A New Insight into Software Reliability Growth Modeling

Transcription

1 Inernaional Journal of Performabiliy Engineering, Vol. 5, No. 3, April, 29, pp RAMS Consulans Prined in India A New Insigh ino Sofware Reliabiliy Growh Modeling P.K. KAPUR 1, ANU G. AGGARWAL 1, and SAMEER ANAND 2 1 Deparmen of Operaional Research, Universiy of Delhi, Delhi S.S.College of Business Sudies, Universiy of Delhi, Delhi (Received on Ocober 24, 28; Final Revision on December 9, 28) Absrac: Several sofware reliabiliy growh models have been presened in he lieraure in he las hree decades. They have been developed for uniform and non-uniform operaional profile. Some of hem are flexible whereas ohers are no. Model selecion becomes an uphill ask. Of lae, some auhors have ried o develop a unifying approach so as o capure differen growh curves, hus easing he model selecion process. Some of hese approaches use (a) Random lag funcion (b) Infinie server queuing heory (c) Hazard rae funcion. The purpose of his paper is o show ha all hese approaches are equivalen and furher show ha hazard rae approach is more general and can handle boh Imperfec Debugging and Faul generaion. This paper hus provides a new insigh ino he model developmen and i is shown ha how a wide variey of exising sofware reliabiliy can be unified. Keywords: Faul deecion, faul correcion, infinie server queue, hazard rae, imperfec debugging, faul generaion 1. Inroducion Wih increased complexiy of producs design, shorened developmen cycles and highly desrucive consequences of sofware failures, a major responsibiliy lies in he areas of sofware debugging, esing and verificaion. Tesing is defined as he execuion of a program o find he fauls, which migh have been inroduced in i during various sages of he developmen cycle. I is also performed o judge he performance, safey, faulolerance or securiy of he sofware. More imporanly, esing provides a mahemaical measure of sofware reliabiliy (i.e., failure/execuion ime) which forms a vial inpu o he release decision. A large number of Sofware Reliabiliy Growh Models (SRGM), which relae he number of failures (fauls idenified/correced) and execuion ime, have been discussed in he lieraure [6,8]. These SRGM assume diverse esing environmen like disincion beween failure and correcion processes, learning of he esing personnel, possibiliy of imperfec debugging and faul generaion, consan or monoonically increasing / decreasing Faul Deecion Rae (FDR) or randomness in he growh curve. Bu no SRGM can be claimed o be he bes as he physical inerpreaion of he esing and debugging changes due o numerous facors e.g., design of es cases, defec densiy, skills and efficiency of esing eam, availabiliy of esing resources ec. The plehora of SRGM makes he model selecion a edious ask. To reduce his difficuly, unified modeling approaches have been proposed by many researchers. These schemes have proved o be successful in obaining several exising SRGM by following single mehodology and *Corresponding auhor s pkkapur1@gmail.com 267

2 268 P.K.Kapur, Anu G. Aggarwal and Sameer Anand hus provide an insighful invesigaion for he sudy of general models wihou making many assumpions. Some of hese approaches are based on (a) Infinie server queuing heory (b) Random lag funcion (c) Hazard rae funcion. The work in his area sared as early as in 198s wih Shanikumar [1] proposing a Generalized birh process model. Gokhale and Trivedi [2] used Tesing coverage funcion o presen a unified framework and showed how NHPP based models can be represened by probabiliy disribuion funcions of faul deecion imes. Dohi e al [1] proposed a unificaion mehod for NHPP models describing es inpu and program pah searching imes sochasically by an infinie server queuing heory. Inoue [3] applied infinie server queuing heory o he basic assumpions of delayed S-shaped SRGM [13] i.e. faul correcion phenomenon consiss of successive failure observaion and deecion/correcion processes and obained several NHPP models describing faul correcion as a wo sage process. Anoher unificaion mehodology is based on a sysemaic sudy of Faul deecion process (FDP) and Faul correcion process (FCP) where FCPs are described by deecion process wih ime delay. The idea of modeling FCP as a separae process following he FDP was firs used by Schneidewind [9]. More general reamen of his concep is due o Xie e al [11] who suggesed modeling of Faul deecion process as a NHPP based SRGM followed by Faul correcion process as a delayed deecion process wih random ime lag. The recen unificaion scheme (due o Kapur e al [4]) is based on Cumulaive Disribuion Funcion for he deecion/correcion imes. They have exended he concep of unified modeling by incorporaing he concep of change poin in Faul deecion rae [5]. In his paper, we discuss above-menioned hree approaches in deail and show ha how hese unifying ools, hough derived under differen ses of assumpions are mahemaically equivalen. The paper also highlighs he imporance of hazard rae funcion based unifying echnique. This paper has been organized as follows: Secion 2 discusses hree ypes of unificaion schemes repored in he lieraure. This secion has been divided ino hree subsecions. Subsecion 2.1 describes unificaion approach for modeling sofware reliabiliy Growh using infinie server queuing heory (Inoue [3]) In subsecion 2.2 we describe he unificaion approach based on random ime lag funcions (Xie e al [11]). We include Hazard rae funcion based unifying echnique in subsecion 2.3 (Kapur e al [4]). Secion 3 proves he equivalence of hese unifying models. Secion 4 highlighs he imporance of Hazard rae funcion scheme wih respec o is capabiliy o incorporae Imperfec Debugging and Faul generaion. Finally, he paper concludes wih a brief summary and direcions for fuure research in Secion 5. Noaion m d (), m c () Mean value funcion (MVF) or he expeced number of fauls deeced and correced by ime. a Consan, represening he iniial number of fauls lying dorman in he sofware when he esing sars. λ d (), λ c ( ) Inensiy funcion for FDP and FCP or Faul Deecion and Correcion rae per uni ime. F d (), F c () Disribuion Funcion for Faul Deecion and Correcion Times f c () Probabiliy Densiy Funcion for Faul Correcion Time * Convoluion. Seiljes convoluion.

3 A New Insigh ino Sofware Reliabiliy Growh Modeling Unificaion Approaches for Modeling Sofware Reliabiliy Growh 2.1 Unificaion Approach for Modeling Sofware Reliabiliy Growh Using Infinie Server Queuing Theory (Inoue [3]) The model is based on he following assumpions 1. Sofware sysem is subjec o failure during execuion caused by fauls remaining in he sysem. 2. The number of fauls deeced a any ime insan is proporional o he remaining number of fauls in he sofware. Furher, All fauls are muually independen from failure deecion poin of view. 3. On a failure, correcion effor sars and faul causing he failure is correced wih cerainy. Number of correced fauls lags behind he oal number of deeced fauls, hence here is a ime lag beween he Faul correcion and deecion processes. 4. The faul deecion process is modeled by NHPP. The faul correcion imes are assumed o be independen wih probabiliy disribuion F c (). The represenaion of Sofware Faul correcion process as a infinie server queuing model is depiced in Fig. 1. Fig 1: Sofware Faul Correcion Process Le he couning processes {, },{ N( ), } X represen he cumulaive number of sofware faul deeced, fauls correced respecively up o ime and he es begun a ime =. Then he disribuion of N() is given by j m ( md e d ) Pr{ N( ) = n} = Pr{ N( ) = n X = j} (1) j= j! If failure observaions coun is j hen probabiliy ha n fauls are correced via he faul correcion process is given as j n j n Pr{ N( ) = n X = j} = ( n )( p( )) (1 p( )) (2) where p() is he probabiliy ha an arbirary faul is correced by ime, which can be defined using he Sieljes convoluion and he concep of he condiional disribuion of arrival imes, given as p = Fc ( u) (3) md

4 27 P.K.Kapur, Anu G. Aggarwal and Sameer Anand The disribuion funcion of cumulaive number of fauls correced up o ime using equaions (2) and (3) is given as n Fc ( u) e Pr{ N( ) = n} = Fc ( u) (4) n! Equaion (4) describes ha N() follows an NHPP wih MVF Fc ( u) i.e., mc = Fc ( u) (5) Hence knowing he MVF for deecion imes m d () and disribuion of correcion imes F c (.) we can compue he MVF m c () of a wo sage faul isolaion/deecion and correcion process for he various exising SRGM. 2.2 Unificaion Approach For Modeling of Sofware Faul Deecion and Faul Correcion Process (Xie e al [11]) This approach is based upon separae analysis of faul deecion process (FDP) and faul correcion process (FCP). Given λ d (), he mean value funcion (MVF) m d () saisfies d = d m λ (6) Specifically, a faul can be correced only afer is deecion, and a FCP can be modeled as a delayed FDP. Such delay could be modeled as deerminisic or random, bu he deerminisic assumpions on correcion ime are quie simplisic. In fac, i is more pracical o model he correcion ime wih random variables. Given he faul deecion inensiy funcion λ d (), he faul correcion inensiy funcion is he expecaion of λ d ( x) i.e., λ c = E [ λd ( x) ] or λ c = λ d ( x) fc (7) Then FCP can be described by he following MVF c = m λ du (8) c 2.3 A Unified Approach For Developing Sofware Reliabiliy Growh Models Using Hazard Rae (Kapur e al [4]) Mos of he SRGM repored in he las few decades assume deecion process is followed by immediae faul correcion. While in realiy, each deeced fauls is repored, diagnosed, correced and hen verified. Therefore, he ime from deecion o correcion should no be negleced in sofware esing process. When he correcion in done in wo sages, hen he faul correcion process is given by he following differenial equaion: dmc ( f * f ) [ a m ] d = c d c [1 ( Fc )] (9)

5 A New Insigh ino Sofware Reliabiliy Growh Modeling 271 ( fc * fd ) where is faul deecion - correcion rae per faul or he hazard rae [1 ( Fc )] funcion. Solving he above, we have mc = a[( Fc )] (1) I is an exension of one sage failure-deecion/correcion process given in Musa [7]. 3. Equivalence of Three Unificaion Approaches In firs sep we show he equivalence of scheme due o Xie e al [11] wih Infinie server approach due o Inoue [3]. Considering mc = λ c ( y) dy (equaion 8) From (7) and (8) we have y λ c ( y) = λd ( y x) f c So, y mc = λ d ( y x) fc dy = λ d ( y x) dy fc x or mc = md ( x) fc = Fc ( x) which is he mean value funcion of infinie server model as given in equaion (5) The nex sep esablishes he equivalence of infinie server queuing model o unificaion scheme based on hazard rae (Kapur e al [4]) Considering mc = md ( x) dfc (equaion 5) = F c = Fc md mc = a( Fc ) Using md = afd [Musa, 7], we ge which is he mean value funcion of hazard rae approach as given in equaion (1) Hence we show ha he above-described hree approaches are equivalen 4. Advanages of Hazard Rae Funcion Based Unificaion Approach This secion highlighs he advanages of he unifying echnique wih hazard rae funcion. Firs, le us observe he unifying models as given by infinie server queuing heory and

6 272 P.K.Kapur, Anu G. Aggarwal and Sameer Anand Xie e al [11] These wo schemes express MVF for Faul correcion process in erms of MVF for Faul deecion process and probabiliy disribuion funcion for faul correcion imes. These wo schemes fall shor of generalizaion when we wish o incorporae he possibiliy of imperfec debugging and/or faul generaion. I should be noed ha imperfec debugging is possible when aemps are made o correc he cause of he failure. During he correcion process we can have imperfec debugging under following hree cases: (i) (ii) (iii) The faul is wrongly isolaed followed by inaccurae correcion, The underlying faul is parially removed, Few addiional fauls are inroduced while correcing he underlying faul. All hese cases are possible only when effors are made o remove he faul. This concep canno be incorporaed in Faul deecion process. So m d () canno be aken as imperfec debugging NHPP model. We consider a unified model based on hazard rae wih Imperfec Debugging and Faul Generaion, which is given by he following differenial equaion: dm( ) ( f * f ) p[ a m( ) m( )] d = c d + [1 ( Fc )] α (11) where p is he probabiliy of perfec debugging and α is he rae a which fauls are inroduced while removing/correcing a faul from sofware. Solving he above differenial equaion, we ge he soluion as: a p( 1 α ) m = [1 (1 ( Fc )) ] (12) (1 α ) Here, regardless of he probabiliy disribuion followed by he deecion and correcion imes, we have MVF for faul correcion process wih boh imperfec debugging and faul generaion. Above has been ermed as Generalized Non-homogeneous Poisson Process (GINHPP) SRGM in he presence of Imperfec Debugging and Faul Generaion. (Kapur e al [4]). For p=1 and α=, we obain model for perfec debugging and no faul Generaion as given in equaion (1). 5. Conclusions In his paper, we have discussed hree differen approaches o unify a wide range of sofware reliabiliy growh models under a common modeling framework. Though hese hree schemes have been derived under differen ses of assumpions bu hey are proved o be mahemaically equivalen. Furher, we have menioned cerain advanages ha come wih hazard rae based unifying mehodology. This scheme provides an inegraed common plaform for no only growh models wih perfec debugging bu also for imperfec debugging and faul generaion. Here, we have considered he unificaion of coninuous ime NHPP based models bu research can be done o work ou he unificaion plaform for he discree ime models. The Tesing effors funcion based models can also be inegraed wih one of he approaches described in his paper. So far, we have resriced ourselves o wo-sage faul deecion followed by faul correcion process. The sudy can be exended o siuaions where faul correcion akes place in hree sagesfailure observaion, faul isolaion/deecion and finally faul correcion. We are also

7 A New Insigh ino Sofware Reliabiliy Growh Modeling 273 working on he unificaion of sochasic calculus based models. The unificaion framework of modeling seems o be quie ineresing and promising for easing he problem of model selecion. References [1] Dohi, T., S. Osaki, and K.S. Trivedi, An Infinie Server Queuing Approach For Describing Sofware Reliabiliy Growh ~ Unified Modeling And Esimaion Framework ~, Proceedings Of The 11h Asia-Pacific Sofware Engineering Conference (APSEC 4), pp , 24. [2] Gokhale S.S., T. Philip, P.N. Marinos.and K.S. Trivedi, Unificaion of Finie Failure Non- Homogeneous Poisson Process Models hrough Tes Coverage, In Proc. Inl. Symposium on Sofware Reliabiliy Engineering (ISSRE 96), Whie Plains, NY, pp , [3] Inoue, S., A sudy on Sochasic Modelling for Accurae Sofware Reliabiliy Assessmen, Ph.D. Thesis, Docoral Program of Graduae School of Engineering, Toori Universiy, Japan, 26. [4] Kapur P. K., H. Pham, S. Anand and K. Yadav, A Unified Approach for Developing Sofware Reliabiliy Growh Models in he Presence of Imperfec Debugging and Error Generaion, 28, submied o IEEE Transacions on Sofware Reliabiliy. [5] Kapur P.K., J. Kumar and R. Kumar, A Unified Modeling Framework Incorporaing Change Poin For Measuring Reliabiliy Growh During Sofware Tesing o appear in OPSEARCH. [6] Kapur P.K., R.B. Garg and S. Kumar, Conribuions o hardware and sofware reliabiliy, World Scienific, Singapore, [7] Musa, J.D., Sofware Reliabiliy: Measuremen, Predicion, Applicaions, New York: Mc Graw Hill, [8] Pham, H., Sysem Sofware Reliabiliy, Reliabiliy Engineering Series, Springer, 26. [9] Schneidewind, N.F., Analysis Of Error Processes In Compuer Sofware, Sigplan Noices, Vol. 1, pp , [1] Shanhikumar, J. G., A General Sofware Reliabiliy Model For Performance Predicion, Microelecronics Reliabiliy, Vol. 21, pp , [11] Xie, M., Q.P. Hu, Y.P. Wu and S.H. Ng., A Sudy Of The Modeling And Analysis Of Sofware Faul-Deecion And Faul-Correcion Processes, Qualiy And Reliabiliy Engineering Inernaional, Vol. 23, No. 4, pp , 27. [12] Xie, M. and M. Zhao, The Schneidewind Sofware Reliabiliy Model Revisied, Proceedings of The 3rd Inernaional Symposium On Sofware Reliabiliy Engineering, May, pp , [13] Yamada, S., M. Obha and S. Osaki, S-Shaped Sofware Reliabiliy Growh Modeling For Sofware Error Deecion, IEEE Trans. On Reliabiliy, Vol. R-32, No. 5, pp , P. K. Kapur is Professor and former Head in he Deparmen of Operaional Research, Universiy of Delhi. He has edied special issues of Inernaional Journal of Reliabiliy, Qualiy and Safey Engineering (24, 27), OPSEARCH, India (25), Inernaional Journal of Performabiliy Engineering (IJPE India, 26) and a special issue of Communicaions on Dependabiliy and Qualiy Managemen, Belgrade, Serbia. He has organized hree Inernaional Conferences in he year 2, 23 and 26 on Qualiy Reliabiliy and Informaion Technology and has published more han 15 research papers in he areas of hardware reliabiliy, opimizaion, queuing heory, mainenance and sofware reliabiliy. He has edied four volumes on Qualiy, Reliabiliy and IT published in he year 24, 25, 27 and 28 and co-auhored a book "Conribuions o Hardware and Sofware Reliabiliy" published from World Scienific, Singapore. He has guided M.Tech. / Ph.D. heses in Compuer Science as well as in Operaions Research. He

8 274 P.K.Kapur, Anu G. Aggarwal and Sameer Anand has raveled exensively in India and abroad and delivered invied alks. His research ineress include Hardware and Sofware Reliabiliy, Opimizaion, Queuing Theory, Innovaion Diffusion Modeling, Numerical Compuaion of Sochasic models and Markeing. He is former Presiden of Operaional Research Sociey of India. Currenly he is presiden of Sociey for reliabiliy Engineering, Qualiy and Operaions Managemen (SREQOM). Anu G. Aggarwal received her Ph.D. degree in Sofware Reliabiliy from Universiy of Delhi, Delhi. She is presenly on he faculy in he Deparmen of Operaional Research, Universiy of Delhi. She has published several papers in he area of sofware reliabiliy. Her curren research ineress include sof compuing. She is a member of Sociey for Reliabiliy Engineering, Qualiy and Operaions Managemen (SREQOM). Sameer Anand received his M.Sc. and M.Phil. degree in Operaional Research from Universiy of Delhi, Delhi. He is presenly lecurer in he Deparmen of Compuer Science a S.S. College of Business Sudies, Universiy of Delhi. He has published several papers in he area of sofware reliabiliy and Markeing. He is a member of Sociey for Reliabiliy Engineering, Qualiy and Operaions Managemen (SREQOM).