andReliability Growth Rate Softwarereliability 119877(119905 119879) is broadly defined as the probability that nosoftware fault is detected in the time interval [119879 119879 + 119905] giventhat the last fault occurred at time 119879 The software reliabilitycan be expressed as

119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)

G-O modelWhite-noise model

minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE

PDE modelG-O modelWhite-noise model

Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)

Figure 2 Predicted relative error (PRE) for the PDEmodel G-Omodel and white-noise model are compared with four data sets in (a) DS1(b) DS2 (c) DS3 and (d) DS4 respectively

Thus the software reliability growth rate RGR(119905 119879) ameasure of reliability change rate can be computed as

RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)

The temporal derivative of the variance d120590119868d119905 is listed in

Table 2Three snapshots of software reliability119877(119905 119879) at time119879 =

2nd 10th 50th weeks are presented in Figure 7 using 119886 =

156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092

and 120591 = 01 Reliability at all three timeframes exhibits initialdrop but such deterioration gradually slows down and119877(119905 119879)eventually approaches zero as 119905 rarr infin This is expected


Table 3 The MSE 1198772 Bias PRV and RMSPE results of PDE model G-O model and white-noise model for DS1 DS2 DS3 and DS4

Models Data set Setting MSE 119877-square Bias PRV RMSPE

PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model

DS1 119886 = 249 and 119887 = 00146 8479599 02858 256179 139619 291755DS2 119886 = 94 and 119887 = 0146 4956554 00701 193977 110395 223191DS3 119886 = 101 and 119887 = 0246 3185772 06081 135753 118893 122783DS4 119886 = 173 and 119887 = 00146 153147 09178 minus27146 28961 39694

white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE

G-O modelWhite-noise modelPDE model with 120591 = 005

PDE model with 120591 = 01

PDE model with 120591 = 015PDE model with 120591 = 02


Figure 3 Sensitivity analysis of PRE for PDE model G-O modeland white-noise model with various configurations of 120591

since at earlier time most of the faults are yet removed withelapsing time more are debugged and this leads to a fall insoftware reliability when the system is cleared of all faults119877(119905 119879) reaches a stationary state Therefore later timeframe(119879 = 50th week) also presents a sharper drop to zero

For a closer examination we now look at the softwarereliability growth rate RGR(119905 119879) at 119879 = 2nd 10th 50th

0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE

G-O modelWhite-noise model with 1205902r = 0

White-noise model with 1205902r = 00002228

White-noise model with 1205902r = 02228PDE model with 1205902r = 0

PDE model with 1205902r = 00002228

PDE model with 1205902r = 02228


119903

weeks As shown in Figure 7 RGR in all three timeframesdemonstrates an initial rise to maximum and late fall tostationarity of zero albeit earlier timeframe (119879 = 2nd week)exhibits higher zenith (RGR = 600) since fewer faults areremoved at earlier stage


Table 4 Various configurations of 120591 in obtainingMSE 1198772 Bias PRV and RMSPE values of PDEmodel G-Omodel and white-noise model

Model MSE 1198772 Bias PRV RMSPE

G-O model 4956554 00701 193977 110395 223191White-noise model 1718446 06776 109225 73238 131506PDE model with 120591 = 005 1265667 07625 51685 100962 113423PDE model with 120591 = 010 611382 08853 minus69274 36637 78366PDE model with 120591 = 015 1467881 07244 minus119221 21791 121196PDE model with 120591 = 020 2225672 05824 minus146230 29862 149248PDE model with 120591 = 025 2806167 04735 minus163072 38729 167608

Table 5 Various configurations of 1205902119903in obtainingMSE1198772 Bias PRV and RMSPE values of PDEmodel G-Omodel and white-noisemodel


G-O model 4956554 00701 193977 110395 223191White noise model with 120590

2

119903= 0 1716587 06779 109154 73217 131435

White noise model with 1205902

119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861

PDE model with 1205902

119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE

G-O modelWhite-noise modelPDE model with exponential correlationPDE model with besselj correlationPDE model with normal correlationPDE model with cosine correlationPDE model with cubic correlation

Figure 5 Sensitivity analysis of PRE for PDE model G-O modeland white-noise model with various correlation functions

5 Conclusion

We presented a novel model to quantify uncertainties associ-ated with the perfect or imperfect debugging process Treat-ing the random environmental impacts collectively as a noise

with arbitrary correlation an equation of the distributionof fault detection that is its probabilistic density function(PDF) is derived Our analysis leads to the following majorconclusions

(i) Under the conventional evaluation criteria on meanvalue of detected fault in the perfect or imperfectdebugging process our model provides best fit toobservation data comparing to the classic G-Omodeland white-noise model

(ii) The proposed approach also goes beyond uncertaintyquantification based on mean and variance of systemstates by computing its PDF and cumulative den-sity function (CDF) This enables one to evaluateprobabilities of rare events which are necessary forprobabilistic risk assessment

(iii) Predictive uncertainty of fault detection initiallyincreases but gradually diminishes as time elapseshencemoremaintenance is needed at the initial stage

(iv) Software developers could also use CDF as a newcriteria to assess system reliability for example bycomputing the minimum number of faults detectedfor an prescribed confidence level

(v) The proposed correlated model describes an imper-fect debugging process for constant 119896 which enablesus to obtain an analytical solution ⟨119872⟩The approachcould also be extended to the imperfect debuggingprocess with time-dependent 119896(119905) to help us drive anumerical solution by computing its PDFThis wouldbe the focus of our future work


0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)

t = 2 weekst = 10 weekst = 50 weeks

0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)


Figure 6 Snapshots of the probabilistic density function (PDF)119891119872(119898 119905) and the cumulative density function (CDF)119865

119872(119898 119905) at time 119905 = 2

10 and 50 weeks using 119886 = 156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092 and 120591 = 01

0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)

T = 2nd weekT = 10th weekT = 50th week


Figure 7 Temporal Evolutions of software reliability 119877(119905 119879) and software reliability growth rate RGR(119905 119879) at time 119879 = 2nd 10th 50thweek using 119886 = 156 119887 = 0005446 119889 = 0026 1205902

119903= 00002228 119896 = 05092 and 120591 = 01

Table 6 Various correlation functions in obtaining MSE 1198772 Bias PRV and RMSPE values of PDE model G-O model and white-noisemodel


G-O model 4956554 00701 193977 110395 223191White-noise model 1718446 06776 109225 73238 131506PDE model with exponential correlation 611382 08853 minus69274 36637 78366PDE model with besselj correlation 2464608 05374 minus152180 38966 157089PDE model with normal correlation 6020697 01296 minus231279 82806 245656PDE model with cosine correlation 843103 08418 07364 92473 92766PDE model with cubic correlation 2878605 04599 102405 136676 170784


Table 7 The failure data of web-based and integrated accounting ERP system (DS1) from August 2003 to July 2008 [37 38]

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults Corrected faults

1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136

Table 8 The failure data of open source project management software (DS2) from August 2007 to July 2008 [37 38]

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)


1 9 0 18 55 31 35 82 412 15 2 19 57 31 36 83 413 19 11 20 58 31 37 83 444 24 12 21 61 31 38 84 445 28 12 22 61 31 39 84 446 29 13 23 64 33 40 85 447 32 14 24 66 33 41 85 458 36 15 25 67 33 42 87 459 36 20 26 69 33 43 87 4610 40 21 27 70 33 44 87 4611 41 22 28 73 33 45 89 4712 41 22 29 74 33 46 89 6213 45 24 30 75 36 47 91 6214 47 25 31 75 38 48 91 6515 49 30 32 78 38 49 94 6516 52 30 33 79 40 mdash mdash mdash17 52 31 34 80 41 mdash mdash mdash

Appendix

Data Sets

In the Appendix four data sets used are presented Thedetailed information about two data sets DS1 and DS2 can

be referenced for further study at the SourceForge websiteOriginally these two data sets had belonged to time-between-failures data as tabulated in Tables 7 and 8Theother softwaretesting data were collected from two major releases of thesoftware products at Tandem Computers [4] as summarizedin Table 9


Table 9 The failure data of tandem software (DS3 and DS4) from the first and fourth release [32]

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week) Detected faults

Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100

Release 41 1 11 32 6 16 16 392 3 12 32 7 19 17 413 8 13 36 8 25 18 424 9 14 38 9 27 19 425 11 15 39 10 29 mdash mdash

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] IEEE ldquoStandard glossary of software engineering terminologyrdquoIEEE Std STD-729-1991 1991

[2] V Almering M van Genuchten G Cloudt and P J M Son-nemans ldquoUsing software reliability growth models in practicerdquoIEEE Software vol 24 no 6 pp 82ndash88 2007

[3] M LyuHandbook of Software Reliability Engineering McGraw-Hill 1996

[4] H Pham System Software Reliability Springer London UK2006

[5] N Schneidewind and T Keller ldquoApplying reliability models tothe space shuttlerdquo IEEE Software vol 9 no 4 pp 28ndash33 1992

[6] C-Y Huang and M R Lyu ldquoEstimation and analysis of somegeneralized multiple change-point software reliability modelsrdquoIEEETransactions on Reliability vol 60 no 2 pp 498ndash514 2011

[7] J Zhao H-W Liu G Cui and X-Z Yang ldquoSoftware reliabilitygrowth model with change-point and environmental functionrdquoJournal of Systems and Software vol 79 no 11 pp 1578ndash15872006

[8] Z Jelinski and P Moranda ldquoSoftware reliability researchrdquo inProceedings of the Statistical Methods for the Evaluation of Com-puter System Performance pp 465ndash484 Academic Press NewYork NY USA 1972

[9] A L Goel and K Okumoto ldquoTime-dependent error-detectionrate model for software reliability and other performance mea-suresrdquo IEEE Transactions on Reliability vol 28 no 3 pp 206ndash211 1979

[10] S Yamada M Ohba and S Osaki ldquoS-shaped reliability growthmodeling for software error detectionrdquo IEEE Transactions onReliability vol 32 no 5 pp 475ndash484 1983

[11] M Ohba Inflection S-Shaped Software Reliability Growth Mod-els Springer Berlin Germany 1984

[12] S Yamada J Hishitani and S Osaki ldquoSoftware-reliabilitygrowth with a Weibull test-effort a model and applicationrdquoIEEE Transactions on Reliability vol 42 no 1 pp 100ndash106 1993

[13] C-Y Huang M R Lyu and S Osaki ldquoEstimation and analysisof some generalized multiple change-point software reliabilitymodelsrdquo IEEETransactions on Reliability vol 60 no 2 pp 498ndash514 2011

[14] P K Kapur and R B Garg ldquoSoftware reliability growthmodel for an error removal phenomenonrdquo Software EngineeringJournal vol 7 no 4 pp 291ndash294 1992

[15] A L Goel ldquoSoftware reliability models assumptions limita-tions and applicabilityrdquo IEEE Transactions on Software Engi-neering vol 11 no 12 pp 1411ndash1423 1985

[16] M Ohba and X-M Chou ldquoDoes imperfect debugging affectsoftware reliability growthrdquo in Proceedings of the 11th Interna-tional Conference on Software Engineering (ICSE rsquo89) pp 237ndash244 ACM Pittsburgh Pa USA May 1989

[17] H Pham L Nordmann and X Zhang ldquoA general imperfectsoftware debugging model with s-shaped fault detection raterdquoIEEE Transactions on Reliability vol 48 no 2 pp 169ndash175 1999

[18] H Pham ldquoA software cost model with imperfect debuggingrandom life cycle and penalty costrdquo International Journal ofSystems Science vol 27 no 5 pp 455ndash463 1996

[19] X Zhang X Teng and H Pham ldquoConsidering fault removalefficiency in software reliability assessmentrdquo IEEE Transactionson Systems Man and Cybernetics Part A Systems and Humansvol 33 no 1 pp 114ndash120 2003

[20] P K Kapur D Kumar A Gupta and P C Jha ldquoOn how tomodel software reliability growth in the presence of imperfectdebugging and error generationrdquo in Proceedings of the 2ndInternational Conference on Reliability and Safety Engineeringpp 515ndash523 Chennai India December 2006

[21] S Yamada K Okumoto and S Osaki ldquoImperfect debuggingmodels with fault introduction rate for software reliabilityassessmentrdquo International Journal of Systems Science vol 23 no12 pp 2253ndash2264 1992

[22] J Wang Z Wu Y Shu and Z Zhang ldquoAn imperfect softwaredebugging model considering log-logistic distribution faultcontent functionrdquo Journal of Systems and Software vol 100 pp167ndash181 2015


[23] X Zhang and H Pham ldquoAnalysis of factors affecting softwarereliabilityrdquo Journal of Systems and Software vol 50 no 1 pp43ndash56 2000

[24] Y Tamura and S Yamada ldquoA flexible stochastic differentialequation model in distributed development environmentrdquoEuropean Journal of Operational Research vol 168 no 1 pp143ndash152 2006

[25] P K Kapur S Anand S Yamada and V S S YadavallildquoStochastic differential equation-based flexible software relia-bility growth modelrdquo Mathematical Problems in Engineeringvol 2009 Article ID 581383 15 pages 2009

[26] P K Kapur H Pham A Gupta and P C Jha SRGMUsing SDESpringer London UK 2011

[27] C Fang and C Yeh ldquoEffective confidence interval estimation offault-detection process of software reliability growth modelsrdquoInternational Journal of Systems Science 2015

[28] N Kumar and S Banerjee Measuring Software Reliability ATrend Using Machine Learning Techniques Springer New YorkNY USA 2015

[29] MD Pacer andT LGriffiths ldquoUpsetting the contingency tablecausal induction over sequences of point eventsrdquo in Proceedingsof the 37th Annual Conference of the Cognitive Science Society(CogSci rsquo15) Pasadena Calif USA July 2015

[30] M Ohba ldquoSoftware reliability analysis modelsrdquo IBM JournalResearch amp Development vol 4 no 28 pp 328ndash443 1984

[31] H Pham and X Zhang ldquoNHPP software reliability and costmodels with testing coveragerdquo European Journal of OperationalResearch vol 145 no 2 pp 443ndash454 2003

[32] AWood ldquoPredicting software reliabilityrdquoComputer vol 29 no11 pp 69ndash77 1996

[33] T S Lundgren ldquoDistribution functions in the statistical theoryof turbulencerdquo Physics of Fluids vol 10 no 5 pp 969ndash975 1967

[34] PWangAMTartakovsky andDMTartakovsky ldquoProbabilitydensity function method for langevin equations with colorednoiserdquoPhysical Review Letters vol 110 no 14 Article ID 1406022013

[35] R H Kraichnan ldquoEddy viscosity and diffusivity exact formulasand approximationsrdquoComplex Systems vol 1 pp 805ndash820 1987

[36] S P Neuman ldquoEulerian-lagrangian theory of transport inspace-time nonstationary velocity fields exact nonlocal formal-ism by conditional moments and weak approximationrdquo WaterResources Research vol 29 no 3 pp 633ndash645 1993

[37] SourceForgenet ldquoAn open source software websiterdquo 2008httpsourceforgenet

[38] C-J Hsu C-Y Huang and J-R Chang ldquoEnhancing softwarereliability modeling and prediction through the introductionof time-variable fault reduction factorrdquo Applied MathematicalModelling vol 35 no 1 pp 506ndash521 2011

[39] W-CHuang andC-YHuang ldquoSoftware reliability analysis andmeasurement using finite and infinite server queueing modelsrdquoIEEE Transactions on Reliability vol 57 no 1 pp 192ndash203 2008

[40] W-C Huang C-Y Huang and C-C Sue ldquoSoftware reliabilityprediction and assessment using both finite and infinite serverqueueing approachesrdquo in Proceedings of the 12th Pacific RimInternational Symposium on Dependable Computing (PRDCrsquo06) pp 194ndash201 IEEE Riverside Calif USA December 2006

[41] M Xie Software Reliability Modeling World Scientific Publish-ing Singapore 1991

[42] C-YHuang andW-CHuang ldquoSoftware reliability analysis andmeasurement using finite and infinite server queueing modelsrdquoIEEE Transactions on Reliability vol 57 no 1 pp 192ndash203 2008

[43] J Musa A Iannino and K Okumoto Software Reliability Mea-surement Prediction Application McGraw-Hill New York NYUSA 1989

[44] K Pillai and V S S Nair ldquoA model for software developmenteffort and cost estimationrdquo IEEE Transactions on Software Engi-neering vol 23 no 8 pp 485ndash497 1997

[45] C-Y Huang and C-T Lin ldquoSoftware reliability analysis by con-sidering fault dependency and debugging time lagrdquo IEEETrans-actions on Reliability vol 55 no 3 pp 436ndash450 2006

[46] K Pillai and V S Nair ldquoA model for software developmenteffort and cost estimationrdquo IEEE Transactions on Software Engi-neering vol 23 no 8 pp 485ndash497 1997

[47] W AMassey G A Parker andWWhitt ldquoEstimating the para-meters of a nonhomogeneous Poisson process with linear raterdquoTelecommunication Systems vol 5 no 4 pp 361ndash388 1996

[48] J G Leite J Rodrigues and L A Milan ldquoA bayesian analysisfor estimating the number of species in a population using non-homogeneous poisson processrdquo Statistics amp Probability Lettersvol 48 no 2 pp 153ndash161 2000

[49] Z Yang and C Liu ldquoEstimating parameters of Ohba three-parameters NHPP Modelsrdquo in Proceedings of the InternationalConference on Computer Science and Service System (CSSS rsquo11)pp 1259ndash1261 IEEE Nanjing China June 2011

[50] CWeiss R Premraj T Zimmermann andA Zeller ldquoHow longwill it take to fix this bugrdquo in Proceedings of the 4th InternationalWorkshop on Mining Software Repositories (ICSE WorkshopsMSR rsquo07) p 1 IEEE Minneapolis Minn USA May 2007

[51] P Wang D M Tartakovsky K D Jarman and A M Tar-takovsky ldquoCdf solutions of buckley-leverett equation withuncertain parametersrdquoMultiscaleModeling and Simulation vol11 no 1 pp 118ndash133 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


software during developmenttesting phase and faults maynot be corrected perfectly This phenomenon is known asimperfect debugging According to the above two phe-nomenons SRGMs can be divided into two categoriesperfect and imperfect debugging model Kapur and Garg[14] discussed their fault removal phenomenon that as theteam gainsmore experience theymay remove detected faultswithout causing new ones However the testing team maynot correct a fault perfectly and the original fault may remainor generate new faults leading to a phenomenon known asimperfect debugging It was Goel [15] who first introducedthe concept of imperfect debugging Latter Ohba and Chou[16] developed an error generation model based on the G-Omodel named as an imperfect debugging model Pham et al[17] proposed a general imperfect software debugging model(P-N-Z model) Pham [18] also developed an SRGM formultiple failure types incorporating error generation Zhanget al [19] proposed a testing efficiency model that includesboth imperfect debugging and error generation Kapur et al[20] proposed a flexible SRGM with imperfect debuggingand error generation using a logistic function for the faultdetection rate which reflects the efficiency of the testingand removal team Employing the learning phenomenon insoftware fault detection process Yamada et al [21] later devel-oped an imperfect debugging model In imperfect softwaredebugging models the fault introduction rate per fault isgenerally assumed to be constant or decreasing changes overtime [22] Recently Wang et al [22] also proposed a modelto represent the imperfect debugging process with a log-logistic distribution To sum up most models presume adeterministic relationship between the cumulative numberof detected faults and the time span of the software faultdebugging process

However software debugging is a stochastic and uncer-tain process which can be effected by many environmentalfactors such as the distribution of resources strategy andrunning environment [23] In particular in the uncertainnetwork environment the aforementioned assumptions ofdeterministic model would become problematic The envi-ronmental noise introduces great uncertainties that signifi-cantly affects traditional debugging process To address suchchallenges new stochastic models [24ndash27] were proposedwhich treat the debugging process as perfect and stochas-tic they assume each failure occurrence is independentand follows identical random distribution By employingthe white noise that is temporally uncorrelated randomvariables for the environmental factors collectively theyused a flexible stochastic differential equation to model theirregular changes Comparing to conventional models suchwhite-noise-based approach assumes the perfect debuggingand no doubt provides closer approximation to uncertainfluctuations in reality but with great mathematical simplicityDebugging activity is usually imperfect in piratical softwaredevelopment and recent data [28 29] show that the faultdetection is highly susceptible to noise and is generally cor-related thus earlier assumptions become problematic Thusbecause of its mathematical simplicity it may also consider-ably underestimate the imperfect debugging process and thetemporal correlation in a dynamic environment

In this study we propose an alternative stochastic frame-work based on partial differential equation (PDE) to describethe perfectimperfect debugging processes in which theenvironmental noise is of arbitrary correlation Details of themodel is provided in Section 2 with an equation for the prob-abilistic density function (PDF) of system states In Section 3we validate our approach with both historical failure dataand results from conventional methods main features of themodel including the temporal evolutions of its full statisti-cal information are later addressed in Section 4 The finalconclusions are given in Section 5

2 Problem Formulation

21Model Formulation Widely used inmany practical appli-cations NHPP model assumes the following

(1) the fault debugging process is modeled as a stochasticprocess within a continuous state space

(2) the system is subject to failures at random timescaused by the remaining errors in the system

(3) the mean number of detected faults in a time intervalis proportional to the mean number of remainingfaults

(4) when a fault is detected it may be removed with theprobability 119901

(5) when a fault is removed new faults may be generatedwith a constant probability 119902

In order to describe the stochastic and uncertain behaviorof the fault debugging process in a continuous state space thetemporal evolution of119872(119905) is routinely described as [12]

d119872d119905

= 119903 (119905) (119886 + 119902119872 minus 119901119872) (1)

where 119872(119905) is the expected number of faults detected in thetime interval (0 119905] 119886 denotes the number of total faults and119903(119905) is a nonnegative function representing the fault detectionrate per remaining fault at testing time 119905 We further notethat 119902(119905)119872 is the number of newly introduced bugs while119901(119905)119872 means the number of successful removal togetherthey represent a (perfectimperfect) debugging process inwhich probabilities 119902 and 119901 are assigned [30ndash32] Without aloss of generality we denote 119896(119905) = 119901(119905) minus 119902(119905) to facilitatesubsequent presentation and 119896 = 1 indicates a perfectdebugging process

Since 119903(119905) is subject to a number of random environmen-tal effects [23] we follow the previous studies [24 25 27] andrepresent 119903(119905) as a sum of its mean value ⟨119903(119905)⟩ and a zero-mean (random) fluctuation 119903

1015840(119905)

119903 (119905) = ⟨119903 (119905)⟩ + 1199031015840

(119905) ⟨1199031015840

(119905)⟩ = 0 (2)

By substituting (2) into (1) we obtain a stochastic differ-ential equation

d119872d119905

= [⟨119903 (119905)⟩ + 1199031015840

(119905)] [119886 minus 119896 (119905)119872] (3)





⟨1199031015840

(119905) 1199031015840

(119904)⟩ = 1205902

119903120588 (

119905 minus 119904

120591) (4)




119872(119898 119905) To be


Π (119898119872 119905) = 120575 [119872 (119905) minus 119898] (5)


⟨Π⟩ equiv int120575 (1198721015840minus 119898)119891

119872(1198721015840 119905) d1198721015840 = 119891

119872(119898 119905) (6)



120597Π

120597119905= minus

120597

120597119898[⟨119903 (119905)⟩ + 119903

1015840

(119905)] [119886 minus 119896 (119905)119898]Π (7)



120597119891119872

120597119905= minus

120597

120597119898(V119891119872) +

120597

120597119898(D

120597119891119872

120597119898) (8a)


D (119898 119905)

asymp int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩ [119886 minus 119896 (119905)119898] [119886 minus 119896 (119904) 119910]

sdotG119889d119910 d119904

(8b)

V (119898 119905) asymp [119886 minus 119896 (119905)119898] ⟨119903 (119905)⟩ + int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩

sdot [119886 minus 119896 (119905)119898] 119896 (119904)G119889d119910 d119904

(8c)



120597G119889

120597119904+ [119886 minus 119896 (119904) 119910] ⟨119903 (119904)⟩

120597G119889

120597119910


(8d)


3 Model Validation





2




⟨119903 (119905)⟩ = 119887119905119889 (9)



119891119872(119898 119905) =

exp [minus ( + )2

21205902

119868]

120590119868(119886 minus 119896119898)radic2120587

119898 lt 119886

0 119898 ge 119886

(10a)









01199031015840(1199051015840)d1199051015840 = sum

119873

119894=1int119894Δ

(119894minus1)Δ1199031015840(1199051015840)d119905 where Δ = 119905119873

Cases 119905 ≪ 120591 119905 ≫ 120591 Others

Variance 1205902119868

1205902

1199031199052

21205902

119903119905 2119873120590

2

119903[int

Δ

0

(Δ minus 1199051015840

) 120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

int

Δ

0

int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus 11990510158401015840

) d1199051015840d11990510158401015840]

d120590119868

d119905(119905 119879) 120590

119903

1

radic2(119879 + 119905)

1

120590119868

int

Δ

0

120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

[int

Δ

0

119894120588 (119894Δ minus 11990510158401015840

) d11990510158401015840 minus int

Δ

0


] d11990510158401015840 + int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus Δ) d1199051015840]

where

=ln (1 minus 119896119898119886)

119896

=119887119905119889+1

119889 + 1

(10b)

and 1205902


1199031015840 = int119905

01199031015840(119904)d119904


2

119868)


119872(119898 119905) can be

derived as 119865119872(119898 119905) = int119891

119872(1198981015840 119905)d1198981015840

119865119872(119898 119905) =

1

2[1 minus erf ( +

120590119868radic2

)] 119898 lt 119886

1 119898 ge 119886

(11)


=119886

119896int

infin

minusinfin

1 minus exp[minus119896(1198681199031015840 +

119887119905119889+1

119889 + 1)]119891

1198681199031015840d1198681199031015840

(12)




PRE (119905) = 119872 (119905) minus 119873 (119905)

119873 (119905) (13)



MSE (119905) =sum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

119873119900

(14)

where 119872(119905119894) and 119873(119905





1198772

(119905) = 1 minussum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

sum119870

119894=1[119873 (119905119894) minus 119873]

2 (15)


119894=1119873(119905119894)119870

1198772





Given

PE (119905119894) = 119872(119905

119894) minus 119873 (119905

119894) (17a)

Bias =119870

sum

119894=1

[PE (119905119894)

119870] (17b)

PRV (119905) = radicsum119870

119894=1(PE119894minus Bias)2

119870 minus 1 (17c)



2



2




2

119903)








2



119903= 00002228 and 119896 =








119903= 00002228



97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903= 00002228






0 10 20 30 40 50 60

PDE modelG-O model


0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)




119872(119898 119905)






119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













⟨1199031015840

(119905) 1199031015840

(119904)⟩ = 1205902

119903120588 (

119905 minus 119904

120591) (4)




119872(119898 119905) To be


Π (119898119872 119905) = 120575 [119872 (119905) minus 119898] (5)


⟨Π⟩ equiv int120575 (1198721015840minus 119898)119891

119872(1198721015840 119905) d1198721015840 = 119891

119872(119898 119905) (6)



120597Π

120597119905= minus

120597

120597119898[⟨119903 (119905)⟩ + 119903

1015840

(119905)] [119886 minus 119896 (119905)119898]Π (7)



120597119891119872

120597119905= minus

120597

120597119898(V119891119872) +

120597

120597119898(D

120597119891119872

120597119898) (8a)


D (119898 119905)

asymp int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩ [119886 minus 119896 (119905)119898] [119886 minus 119896 (119904) 119910]

sdotG119889d119910 d119904

(8b)

V (119898 119905) asymp [119886 minus 119896 (119905)119898] ⟨119903 (119905)⟩ + int

119905

0

intΩ

⟨1199031015840

(119905) 1199031015840

(119904)⟩

sdot [119886 minus 119896 (119905)119898] 119896 (119904)G119889d119910 d119904

(8c)



120597G119889

120597119904+ [119886 minus 119896 (119904) 119910] ⟨119903 (119904)⟩

120597G119889

120597119910


(8d)


3 Model Validation





2




⟨119903 (119905)⟩ = 119887119905119889 (9)



119891119872(119898 119905) =

exp [minus ( + )2

21205902

119868]

120590119868(119886 minus 119896119898)radic2120587

119898 lt 119886

0 119898 ge 119886

(10a)









01199031015840(1199051015840)d1199051015840 = sum

119873

119894=1int119894Δ

(119894minus1)Δ1199031015840(1199051015840)d119905 where Δ = 119905119873

Cases 119905 ≪ 120591 119905 ≫ 120591 Others

Variance 1205902119868

1205902

1199031199052

21205902

119903119905 2119873120590

2

119903[int

Δ

0

(Δ minus 1199051015840

) 120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

int

Δ

0

int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus 11990510158401015840

) d1199051015840d11990510158401015840]

d120590119868

d119905(119905 119879) 120590

119903

1

radic2(119879 + 119905)

1

120590119868

int

Δ

0

120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

[int

Δ

0

119894120588 (119894Δ minus 11990510158401015840

) d11990510158401015840 minus int

Δ

0


] d11990510158401015840 + int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus Δ) d1199051015840]

where

=ln (1 minus 119896119898119886)

119896

=119887119905119889+1

119889 + 1

(10b)

and 1205902


1199031015840 = int119905

01199031015840(119904)d119904


2

119868)


119872(119898 119905) can be

derived as 119865119872(119898 119905) = int119891

119872(1198981015840 119905)d1198981015840

119865119872(119898 119905) =

1

2[1 minus erf ( +

120590119868radic2

)] 119898 lt 119886

1 119898 ge 119886

(11)


=119886

119896int

infin

minusinfin

1 minus exp[minus119896(1198681199031015840 +

119887119905119889+1

119889 + 1)]119891

1198681199031015840d1198681199031015840

(12)




PRE (119905) = 119872 (119905) minus 119873 (119905)

119873 (119905) (13)



MSE (119905) =sum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

119873119900

(14)

where 119872(119905119894) and 119873(119905





1198772

(119905) = 1 minussum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

sum119870

119894=1[119873 (119905119894) minus 119873]

2 (15)


119894=1119873(119905119894)119870

1198772





Given

PE (119905119894) = 119872(119905

119894) minus 119873 (119905

119894) (17a)

Bias =119870

sum

119894=1

[PE (119905119894)

119870] (17b)

PRV (119905) = radicsum119870

119894=1(PE119894minus Bias)2

119870 minus 1 (17c)



2



2




2

119903)








2



119903= 00002228 and 119896 =








119903= 00002228



97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903= 00002228






0 10 20 30 40 50 60

PDE modelG-O model


0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)




119872(119898 119905)






119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















01199031015840(1199051015840)d1199051015840 = sum

119873

119894=1int119894Δ

(119894minus1)Δ1199031015840(1199051015840)d119905 where Δ = 119905119873

Cases 119905 ≪ 120591 119905 ≫ 120591 Others

Variance 1205902119868

1205902

1199031199052

21205902

119903119905 2119873120590

2

119903[int

Δ

0

(Δ minus 1199051015840

) 120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

int

Δ

0

int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus 11990510158401015840

) d1199051015840d11990510158401015840]

d120590119868

d119905(119905 119879) 120590

119903

1

radic2(119879 + 119905)

1

120590119868

int

Δ

0

120588 (1199051015840

) d1199051015840 +119873

sum

119894=2

[int

Δ

0

119894120588 (119894Δ minus 11990510158401015840

) d11990510158401015840 minus int

Δ

0


] d11990510158401015840 + int

119894Δ

(119894minus1)Δ

120588 (1199051015840

minus Δ) d1199051015840]

where

=ln (1 minus 119896119898119886)

119896

=119887119905119889+1

119889 + 1

(10b)

and 1205902


1199031015840 = int119905

01199031015840(119904)d119904


2

119868)


119872(119898 119905) can be

derived as 119865119872(119898 119905) = int119891

119872(1198981015840 119905)d1198981015840

119865119872(119898 119905) =

1

2[1 minus erf ( +

120590119868radic2

)] 119898 lt 119886

1 119898 ge 119886

(11)


=119886

119896int

infin

minusinfin

1 minus exp[minus119896(1198681199031015840 +

119887119905119889+1

119889 + 1)]119891

1198681199031015840d1198681199031015840

(12)




PRE (119905) = 119872 (119905) minus 119873 (119905)

119873 (119905) (13)



MSE (119905) =sum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

119873119900

(14)

where 119872(119905119894) and 119873(119905





1198772

(119905) = 1 minussum119870

119894=1[119872 (119905

119894) minus 119873 (119905

119894)]2

sum119870

119894=1[119873 (119905119894) minus 119873]

2 (15)


119894=1119873(119905119894)119870

1198772





Given

PE (119905119894) = 119872(119905

119894) minus 119873 (119905

119894) (17a)

Bias =119870

sum

119894=1

[PE (119905119894)

119870] (17b)

PRV (119905) = radicsum119870

119894=1(PE119894minus Bias)2

119870 minus 1 (17c)



2



2




2

119903)








2



119903= 00002228 and 119896 =








119903= 00002228



97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903= 00002228






0 10 20 30 40 50 60

PDE modelG-O model


0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)




119872(119898 119905)






119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Given

PE (119905119894) = 119872(119905

119894) minus 119873 (119905

119894) (17a)

Bias =119870

sum

119894=1

[PE (119905119894)

119870] (17b)

PRV (119905) = radicsum119870

119894=1(PE119894minus Bias)2

119870 minus 1 (17c)



2



2




2

119903)








2



119903= 00002228 and 119896 =








119903= 00002228



97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903= 00002228






0 10 20 30 40 50 60

PDE modelG-O model


0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)




119872(119898 119905)






119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60

PDE modelG-O model


0

50

100

150Fa

ults

Time (month)

(a)

PDE modelG-O model


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

Faul

ts

Time (week)

(b)

PDE modelG-O model


0 5 10 15 200

20

40

60

80

100

120

Faul

ts

Time (week)

(c)

PDE modelG-O model


0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

40Fa

ults

Time (week)

(d)




119872(119898 119905)






119877 (119905 119879) = 119890minus[⟨119872(119879+119905)⟩minus⟨119872(119879)⟩]

(18)


0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60

1

PDE model

Time (month)


minus08

minus06

minus04

minus02

0

02

04

06

08

12

PRE

(a)

0 5 10 15 20 25 30 35 40 45minus08

minus06

minus04

minus02

0

02

04

06

08

1

PRE

PDE model

Time (week)


(b)

0 5 10 15 20minus04

minus02

0

02

04

06

08

1

PRE


Time (week)

(c)

0 2 4 6 8 10 12 14 16 18minus04

minus02

0

02

04

06

08

1

12

14

16PR

E


Time (week)

(d)



RGR (119905 119879) =119889

119889119905119877 (119905 119879)

= 119886119877int

infin

minusinfin

[d120590119868

d1199051205902

119868minus 1198682

1199031015840

1205903119868119896

(1 minus 119890minus120573)

minus 119887 (119879 + 119905)119889119890minus120573]1198911198681199031015840d1198681199031015840

(19a)

where

120573 = 119896(1198681199031015840 +

119887119905119889+1

119889 + 1) (19b)




156 119887 = 0005446 119889 = 0026 1205902119903= 00002228 119896 = 05092





PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












PDE model

DS1 119886 = 432 119887 = 004047 119889 = 0028 1205902119903= 00002236 119896 =

09536 and 120591 = 014196179 06466 125213 163490 131506

DS2 119886 = 156 119887 = 05446 119889 = 0026 1205902119903= 00002228 119896 =

05092 and 120591 = 01611382 08853 minus69274 36637 68133

DS3 119886 = 605 119887 = 0846 119889 = 0025 1205902119903= 00002219 119896 =

09824 and 120591 = 01821784 08989 minus69221 60056 91642

DS4 119886 = 269 119887 = 03046 119889 = 0024 1205902119903= 00002207 119896 = 1

and 120591 = 0159284 09682 minus09743 22926 24910

G-O model


white-noise model

DS1 119886 = 510 119887 = 00126 119889 = 0028 119896 = 09536 and 1205902

119903=

000022367609538 03591 236059 143934 276479

DS2 119886 = 97 119887 = 016246 119889 = 0026 119896 = 05092 and 1205902

119903=

000022281718446 06776 109225 73238 131506

DS3 119886 = 199 119887 = 0226 119889 = 0025 119896 = 09824 and 1205902

119903=

000022191466972 08195 88329 91706 127326

DS4 119886 = 377 119887 = 00126 119889 = 0024 119896 = 10000 and 1205902

119903=

00002207204094 08905 minus36981 26661 45589

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

Time (week)

PRE








0 5 10 15 20 25 30 35 40 45minus05

0

05

1

15

2

Time (week)

PRE




PDE model with 1205902r = 00002228



119903









2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















2

119903= 0 1716587 06779 109154 73217 131435


119903= 00002228 1718446 06776 109225 73238 131506


119903= 02228 3436834 03552 161015 92835 185861


119903= 0 611445 08853 minus69277 36640 78370


119903= 00002228 611382 08853 minus69274 36637 78366


119903= 02228 565464 08939 minus66843 34804 75362

0 5 10 15 20 25 30 35 40 45

minus04

minus02

0

02

04

06

08

1

12

14

Time (week)

PRE



5 Conclusion









0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 500

01

02

03

04

05

m

fM(m

t)


0 10 20 30 40 500

02

04

06

08

1

12

m

FM(m

t)



119872(119898 119905) at time 119905 = 2


0 5 10 15 20 25 300

01

02

03

04

05

06

07

08

09

1

t (week) t (week)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

R(tT

)

RGR(tT)




119903= 00002228 119896 = 05092 and 120591 = 01






Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)

Detectedfaults

Correctedfaults

Time unit(month)


1 1 1 21 53 50 41 72 652 7 7 22 53 50 42 74 673 7 7 23 53 50 43 74 674 9 9 24 53 50 44 80 675 9 9 25 53 53 45 84 696 9 9 26 53 53 46 84 767 12 11 27 53 53 47 84 798 18 15 28 53 53 48 84 829 18 15 29 53 53 49 85 8310 18 18 30 53 53 50 86 8411 18 18 31 53 53 51 89 8712 21 21 32 54 54 52 90 9013 22 22 33 56 54 53 90 9014 22 22 34 58 56 54 92 9115 27 26 35 59 56 55 108 10616 30 28 36 60 56 56 120 11917 45 39 37 63 56 57 128 12618 47 44 38 70 60 58 129 12819 49 47 39 71 61 59 139 13620 51 47 40 71 64 60 146 136


Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)

Detectedfaults

Correctedfaults

Time unit(week)



Appendix

Data Sets





Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults

Time unit(week)

Detectedfaults


Release 11 16 11 81 6 49 16 982 24 12 86 7 54 17 993 27 13 90 8 58 18 1004 33 14 93 9 69 19 1005 41 15 96 10 75 20 100




References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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