Unified Framework For Developing Testing Effort Dependent Software Reliability Growth Models With Change Point And Imperfect Debugging

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1 Proceedings of he 4 h Naional Conference; INDIACom-00 Compuing For Naion Developmen, February 5 6, 00 Bharai Vidyapeeh s Insiue of Compuer Applicaions and Managemen, New Delhi Unified Framework For Developing Tesing Effor Dependen Sofware Reliabiliy Growh Models Wih Change Poin And Imperfec Debugging V.S.S.Yadavalli, Anu G. Aggarwal, P.K.Kapur 3 and Jyoish Kumar 4 Deparmen of Indusrial and Sysems Engineering, Universiy of Preoria, Preoria, Souh Africa,3,4 Deparmen of Operaional Research, Universiy of Delhi, Delhi sarma.yadavalli@up.ac.za; anuagg7@gmail.com; 3 pkkapur@gmail.com ABSTRACT In order o address he coninuing demand for high qualiy Low cos reliable sofware, Hundreds of sofware reliabiliy growh models (SRGMs) have been proposed in recen years. In spie of he diversiy and elegance of many of hese, no single model can be readily recommended as bes o represen he challenging naure of he sofware esing. Of lae, some auhors have ried o develop a unifying approach so as o capure differen growh curves, hus easing he model selecion process. The work in his area done so far relaes he faul removal process o he esing / execuion ime and does no consider he consumpion paern of resources such as compuer ime, manpower and number of execued es cases ec. More realisic modeling can resul if he reliabiliy growh process is sudied wih respec he amoun of expended esing effors. Due o he complexiy of sofware sysem and incomplee undersanding of sofware, he esing process may no be perfec or he faul deecion /correcion rae may change a any ime momen. In his paper, we propose a generalized framework for deriving several exising as well as new esing effor dependen sofware reliabiliy growh models incorporaing change poin and he possibiliy of imperfec debugging and error generaion. The proposed framework is based on sandard probabiliy disribuion funcions. The models developed have been validaed and verified using real daa ses. Esimaed Parameers and comparison crieria resuls have also been presened KEYWORDS Non-homogenous Poisson process, Sofware Reliabiliy Growh Model, Hazard Rae, Imperfec Debugging, Changepoin. INTRODUCTION The concep of "sofware reliabiliy" and is measuremen is receiving a lo of aenion in he sofware developmen communiy. Wih he ever increasing role ha sofware is playing in odays and omorrows world, he sofware developers and users are asking: "Jus how good is he sofware?" and "How much esing should be done before he sofware is released?" The sofware reliabiliy mehodology aemps o provide quaniaive measures o help answer hese quesions. Sofware reliabiliy is one of he imporan parameers of sofware qualiy and sysem dependabiliy. I is defined as he probabiliy of failure-free sofware operaion in a specified environmen for a specified period of ime The developmen of high qualiy sofware saisfying cos, a schedule and resource requiremen is an essenial prerequisie for improved compeiiveness of any organizaion. One major difficuly o maser his challenge is he ineviabiliy of defecs in sofware producs. The esing of sofware sysems is subjec o srong conflicing forces. One of he mos effecive ways o do his is o apply sofware reliabiliy engineering o esing (and developmen). Sofware reliabiliy engineering delivers he desired funcionaliy for a produc much more efficienly by quaniaively characerizing is expeced use. The sofware reliabiliy engineering ends o increase reliabiliy while decreasing developmen ime and cos. Thus sofware reliabiliy engineering balances cusomer needs for he major qualiy characerisics of reliabiliy, availabiliy, delivery ime and life cycle cos more effecively. Hundreds of Sofware Reliabiliy Growh models (SRGMs), which relae he number of failures (fauls idenified/correced) and execuion ime, have been discussed in he sofware reliabiliy engineering lieraure. Mos of he exising growh models belong o eiher of following wo caegories- Exponenial Goel and Okumoo [] and S-shaped Yamada and Osaki [8], Ohba [8]. According o heir caegory, hey provide fi on he differen ypes of failure Daa ses. Laer aemps were made o presen he flexible SRGMs which can work on boh ypes of failure daases e.g. Obha [7], Biani e al. [7] and Kapur and Garg models [3]. 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, complexiy of he sofware ec. The esing eam 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 error 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. However, in case of error generaion, he oal faul conen increases as he esing progresses because

2 Proceedings of he 4 h Naional Conference; INDIACom-00 new fauls are inroduced in he sysem while removing he old original fauls. Model due o Obha and Chou [6] is an error generaion model applied on G-O model and has been also named as Imperfec debugging model. Kapur and Garg [4,5] 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. 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. [4]. Bu he failure disribuion can be affeced by many facors such as running environmen, esing sraegy, defec densiy and resource allocaion. 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. Thus, he faul deecion rae may no be smooh and can be changed a some ime momen τ called change-poin. Many researchers have incorporaed change poin in sofware reliabiliy growh modeling. Firsly Zhao [9] incorporaed change-poin in sofware and hardware reliabiliy. Huang e al. [0] 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 [3]. Kapur e al. [,] inroduced various esing effor funcions and esing effor conrol wih change-poin in sofware reliabiliy growh modeling. These SRGM have been developed for diverse esing environmen like esing effor expendiure, imperfec debugging, change-poin ec. 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 hus provide an insighful invesigaion for he sudy of general models wihou making many assumpions [0]. In his paper, we presen a unified framework for Sofware reliabiliy growh modeling wih respec o esing effor expendiure and incorporae he concep of change poin wih imperfec debugging and error generaion. This unified scheme is based on Probabiliy disribuion funcions. I is also shown ha previously repored Non-Homogeneous Poisson Process (NHPP) based SRGMs wih imperfec debugging and error generaion are special cases of he proposed framework. From his approach, we can no only obain exising models bu also develop some new NHPP models. The exising and new models derived here have been validaed and evaluaed on wo acual sofware failure daa ses. Nonlinear regression based on leas square mehod has been used for Parameer esimaion and MSE (Mean Squared Error) and R has been used as he comparison crieria. The goodness of fi curves have been drawn o illusrae he fiing of he models o he daa graphically. NOTATIONS m( W ): The mean value funcion or he expeced number of fauls deeced or removed by ime. a( W ) : Toal faul conen of sofware dependen on ime. p : The probabiliy of faul removal on a failure (i.e., he probabiliy of perfec debugging). α : The rae a which he fauls/errors may be inroduced during he debugging process. b : Faul removal/correcion rae. λ( W ) : Inensiy funcion for NHPP models or faul deecion rae per uni ime. F( W ): Disribuion funcions for faul removal/correcion imes. f( W ) : Densiy funcions for faul removal/correcion imes. s( W ) : Hazard rae funcion. : Learning parameer in logisic funcion. τ :Change Poin. PROPOSED UNIFIED MODELING FRAMEWORK 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 ( ), 0of an NHPP process is given as follows. Pr N ( ) k and m( ) k! k e m( ), k 0,,... () m ( ) ( 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.

3 Unified Framework For Developing Tesing Effor Dependen Sofware Reliabiliy Growh Models Wih Change Poin and Imperfec Debugging 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 probabiliy. 6. Faul deecion / removal rae may change a any ime momen τ. Assumpion 4 and 5 capures he effec of imperfec debugging and error generaion respecively. MODEL DEVELOPMENT Le he couning processes X( W ), 0and N( W ), 0 represen he cumulaive number of failures observed and fauls correced up o ime respecively and le he es begun a ime =0. Then he disribuion of N W ) is given by ( Pr{ N ( W ) n} Pr{ N ( W ) j 0 n X (0) j}pr X (0) j Here i can be noed ha he condiional probabiliy Pr{ N( W ) n X (0) j} is zero for j n. For j n i is given by Pr{ NW ( ) n X(0) j} j n j n ( FW ( )) ( FW ( )) n (4) Therefore, we have (3) jn exp( a) n! j0 ( j n)! n afw ( ) a( FW ( )) Or we can wrie n a F( W ) exp a F( W ) Pr{ N( W ) n} n! (5) Hence we can conclude ha he faul correcion process is poisson wih mean value funcion (MVF) as given by: N( W ) a F( W ) m( W ) E (6) F As specified before, here W is he esing effor dependen probabiliy disribuion funcion for faul correcion F imes. I can be noed ha W so defined saisfy all he properies of probabiliy disribuion funcions.. A 0, W 0and F( W) 0. In his paper, we have used hree ypes of esing effor funcion namely Exponenial, Rayleigh and Weibull ype. All hese funcions saisfy he propery ha a 0, W 0. I can be verified from heir expressions, discussed in deail in appendix a he end of he paper.. For 0, W 0and F( W ) In his paper we have assumed F W o be eiher of Exponenial, Erlang, Logisic ype. As increases, W also increases indicaing monoonically increasing naure of F W.Similarly he coninuiy of F W can also be explained. 4. As esing coninues for an infiniely large ime i.e., W W disribuion funcion F W is, he corresponding value of F W Here W is a very large posiive number represening he upper bound on he availabiliy of esing resources. Therefore, F W can be assumed o be of order Pr{ NW ( ) n} j j n jn a n ( FW ( )) ( FW ( )) j 0 exp( a) j! From Equaion (6), he insananeous failure inensiy funcion W ) is given by: ( ( W ) af Or we can wrie W

4 Proceedings of he 4 h Naional Conference; INDIACom-00 Le us define s d F W ( W ) a m ( W ) dw F W d W sw ( ) F F W W Here represens hazard rae funcion or faul deecion/correcion rae per remaining faul of he sofware, or he rae a which he individual fauls manifes hemselves as failures during esing. Now, Equaion (7) can be wrien as: d dw d (7) F W am( W) am( W) s( W) FW MODEL DEVELOPMENT WITH CHANGE POINT Incorporaing change-poin concep, s ( W ) i.e. faul deecion/ correcion rae per remaining faul of he sofware can be wrien as sw F W W F W F W F (9) d d where F W FW and F W FW d d Furher, incorporaing he change-poin concep in modeling, Equaion (8) becomes: For 0 d F W am( W ) dw F W d (8) mw ( ) a F( W) For dw d d F W am( W ) F W Solving above equaion wih iniial condiion a, W W and m( W ) m( W ), we ge i.e. mw a FW FW F W FW FW a F W m (0) a F ( W ) for for () MODEL DEVELOPMENT WITH CHANGE POINT & TWO TYPE OF IMPERFECT DEBUGGING 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 sysems are deeced and eliminaed during he esing phase, he number of fauls remaining in he sofware sysem gradually decreases as he esing procedure go on. Thus under he common assumpions for sofware reliabiliy growh modeling, we consider he following linear differenial equaion. d bw amw () dw d Where b(w ) is a faul deecion rae per remaining fauls a esing ime. Here we consider he faul deecion rae as hazard rae s(w ), iniial faul is no he consan bu he funcion of ime and incorporaing he imperfec debugging. So he above equaion can be wrien as Solving he above equaion wih iniial condiion a W m W 0, 0 and ( ) 0, we ge

5 Unified Framework For Developing Tesing Effor Dependen Sofware Reliabiliy Growh Models Wih Change Poin and Imperfec Debugging d sw pw a W mw dw d ( ) ( ) (3) We assume ha fauls can be inroduced during he debugging phase wih a consan faul inroducion rae. Therefore, he faul conen rae funcion, a( ), is a linear funcion of he W expeced number of fauls deeced by W and i is defined by: am W for aw a m W m W m W for (4) and Probabiliy of perfec debugging rae will be p for pw ( ) p for (5) Now using equaion (9), (4) and (5), he equaion (3) can be rewrien as, f( W ) F W W f ( W ) d F W p a m W m W for p a m W m W m W m( W ) for Afer solving he above equaions, we ge he following soluions a p F W for p p F W a m W F W FW mw for (6) DERIVATION OF NEW AND EXISTING MODELS SRGM- Le FW ( ) exp bw for and F ( W ) exp bw for Subsiuing and F W ino Equaion (6), we ge: FW ( ) ( ) a exp bp W for a mw ( ) expbp W b p W W ( ) mw ( ) for ( ) (7) The above model can be reduced o he model given by Shyur [3] if we consider he perfec debugging and no faul generaion. SRGM- Le F( W ) be a wo-sage Erlangian disribuion funcion i.e., W for F( W ) bw exp b and F( W) bw exp bw for Subsiuing and F W ino Equaion (6), we ge: FW ( ) ( ) a p bw exp bw for p p bw bw a mw bw exp bp W bp W W mw for SRGM-3 Le F( W FW ( ) And ) be a logisic disribuion funcion exp exp bw bw for (8)

6 Proceedings of he 4 h Naional Conference; INDIACom-00 F ( W ) exp exp bw bw for Then he corresponding mean value funcion is given by: p a bw expbp W for exp a mw e b p W b p W W p p exp bw exp bw exp bw xp mw for (9) For furher simplifying he esimaion procedure we may assume and p p p. MODEL VALIDATION, COMPARISON CRITERIA AND DATA ANALYSES Model Validaion To illusrae he esimaion procedure and applicaion of he SRGM (exising as well as proposed) we have carried ou he daa analysis of real sofware daa se. The parameers of he models have been esimaed using saisical package SPSS and he change-poin of he daa ses have been judged by using change-poin analyzer. Daa se (DS-) The firs daa se (DS-) had been colleced during 35 monhs of esing a radar sysem of size 4 KLOC and 30 fauls were deeced during esing. This daa is cied from Brooks and Moley [9]. 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 of esing a real ime command and conrol sysem and 38 fauls were deeced during esing. This daa is cied from Ohba [7]. The change-poin for his daa se is 6 h week. Comparison Crieria 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 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 observed daa yi is measured 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 [6]. Coefficien of Muliple Deerminaion (R ): We define his coefficien as he raio of he sum of squares resuling from he rend model o ha from consan model subraced from. R residual SS i.e. -. 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 [6]. 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 [5]. Variaion The sandard deviaion of predicion error is known as variaion. Variaion PE Bias i N Lower he value of Variaion beer is he goodness of fi [5]. Roo Mean Square Predicion Error I is a measure of closeness wih which a model predics he observaion. RMSPE Bias Variaion

7 Unified Framework For Developing Tesing Effor Dependen Sofware Reliabiliy Growh Models Wih Change Poin and Imperfec Debugging Lower he value of Roo Mean Square Predicion Error beer is he goodness of fi [5]. Daa Analyses The SRGM wih mean value funcion m(w ) are esimaed for finding heir unknown parameers. For esing effor esimaion we have worked ou resuls on all hree effor funcions namely Exponenial, Rayleigh and Weibull (Effor funcion are defined in Appendix). Bu for model parameer esimaion we have used Weibull funcion as i gives bes resuls as compared o oher wo effor funcions. For DS- The parameer esimaion and comparison crieria resuls for DS- of all he models under consideraion can be viewed hrough Table III(a) and III(b). I is clear from he able ha he value of R for SRGM- 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 IV(a) and IV(b). I is clear from he able ha he value of R for SRGM- is higher and value of MSE is lower in comparison wih oher models and provides beer goodness of fi for DS-. Table I: Esimaion of esing Effor Funcion Parameers for DS- Tesing Parameer Esimaion Effor Funcion W v l Exponenial Rayleigh Weibull Table II: Esimaion of esing Effor Funcion Parameers for DS- Tesing Effor Funcion Parameer Esimaion W v l Exponenial Rayleigh Weibull Table III(a): Model Parameer Esimaion Resuls (DS-) Models a b b SRGM- SRGM SRGM Table III(b): Model Comparison Resuls (DS-) Models R MSE BIAS VARAITION RMSPE SRGM SRGM SRGM Table IV(a): Model Parameer Esimaion Resuls (DS-) Models a b b SRGM SRGM SRGM Table IV(b): Model Comparison Resuls (DS-) Models R MSE BIAS VARAITION RMSPE SRGM SRGM SRGM For DS- Cumulaive fauls GOODNESS OF FIT CURVES Goodness of fi curves Time(monhs) p p Acual Daa SRGM - SRGM - SRGM -3

8 Proceedings of he 4 h Naional Conference; INDIACom-00 For DS- Cumulaive fauls Goodness of fi curves Time(monhs) Acual Daa SRGM- SRGM- SRGM-3 CONCLUSION In his paper we have developed a unified framework for esing effor dependen sofware reliabiliy growh models incorporaing change-poin concep wih wo ypes of imperfec debugging. The framework presened here proves o be excellen for deriving a wide variey of effor dependen models by using differen probabiliy disribuion funcions. The echnique is simple and presens a unique mehodology for developing many new as well as exising models for differen design environmen. Wih his approach, we can derive exising models and propose new model. In his paper we have resriced ourselves o he sandard disribuions e.g. Exponenial, Weibull and Erlang k-ype for correcion imes. Their validiy and accuracy have been carried ou on wo real sofware failure daases. The resuls obained are quie encouraging as can be viewed hrough he numerical illusraions shown in ables obained afer he parameer esimaion FUTURE SCOPE In fuure work we are working upon he possibiliy of including some new disribuion funcions like Normal, Gamma ec. for correcion imes. Their capabiliy o represen he severiy and delays in he faul correcion need o be numerically worked ou and checked. The concep of unificaion provides an area of ineresing sudy which can ease ou he problem of model selecion for he sofware developer and hus make hese echniques more accessible and applicable. REFERENCE () A.L. Goel and Okumoo, K, Time dependen error deecion rae model for sofware reliabiliy and oher performance measure, IEEE Transacions on Reliabiliy, 979. () H. Pham, Sysem sofware reliabiliy, Reliabiliy Engineering Series, Springer, 006. (3) H.J Shyur A Sochasic Sofware Reliabiliy Model wih Imperfec-Debugging and Change-Poin, Journal of Sysems and Sofware, 66, pp. 35-4, 003. (4) J.D. Musa, A Iannino and K Okumoo, Sofware reliabiliy: measuremen, predicion, applicaions New York: Mc Graw Hill, 987. (5) 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. (6) M. Ohba and X.M. Chou, Does Imperfec Debugging Effec Sofware Reliabiliy Growh, proceedings of h Inernaional Conference of Sofware Engineering, pp 37-44,989. (7) M. Ohba, Sofware Reliabiliy Analysis Models, IBM Journal of Research and Developmen, Vol. 8, pp , 984. (8) M. Ohba, Inflecion S-Shaped Sofware Reliabiliy Growh Models, In: Osaki, S. and Haoyama, Y. (Eds.), Sochasic Models in Reliabiliy Theory, Berlin, Germany: Springer-Varlag, pp. 44-6, 984. (9) M. Zhao, Change-Poin Problems In Sofware And Hardware Reliabiliy, Communicaions in Saisics- Theory and Mehods, (3), pp , 993. (0) P. K Kapur, 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, communicaed in IEEE Transacions on Reliabiliy, 008. () P. K Kapur, V. B Singh, Sameer Anand and V.S.S. Yadavalli, 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. () P.K Kapur, A. Gupa, O. Shanawi, and V.S.S. Yadavalli 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 (3) P.K. Kapur and R.B. Garg A sofware reliabiliy growh model for an error removal phenomenon, Sofware Engineering Journal, Vol. 7, pp. 9-94, 99. (4) P.K. Kapur, and R.B. Garg, Opimal Sofware Release Policies for Sofware Reliabiliy Growh Model under Imperfec Debugging, RAIRO, Vol. 4, pp , 990. Coninued on Page No. 5

9 Unified Framework For Developing Tesing Effor Dependen Sofware Reliabiliy Growh Models Wih Change Poin and Imperfec Debugging Coninued from Page No. 56 (5) P.K. Kapur, D. Kumar, A.Gupa and P.C. Jha, 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. (6) P.K. Kapur, R.B.Garg and S. Kumar Conribuions o hardware and sofware reliabiliy, World Scienific, Singapore 999. (7) S. Biani, P. Bolzern, E. Pedroi, Pozzi, N. and R. Scaolini, 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. (8) S. Yamada, M. Obha, and S. Osaki, S-shaped sofware reliabiliy growh modeling for sofware error deecion, IEEE Trans. On Reliabiliy, Vol. R-3 No. 5, pp , 983. (9) W. D. Brooks and R.W. Moley, Analysis of Discree Sofware Reliabiliy Models-Technical Repor (RADC- TR-80-84), Rome Air Developmen Cener, New York, 980. (0) Yu Chin Huang 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. Where ν() is he ime dependen rae a which esing resources are consumed, wih respec o remaining available resources. Case : When ν()=ν, a consan, we ge Exponenial funcion: W v e Case : If ν()= ν., we ge Rayleigh ype v / curve: W e Case 3: If ν()=ν.l. l-, we ge Weibull l funcion: v W e To sudy he esing effor process, one of he above funcions can be seleced. APPENDIX MODELING TESTING EFFORT The esing resources spen during esing of any sofware basically, include manpower used for faul deecion / removal and CPU ime spen in execuing sofware under es. Greaer he amoun of esing effor faser is he esing process. The esing effor (resources) ha govern he pace of esing for almos all he sofware projecs are [4]:. Manpower. Compuer ime. The key funcion of manpower engaged in sofware esing is o design and run es cases and compare he es resuls wih desired specificaions. Any deparure from he specificaions is ermed as a failure. On a failure he faul causing i is idenified and hen removed by failure correcion personnel. During esing coninuous monioring is done o analyze he progress of esing and qualiy achieved. The compuer faciliies represen he compuer ime, which is necessary for failure idenificaion and correcion. The Funcions which will be used in his paper o explain he esing effor are- Exponenial, Rayleigh and Weibull. They can be derived from he assumpion ha, "The esing effor rae is proporional o he esing resources available". dw d v( ) W