Genetic Algorithm Based Optimal Testing Effort Allocation Problem for Modular Software

Transcription

1 BIJIT - BVICAM s Inernaional Journal of Informaion Technology Bharai Vidyapeeh s Insiue of Compuer Applicaions and Managemen (BVICAM, ew Delhi Geneic Algorihm Based Opimal Tesing Effor Allocaion Problem for Modular Sofware Anu G. Aggarwal, P. K. Kapur 2, Gurjee Kaur 3 and Ravi Kumar 4 Absrac - Sofware reliabiliy growh models (SRGM are used o assess modular sofware quaniaively and predic he reliabiliy of each of he modules during module esing phase. In he las few decades various SRGM s have been proposed in lieraure. However, i is difficul o selec he bes model from a plehora of models available. To reduce his difficuly, unified modeling approaches have been proposed by many researchers. In his paper we presen a generalized framework for sofware reliabiliy growh modeling wih respec o esing effor expendiure and incorporae he fauls of differen severiy. e have used differen sandard probabiliy disribuion funcions for represening failure observaion and faul deecion/ correcion imes. The fauls in he sofware are labeled as simple, hard and complex fauls. Developing reliable modular sofware is necessary. Bu, a he same ime he esing effor available during he esing ime is limied. Consequenly, i is imporan for he projec manager o allocae hese limied resources among he modules opimally during he esing process. In his paper we have formulaed an opimizaion problem in which he oal number of fauls removed from modular sofware is (which include simple, hard and complex fauls maximized subjec o budgeary and reliabiliy consrains. To solve he opimizaion problem we have used geneic algorihm. One numerical example has been discussed o illusrae he soluion of he formulaed opimal effor allocaion problem. Index Terms - on-homogenous Poisson process, sofware reliabiliy growh model, Probabiliy Disribuion Funcions, Faul Severiy, Geneic Algorihm. ITRODUCTIO owadays large and complex sofware sysems are developed by inegraing a number of small and independen modules. Modules can be visualized as independen sofwares performing predefined asks, mosly developed by separae eams of programmers and someimes a differen geographical locaions. During he developmen of modular sofware, fauls can crop in he modules due o human imperfecion. These,2,3,4 Deparmen of Operaional Research, Universiy of Delhi, Delhi anuagg7@gmail.com, 2 pkkapur@gmail.com, 3 gurjeekaur85@gmail.com and 4 sianaravi@gmail.com Submied in April 200; Acceped in Sepember 20 fauls manifes hemselves in erms of failures when he modules are esed independenly during he module esing phase of sofware developmen life cycle. However, in oday s compuer invaded world hese failures can lead o big losses in erms of money, ime and life. Thus i is very imporan o evaluae sofware reliabiliy of each module during modular esing phase. To assess modular sofware quaniaively and predic he reliabiliy of each of he modules during module esing, sofware reliabiliy growh models (SRGM are used. umerous SRGM s, which relae he number of failures (faul idenified and he Execuion ime (CPU ime/calendar ime have been discussed in he lieraure [9,5,3]. All hese SRGMs assume ha he fauls in he sofware are of he same ype. However, his assumpion is no ruly represenaive of realiy. The sofware includes differen ypes of fauls, and each faul requires differen sraegies and differen amouns of esing effor for removal. Ohba [8] refined he Goel-Okumoo[] model by assuming ha he faul deecion/removal rae increases wih ime and ha here are wo ypes of fauls in he sofware. SRGM proposed by Biani e al. [22] and Kapur and Garg [3] has similar forms as ha of Ohba [8] bu hey developed under differen se of assumpions. These models can describe boh exponenial and S-shaped growh curves and herefore are ermed as flexible models [22, 8, 3]. Kapur e al. [6] developed Flexible sofware reliabiliy growh model wih esing effor dependen learning process in which wo ypes of sofware fauls were aken. Furher, hey proposed an SRGM wih hree ypes of fauls [9]. The firs ype of faul was modeled by an Exponenial model of Goel and Okumoo []. The second ype was modeled by Delayed S-shaped model of Yamada e al. [2]. The hird ype was modeled by a hreesage Erlang model proposed by Kapur e al. [9]. The oal removal phenomenon was modeled by he superposiion of he hree SRGMs. Shanawi and Kapur [] laer proposed a generalized model based on classificaion of he fauls in he sofware sysem according o heir removal complexiy. The above lieraure review reveals ha in he las few decades several SRGM s have been proposed. This plehora of SRGM s makes he model selecion a edious ask. To reduce his difficuly, unified modeling approaches have been proposed by many researchers. The work in his area sared as early as in 980s wih Shanikumar [4] proposing a Generalized birh process model. Gokhale and Trivedi [23] used Tesing coverage funcion o presen a unified framework and showed how HPP based models can be represened by probabiliy Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

2 Geneic Algorihm Based Opimal Tesing Effor Allocaion Problem for Modular Sofware disribuion funcions of faul deecion imes. 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 [0]. More general reamen of his concep is due o Xie e al [9] who suggesed modeling of Faul deecion process as a HPP based SRGM followed by Faul correcion process as a delayed deecion process wih random ime lag. The unificaion scheme due o Kapur e al [7] is based on Cumulaive Disribuion Funcion for he deecion/correcion imes and incorporaes he concep of change poin in Faul deecion rae. These schemes have proved o be fruiful in obaining several exising SRGM by following single mehodology and hus presen a percepive invesigaion for he sudy of general models wihou making many assumpions. In his paper we made use of such unified scheme for presening a generalized framework for sofware reliabiliy growh modeling wih respec o esing effor expendiure and incorporae he fauls of differen severiy. e have used differen sandard probabiliy disribuion funcions for represening failure observaion and faul correcion imes Also, he oal number of fauls in he sofware are labeled as simple, hard and complex fauls.i is assumed ha he esing phase consiss of hree differen processes, namely failure observaion, faul isolaion and faul removal. The ime delay beween he failure observaion and subsequen removal is assumed o represen he severiy of he faul. Developing reliable modular sofware is necessary. Bu, a he same ime he esing effor available during he esing ime is limied. These esing effors include resources like human power, CPU hours, and elapsed ime, ec. Hence, o develop a good reliable sofware sysem, a projec manager mus deermine in advance how o effecively allocae hese resources among he various modules. Such opimizaion problems are called Resource Allocaion problems. Many auhors have invesigaed he problem of resource allocaion [2, 7]. Kapur e al [20, 5] sudied various resource allocaion problems maximizing he number of fauls removed form each module under consrain on budge and managemen aspiraions on reliabiliy for exponenial and S-shaped SRGMs [,9,8].In his paper we have formulaed an opimizaion problem in which he oal number of fauls removed from modular sofware is (which include simple, hard and complex fauls maximized subjec o budgeary and reliabiliy consrains. To solve he effor allocaion problem formulaed in his research paper we use Geneic Algorihm(GA. GA sands up a powerful ool for solving search & opimizaion problems. The complex non linear formulaion of he opimal effor allocaion problem is he reason behind choosing geneic algorihm as he solving ool. GA always considers a populaion of soluions. There is no paricular requiremen on he problem before using GA s, as i can be applied o solve any kind of problem. The paper is organized as follows. Secion 2 gives he generalized framework for developing he sofware reliabiliy growh model for fauls of differen severiy. In secion 3 parameer esimaion and model validaion of he proposed model is done hrough SPSS. The esing effor allocaion problem is formulaed in secion 4. In secion 5 geneic algorihm is presened for solving he discussed problem. Secion 6 illusraes he opimizaion problem soluion hrough a numerical example. Finally, conclusions are drawn and are given in secion oaions ( : Cumulaive esing effor in he inerval (0.]. w( : Curren esing-effor expendiure rae a esing ime. d ( = w ( d m j ( : Expeced number of fauls removed of ype j(j=simple, Hard, Complex Fauls. m( : Expeced number of oal fauls removed. b : Consan faul deecion rae. : rae of consumpion of esing-effor β λ ( : Inensiy funcion for Faul correcion process (FCP or Faul correcion rae per uni ime. G (, F (, H ( : Tesing effor dependen Probabiliy Disribuion Funcion for Failure observaion, Faul Deecion and Faul Correcion Times g (, f (, h ( : Tesing effor dependen Probabiliy Densiy Funcion for Failure observaion, Faul Deecion and Faul Correcion Times * : Convoluion. : Seiljes convoluion. 2.2 Basic Assumpions The proposed model is based upon he following basic assumpions:. Failure occurrence, faul deecion, or faul removal phenomenon follows HPP. 2. Sofware is subjec o failures during execuion caused by fauls remaining in he sofware. 3. The fauls exising in he sofware are of hree ypes: simple, hard and complex. They are disinguished by he amoun of esing effor needed o remove hem 4. Faul removal process is prefec and failure observaion/faul isolaion/ faul removal rae is consan. 5. Each ime a failure occurs, an immediae effor akes place o decide he cause of he failure in order o remove i. The ime delay beween he failure observaion and is subsequen faul removal is assumed o represen he severiy of he fauls. The more severe he faul, more he ime delay. Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

3 BIJIT - BVICAM s Inernaional Journal of Informaion Technology 6. The faul isolaion/removal rae wih respec o esing effor inensiy is proporional o he number of observed failures. 2.3 Modeling Tesing Effor The proposed SRGM in his paper akes ino accoun he ime dependen variaion in esing effor. The esing effor (resources ha govern he pace of esing for almos all he sofware projecs are Manpower and Compuer ime. To describe he behavior of esing effor, Exponenial, Rayleigh, or eibull funcion has been used. The esing-effor described by a eibull-ype disribuion is given by: ( = α exp( g( τ dτ ( 0 In equaion (, if g(=β. Then, here is an exponenial curve, and he cumulaive esing- ( = α exp( β. (2 effor in (0,] is [ ] Similarly in ( if g ( = β. Then, here is a Rayleigh curve and he cumulaive esingeffor is given by: ( = α exp β 2. 2 (3 And if g ( γβ.. γ = in (, hen ( β γ ( = α exp. (4 which is cumulaive esing effor of eibull curve. 2.4 Model Developmen Le a, a 2 and a 3 be he simple, hard and complex fauls respecively a he beginning of esing. Also a is he oal faul conen i.e. a= a + a 2+ a Modeling Simple Fauls Simple fauls are he fauls which can be removed insanly as soon as hey are observed. The mean value funcion for he simple fauls of he sofware reliabiliy growh model wih respec o esing effor expendiure can be wrien as [8]: m( = a F( (5 where, F( is esing effor dependen disribuion funcion. From Equaion (5, he insananeous failure inensiy funcion λ ( is given by: ' λ ( = a F ( (6 Or we can wrie dm ' d F ( λ ( = = [ a m( ] (7 d F ( d Modeling Hard Fauls The hard fauls consume more esing ime for he removal. This means ha he esing eam will have o spend more ime o analyze he cause of he failure and herefore requires greaer ime o remove hem. Hence he removal process for hard fauls is modeled as a wo-sage process and is given by[8]: m2( = a2( F G(, and (8 ( f * g( ( ( [ 2 ] F G λ ( = a m( Modeling Complex Fauls These fauls require more esing ime for removal afer isolaion as compared o hard faul removal. Hence hey need o be modeled wih greaer ime lag beween failure observaion and removal. Thus, he removal process for complex fauls is modeled as a hree-sage process: m3( = a3( F G H( (0 And he insananeous failure inensiy funcion λ( is: ( f * g* h( λ ( = ( ( [ a3 m ( ] ( F G H Modeling Toal Fauls The oal faul removal phenomenon is he superimposiion of he simple, hard and complex fauls, and is herefore given as: m ( = m( + m2( + m3( (2 = a F( + a2( F G( + a3( F G H( A paricular case of he proposed model is abulaed in Table 2. Fauls F( G( H( m( Simple Hard exp( b - - exp( b2 exp( b Complex I( I( 2 - m ( = a b e m 2( b = a2. (( + b e m3 ( [,, ] (µ,σ 2 2 = a3 Φ( µσ (9 Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

4 Geneic Algorihm Based Opimal Tesing Effor Allocaion Problem for Modular Sofware MVF of Toal Faul b (( b m ( = a e a b e 2 + a 3 ( Φ(, µσ, Table 2.: A Paricular Case 2.5 Reliabiliy Evaluaion Using he SRGM we can evaluae he reliabiliy of he sofware during he progress of esing and predic he reliabiliy a he release ime. Reliabiliy of sofware is defined as given ha he esing has coninued up o ime, he probabiliy ha a sofware failure does no occur in ime inerval (, + ( 0. Hence he reliabiliy of sofware is represened mahemaically as ( m( m( R( R( + = exp + (3 Anoher measure of sofware reliabiliy a ime is defined as he raio of he cumulaive number of deeced fauls a ime o he expeced number of iniial faul conen of he sofware given by[4]: m( R( = (4 a To incorporae he effec of esing effor in he reliabiliy esimaion of each module Equaion (4 can be modified as: m( R( = (5 a 3. PARAMETER ESTIMATIO AD MODEL VALIDATIO To measure he performance of he proposed model we have carried ou he parameer esimaion on he daa se cied in M.Ohba [8](DS-I. The sofware was esed for 9 weeks during which compuer hours were used and 328 fauls were removed. The esimaion resuls for Exponenial, Rayleigh, and eibull funcion are given in able 3. Tesing Effor Funcion Exponenial funcion Rayleigh funcion Parameer Esimaion for DS-I α β γ R eibull funcion Table 3.: Tesing Effor Funcion Parameer Esimaes eibull effor funcion is chosen o represen he esing effor as i provided he bes fi on he esing effor daa (based on he highes value of R 2. Based upon hese esimaed parameers, parameers of proposed SRGM were esimaed. The goodness of fi measures used are Mean Square Error (MSE and Coefficien of muliple deerminaion (R 2. The resuls are compared wih SRGM proposed by Kapur e al. [9] wih hree ypes of faul. The resuls are abulaed in able 3.2 (Leing b =b 2 =b 3 =bthe goodness of fi curves for DS-I is given in Figure: 3. Paramer Esimaes Proposed Model Kapur e al. Model [9] a b µ σ m( R MSE Table 3.2: Parameer Esimaes for DS-I Goodness of Fi Curve Acual Values Esimaed 200 Values wih 50 3 sage 00 Erlang 50 Esimaed 0 Values of proposed Time Model Figure3.: Goodness of Fi Curve for DS-I 4. TESTIG RESOURCE ALLOCATIO PROBLEM 4. oaions: j :,2,3; Simple fauls-;hard Fauls-2, Complex Fauls-3 i : Module,,2.. : Toal number of modules m i ( : Mean value funcion for i h module b j i : Consan faul deecion rae for jh faul ype in i h module a ji : Consan, represening he number of j h faul ype lying dorman in i h module a he beginning of esing, c ji : Cos of removing j h faul from i h module i : Tesing effor for i h module R i : Reliabiliy of each module B : Toal cos of removing differen ypes of fauls Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

5 BIJIT - BVICAM s Inernaional Journal of Informaion Technology : Toal esing effor expendiure 4.2 Mahemaical Formulaion Consider sofware wih modules where each module is differen in size, complexiy, he funcions hey perform ec. In each module here are hree ypes of fauls; simple, hard and complex. The sofware has o be released in he marke a a predefined sofware release ime wih limied availabiliy of esing resources expendiure. Furher he cos of removing he faul from each module is dependen on is severiy. Therefore, he problem of maximizing he fauls of each of independen modules such ha reliabiliy of each module is a leas R 0 is formulaed as: Maximize m ( = m( i i i i= b ( ( i i b a ( 2 ( ( 2 i i i e a i b ii e = + + i= i= a 3i i= i µ i σ 2 ( (,, i i + Φ Subjec o: ( Cimi( i + C2im2i( i + C3im3i( i B i=,2... i= i i =, 2... i= Ri R0 i=,2... (P i 0 i =, GEETIC ALGORITHM FOR TESTIG RESOURCE ALLOCATIO The above opimizaion problem is solved by a powerful compuerized heurisic search and opimizaion mehod, viz. geneic algorihm (GA ha is based on he mechanics of naural selecion and naural geneics. In each ieraion (called generaion, hree basic geneic operaions i.e., selecion /reproducion, crossover and muaion are execued. For implemening he GA in solving he allocaion problem, he following basic elemens are o be considered. 5. Chromosome Represenaion Geneic Algorihm sars wih he iniial populaion of soluions represened as chromosomes. A chromosome comprises genes where each gene represens a specific aribue of he soluion. Here he soluion of he esing-effor allocaion problem in modular sofware sysem includes he effor resources consumed by individual modules. Therefore, a chromosome is a se of modular esing effor consumed as par of he oal esing effor availabiliy. 5.2 Iniial Populaion For a given oal esing ime, GA generaes he iniial populaion randomly. I iniialize o random values wihin he limis of each variable. 5.3 Finess Of A Chromosome The finess is a measure of he qualiy of he soluion i represens in erms of various opimizaion parameers of he soluion. A fi chromosome suggess a beer soluion. In he effor allocaion problem, he finess funcion is he objecive of esing effor opimizaion problem along wih he penalies of he consrains ha are no me. 5.4 Selecion Selecion is he process of choosing wo parens from he populaion for crossover. The higher he finess funcion, he more chance an individual has o be seleced. The selecion pressure drives he GA o improve he populaion finess over he successive generaions. Selecion has o be balanced wih variaion form crossover and muaion. Too srong selecion means sub opimal highly fi individuals, will ake over he populaion, reducing he diversiy needed for change and progress; oo weak selecion will resul in oo slow evoluion. e use Tournamen selecion here. 5.5 Crossover Crossover is he process of aking wo paren soluions and producing wo similar chromosomes by swapping ses of genes, hoping ha a leas one child will have genes ha improve is finess. In he esing resource allocaion problem, crossover diversifies he populaion by swapping modules wih disinc ime consuming, paricularly when he populaion size is small. 5.6 Muaion Muaion prevens he algorihm o be rapped in a local minimum. Muaion plays he role of recovering he los geneic maerials as well as for randomly disurbing geneic informaion. The imporan parameer in he muaion echnique is he muaion probabiliy. The muaion probabiliy decides how ofen pars of chromosome will be muaed. If here is no muaion, offspring are generaed immediaely afer crossover (or direcly copied wihou any change. In our problem of esing resource allocaion, we have used a muaion probabiliy of 0%. ih he basic modules of geneic algorihm described above, he procedure for solving he opimal effor allocaion problem is as follows [6]: Sep : Sar Sep 2: Generae random populaion of chromosomes Sep 3: Evaluae he finess of each chromosome in he populaion Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

6 Geneic Algorihm Based Opimal Tesing Effor Allocaion Problem for Modular Sofware Sep 4: Creae a new populaion by repeaing following seps unil he new populaion is complee: [Selecion] Selec wo paren chromosomes from a populaion according o heir finess [Crossover] ih a crossover probabiliy, cross over he parens o form new offspring (children. If no crossover is performed, offspring is he exac copy of parens. [Muaion] ih a muaion probabiliy, muae offspring a each locus (posiion in chromosome [Acceping] Place new offspring in he new populaion [Replace] Use new generaed populaion for furher sum of he algorihm. [Tes] If he end condiion is saisfied, sop and reurn he bes soluion in he curren populaion [Loop] Go o sep 3 for finess evaluaion 6. UMERICAL EXAMPLE The Effor Allocaion Problem described in secion 4 is illusraed numerically in his secion. Consider a sofware sysem consising of hree modules, whose parameers have already been esimaed using sofware failure daa. These parameer esimaes for each module is shown in Table 6.. The oal esing resources available is assumed o be 5000 unis. Toal cos for removing he differen ypes of fauls is 0000 unis. Also, i is desired ha he reliabiliy of each module is a leas 0.9. odule a a 2 a 3 b c c 2 c 3 µ σ Table 6.: Parameer Esimaes for effor allocaion problem Based on he above informaion, he problem (P is solved using geneic algorihm. The parameers used in GA evaluaion are given in able 6.2. Parameer Value Populaion Size 06 umber of Generaions 26 Selecion Mehod Tournamen Crossover Probabiliy 0.9 Muaion Probabiliy 0. Table 6.2: Parameer of he GA The opimal esing ime allocaion o each ype of faul in module and hence oal faul removed from each module and heir corresponding cos of removing is shown in able 6.3. Module m m 2 m 3 m Cos of Reliabiliy removing fauls Toal Table 6.3: The opimal esing effor expendiure wih he corresponding cos of each module 7. COCLUSIO In his paper we have discussed he problem for modular sofware a he uni esing sage. e have made use of unified scheme for presening a generalized framework for Sofware reliabiliy growh modeling wih respec o esing effor expendiure and incorporaed he fauls of differen severiy. The fauls in each module are of hree ypes-simple, hard and complex. Furher we have opimally allocaed he esing effor o each ype of faul and he modules and have found ou he differen ypes of fauls removed in he modules wih a fixed budge and a prerequisie level of reliabiliy. Geneic Algorihm is developed o solve he problem of resource allocaion. umerical example is discussed o illusrae he solving of he discussed opimizaion problem hrough GA. FUTURE SCOPE The presen sudy is done under he assumpion of independence of he failures of differen modules. In fuure, dependence of he failures from differen modules as well as he archiecure syles and connecors reliabiliy can also be sudied. REFERECES []. A. L. Goel and K. Okumoo, Time-dependen errordeecion rae model for sofware reliabiliy and oher performance measures, IEEE Transacions on Reliabiliy, Vol. 28, o. 3, pp 206 2, 979. [2]. H. Oheera and S Yamada., Opimal allocaion and conrol problems for sofware esing resources, IEEE Transacions on Reliabiliy, Vol. 39, o. 2, p. 7-76, 990. [3]. H. Pham, Sysem Sofware Reliabiliy, Reliabiliy Engineering Series, Springer, [4]. J. G. Shanhikumar,, A General Sofware Reliabiliy Model For Performance Predicion, Microelecronics Reliabiliy, Vol. 2, pp , 98. [5]. J.D. Musa, A. Iannino and K. Okumoo, Sofware Reliabiliy: Measuremen, Predicion, Applicaions, McGraw Hill, 987. [6]. K. Sasry, Single and Muliobjecive Geneic Algorihm Toolbox for Malab in C++ (IlliGAL Repor o Urbana, IL: Universiy of Illinois a Urbana- Champaign, 2007 Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS

7 BIJIT - BVICAM s Inernaional Journal of Informaion Technology [7]. Lo. Huang and Lyu Kuo, Opimal allocaion of esing resources considering cos, reliabiliy, and esing effor, Proceedings of he 0h IEEE Pacific Inernaional Symposium on dependable Compuing, 2004 [8]. M. Ohba, Sofware reliabiliy analysis models, IBM Journal of Research and Developmen, vol. 28, no. 4, pp , 984. [9]. M. Xie, and B. Yang, Opimal Tesing ime Allocaion for Modular Sysems, Inernaional Journal of Qualiy and Reliabiliy Managemen, Vol. 8, o. 4, ,200. [0]..F. Schneidewind,, Analysis Of Error Processes In Compuer Sofware, Sigplan oices, Vol. 0, pp , 975. []. O. Shanawi, P.K. Kapur, A Generalized Sofware Faul Classificaion Model, SEAS Transacions on Compuers, Vol. 2, o. 9, pp , 2008 [2]. P. K Kapur., S.Younes, and S. Agarwala, Generalised Erlang model wih n ypes of fauls, ASOR Bullein, Vol. 4, o., pp. 5, 995. [3]. P. K. Kapur and. R. B. Garg, Sofware reliabiliy growh model for an error-removal phenomenon, Sofware Engineering Journal, Vol. 7, o. 4, pp , 992. [4]. P. K., Kapur. V. B. Singh, and B Yang., Sofware reliabiliy growh model for deermining faul ypes, in Proceedings of he 3rd Inernaional Conference on Reliabiliy and Safey Engineering (ICRESE '07,pp , [5]. P.K Kapur., P.C. Jha, A.K. Bardhan,, Dynamic programming approach o esing resource allocaion problem for modular sofware, in Raio Mahemaica, Journal of Applied Mahemaics, Vol. 4, pp , [6]. P.K. Kapur, D..Goswami, A. Bardhan, O. Singh, Flexible sofware reliabiliy growh model wih esing effor dependen learning process, Applied Mahemaical Modelling, Vol. 32, o. 7, pp ,2008 [7]. P.K. Kapur, J. Kumar and R. Kumar, A Unified Modeling Framework Incorporaing Change Poin For Measuring Reliabiliy Growh During Sofware Tesing. To appear in OPSEARCH. [8]. P.K. Kapur, O. Shnawi, A Aggarwal., R. Kumar, Unified Framework for Developing Tesing Effor Dependen Sofware Relaibiliy Growh Models, SEAS Transacions on Sysems, Vol. 8 o. 4,pp 52-53,2009 [9]. P.K. Kapur, R. B. Garg and S. Kumar, Conribuions o Hardware and Sofware Reliabiliy, Singapore, orld Scienific Publishing Co. Ld., 999. [20]. P.K.Kapur, P.C. Jha, A.K. Bardhan, Opimal allocaion of esing resource for a modular sofware, Asia Pacific Journal of Operaional Research, Vol. 2, o. 3, pp , [2]. S. Yamada, M. Ohba, and S. Osaki, S-shaped sofware reliabiliy growh models and heir applicaions, IEEE Transacions on Reliabiliy, vol. 33, o. 4, pp , 984. [22]. S. Biani, P.Bolzern, E.Pedroi, and R. Scaolini, A flexible modeling approach for sofware reliabiliy growh, in Sofware Reliabiliy Modelling and Idenificaion, G. Goos and J. Harmanis, Eds., Springer, Berlin, Germany, pp 0 40, 998. [23]. S.S Gokhale., T. Philip, P.. Marinos and K.S. Trivedi, Unificaion of Finie Failure on-homogeneous Poisson Process Models hrough Tes Coverage, In Proc. Inl. Symposium on Sofware Reliabiliy Engineering (ISSRE 96, pp , Ocober 996. Copy Righ BIJIT 202; January - June, 202; Vol. 4 o. ; ISS