ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION

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1 Joural of Data Sciece , DOI: /JDS _16(4) ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION David D. Haagal *, Nileema N. Bhalerao 2 Departmet of Statistics, Savitribai Phule Pue Uiversity Pue , Idia ABSTRACT May software reliability growth models based upo a o-homogeeous Poisso process (NHPP) have bee proposed to measure ad asses the reliability of a software system quatitatively. Geerally, the error detectio rate ad the fault cotet fuctio durig software testig is cosidered to be depedet o the elapsed time testig. I this paper we have proposed three software reliability growth models (SRGM s) icorporatig the otio of error geeratio over the time as a extesio of the delayed S-shaped software reliability growth model based o a o-homogeeous Poisso process (NHPP). The model parameters are estimated usig the maximum likelihood method for iterval domai data ad three data sets are provided to illustrate the estimatio techique. The proposed model is compared with the existig delayed S-shaped model based o error sum of squares, mea sum of squares, predictive ratio risk ad Akaike s iformatio criteria usig three differet data sets. We show that the proposed models perform satisfactory better tha the existig models. Keywords:Error detectio rate; Fault Cotet Rate Fuctio; No-homogeeous Poisso process; Predictive ratio risk. * Correspodig author: David D. Haagal david haagal@yahoo.co.i

2 858 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION 1. Itroductio Over the past four decades several software models (SRGM) have bee proposed by various researchers [See Pham (1993, Teg ad Pham (2006), Pham (1996), Obha ad Yamada (1984), Teg ad Pham (2004), Zhag et al. (2003)]. The pioeerig attempt i the o-homogeeous Poisso process (NHPP) software reliability growth model was by Goel ad Okumoto (1979) (GO). The model describes the failure observatio pheomeo by a expoetial curve. There are other SRGM s describig either S-shaped curves or a mixture of expoetial ad S-shaped curves (flexible). Some of the importat models fallig uder this category are due to Yamada et al. (1983), Obha (1984), Bittati et al. (1988), Kapur ad Garg (1992), Kapur et al. (1999), Pham (2006), Che (2010) ad Lai ad Garg (2012). These models were developed based o the assumptio that faults detected i the testig phase are removed immediately with o debuggig time delay ad o ew faults are itroduced ito the software. I other words, it is assumed that wheever a attempt is made to remove a fault, it is removed with certaity ad this is referred as perfect because of a umber of factors like tester s skill ad expertise. The testig is observed ad the origial fault may remai, leadig to a pheomeo kow as imperfect debuggig. Aother possibility is that while correctig a software error additioal errors may be geerated ad these errors may get ito software. Such models may be referred as error geeratio models. I the case of error geeratio, the total fault cotet icreases as testig progresses because ew faults are itroduced ito the system while removig the origial faults. Goel ad Okumoto (1979) have itroduced the cocept of imperfect debuggig for the first time i literature. Obha ad Cho (1989) exteded this work by itroducig the error geeratio ito the GO model ad amed it as imperfect debuggig model. Kapur ad Grag (1990) itroduced the imperfect debuggig i GO model ad they cosidered fault itroductio rate per remaiig faults beig reduced due to the imperfect debuggig. Thus the umber of failure observed ad detected by ifiite time is more tha the iitial fault cotet. Yamada et al. (1992) exteded GO model by assumig that fault itroductio rate is liear ad expoetial time depedet. Further several authors have used this approach to model error geeratio activity i software reliability model. Pham ad Zhag (1997) have developed a software reliability model cosiderig error itroductio rate as expoetial ad error itroductio rate as o-decreasig type. Pham et al. (1999) have discussed liear fault itroductio rate fuctio ad fault itroductio is of o-decreasig time depedet type. Tokuo ad Yamada (2000) proposed imperfect debuggig SRGM with two types of hazard rates for software reliability measuremet ad assessmet. Zhag ad Pham (2000) developed NHPP model with imperfect debuggig ad time depedet fault-detectio rate. Zhag et al. (2003) proposed SRGM model to itegrate fault removal efficiecy, failure rate, ad fault itroductio rate ito software reliability assessmet. Chatterjee ad Shukla (2016) proposed a chage poit-based SRGM uder imperfect debuggig with revised cocept of fault depedecy. The above discussed models does ot always reflect real testig eviromets. Recetly Haagal ad Bhalerao (2016a, 2016b, 2017) obtaied software reliability growth models based o iverse Weibull, geeralized iverse Weibull, exteded iverse Weibull distributios. It is of iterest to costruct a SRM by cosiderig

3 David D. Haagal ad Nileema N. Bhalerao 859 itroductio of error geeratio durig testig ad cetral processig uit (CPU) executio time. The remaider of the paper is orgaized as follows. I Sectio 2, we describe NHPP delayed S-shaped models. I sectio 3, we discuss the proposed models with three differet fault cotet fuctios ad software reliability assessmet measures ad the estimatio of ukow model parameters usig maximum likelihood method. Parameter estimatio of is of primary importace i software reliability predictio. Oce the aalytical solutio for m(t) is kow for a give model, the parameters i the solutio eed to be determied. Parameter estimatio is achieved by applyig a techique of maximum likelihood estimate (MLE), for the iterval domai area data. I may cases, the maximum likelihood estimators are cosistet ad asymptotically ormally distributed as the sample size icreases; see Zhao ad Xie (1996). I Sectio 4, we represet the aalysis of three data sets DS I, DS II ad DS III. Sectio 5 cotais the major coclusios ad remarks of the study. 2. NHPP Delayed S-Shaped Model I the NHPP S-shaped mode, the software reliability growth curve is a S-shaped curve. The detectio rate of faults, where the error detectio rate chages with time, because the greatest at a certai time after testig begis, after which it decreases expoetially. I other words, some faults are covered by other faults at the begiig of the testig phase, before these faults are actually removed, the covered faults remai udetected. Yamada et al. (1984) also determied that the software testig process usually ivolves a learig process where testers becomes familiar with the software products, eviromets, ad software specificatios. Several S-shaped models (Yamada et al. 1984; Pham 1997a) such as delayed S-shaped, iflectio S-shaped, etc., exists i literature. Kareer et al. (1990) proposed a S-shaped SRGM with two types of errors. The errors have bee classified depedig upo their survey. Gupta et al. (2011) obtaied software reliability estimatio usig delayed S-shaped model uder imperfect debuggig ad time lag. Ahmed et al. (2011) developed a iflatio SRGM cosiderig log-logistic testig-effort ad imperfect debuggig. Kaur ad Sharma (2015) predict the time betwee failure ad accuracy by usig CASRE tool for SRG models. Bokhari et al. (2017) proposed delayed S-shaped SRGM with imperfect debuggig ad ew modified Weibull testig effort fuctio. We ow discuss a stochastic model for a software error detectio process based o NHPP i which the growth curve of the umber of detected software errors for the observed failure data is S-shaped, called delayed S-shaped NHPP model (Yamada et al 1984). The software error detectio procedure described by a S-shaped curve ca be characterized as a learig process i which the test team members become familiar with the test eviromet, testig tools or project requiremets, i.e., their test skills gradually improve. The delayed S-shaped model is based o the followig assumptios: 1. All the faults i the program are mutually idepedet from the failure detectio poit of view.

4 860 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION 2. The probability of failure detectio at ay time is proportioal to the curret umber of faults i a software. 3. The proportioality of failure detectio i costat. 4. The iitial error cotet of the software is a radom variable. 5. A software system is subject to failures at radom times caused by errors preset i the system. 6. The time betwee (i 1) th ad i th failures depeds o the time to the (i 1) th failure. 7. Each time a failure occurs, the error which caused it is simultaeously removed ad o other errors are itroduced. O the basis of these assumptios we have the followig differetial equatio: m(t) = b(t)[a m(t)]. a > 0, b(t) > 0, t 0 (2.1) t a=expected total umber of faults that exists i the software before testig. b(t) = failure detectio rate per fault, which also represets the average failure rate of a fault. I delayed S-shaped NHPP model, b(t) = b2 t 1+bt,where b is the error detectio rate per error i the steady state. Solvig the differetial equatio give i (2.1) with respect to m(t) uder the iitial coditio m(0) = 0. Yamada et al. (1984) obtaied the mea value fuctio give by: m(t) = a[1 (1 + bt)e bt ] (2.2) which shows a S-shaped curve. This model is called delayed S-shaped NHPP model. Further, the coutig process N(t), t 0 represetig the cumulative umber of software faults detected up to testig time t is a stochastic process. Basic assumptio about this process lead to the commoly accepted coclusio that, for ay t 0, N(t) is a Poisso distributed with time depedet Poisso parameter m(t), the mea value fuctio (MVF). P{N(t) = } = [m(t)] e m(t), = 0,1,2,, (2.3)! where m(t) is give by : t m(t) = λ(x)dx (2.4) 0 The MVF represets the expected umber of software errors that have accumulated up to time t, or estimated cumulative faults up to time t. The itesity fuctio, λ(t) of the NHPP give by: λ(t) = ab 2 te bt (2.5) This represets the umber of faults represeted per uit testig time.

5 David D. Haagal ad Nileema N. Bhalerao 861 I geeral the secod assumptio (2) i.e. the detected faults are removed with certaity is ot always true [See Yamada et al. (1992), Pham ad Zhag (1997) ad Pham (1999)]. If additioal itroduced faults affect the fractio of the fault cotet, we have to modify this assumptio by itroducig the cocept of error geeratio while debuggig (or testig) process is a actio. That is, total fault cotet rate fuctio, a(t) represetig the sum of expected umber of iitial software faults ad itroduced faults by time t. Replacig a by a(t) i equatio (2.1), we have differetial equatio m(t) = b(t)[a(t) m(t)]. a(t) > 0, b(t) > 0, t 0 (2.6) t The above differetial equatio was give by Pham (2007). The geeralized mea value fuctio (MVF) solutio of the above differetial equatio (2.6) give by (Pham ad Zhag 1997) is as follows: m(t) = e B(t) [m 0 + a(τ)b(τ)e B(τ) 0 t dτ] (2.7) t where B(t) = b(τ)dτ with iitial coditio m(t t 0 ) = m 0 ad t 0 is the time to begi the 0 debuggig process. Pham (2007) obtaied NHPP SRGM with a(t) = α(1 + γt) 2 ad b(t) = γ2 t (1+γt) i this case both a(t) ad b(t) have commo parameter γ. I our proposed models we cosider the parameters are idepedet istead of depedet case. May existig software reliability growth models [Pham (2006)], Pham ad Zhag (2003), Yamada ad Osaki (1985) ca be cosidered as a special case of the above geeral model. A icreasig fuctio a(t) implies a icreasig total umber of faults ad reflects imperfect debuggig. A icreasig b(t) implies a icreasig fault detectio rate, which could be either attributed to imperfect debuggig, or to software process fluctuatios, or a combiatio of both. Differet a(t) ad b(t) fuctios also reflect differet assumptios of the software testig processes. I the simplest model, the fuctio a(t) ad b(t) are both costats. A costat a(t) = a stads for the assumptio that o ew errors are itroduced durig the debuggig process. A costat b(t) = b implies that the proportioal factor relatig the error detectio rate λ(t) to the total umber of remaiig errors is costat. I a geeral model, the fuctios a(t) ad b(t) are both fuctios of time. I the followig sectio we propose geeral models with three differet types of fault cotet fuctio a(t) ad discuss the methods of estimatig parameters of the models.

6 862 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION 3. Proposed Models with Three Differet Fault Cotet Rate Fuctios i Delayed S-shaped NHPP 3.1.1Liear Fault Cotet Rate Fuctio Here we assume that fault itroductio rate is a liear fuctio ad depedet o time, a(t) = a 1 (t) = a(1 + αt) a > 0, α > 0, t 0 where a is the umber of iitial fault cotet i the system ad α is a icreasig rate of the umber of itroduced faults to the iitial fault cotet fuctio. Let m 1 (t) be the MVF whe fault cotet fuctio a 1 (t) is substituted for a(t) i (2.6) thus we have: m 1 (t) t = b(t)[a(1 + αt) m 1 (t)]. a > 0, α > 0, b(t) > 0 t 0 (3.1) with b(t) = b2 t 1+bt. Solvig the differetial equatio give i (3.1) with respect to m 1(t) uder the iitial codtitio m 1 (0) = 0 we derived the mea value fuctio give by: m 1 (t) = ( a 2aα b ) e bt (1 + bt) + aα b (1 + bt) + (a 2aα b ) + 3aα b(1 + bt) (3.2) I this paper we call NHPP model with MVF m_1(t) give i (3.2) as delayed S-shaped liear model. The itesity fuctio of this model is: λ 1 (t) = (a + 2aα b ) b2 te bt 3aα + aα (3.3) (1 + bt) 2 Software Reliability Assessmet Measures Based o delayed S-shaped liear model, we ca derive the followig quatitative measures useful for software reliability assessmet. (a) Software Reliability Let S i, (i = 1,2, 3 ) be a radom variable represetig the i th software failure occurrece time. Defie X i = S i S i 1, (i = 1,2,3 ); S 0 = 0, is a radom variable represetig the time iterval betwee (i 1) th ad i th software failure occurrece. The coditioal probability that the i th software failure does ot occur betwee (t, t + x], (x 0) o the coditio that the (i 1) th software failure has occurred at testig time t, is give by: R(x t) = P{X i > x S i 1 = t} = exp[ {m(x + t) m(t)}], (3.4) t 0, x 0. substitutig (3.2) i (3.4) we have delayed S-shaped liear software reliability R 1 (x t) = exp [ {(a + 2aα b ) e bt {(1 + bt) (1 + b(t + x)e bx } + aα b (1 + bx) 3axα (1 + bt)(1 + b(t + x)) }], x, t 0 (3.5)

7 David D. Haagal ad Nileema N. Bhalerao 863 (b) Expected Residual Fault Cotet Cosider the expectatio of {a(t) N(t)} for the NHPP model i (3.2). Sice N(t) is a radom variable, the umber of residual fault cotet at testig time t is: (t) = a(t) E[N(t)] = a(t) m(t) (3.6) Let 1 (t) deote the expected residual fault cotet for delayed S-shaped liear SRM, that is 1 (t) = a 1 (t) m 1 (t) (3.7) 1 (t) = a(1 + αt) [( a 2aα b ) e bt (1 + bt) + aα b (1 + bt) +(a 2aα b + 3aα b(1 + bt) )]. (3.8) 3.1.1Parameter Estimatio I this sectio we discuss the method of estimatig the parameters of the models metioed above Estimatio usig iterval Domai Data Suppose that pairs of observatios (t i, y i ); (i = 1,2,3,, ; 0 < t 1 < t 2 < < t ) are made durig the testig phase where y i deote the umber of software faults detected up to the testig time t i, i = 1,2,,. The correspodig probability mass fuctio is P[N(t 1 ) = y 1, N(t 2 ) = y 2,, N(t ) = y ] ad hece the likelihood fuctio for the iterval domai is give by L = (m(t i) m(t i 1 )) y i y i 1 exp[ m(t (y i y i 1 )! i ) m(t i 1 )] (3.9) i=1 where t 0 = 0 ad y 0 = 0. Takig the atural logarithm of equatio (3.9) yields loglikelihood fuctio, that is ll = (y i y i 1 ) l[m(t i ) m(t i 1 )] m(t ) l[(y i y i 1 )!] i=1 i=1 (3.10)

8 864 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION Substitutig mea value fuctio m 1 (t) ito the equatio (3.10), we get logarithmic of likelihood fuctio excludig the costat terms, ll = (y i y i 1 ) l [( a 2aα b ) e bt i (1 + bti ) + aα b (1 + bt i) i=1 + (a 2aα b ) + 3aα b(1 + bt i ) ( a 2aα b ) e bt i 1 (1 + bti 1 ) + aα b (1 + bt i 1) + (a 2aα b ) + 3aα b(1 + bt i 1 ) ] ( a 2aα b ) e bt (1 + bt ) + aα b (1 + bt ) + (a 2aα b ) + 3aα b(1 + bt ) (3.11) Closed form expressios of MLEs of a, α ad b caot be obtaied. However, the MLEs ca be obtaied by iterative solutio procedure. Settig the derivatives of the loglikelihood fuctio for (a, α, β, b) to zero, the MLEs (a, α, b ) are obtaied by iterative solutio procedure i.e., Newto Raphso method. We have used R-software for the iterative solutio procedure. Let a, α, b be the MLEs of a, α, b respectively. 3.2 Quadratic Fault Cotet Rate Fuctio Here we assume that fault itroductio rate is a quadratic fuctio ad depedet o time, a(t) = a 2 (t) = δ + αt βt 2, δ > 0, α > 0, β > 0 t 0 where δ is the umber of iitial fault cotet i the system prior to the testig, ad α ad β are icreasig ad decreasig rate of the umber of itroduced faults to the iitial fault cotet fuctio. Let m 2 (t) be the MVF whe fault cotet fuctio a 2 (t) is substituted for a(t) i (2.6) thus have m 2 (t) = b(t)[δ + αt βt 2 m t 2 (t)]. δ > 0, α > 0, β > 0, b(t) > 0, t 0 (3.12) b2 t with b(t) =. Solvig the differetial equatio with respect to m (1+bt) 2(t) uder the iitial coditio m 2 (0) = 0 we derived the mea value fuctio give by : m 2 (t) = α b (1 + bt) + (δ 2α b + 3α b(1 + bt) + 4β b(1 + bt) ) ( 2α b + 4β b + δ) e bt (1 + bt) (3.13)

9 David D. Haagal ad Nileema N. Bhalerao 865 We shall call the preset NHPP model with MVF give i (3.13) as delayed S-shaped quadratic model. The itesity fuctio of this model is give by: 3α + 4β + 4β λ 2 (t) = α + (2α + δ) e (1 + bt) bt b 2 t (3.14) 2 b Based o delayed S-shaped quadratic model, we ca derive the followig quatitative measures useful for software reliability assessmet. (a) Software Reliability R(x t) = P{X i > x S i 1 = t} = exp[ {m(x + t) m(t)}], t 0, x 0. (3.15) Substitutig (3.13) i (3.15) we have delayed S-shaped quadratic software reliability R 2 (x t) = exp [ { α (3α + 4β)x (1 + bx) b (1 + b(t + x))(1 + bt) 2α + 4β ( + δ) e bt [e bx (1 + b(t + x)) b (1 + bt)]}], x, t 0. (3.16) (b) Expected Residual Fault Cotet Cosider the Expectatio of {a(t) N(t)} for the NHPP model i (3.13). Sice N(t) is a radom variable, the umber of residual fault cotet at testig time t is: (t) = a(t) E[N(t)] = a(t) m(t) (3.17) Let 2 (t) deote the expected residual fault cotet for delayed S-shaped quadratic SRM, that is: 2 (t) = a 2 (t) m 2 (t) (3.18) 2(t) = δ + αt βt 2 α b (1 + bt) + (δ 2α b + ( 2α b + 4β b + δ) e bt (1 + bt) 3α b(1 + bt) + 4β b(1 + bt) ) (3.19) 3.2.1Parameter Estimatio I this sectio we discuss the method of estimatig the parameters of the method metioed above Estimatio usig iterval Domai Data The likelihood for the iterval domai data is give by L = (m(t i) m(t i 1 )) y i y i 1 exp[ m(t (y i y i 1 )! i ) m(t i 1 )] (3.20) i=1

10 866 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION where t 0 = 0 ad y 0 = 0. Takig the atural logarithm of equatio (3.20) yields log-likelihood fuctio, that is ll = (y i y i 1 ) l[m(t i ) m(t i 1 )] m(t ) i=1 l [(y i y i 1 )!] i=1 (3.21) substitutig mea value fuctio m 1 (t) ito the equatio (3.21), we get logarithmic of likelihood fuctio excludig the costat terms. ll = (y i y i 1 ) l [ α b (1 + bt i) i=1 + (δ 2α b + 3α b(1 + bt i ) + 4β b(1 + bt i ) ) ( 2α b + 4β b + δ) e bt i (1 + bti ) α b (1 + bt i 1) + (δ 2α b + 3α b(1 + bt i 1 ) + 4β b(1 + bt i 1 ) ) ( 2α b + 4β b + δ) e bt i 1 (1 + bti 1 )] α b (1 + bt ) + (δ 2α b + 3α b(1 + bt ) + 4β b(1 + bt ) ) ( 2α b + 4β b + δ) e bt (1 + bt ) (3.22) Closed form expressios for MLEs of a, α ad b caot be obtaied. However, the MLEs ca be obtaied by iterate solutio procedure. Settig the derivatives of the loglikelihood fuctio for (a, α, β, b) to zero, the MLEs (a, α, b ) are obtaied by iterative solutio procedure i.e., Newto Raphso method. We have used R-software for the iterative solutio procedure. Let a, α ad b be the MLEs of a, α, b respectively. 3.3 Expoetial Fault Cotet Rate Fuctio Here we assume that the fault itroductio rate is expoetial fuctio depedet o time. We chose the followig fault cotet fuctio: a(t) = a 3 (t) = αe βt, α > 0, β > 0, t 0 where α is a umber of iitial fault cotet i the system prior to the testig ad b is a icreasig rate of the umber of itroduced faults to the iitial fault cotet fuctio. Let m 3 (t) be the MVF whe fault cotet fuctio a 3 (t) is substituted for a(t) i (2.6) thus we have m 3 (t) = b(t)[αe βt m t 3 (t)], a > 0, b(t) > 0, t 0 (3.23)

11 David D. Haagal ad Nileema N. Bhalerao 867 with b(t) = b2 t. Solvig the differetial equatio give i (3.23) with respect to m 1+bt 3(t) uder the iitial coditio m 3 (0) = 0 we derived the mea value fuctio give by: αβb 2 e βt αβb(1 + bt) m 3 (t) = β(β b) 2 (1 βt + bt) (1 + bt) (β b) 2 e bt (3.24) We shall call the preset model with MVF give i (3.24) as delayed S-shaped expoetial model. The itesity fuctio of this model is give by: λ 3 (t) = αbβ e βt (β b) [e βt β (β b)(1 + bt) b e βt (β b)(1 + bt) 2 b2 te bt (β b) ] (3.25) Based o delayed S-shaped expoetial model, we ca derive the followig quatitative measures useful for software reliability assessmet. (a) Software Reliability R(x t) = P{X i > x S i 1 = t} (3.26) = exp[ {m(x + t) m(t)}], t 0, x 0. substitutig (3.24) i (3.26) we have delayed S-shaped expoetial software reliability αb 2 β(t + x) + b(t + x))e β(t+x) R 3 (x t) = exp [ { [(1 (β b) 2 (1 + b(t + x)) (1 βt + bt)e βt ] (1 + bt) αβb (β b) 2 [(1 + b(t + x))e b(t+x) (1 + bt)e bt ]}], x, t 0 (3.27) (b) Expected Residual Fault Cotet Cosider the Expectatio of {a(t) N(t)} for the NHPP model i (3.24). Sice N(t) is a radom variable, the umber of residual fault cotet at testig time t is: (t) = a(t) E[N(t)] = a(t) m(t) (3.28) Let 3 (t) deote the expected residual fault cotet for delayed S-shaped expoetial SRM, that is: 3 (t) = a 3 (t) m 3 (t) (3.29) 3 (t) = αe bt αbβ e βt [ β b β + e βt (β b)(1 + bt) e bt (1 + bt) ] (3.30) β b 3.3.1Estimatio of Parameters Usig Iterval Domai Data The likelihood fuctio metioed above for the iterval domai data is give by L = (m(t i) m(t i 1 )) y i y i 1 exp[ m(t (y i y i 1 )! i ) m(t i 1 )] (3.31) i=1 where t 0 = 0 ad y 0 = 0. Takig the atural logarithm of equatio (3.31) yields loglikelihood fuctio, that is

12 868 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION ll = (y i y i 1 ) l[m(t i ) m(t i 1 )] m(t ) l [(y i y i 1 )!] i=1 i=1 (3.32) substitutig mea value fuctio m 1 (t) ito the equatio (3.32), we get logarithmic of likelihood fuctio excludig the costat terms. ll = (y i y i 1 ) l [ αbβ β b i=1 [ e βti β + e βti αbβ e βti 1 e βt i 1 [ + (β b) β (β b)(1 + bt i 1 ) ebt i 1(1 + bt i 1 ) ]] (β b) (β b)(1 + bt i ) e bti (1 + bti ) ] (β b) αbβ e βt [ (β b) β + e βt (β b)(1 + bt ) e bt(1 + bt ) ] (β b) (3.33) Closed form expressios for MLEs of a, α ad b caot be obtaied. However, the MLEs ca be obtaied by iterate solutio procedure. Settig the derivatives of the loglikelihood fuctio for (a, α, β, b) to zero, the MLEs (a, α, b ) are obtaied by iterative solutio procedure i.e., Newto Raphso method. We have used R-software for the iterative solutio procedure. Let a, α, b be the MLEs of a, α, b respectively. 4. Aalysis of Three Data Sets The Data Set I is about US Tactical Data Systems (NTDS) give by [Goel ad Okumoto (1979)a]: the software data set was extracted from iformatio about failures i the developmet of software for the real time multi-computer complex of the US Naval Fleet Computer Programmig Ceter of the US Naval Tactical Data System (NTDS). The software cosists of 38 differet project modules. The time horizo is divided ito four phases: Productio phase, test phase, user phase ad subsequet test phase. The 26 software failures were foud durig the productio phase, five durig the test phase ad the last failure was foud o 1th Ja Oe failure was observed durig the user phase, i September 1971, ad two failures durig the test phase i The Data Set II is about O Lie Data Etry IBM Software Package: The data reported by [Ohba (1984)] are recorded from testig a olie data etry software package developed by IBM. The Data Set III is about Real-Time Commad ad Cotrol System: The data set was reported by [Musa et el. (1987)] based o failure data from a real-time commad ad cotrol system, which represets the failure observed durig system testig for 25 hours of CPU time. Applyig data sets DS I, DS II ad DS III o the models discussed above, we obtai the maximum likelihood (ML) estimates ad goodess of fit measures, such as sum of squares due to error (SSE), mea squares error (MSE), predictive ratio risk (PRR) [Pham ad Deg (2003)] ad Akaike s iformatio criterio (AIC) values for existig ad for the proposed fiite failure models are summarized i Table 1, 2 ad 3 respectively. Further we have

13 David D. Haagal ad Nileema N. Bhalerao 869 plotted the actual data ad estimated values of cumulative faults (or Mea Value Fuctio (MVF)) of fiite failure models agaist time for data sets DS I, DS II ad DS II as show i Figures 1, 2 ad 3 respectively. The S-shaped model, Yamada et al (1984), with fault cotet fuctio as costat is compared with the proposed delayed S-shaped liear, quadratic ad expoetial model. From the Table 1, we observe that the SSE, MSE, PRR, AIC ad BIC values of our proposed delayed S-shaped liear, quadratic ad expoetial models are smaller tha the existig delayed S-shaped ad Pham model. Hece we may coclude that our proposed models perform better. Further we observe that the SSE, MSE, PRR, AIC ad BIC values for delayed S-shaped expoetial model are smaller as compared to the other two proposed models ad existig delayed S-shaped ad Pham model. Hece we coclude that our proposed delayed S-shaped expoetial model performs better tha all other models for Data Set I. Table 1 : ML ad AIC estimates of Models for DS I Model Name Estimates SSE MSE PRR AIC BIC a = Delayed S-shaped b = α = Pham γ =1.84 a = Delayed S-shaped α = Liear b = Delayed S-shaped Quadratic Delayed S-shaped Expoetial δ =4 α =0.1 β =0.2 b =0.1 α =39.03 β =0.001 b =

14 870 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION Figure 1 : Plot of Data ad Estimated cumulative Faults agaist time of DS I From the Figure 1, we observe that for our proposed delayed S-shaped liear, quadratic ad expoetial models, the estimated MVF is closer to the actual data poits as compared to the MVF of existig delayed S-shaped ad Pham software reliability growth models. Hece we coclude that our proposed models explais the data better tha the give existig models. Figure 2 : Plot of Estimated Residual Fault Cotet Fuctio agaist time of DS I

15 David D. Haagal ad Nileema N. Bhalerao 871 The Figure 2 illustrates the behavior of residual fault cotet fuctios of software reliability growth models for delayed S-shaped model ad our proposed delayed S-shaped liear, quadratic ad expoetial models. We observe that i compariso with the existig closed S-shaped model our proposed models residual fault cotet fuctios are gettig closer to zero as time icreases. Hece our proposed models perform better for data set I From the Table 2, we observe that the SSE, MSE, PRR, AIC ad BIC values of our proposed delayed S-shaped liear, quadratic ad expoetial models are smaller tha the existig delayed S-shaped model ad Pham model. Hece we may coclude that our proposed models perform better. Further we observe that the SSE, MSE, PRR, AIC ad BIC values for delayed S-shaped expoetial model are smaller as compared to the other two proposed models ad existig delayed S-shaped model ad Pham model. Hece we coclude that our proposed delayed S-shaped expoetial perform better tha all other models for Data Set II. Table 2 : ML ad AIC estimates of Models for DS II Model Name Estimates SSE MSE PRR AIC BIC Delayed S-shaped a = b = Pham α = γ = a =8 Delayed S-shaped Liear α = b = Delayed S-shaped Quadratic Delayed S-shaped Expoetial δ =4.9 α =0.030 β =0.001 b =0.1 α =23 β = b = From the Figure 3, we observe that for our proposed delayed S-shaped liear, quadratic ad expoetial models, the estimated MVF is closer to the actual data poits as compared to the MVF of existig delayed S-shaped ad Pham software reliability growth models. Hece we coclude that our proposed models explai the data tha the give existig model The Figure 4 illustrates the behavior of residual fault cotet fuctios of software reliability growth models for delayed S-shaped model ad our proposed delayed S-shaped liear, quadratic ad expoetial models. We observe that i compariso with the existig closed S-shaped model our proposed models residual fault cotet fuctios are gettig closer to zero as time icreases. Hece our proposed models perform better for data set II. From the Table 3, we observe that the SSE, MSE, PRR, AIC ad BIC values of our

16 872 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION proposed delayed S-shaped liear, quadratic ad expoetial models are smaller tha the existig delayed S-shaped model ad Pham model. Hece we may coclude that our proposed models perform better. Further we observe that the SSE, MSE, PRR, AIC ad BIC values for delayed S-shaped expoetial model are smaller as compared to the other two proposed models ad existig delayed S-shaped model ad Pham model. Hece we coclude that our proposed delayed S-shaped expoetial perform better tha all other models for Data Set III. Figure 3 : Plot of Data ad Estimated Cumulative Faults agaist Time of DS II From the Figure 5, we observe that for our proposed delayed S-shaped liear, quadratic ad expoetial models, the estimated MVF is closer to the actual data poits as compared to the MVF of existig delayed S-shaped ad Pham software reliability growth models. Hece we coclude that our proposed models explai the data tha the give existig model The Figure 6 illustrates the behavior of residual fault cotet fuctios of software reliability growth models for delayed S-shaped model ad our proposed delayed S-shaped liear, quadratic ad expoetial models. We observe that i compariso with the existig closed S-shaped model our proposed models residual fault cotet fuctios are gettig closer to zero as time icreases. Hece our proposed models perform better for data set III.

17 David D. Haagal ad Nileema N. Bhalerao 873 Figure 4 : Plot Estimated Residual Fault Cotet Fuctio agaist time of DS II Table 3 : ML ad AIC estimates of Models for DS III Model Name Estimates SSE MSE PRR AIC BIC Delayed S-shaped a = b = Pham α = γ = Delayed S-shaped Liear a = α = b = Delayed S-shaped Quadratic Delayed S-shaped Expoetial δ =3 α =0.045 β =0.3 b =0.11 α = β = b =

18 874 ANALYSIS OF DELAYED S SHAPED SOFTWARE RELIABILITY GROWTH MODEL WITH TIME DEPENDENT FAULT CONTENT RATE FUNCTION Figure 5 : Plot of Data ad Estimated Cumulative Faults agaist of DS III. Figure 6 : Plot of Estimated Residual Fault Cotet Fuctio agaist time of DS III

19 David D. Haagal ad Nileema N. Bhalerao Coclusios ad Remarks I this paper we proposed three differet software reliability growth models based o three types of fault cotet rate fuctio a(t). The model parameters are estimated usig maximum likelihood method by usig the iterval domai data type. Three data sets are illustrated for the above estimate techique ad model comparisos. The proposed models are compared with the existig delayed S-shaped models, usig sum of squares due to error, mea squares sum of error, predictive ratio risk, Akaikes iformatio criterio ad Bayesia Iformatio criterio. It may be observed that the proposed models perform better tha the existig models for all the three data sets. Further we also plotted estimated MVF values versus time. Ackowledgemets We thak the referee for the valuable suggestios ad commets which resulted ito the improved versio of the earlier mauscript.

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