Unified Framework for Developing Testing Effort Dependent Software Reliability Growth Models

Transcription

1 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Unified Frmework for Developing Tesing Effor Dependen Sofwre Relibiliy Growh Models P.K. KAPUR 1, OMAR SHATNAI, ANU G. AGGARAL 1, RAVI KUMAR 1 1 Deprmen of Operionl Reserch, Universiy of Delhi, Delhi 117, INDIA Deprmen of Compuer Science, Al l-by Universiy, Mfrq 5113, JORDAN pkkpur1@gmil.com Absrc: - Severl sofwre relibiliy growh models (SRGMs hve been presened in he lierure in he ls hree decdes. These SRGMs ke ino ccoun differen esing environmen depending on size nd efficiency of esing em, ype of componens nd fuls, design of es cses, sofwre rchiecure ec. The plehor of models mkes he model selecion n uphill sk. Recenly, some uhors hve ried o develop unifying pproch so s o cpure differen growh curves, hus esing he model selecion process. The work in his re done so fr reles he ful removl process o he esing/execuion ime nd does no consider he consumpion pern of esing resources such s CPU ime, mnpower nd number of execued es cses. More relisic modeling echniques cn resul if he relibiliy growh process is sudied wih respec o he moun of expended esing effors. In his pper, we propose unified frmework for esing effor dependen sofwre relibiliy growh models incorporing imperfec debugging nd error generion. The proposed frmework represens he relisic cse of ime delys beween he differen sges of ful removl process i.e Filure Observion/Ful Deecion nd Ful Removl/Correcion processes. The Convoluion of probbiliy disribuion funcions hve been used o chrcerize ime differeniion beween hese wo processes. Severl exising nd new effor dependen models hve been derived by using differen ypes of disribuion funcions. e hve lso provided d nlysis bsed on he cul sofwre filure d ses for some of he models discussed nd proposed in he pper. Key-ords: - Sofwre relibiliy growh model, Tesing effor funcion, Imperfec debugging, Error generion, Convoluion, Probbiliy disribuion funcion. 1 Inroducion The role of sofwre is expnding rpidly in mny specs of modern life, rnging from criicl infrsrucures, such s rnsporion, defense, nd elecommunicion sysems, o work-plce uomion, produciviy enhncemen, educion, helh-cre, publishing, on-line services, enerinmen, ec. Given he poenilly cosly impc of sofwre filures for mny of hese pplicions, i is imporn o hve sound mehods of developing relible sofwre s well s ccure mehods of quniively cerifying sofwre relibiliy. Hence i is crucil h sofwre relibiliy engineering echniques should ply cenrl role in he plnning nd conrol of sofwre developmen projecs. In priculr, i is imporn o documen he imes nd nure of bug occurrences, nd heir correcion imes, hroughou he design nd implemenion phses s well s esing phse. ih such d i is possible o esime he ime which he sofwre produc will hve reched rge level of relibiliy, or o devise mehods o decrese h ime. A lrge number of sofwre relibiliy growh models (SRGMs, which rele he number of filures (fuls idenified/correced nd execuion ime, hve been discussed in he lierure [1,13,16,3]. These SRGM ssume diverse esing environmen like disincion beween filure nd correcion processes, lerning of he esing personnel, possibiliy of imperfec debugging nd ful generion, consn or monooniclly incresing / decresing ful deecion re (FDR or rndomness in he growh curve. Bu no SRGM cn be climed o be he bes s he physicl inerpreion of he esing nd debugging chnges due o numerous fcors e.g., design of es cses, defec densiy, skills nd efficiency of esing em, vilbiliy of esing resources ec. The plehor of SRGM mkes he model selecion edious sk. To reduce his difficuly, unified modeling pproches hve been proposed by mny reserchers. These schemes hve proved o be successful in obining severl exising SRGM by following single mehodology nd hus provide n ISSN: Issue 4, Volume 8, April 9

2 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr insighful invesigion for he sudy of generl models wihou mking mny ssumpions. The work in his re sred s erly s in 198s wih Shnikumr [19] proposing Generlized birh process model. Gokhle nd Trivedi [6] used Tesing coverge funcion o presen unified frmework nd showed how NHPP bsed models cn be represened by probbiliy disribuion funcions of ful deecion imes. Dohi e l. [3] proposed unificion mehod for NHPP models describing es inpu nd progrm ph serching imes sochsiclly by n infinie server queuing heory. Inoue [7] pplied infinie server queuing heory o he bsic ssumpions of delyed S-shped SRGM [] i.e. ful correcion phenomenon consiss of successive filure observion nd deecion/correcion processes nd obined severl NHPP models describing ful correcion s wo sge process. Anoher unificion mehodology is bsed on sysemic sudy of ful deecion process (FDP nd ful correcion process (FCP where FCPs re described by deecion process wih ime dely. The ide of modeling FCP s sepre process following he FDP ws firs used by Schneidewind [18]. More generl remen of his concep is due o Xie e l [,1] who suggesed modeling of ful deecion process s NHPP bsed SRGM followed by ful correcion process s delyed deecion process wih rndom ime lg. The recen unificion scheme (due o Kpur e l [11] is bsed on cumulive disribuion funcion for he deecion/correcion imes nd incorpores he concep of chnge poin in ful deecion re. These unificion schemes predic he ful conen nd relibiliy of he sofwre wih respec o he clendr ime nd do no consider he consumpion pern of resources such s compuer ime, mnpower nd number of execued es cses ec. More relisic unifying echniques cn resul if he relibiliy growh process is reled o he moun of expended esing effors. In his pper, we propose generlized frmework for deriving severl exising s well s new esing effor dependen sofwre relibiliy growh models wih he possibiliy of imperfec debugging nd error generion. In prcicl sofwre developmen scenrio, As soon s filure is observed, he effors re mde o correc he cuse of he filure. I is quie possible h he esing em my no be ble o remove/correc ful compleely nd he originl ful my remin leding o phenomenon known s imperfec debugging, or replced by noher ful resuling in error generion. In cse of imperfec debugging he ful conen of he sofwre is no chnged, bu becuse of incomplee removl, he originl deeced ful is no correced perfecly. Bu in cse of error generion, he ol ful conen increses s he esing progresses becuse new fuls re inroduced in he sysem while removing he old originl fuls. I ws Goel [4] who firs inroduced he concep of imperfec debugging. Model due o Obh nd Chou [1] is n error generion model pplied on G-O model nd hs been lso nmed s Imperfec debugging model. Kpur nd Grg [1] inroduced he imperfec debugging in Goel nd Okumoo [5]. They ssumed h he FDR per remining fuls is reduced due o imperfec debugging. Thus he number of filures observed/deeced by ime infiniy is more hn he iniil ful conen. Phm [15] developed n SRGM for muliple filure ypes incorporing error generion. Recenly, Kpur e l. [9] proposed flexible SRGM wih imperfec debugging nd error generion using logisic funcion for ful deecion re which reflecs he efficiency of he esing/removl em. In his pper, we presen unified frmework for sofwre relibiliy growh modeling wih respec o esing effor expendiure nd incorpore he concep of imperfec debugging nd error generion. This unified scheme is bsed on probbiliy disribuion funcions. I is lso shown h previously repored non-homogeneous Poisson process (NHPP bsed SRGMs wih imperfec debugging nd error generion re specil cses of he proposed frmework. From his pproch, we cn no only obin exising models bu lso develop some new NHPP models. The proposed models re formuled for he cse when here is ime differeniion beween filure observion / deecion nd ful removl/correcion processes. Here we hve used differen sndrd probbiliy disribuion funcions for represening filure observion nd ful correcion imes. These disribuion funcions hve been discussed briefly o demonsre heir uiliy nd pplicbiliy for represening hese rndom imes. The exising nd new models derived here hve been vlided nd evlued on wo cul sofwre filure d ses. Non-liner regression bsed on les squre mehod hs been used for prmeer esimion nd MSE (men squred error nd R (coefficien of muliple deerminion hs been used s he performnce comprison crieri. For fser nd ccure clculions, he sisicl pckge SPSS hs been ISSN: Issue 4, Volume 8, April 9

3 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr uilized for he purpose. The goodness of fi curves hve been drwn o illusre he fiing of he models o he d grphiclly. Res of his pper is orgnized s follows: Secion menions he bsic ssumpions mde followed by he model developmen under imperfec debugging nd error generion. This secion describes he unified frmework for esing effor dependen sofwre relibiliy growh models by considering ime differeniion beween filure observion/deecion nd ful removl/correcion processes. In secion 3 we derive mny exising nd new sofwre relibiliy growh models by using differen probbiliy disribuion funcions. Secion 4 shows numericl exmples for he proposed models bsed on wo rel sofwre filure d ses. Finlly, conclusions re drwn in secion 5. Noions m( Men vlue funcion (MVF or he expeced number of fuls correced by ime. Expeced number of fuls lying dormn in he sofwre when he esing srs i.e =. Amoun of esing effor expended by ime. ( Tol ful conen of sofwre dependen on esing effor expended λ( Inensiy funcion for ful correcion process (FCP or ful correcion re per uni ime. G(, F( Tesing effor dependen probbiliy disribuion funcion for filure observion nd ful correcion Times g(, f( Tesing effor dependen probbiliy densiy funcion for filure observion nd ful correcion imes * Convoluion. Seiljes convoluion. Unified Frmework for Modeling Relibiliy Growh wih Time Differeniion beween Filure Observion/Deecion nd Ful Removl/Correcion.1 Bsic Assumpions The model is bsed on he following ssumpions: 1. Sofwre sysem is subjec o filure during execuion cused by fuls remining in he sysem.. The number of fuls deeced ny ime insn is proporionl o he remining number of fuls in he sofwre. Ech ime filure is observed, immedie correcion effor srs nd he following my occur: ( Ful conen is reduced by one wih probbiliy (p. (b Ful conen remins unchnged wih probbiliy (1-p. 3. During he ful correcion process, wheher he ful is removed successfully or no, new fuls re genered wih consn probbiliy α. 4. The Ful correcion imes re i.i.d. rndom vribles wih probbiliy disribuion funcion F( = f ( x dx where F( is esing effor dependen disribuion funcion. 5. The ful correcion process is modeled by NHPP. 6. The iniil number of filure observed in he sofwre sysem = is Poisson rndom vrible wih men of.. Model Developmen Le he couning processes {X(, } nd {N(, } represen he cumulive number of filures observed nd fuls correced up o ime respecively nd le he es begun ime =. Then he disribuion of N( is given by Pr{ N( = n} = Pr{ N( n X( j}pr{ X( j} j = = = = (1 Here i cn be noed h he condiionl probbiliy Pr{N( =n X(=j} is zero for j<n. For j n i is given by j n j n Pr{ N( = n X( = j} = ( F( ( 1 F( ( n Therefore, we hve j Pr{ N( = n} = F( j= n = n ( ( 1 F( n [ F( ] [ ( 1 F( ] exp( n! j= j n j exp( j! ( j n! j n ISSN: Issue 4, Volume 8, April 9

4 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Here i cn be noed h = [ (1 F( ] j n ( ( 1 F( = exp j ( j n! From bove we obin n ( F( exp( F( Pr{ N( = n} = (3 n! Hence we cn conclude h he ful correcion process is poison wih men vlue funcion (MVF s given by: [ N( ] F( m ( = E = (4 As specified before, here F( is he esing effor dependen probbiliy disribuion funcion for ful correcion imes. I my be noed h F( is so defined h i sisfies ll he properies of probbiliy disribuion funcions. 1. A =, = nd F(. In his pper, we hve used hree ypes of esing effor funcion nmely Exponenil, Ryleigh nd eibull ype. All hese funcions sisfy he propery h =, =. I cn be verified from heir expressions, discussed in deil in ppendix he end of he pper.. For >, > nd F( >. In his pper we hve ssumed F( o be eiher Exponenil, Gmm, eibull or Norml ype. As increses, lso increses indicing monooniclly incresing nure of F(. Similrly he coninuiy of F( cn lso be explined. 3. As esing coninues for n infiniely lrge ime, i.e.,,, he corresponding vlue of disribuion funcion F( is F(. Here is very lrge posiive number represening he upper bound on he vilbiliy of he moun of esing resources vilble. Therefore, F( cn be ssumed o be of order 1. From Equion (4, he insnneous filure inensiy funcion λ( is given by: ' λ ( = F ( Or we cn wrie dm ' d F ( λ ( = = [ m ( ] d 1 F( (5 d Le us define ' F s( = 1 F ( ( Here s( represens hzrd re funcion or filure occurrence re per remining ful of he sofwre, or he re which he individul fuls mnifes hemselves s filures during esing or hzrd re funcion. The expression of hzrd re funcion s( in erms of probbiliy disribuion funcion gives he direcions for incorporing he cse of ime differeniion beween he sges of filure observion nd ful correcion..3 Proposed Tesing Effor Dependen Modeling Le us consider he cse when here is ime dely beween he observion of he filure nd he correcion of he underlying ful. This ime dely cn be due o vrious fcors e.g. severiy / complexiy of he fuls, chnge in defec densiy, skill of he esing em ec. Then FCP is no longer one-sge process. The correcion my be wo / hree sge process nmely filure observion, ful deecion followed by he ful removl/correcion. This division of ful correcion ino differen processes defines he complexiy of fuls presen in sofwre. More he dely in removl/correcion of ful on is observion/deecion, more complex is he ful. In h cse, Equion (5 cn be modified s: dm d ( f * g( λ ( = = ( ( [ m( d ] (6 1 F G d or, λ ( = h ( [ m( ] ( f * g( where h = 1 ( F G( ( is he filure observion/deecion-ful removl/correcion re. Upon solving we ge: m( = ( F G( (7 By selecing suible probbiliy disribuion funcions, we cn derive MVF for severl exising nd new Finie filure coun models. This equion represens wo sge ful correcion under perfec debugging condiions. Now le us consider he cse when fuls cn be inroduced during he debugging phse wih consn ful inroducion re α. Therefore, he ful conen re funcion ( is liner funcion of he expeced number of fuls deeced by ime. Th is, = + α m ( ( ISSN: Issue 4, Volume 8, April 9

5 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Now incorporing imperfec debugging nd error generion in proposed modeling, we hve dm d ( f * g( λ( = = p ( ( [ + α m( m( d ] 1 F G d (8 where p is he probbiliy of perfec debugging. Solving Equion (8 wih iniil condiion h =, m(= nd =, we ge: p ( [ ( ( ( 1 m = 1 1 ( ( ] α F G (9 Here If p=1 nd α=, i.e. perfec debugging, Equion (9 is nohing bu Equion (7. The men vlue funcions m( for vrious models cn be derived by using differen ypes of disribuion funcions F( nd G(..4 Priculr Cses Here if we define filure observion imes disribuion, i.e., G( is uni funcion, hen Equion (7 is sme s he Equion (4 nd Equion (9 becomes p( m ( = 1 1 F( (1 [ ( ] This equion defines he removl process s one sge process where no ime is los beween he filure observion nd is removl. This cse hs been discussed in deil in [1]. 3 Derivion of Exising nd New SRGM 3.1 Probbiliy Disribuion Funcions for Modeling Deecion/Correcion Times In his pper we hve used he following probbiliy disribuions funcions for rndom filure observion / deecion nd correcion imes.. Disribuion Exponenil Descripion / Applicion This is he mos simple nd widely used disribuion in relibiliy engineering modeling becuse i hs consn re. I indices he uniform disribuion of fuls in he sofwre code where ech nd every ful hs sme probbiliy for is removl. Though in mos of he sofwre esing projecs, for ske of simpliciy, he removl imes re ssumed o follow exponenil disribuion, bu o chieve more flexible modeling of removl imes, we cn use eibull or Gmm disribuion. Boh of hese disribuions re generlizion of Exponenil disribuion only nd hve very similr shpes. eibull Gmm / Erlng Norml I cn represen differen ypes of curves depending on he vlues of is shpe prmeer nd hence exremely flexible. I is very pproprie for represening he processes wih flucuing re i.e. incresing /decresing res. Gmm nd Erlng disribuions re exensions of Exponenil disribuion where he ful removl consiss of muliple seps e.g. generion of filure repor, is nlysis nd correcion ime followed by verificion nd vlidion. During esing, here re numerous fcors, which ffec he ful correcion process. These fcors cn be inernl e.g. defec densiy, complexiy of he fuls, he inernl srucure of he sofwre or he fcors cn be exernl nd come from he esing environmen iself e.g. design of he es cses, skill of he esers / es cse designers, esing effor vilbiliy/consumpion. ec. This wo-prmeer disribuion cn describe he correcion imes quie well for he cses where correcion ime depends on muliple fcors. By combining bove-menioned disribuions in our proposed UM pproch; we cn explin number of exising SRGM formuled for differen T&D scenrio. In he nex secion we discuss how o obin MVF of he vrious exising SRGM nd propose few new models lso. ISSN: Issue 4, Volume 8, April 9

6 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr 3. MVF for Vrious New nd Exising Models The men vlue funcions m( corresponding o differen forms of disribuion funcions F( nd G( re summrized in Tble I. Tble I Model F( G( m( SRGM-1 ~ exp(b 1( bp ( [ 1 e ] 1 α SRGM- ~ exp(b ~ b p exp(b 1 ((1 b e (1 + α SRGM-3 ~ exp( b 1 ~ exp( SRGM-4 SRGM-5 SRGM-6 SRGM-7 SRGM-8 ~Erlng- (b ~ ei(b,k (eibull Disribuion ~ N(µ,σ (Norml Disribuion ~ N(µ,σ ~ α 1, β γ ( 1 1 b b1 b 1 ( b e b e ~ exp(b 1( b 1 b 1 p( p(1 α b b 1 1 b + + e k bp ( [ 1 e ] 1( 1 α p( [ ] 1 ( 1 φ(, µ, σ ~ exp(b ~ exp(b ( 1 ϕ, µ, σ 1 ( bσ b+ µ b+ + e ϕ (, µ + bσ, σ 1 Γ(, α 1, β 1 b 1 e β 1 + Γ, α 1 1, α ( 1 bβ 1 1 bβ 1 p(1 α p( 4 Model Vlidion, Comprison Crieri nd D Anlyses 4.1 Model Vlidion To illusre he esimion procedure nd pplicion of he SRGM (exising s well s proposed we hve crried ou he d nlysis of he following wo rel sofwre d ses. D se DS-1 DS-1I Sofwre Projec / Progrm Descripion The firs d se (DS-1 hd been colleced during 35 monhs of esing rdr sysem of size 14 KLOC nd 131 fuls were deeced during esing []. The second d se (DS- hd been colleced during 19 weeks of esing rel ime commnd nd conrol sysem of size 1317 KLOC nd 38 fuls were deeced during esing [14]. ISSN: Issue 4, Volume 8, April 9

7 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr 4. Comprison Crieri for SRGM The performnce of SRGM re judged by heir biliy o fi he ps sofwre ful d (goodness of fi nd predicing he fuure behvior of he ful. Goodness of Fi crieri The erm goodness of fi is used in wo differen conexs. In one conex, i denoes he quesion if smple of d cme from populion wih specific disribuion. In noher conex, i denoes he quesion of How good does mhemicl model (for exmple liner regression model fi o he d?. The Men Squre-Error (MSE: The model under comprison is used o simule he ful d, he difference beween he expeced vlues, mˆ ( i nd he observed d y i is mesured by MSE s follows. k ( mˆ( y i i MSE= = i 1 k where k is he number of observions. The lower MSE indices less fiing error, hus beer goodness of fi [1]. b. Coefficien of Muliple Deerminion (R : e define his coefficien s he rio of he sum of squres resuling from he rend model o h from consn model subrced from 1. residul SS R = 1 correced SS R mesures he percenge of he ol vriion bou he men ccouned for he fied curve. I rnges in vlue from o 1. Smll vlues indice h he model does no fi he d well. The lrger R, he beer he model explins he vriion in he d [1]. c. Bis: The difference beween he observion nd predicion of number of filures ny insn of ime i is known s PE i. (predicion error. The verge of PEs is known s bis. Lower he vlue of Bis beer is he goodness of fi [17]. d. Vriion: The sndrd deviion of predicion error is known s vriion. Vriion= ( 1 ( PE N 1 Bis i Lower he vlue of Vriion beer is he goodness of fi [17]. e. Roo Men Squre Predicion Error: I is mesure of closeness wih which model predics he observion. ( Bis Vriion RMSPE = + Lower he vlue of Roo Men Squre Predicion Error beer is he goodness of fi [17]. In oher words, we evlue he performnce of he models under comprison using MSE, R, Bis, Vriion, nd RMSPE merics. For MSE, Bis, Vriion, nd RMSPE, he smller he meric vlue he beer he model fis relive o oher models run on he sme d se. For R, he lrger he meric vlue he beer. 4.3 D Anlyses The SRGM wih men vlue funcion m( given in Tble I re esimed for finding heir unknown prmeers. For esing effor esimion we hve worked ou resuls on ll hree effor funcions nmely Exponenil, Ryleigh nd eibull. Bu for model prmeer esimion we hve used eibull funcion s i gives bes resuls s compred o oher wo effor funcions. The esimed vlues of esing effor funcion prmeers re given in Tbles II nd III for DS-1 nd DS- respecively. For DS-1 The prmeer esimion nd comprison crieri resuls for DS-1 of ll he models under considerion cn be viewed hrough Tble IV nd Tble V. The fiing of he models o DS-1 is grphiclly illusred in Fig. 1.1 nd Fig.1.. SRGM-5 shows poor fiing o he cul vlues of he rel ime d se while ll oher models fi he d excellenly well. For DS- The prmeer esimion nd comprison crieri resuls for DS- of ll he models under considerion cn be viewed hrough Tble VI nd Tble VII. The fiing of he models o DS- is grphiclly illusred in Fig..1 nd Fig... SRGM-4 shows poor fiing o he cul vlues of he rel ime d se while ll oher models fi he d quie well. ISSN: Issue 4, Volume 8, April 9

8 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Tbles: Prmeers Esimion nd Comprison Crieri Tble II: Esimion of Tesing Effor Funcion Prmeers for DS-I Tesing Effor Prmeer Esimion Funcion v l Exponenil Ryleigh eibull Tble III: Esimion of Tesing Effor Funcion Prmeers for DS-II Tesing Effor Prmeer Esimion Funcion v l Exponenil Ryleigh eibull Tble IV: Prmeer Esimes for DS-1 Models b/b 1 b /k p α µ σ α 1 β 1 SRGM SRGM SRGM SRGM SRGM SRGM SRGM SRGM Tble V: Model Comprison Resuls for DS-1 Models R MSE BIAS VARAITION RMSPE SRGM SRGM SRGM SRGM SRGM SRGM SRGM SRGM Tble VI: Prmeer Esimes for DS- Models b/b 1 b /k p α µ σ α 1 β 1 SRGM SRGM SRGM SRGM SRGM SRGM SRGM SRGM ISSN: Issue 4, Volume 8, April 9

9 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Tble VII: Model Comprison Resuls for DS- Models R MSE BIAS VARAITION RMSPE SRGM SRGM SRGM SRGM SRGM SRGM SRGM SRGM Figures: Goodness of Fi Curves Goodness of Fi (DS-I Goodness of Fi (DS-I Cumulive Fuls Cumulive Fuls Tesing Time (monh Acul D SRGM-1 SRGM- SRGM-3 SRGM Tesing Time (monh Acul D SRGM-5 SRGM-6 SRGM-7 SRGM-8 Figure.1.1 Figure.1. Goodness of Fi (DS-II Goodness of Fi (DS-II Cumulive Fuls Cumulive Fuls Tesing Time (week Acul D SRGM-1 SRGM- SRGM-3 SRGM Tesing Time (week Acul D SRGM-5 SRGM-6 SRGM-7 SRGM-8 Figure..1 Figure.. ISSN: Issue 4, Volume 8, April 9

10 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr 5 Conclusion In his pper, unified frmework for esing effor dependen sofwre relibiliy growh models hs been discussed under he ssumpion h filure observion/deecion hs ime difference o he ful correcion process. More relisic sofwre esing scenrio hs been modeled by incorporing he possibiliy of wo ypes of imperfec debugging i.e. imperfec debugging nd error generion. The frmework presened here proves o be excellen for deriving wide vriey of effor dependen models by using differen probbiliy disribuion funcions. The echnique is simple nd presens unique mehodology for developing mny new s well s exising models for differen design environmen. The scope for fuure reserch in his re lies for he cse when relibiliy growh is sudied wih respec o number of es-cses execued i.e. discree ime unified modeling frmework. In his pper we hve used sndrd disribuions e.g. Exponenil, eibull, Erlng k- ype, Norml nd Gmm for correcion imes. Their vlidiy nd ccurcy hve been crried ou on wo rel sofwre filure dses. The resuls obined re quie encourging s cn be viewed hrough he numericl illusrions shown in bles obined fer he prmeer esimion. In fuure work he possibiliy of including chnge poin or he modeling using sochsic differenil equions cn be worked ou. The concep of unificion provides n re of ineresing sudy which cn ese ou he problem of model selecion for he sofwre developer nd hus mke hese echniques more ccessible nd pplicble. References: [1] Aggrwl G Anu, Kumr Rvi nd PK Kpur, A New Approch for Developing Tesing Effor Dependen Sofwre Relibiliy Growh Models, In: Proc. of he 3 rd Nionl Conference on Compuing for Nion Developmen (INDIACoM-9, New-Delhi, Indi, 9, pp [] Brooks D nd Moley R, Technicl Repor, Rome Air Developmen Cener, New York, 198. [3] Dohi T, Oski S, nd Trivedi KS, An Infinie Server Queuing Approch for Describing Sofwre Relibiliy Growh: Unified Modeling nd Esimion Frmework, In: Proc. of he 11 h Asi-Pcific Sofwre Engineering Conference (APSEC 4, Pusn, Kore, 4, pp [4] Goel AL, Sofwre Relibiliy Models: Assumpions, Limiions nd Applicbiliy, IEEE Trnscions on Sofwre Engineering, Vol.11, No.1, 1985, pp [5] Goel AL nd Okumoo K, Time Dependen Error Deecion Re Model for Sofwre Relibiliy nd oher Performnce Mesures, IEEE Trnscions on Relibiliy, Vol.8, No.3, 1979, pp [6] Gokhle SS, Philip T, Mrinos PN nd Trivedi KS, Unificion of Finie Filure Non-Homogeneous Poisson Process Models hrough Tes Coverge, In: Proc. In l Symposium on Sofwre Relibiliy Engineering (ISSRE 96, NY, USA, 1996, pp [7] Inoue S, A sudy on Sochsic Modelling for Accure Sofwre Relibiliy Assessmen, Ph.D. Thesis, Docorl Progrm of Grdue School of Engineering, Toori Universiy, Jpn, 6. [8] Kpur PK, Aggrwl G Anu nd Annd Smeer, A New Insigh ino Sofwre Relibiliy Growh Modeling, Inernionl Journl of Performbiliy Engineering, Vol.5, No.3, 9, pp [9] Kpur PK, Kumr D, Gup A nd Jh PC, On How To Model sofwre Relibiliy Growh in he Presence Of Imperfec Debugging nd error Generion, In: Proc. of he nd In l Conference on Relibiliy nd Sfey Engineering, Chenni, Indi, 6, pp [1] Kpur PK, Grg RB nd Kumr S, Conribuions o Hrdwre nd Sofwre Relibiliy, orld Scienific, [11] Kpur PK, Kumr J nd Kumr R, A Unified Modeling Frmework Incorporing Chnge Poin for Mesuring Relibiliy Growh During Sofwre Tesing, OPSEARCH, Specil Issue on Quniive Assessmen of Sofwre Relibiliy (Eds. Kpur PK nd Phm H, Vol.45, No.4, 8, pp [1] Ohb M nd Chou XM, Does Imperfec Debugging Effec Sofwre Relibiliy Growh, In: Proc. of 11 h In l Conference of Sofwre Engineering, Pisburgh, Pennsylvni, USA, 1989, pp [13] Mus JD, Innino A nd Okumoo K, Sofwre Relibiliy: Mesuremen, Predicion, Applicions, McGrw-Hill, [14] Ohb M, Sofwre Relibiliy Anlysis Models, IBM Journl of Reserch nd ISSN: Issue 4, Volume 8, April 9

11 P. K. Kpur, Omr Shnwi, Anu G. Aggrwl, Rvi Kumr Developmen, Vol.8, No.4, 1984, pp [15] Phm H nd Zhng X, An NHPP Sofwre Relibiliy Models nd is Comprison, Inernionl Journl of Relibiliy, Quliy nd Sfey Engineering, Vol.4, No.3, 1997, pp [16] Phm H, Sysem Sofwre Relibiliy, Relibiliy Engineering Series, Springer, 6. [17] Pilli K nd Nir VSS, A Model for Sofwre Developmen Effor nd Cos Esimion, IEEE Trnscions on Sofwre Engineering, Vol.3, No.8, 1997, pp [18] Schneidewind NF, Anlysis of Error Processes In Compuer Sofwre, Sigpln Noices, Vol.1, No.6, 1975, pp [19] Shnhikumr JG, A Generl Sofwre Relibiliy Model for Performnce Predicion, Microelecronics Relibiliy, Vol.1, No.5, 1981, pp [] Xie M nd Zho M, The Schneidewind Sofwre Relibiliy Model Revisied, In: Proc. of he 3 rd In l Symposium on Sofwre Relibiliy Engineering, Reserch Tringle Prk, NC, USA 199, pp [1] Xie M, QP Hu, u YP nd Ng SH, A Sudy of he Modeling nd Anlysis of Sofwre Ful-Deecion nd Ful-Correcion Processes, Quliy nd Relibiliy Engineering Inernionl, Vol.3, No.4, 7, pp [] Ymd S, Ohb M nd Oski S, S-shped Relibiliy Growh Modelling for Sofwre Error Deecion, IEEE Trnscions on Relibiliy, Vol.3, No.5, 1983, pp [3] Shnwi Omr nd Kpur PK, A Generlized Sofwre Ful Clssificion Model, SEAS Trnscions on Compuers, Vol.7, No.9, 8, pp [4] Junhong G, Hongwei L, Xiozong Y, nd Cheng ZD, A Sofwre Relibiliy Time Series Growh Model wih Klmn Filer, SEAS Trnscions on Compuers, Vol.5, No.1, 6, pp.1-8. [5] Junhong G, Hongwei L, Xiozong Y, A Sofwre Relibiliy Time Series Growh Model Trnsformed from Goel-Okumoo Model, SEAS Trnscions on Signl Process, Vol.1, No.1, 5, pp Appendix Descripion of Tesing Effor funcion The esing resources spen during esing of ny sofwre bsiclly, include mnpower used for ful deecion/removl nd CPU ime spen in execuing sofwre under es. Greer he moun of esing effor fser is he esing process. The esing effor (resources h govern he pce of esing for lmos ll he sofwre projecs re [13]: 1. Mnpower. Compuer ime. The key funcion of mnpower engged in sofwre esing is o design nd run es cses nd compre he es resuls wih desired specificions. Any deprure from he specificions is ermed s filure. On filure he ful cusing i is idenified nd hen removed by filure correcion personnel. During esing coninuous monioring is done o nlyze he progress of esing nd quliy chieved. The compuer fciliies represen he compuer ime, which is necessry for filure idenificion nd correcion. The Funcions which hve been used in his pper o explin he esing effor re- Exponenil, Ryleigh nd eibull. They cn be derived from he ssumpion h, "The esing effor re is proporionl o he esing resources vilble". d = v ( d where ν( is he ime dependen re which esing resources re consumed, wih respec o remining vilble resources. For solving his differenil equion, we use iniil condiion h Cse 1: hen ν(=ν, consn, we ge Exponenil funcion: ( 1 v = e Cse : If ν(= ν., we ge Ryleigh ype curve: v 1 / e = Cse 3: If ν(=ν.l. l-1, we ge eibull funcion: l 1 v e = To sudy he esing effor process, one of he bove funcions cn be seleced. ISSN: Issue 4, Volume 8, April 9