Distribution Based Change-Point Problem with Two Types of Imperfect Debugging in. Software Reliability

Transcription

1 BIJIT - BVICAM s Inernaional Journal of Informaion Technology Bharai Vidyapeeh s Insiue of Compuer Applicaions and Managemen (BVICAM, New Delhi Disribuion Based Change-Poin Problem wih Two Types of Imperfec Debugging in Absrac - Sofware esing is an imporan phase of sofware developmen life cycle. I conrols he qualiy of sofware produc. Due o he complexiy of sofware sysem and incomplee undersanding of sofware, he esing eam may no be able o remove/correc he faul perfecly on observaion/deecion of a failure and he original faul may remain resuling in a phenomenon known as imperfec debugging, or ge replaced by anoher faul causing faul generaion. In case of imperfec debugging, he faul conen of he sofware remains same while in case of faul generaion, he faul conen increases as he esing progresses and removal/correcion resuls in inroducion of new fauls while removing/correcing old ones. During sofware esing faul deecion /correcion rae may no be same hroughou he whole esing process, bu i may change a any ime momen. In he lieraure various sofware reliabiliy models have been proposed incorporaing changepoin concep. In his paper we propose a disribuion based change-poin problem wih wo ypes of imperfec debugging in sofware reliabiliy. The models developed have been validaed and verified using real daa ses. Esimaed Parameers and comparison crieria resuls have also been presened Index Terms - Non-homogenous Poisson process, sofware reliabiliy growh model, hazard rae, imperfec debugging. NOTATION m( : he mean value funcion or he expeced number of fauls deeced or removed by ime. a( : oal faul conen of sofware dependen on ime. p : he probabiliy of faul removal on a failure (i.e., he probabiliy of perfec debugging. α : he rae a which he fauls/errors may be inroduced during he debugging process. b : faul removal/correcion rae. λ( : inensiy funcion for NHPP models or faul deecion rae per uni ime. F( : disribuion funcions for faul removal/correcion imes. f( : densiy funcions for faul removal/correcion imes. z( : hazard rae funcion. β : learning parameer in logisic funcion. Deparmen of Operaional Research, Universiy of Delhi, India S. S. College of Business Sudies, Universiy of Delhi, India 3 Delhi College of Ars & Commerce, Universiy of Delhi, India pkkapur@gmail.com, sanand_or@yahoo.com and 3 singh_vb@rediffmail.com Sofware Reliabiliy P. K. Kapur, Sameer Anand and V. B. Singh 3. INTRODUCTION Compuer sofware is embedded in sysems of all kinds: ransporaion, medical, elecommunicaions, miliary, indusrial processes, enerainmen, office producs.he lis is almos endless. Sofware is virually inescapable in a modern world. And as we move ino he weny-firs cenury, i will become he driver for new advances in everyhing from elemenary educaion o geneic engineering. Sofware developmen consiss of differen phases: requiremen analysis, design, coding, esing, implemenaion and mainenance called SDLC. Research has been conduced in sofware reliabiliy engineering over he pas hree decades and many sofware reliabiliy growh models (SRGM have been proposed. The Sofware Reliabiliy Growh Model (SRGM is he ool, which can be used o evaluae he sofware quaniaively, develop es saus, schedule saus and monior he changes in reliabiliy performance. Research has been conduced in sofware reliabiliy engineering over he pas hree decades and many sofware reliabiliy growh models (SRGM have been proposed. The pioneering aemp in non-homogenous Poisson process based on SRGM was made by Goel and Okumoo (G-O []. The model describes he failure observaion phenomenon by an exponenial curve. There are also SRGM ha describe eiher S- shaped curves or a mixure of exponenial and S-shaped curves (flexible. Some of he imporan conribuions of hese ype of models are due o Yamada e al. [7], Ohba [8], Biani e al. [3], Kapur and Garg [6], Kapur e al. [7], Pham [4] ec. In mos of he models discussed above i is assumed ha whenever an aemp is made o remove a faul, i is removed wih cerainy i.e. a case of perfec debugging. Bu he debugging aciviy is no always perfec because of number of facors like eser s skill/experise ec. In pracical sofware developmen scenario, he number of failures observed/deeced may no be necessarily same as he number of errors removed/correced. Kapur and Garg [6] have discussed in heir error removal phenomenon model ha as esing grows and esing eam gains experience, addiional numbers of fauls are removed wihou hem causing any failure. The esing eam, however, may no be able o remove/correc faul perfecly on observaion/deecion of a failure and he original faul may remain leading o a phenomenon known as imperfec debugging, or replaced by anoher faul resuling in faul generaion. In case of imperfec debugging he faul conen of he sofware is no changed, bu because of incomplee undersanding of he sofware, he original deeced faul is no removed perfecly. Bu in case of faul generaion, he oal faul conen increases as he esing progresses Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN

2 Disribuion Based Change-Poin Problem wih Two Types of Imperfec Debugging in Sofware Reliabiliy because new fauls are inroduced in he sysem while removing he old original fauls. Model due o Obha and Chou [8] is an faul generaion model applied on G-O model and has been also named as Imperfec debugging model. Kapur and Garg [] inroduced he imperfec debugging in G-O model. They assumed ha he FDR per remaining fauls is reduced due o imperfec debugging. Thus he number of failures observed/deeced by ime infiniy is more han he iniial faul conen. Alhough hese wo models describe he imperfec debugging phenomenon ye he sofware reliabiliy growh curve of hese models is always exponenial. Moreover, hey assume ha he probabiliy of imperfec debugging is independen of he esing ime. Thus, hey ignore he role of he learning process during he esing phase by no accouning for he experience gained wih he progress of sofware esing. Pham [4] developed an SRGM for muliple failure ypes incorporaing faul generaion. Zhang e al. [6] proposed a esing efficiency model which includes boh imperfec debugging and faul generaion, modeling i on he number of failures experienced/observed/deeced, however boh imperfec debugging and faul generaion are acually seen during faul removal/correcion. Recenly, Kapur e al. [] proposed a flexible SRGM wih imperfec debugging and faul generaion using a logisic funcion for faul deecion rae which reflecs he efficiency of he esing/removal eam. We execue he program in specific environmen and improve is qualiy by deecing and correcing fauls. Many SRGM assume ha, during he faul deecion process, each failure caused by a faul occurs independenly and randomly in ime according o he same disribuion Musa e al. []. Bu he failure disribuion can be affeced by many facors such as running environmen, esing sraegy, defec densiy and resource allocaion. On he oher hand, in pracice, if we wan o deec more fauls for a shor period of ime, we may inroduce new echniques or ools ha are no ye used, or bring in consulans o make a radical sofware risk analysis. In addiion, here are newly proposed auomaed esing ools for increasing es coverage and can be used o replace radiional manual sofware esing regularly. The benefis o sofware developers/esers include increased sofware qualiy, reduced esing coss, improved release ime o marke, repeaable es seps, and improved esing produciviy. These echnologies can make sofware esing and correcion easier, deec more bugs, save more ime, and reduce much expense. Alogeher, we wish ha he consulans, new auomaed es ools or echniques could grealy help us in deecing addiional fauls ha are difficul o find during regular esing and usage, in idenifying and correcing fauls mos cos effecively and in assising cliens o improve heir sofware developmen process. Thus, he faul deecion rae may no be smooh and can be changed a some ime momen called change-poin. Many researchers have incorporaed change poin in sofware reliabiliy growh modeling. Firsly Zhao [8] incorporaed change-poin in sofware and hardware reliabiliy. Huang e al. [7] used change-poin in sofware reliabiliy growh modeling wih esing effor funcions. The imperfec debugging wih change-poin has been inroduced in sofware reliabiliy growh modeling by Shyur [5]. Kapur e al. [9,4] inroduced various esing effor funcions and esing effor conrol wih changepoin in sofware reliabiliy growh modeling. Goswami e al.[6] and Kapur e al.[5] proposed a sofware reliabiliy growh model for errors of differen severiy using changepoin. The muliple change-poins in sofware reliabiliy growh modeling for fielded sofware has been proposed by Kapur e al. []. Laer on SRGM based on sochasic differenial equaions incorporaing change-poin concep has been proposed by Kapur e al. [0].. BASIC ASSUMPTION The NHPP models are based on he assumpion ha he sofware sysem is subjec o failures a random imes caused by manifesaion of remaining fauls in he sysem. Hence NHPP are used o describe he failure phenomenon during he esing phase. The couning process { N(, 0} of an NHPP process is given as follows. Pr and { N ( = k} = ( m( k! k e m(, k = 0,,... m ( = λ ( x dx 0 ( The inensiy funcion λ(x (or he mean value funcion m( is he basic building block of all he NHPP models exising in he sofware reliabiliy engineering lieraure. The proposed models are based upon he following basic assumpions:. Failure faul removal phenomenon is modeled by NHPP.. Sofware is subjec o failures during execuion caused by fauls remaining in he sofware. 3. Failure rae is equally affeced by all he fauls remaining in he sofware. 4. When a sofware failure occurs, an insananeous repair effor sars and he following may occur: (a Faul conen is reduced by one wih probabiliy p (b Faul conen remains unchanged wih probabiliy -p. 5. During he faul removal process, wheher he faul is removed successfully or no, new fauls are generaed wih a consan probabiliyα. 6. Faul deecion / removal rae may change a any ime momen. Assumpion 4 and 5 capures he effec of imperfec debugging and faul generaion respecively. 3. MODEL DEVELOPMENT In his secion, we formulae disribuion based sofware reliabiliy growh models incorporaing change-poin and wo ypes of imperfec debugging. Since he fauls in he sofware ( Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN

3 BIJIT - BVICAM s Inernaional Journal of Informaion Technology sysems are deeced and eliminaed during he esing phase, he number of fauls remaining in he sofware sysem gradually decreases as he esing procedure goes on. Thus under he common assumpions for sofware reliabiliy growh modeling, we consider he following linear differenial equaion. dm( = b (( a m ( (3 d Where b( is a faul deecion rae per remaining fauls a esing ime. Here we consider he faul deecion rae as hazard rae z(, iniial faul is no he consan bu he funcion of ime and incorporaing he imperfec debugging. So he above equaion can be wrien as dm( = z ( p a m ( ( ( d We assume ha fauls can be inroduced during he debugging phase wih a consan faul inroducion raeα. Therefore, he faul conen rae funcion, a(, is a linear funcion of he expeced number of fauls deeced by ime. Tha is a ( = a+ αm, above equaion becomes ( ( dm = zpa ( ( + αm ( m ( (4 d In he proposed model we assume ha he hazard f ( rae z ( =, he faul inroducion rae α and F( probabiliy of perfec debugging p, may be changed a some ime momen called change poin. Afer incorporaing change-poin, we ge he following form of faul deecion /removal, probabiliy of perfec debugging and faul generaion rae. f( for F ( z ( = (5 f( for > F ( Where F, f and F, fare he disribuions, densiy funcions before and afer change poin respecively. The equaion (5 can be rewrien as Probabiliy of perfec debugging rae will be p for p = (6 p for > and faul conen rae a+ αm( for a ( = a + αm( + α( m( m( for > (7 The Equaion (5 can be rewrien as f ( f ( z U U ( = ( + ( F( F( Using uni sep funcion give by 0 if x < 0 U( x = if x > Similarly equaion (6 & (7 can also be rewrien as p= pu + pu ( ( ( ( ( ( ( ( ( & a ( = a+ αmu ( + a+ αm + α m m U Now using equaion (5, (6 and (7, he equaion (4 can be rewrien as, f ( p( a+ αm ( m ( for dm( F ( = (8 d f( p( a+ αm( + α( m( m( m ( for> F ( Afer solving he above equaions, we ge he following soluions a p ( ( ( α F for ( α m ( = p( α (9 a p ( ( ( ( ( F α ( α α F + m( for> ( α F ( ( α SRGM- The following exponenial disribuion funcion is used o model SRGM-: Le T~ exp(b for ie. F( = exp( b for (0 and Le T~ exp(b for > F ( = exp ( b for > ( Subsiuing he value of F( and F ( from Equaion (0 and ( ino Equaion (9, we ge: a exp( bp ( α for ( α a m ( = exp( bp ( α bp ( α( ( ( α ( α α + m( for > ( α The above model can be reduced o he model given by Shyur [5] if we consider he perfec debugging and no faul generaion. Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN

4 Disribuion Based Change-Poin Problem wih Two Types of Imperfec Debugging in Sofware Reliabiliy SRGM- For furher simplifying he esimaion procedure we may Le F( be a wo-sage Erlangian disribuion funcion i.e., assumeα = α = α and p = p = p. T~ Erlang-(b for ie.. F( = ( + b exp( b for (3 4. MODEL VALIDATION, (3 COMPARISON CRITERIA AND DATA ANALYSES And Model Validaion T~ Erlang-(b for > To illusrae he esimaion procedure and applicaion of he ie.. F( = ( + b exp ( b for > (4 SRGM (exising as well as proposed we have carried ou he Subsiuing he value of F( and F ( daa analysis of real sofware daa se. The parameers of he from Equaion (3 models have been esimaed using saisical package SPSS and and (4 ino Equaion (9, we ge: he change-poin of he daa ses have been judged by using change-poin analyzer. a p (( ( ( α b Daa se (DS- + exp b for ( α The firs daa se (DS- had been colleced during 35 monhs of esing a radar sysem of size 4 KLOC and 30 fauls p ( ( ( α a p α + b were deeced during esing. This daa is cied from Brooks m ( = ( + b exp( bp ( α bp ( α( ( α ( + b and Moley [4]. The change-poin for his daa se is 7 h monh. Daa se (DS- The second daa se (DS- had been colleced during 9 weeks ( α α + m( for > of esing a real ime command and conrol sysem and 38 ( α fauls were deeced during esing. This daa is cied from Ohba [9]. The change-poin for his daa se is 6 h week. (5 The above model can be reduced o he model given by Archana [] if we consider he perfec debugging and no faul generaion. SRGM-3 Le F( be a logisic disribuion funcion i.e., T~ logisic disribuion (b, β for exp b ie.. F( = ( ( + β ( b ( exp for And T~ logisic disribuion (b, β for ( exp( b ie.. F ( = for ( + βexp( b Subsiuing he value of F( and F ( from Equaion (6 and (7 ino Equaion (9, we ge: p ( α ( a + β exp( bp ( α for 0 ( α ( + βexp( b p ( α p ( ( ( ( α a + β + βexp b m ( = ( α ( + βexp( b ( + βexp( b *xp e ( b p ( α bp( α( ( α α + m( for > ( α 5. COMPARISON CRITERIA FOR SRGM The performance of SRGM are judged by heir abiliy o fi he pas sofware faul daa (goodness of fi and predicing he fuure behavior of he faul. Goodness of Fi crieria The erm goodness of fi is used in wo differen conexs. In one conex, i denoes he quesion if a sample of daa came from a populaion wih a specific disribuion. In anoher conex, i denoes he quesion of How good does a (6 mahemaical model (for example a linear regression model fi o he daa? The Mean Square -Error (MSE: The model under comparison is used o simulae he faul daa, he difference beween he expeced values, mˆ ( i and he > (7 observed daa y i is measured (7 by MSE as follows. MSE = k = i ( mˆ ( y i i k where k is he number of observaions. The lower MSE indicaes less fiing error, hus beer goodness of fi [7]. Coefficien of Muliple Deerminaion (R: We define his coefficien as he raio of he sum of squares resuling from he rend (8 model o ha from consan model subraced from. residual SS i.e. R = -. correced SS R measures he percenage of he oal variaion abou he mean accouned for he fied curve. I ranges in value from 0 o. Small values indicae ha he model does no fi he daa well. The larger R, he beer he model explains he variaion in he daa [7]. Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN

5 BIJIT - BVICAM s Inernaional Journal of Informaion Technology Bias The difference beween he observaion and predicion of number of failures a any insan of ime i is known as PE i. (predicion error. The average of PEs is known as bias. Lower he value of Bias beer is he goodness of fi [8]. Variaion The sandard deviaion of predicion error is known as variaion. Variaion = ( ( PE N Bias i Lower he value of Variaion beer is he goodness of fi [8]. Roo Mean Square Predicion Error I is a measure of closeness wih which a model predics he observaion. ( RMSPE = Bias + Variaion Lower he value of Roo Mean Square Predicion Error beer is he goodness of fi [8]. Daa Analyses For DS- The parameer esimaion and comparison crieria resuls for DS- of all he models under consideraion can be viewed hrough Table I(a and I(b. I is clear from he able ha he value of R for SRGM-3 is higher and value of MSE is lower in comparison wih oher models and provides beer goodness of fi for DS-. For DS- The parameer esimaion and comparison crieria resuls for DS- of all he models under consideraion can be viewed hrough Table II(a and II(b. I is clear from he able ha he value of R for SRGM-3 is higher and value of MSE is lower in comparison wih oher models and provides beer goodness of fi for DS-. Models a b b β β p α SRGM SRGM SRGM Table I(a: Model Parameer Esimaion Resuls (DS- Models R MSE SRGM SRGM SRGM Table I(b: Model Comparison Resuls (DS- Models a b b β β p α SRGM SRGM SRGM Table II(a: Model Parameer Esimaion Resuls (DS- Models R MSE SRGM SRGM SRGM Table II(b: Model Comparison Resuls (DS- 6. GOODNESS OF FIT CURVES For DS- Cumulaive fauls For DS- Cumulaive fauls Goodness of fi curves Time(monhs Acual SRGM- SRGM- SRGM-3 0 Goodness of fi curves Time(weeks Acual SRGM- SRGM- SRGM-3 7. CONCLUSION In his paper we have developed a disribuion based changepoin problem wih wo ypes of imperfec debugging in sofware reliabiliy. Wih his approach, we can derive exising models and propose new model. All hese models have been validaed and verified using real daa ses. Parameer esimaes, comparison resuls and goodness of fi curves have also been presened. 8. FUTURE SCOPE In fuure, we will ry o develop more models in he same line by using Erlang normal, weibull and gamma disribuion funcions. Models can be exended for muliple change-poins problem.. Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN

6 Disribuion Based Change-Poin Problem wih Two Types of Imperfec Debugging in Sofware Reliabiliy REFERENCES [] A.L. Goel, Sofware Reliabiliy Models: Assumpions, Limiaions and Applicabiliy, IEEE Transacions on Sofware Engineering; SE-, pp. 4-43, 985. [] Archana Kumar, Ph.D. Thesis A Sudy in Sofware Reliabiliy Growh Modelling Under Disribued Developmen Environmen, Deparmen of Operaional Research, Universiy of Delhi, Delhi, 007. [3] Biani S., Bolzern, P., Pedroi, E., Pozzi, N. and Scaolini R., A flexible modeling approach for sofware reliabiliy growh, Goos, G., and Harmanis, J., (Ed., Sofware Reliabiliy Modelling and Idenificaion, Springer Verlag, Berlin, pp. 0-40, 988. [4] Brooks W. D. and Moley R.W., Analysis of Discree Sofware Reliabiliy Models-Technical Repor (RADC- TR-80-84, Rome Air Developmen Cener, New York, 980. [5] Goel, A.L. and Okumoo, K, Time dependen error deecion rae model for sofware reliabiliy and oher performance measure, IEEE Transacions on Reliabiliy, 979. [6] Goswami D.N., Khari Sunil K, and Kapur Reecha Discree Sofware Reliabiliy Growh Modeling for Errors of Differen Severiy incorporaing Change-Poin Concep Inernaional Journal of Auomaion and Compuing Vol. 4(4 pp ,007. [7] Huang Yu Chin Performance Analysis of Sofware Reliabiliy Growh Models wih Tesing Effor and Change-Poin, Journal of Sysems and Sofware Vol. 76, Issue, pp 8-94, 005. [8] K. Pillai and V.S.S. Nair, A Model for Sofware Developmen effor and Cos Esimaion, IEEE Transacions on Sofware Engineering; vol. 3(8, pp , 997. [9] Kapur P. K, Singh V. B, Anand Sameer and Yadavalli V.S.S., Sofware Reliabiliy Growh Model wih Change- Poin and Effor Conrol Using a Power Funcion of Tesing Time Inernaional Journal of Producion Research, Vol. 46 Issue no. 3 pp ,008. [0] Kapur P. K, Singh V. B, Anand Sameer Effec of Change-Poin on Sofware Reliabiliy Growh Models Using Sochasic Differenial Equaions published in he proceedings of 3rd Inernaional Conference on Reliabiliy and Safey Engineering, (Eds. R.B.Misra, V.N.A. Naikan, S.K.Chaurvedi, and N.K.Goyal, (INCRESE-007, Udaipur, pp , 007. [] Kapur P. K, Singh V. B, Anand Sameer, Sofware Reliabiliy Growh Model of Fielded Sofware Based on Muliple Change-Poin Concep Using a Power Funcion of Tesing Time, Qualiy Reliabiliy and Infocom Technology, (Eds. P.K.Kapur and A.K.Verma, MacMillan India Ld., pp.7-78, 007. [] Kapur P. K., Kumar D., Gupa A. and Jha P. C., On How To Model sofware Reliabiliy Growh in he Presence Of Imperfec Debugging and error Generaion, Proceedings of nd Inernaional Conference on Reliabiliy and safey Engineering, pp , 006. [3] Kapur P. K., Pham H., Anand S. and Yadav K. A Unified Approach for Developing Sofware Reliabiliy Growh Models in he Presence of Imperfec Debugging and Error Generaion, communicaed in IEEE Transacions on Reliabiliy, 008. [4] Kapur P.K, Gupa A., Shanawi O, and Yadavalli V.S.S Tesing Effor Conrol Using Flexible Sofware Reliabiliy Growh Model wih Change Poin Inernaional Journal of Performabiliy Engineering- Special issue on Dependabiliy of Sofware/ Compuing Sysems, Vol., No. 3, pp 45-63, 006 [5] Kapur P.K. Kumar Archana,Yadav Kalpana and Khari Sunil Sofware Reliabiliy Growh Modelling for Errors of Differen Severiy using Change Poin Inernaional Journal of Qualiy,Reliabiliy and Safey Engineering,Vol.4,No.4, pp 3-36,007 Coninued on page no. 4 Copy Righ BIJIT 009; July December, 009; Vol. No. ; ISSN