Software Reliability using SPRT: Inflection S- shaped Model

Transcription

1 Volum 2, Issu 6, Jun 23 ISSN Sofwar Rliabiliy using SPRT: Inflcion S- shapd Modl Dr. R. Saya Prasad, K. Prasada Rao 2 and G. Krishna Mohan3 Associa Profssor, Dp. of Compur Scinc & Engg., Acharya Nagrjuna Univrsiy Nagarjuna Nagar, Gunur, Andhrapradsh, India Profssor & Dircor, Dp. of MCA, CMRIT, Kundalahalli, Bangalor. 3 Radr, Dp. of Compur Scinc, P.B.Siddharha collg Vijayawada, Andhrapradsh, India Absrac In Classical Hypohsis sing volums of daa is o b collcd and hn h conclusions ar drawn, which may nd mor im. Bu, Squnial Analysis of Saisical scinc could b adopd in ordr o dcid upon h rliabiliy / unrliabiliy of h dvlopd sofwar vry quickly. Th procdur adopd for his is, Squnial Probabiliy Raio Ts (SPRT). I is dsignd for coninuous monioring. Th liklihood basd SPRT proposd by Wald is vry gnral and i can b usd for many diffrn probabiliy disribuions. In h prsn papr w propos h prformanc of SPRT on 5 daa ss of Tim domain daa and analyzd h rsuls. Th paramrs ar simad using Maximum Liklihood Esimaion mhod. Kywords: Inflcion S-shapd modl, Maximum Liklihood Esimaion, SPRT, Sofwar sing, Sofwar failur daa.. INTRODUCTION Wald's procdur is paricularly rlvan if h daa is collcd squnially. Squnial Analysis is diffrn from Classical Hypohsis Tsing wr h numbr of cass sd or collcd is fixd a h bginning of h xprimn. In Classical Hypohsis Tsing h daa collcion is xcud wihou analysis and considraion of h daa. Afr all daa is collcd h analysis is don and conclusions ar drawn. Howvr, in Squnial Analysis vry cas is analyzd dircly afr bing collcd, h daa collcd upo ha momn is hn compard wih crain hrshold valus, incorporaing h nw informaion obaind from h frshly collcd cas. This approach allows on o draw conclusions during h daa collcion, and a final conclusion can possibly b rachd a a much arlir sag as is h cas in Classical Hypohsis Tsing. Th advanags of Squnial Analysis ar asy o s. As daa collcion can b rminad afr fwr cass and dcisions akn arlir, h savings in rms of human lif and misry, and financial savings, migh b considrabl. In h analysis of sofwar failur daa w ofn dal wih ihr Tim Bwn Failurs or failur coun in a givn im inrval. If i is furhr assumd ha h avrag numbr of rcordd failurs in a givn im inrval is dircly proporional o h lngh of h inrval and h random numbr of failur occurrncs in h inrval is xplaind by a Poisson procss hn w know ha h probabiliy quaion of h sochasic procss rprsning h failur occurrncs is givn by a Homognous Poisson Procss wih h xprssion n P N n (.) n! Sibr (997) obsrvs ha if classical sing sragis ar usd, h applicaion of sofwar rliabiliy growh modls may b difficul and rliabiliy prdicions can b mislading. Howvr, h obsrvs ha saisical mhods can b succssfully applid o h failur daa. H dmonsrad his obsrvaion by applying h wll-known squnial probabiliy raio s (SPRT) of Wald (947) for a sofwar failur daa o dc unrliabl sofwar componns and compar h rliabiliy of diffrn sofwar vrsions. In his papr w considr popular modl Inflcion S-shapd and adop h principl of Sibr (997) in dcing unrliabl sofwar componns in ordr o accp or rjc h dvlopd sofwar. Th hory proposd by Sibr (997) is prsnd in Scion 2 for a rady rfrnc. Exnsion of his hory o h SRGM Inflcion S-shapd modl is prsnd in Scion 3. Applicaion of h dcision rul o dc unrliabl sofwar componns wih rspc o h proposd SRGM is givn in Scion 4. Analysis of h applicaion of h SPRT on 5 daa ss and conclusions drawn ar givn in Scion 5 and 6 rspcivly. Volum 2, Issu 6, Jun 23 Pag 349

2 Volum 2, Issu 6, Jun 23 ISSN WALD'S SEUENTIAL TEST FOR A POISSON PROCESS Th squnial probabiliy raio s (SPRT) was dvlopd by A.Wald a Columbia Univrsiy in 943. Du o is usfulnss in dvlopmn work on miliary and naval quipmn i was classifid as Rsricd by h Espionag Ac (Wald, 947). A big advanag of squnial ss is ha hy rquir fwr obsrvaions (im) on h avrag han fixd sampl siz ss. SPRTs ar widly usd for saisical qualiy conrol in manufacuring procsss. An SPRT for homognous Poisson procsss is dscribd blow. L {N(), } b a homognous Poisson procss wih ra. In our cas, N() = numbr of failurs up o im and is h failur ra (failurs pr uni im ). Suppos ha w pu a sysm on s (for xampl a sofwar sysm, whr sing is don according o a usag profil and no fauls ar corrcd) and ha w wan o sima is failur ra. W can no xpc o sima prcisly. Bu w wan o rjc h sysm wih a high probabiliy if our daa suggs ha h failur ra is largr han and accp i wih a high probabiliy, if i s smallr han. As always wih saisical ss, hr is som risk o g h wrong answrs. So w hav o spcify wo (small) numbrs α and β, whr α is h probabiliy of falsly rjcing h sysm. Tha is rjcing h sysm vn if λ. This is h "producr s" risk. β is h probabiliy of falsly accping h sysm.tha is accping h sysm vn if λ. This is h consumr s risk. Wih spcifid choics of and such ha < <, h probabiliy of finding N() failurs in h im span (, ) wih, as h failur ras ar rspcivly givn by N (2.) N( )! N (2.2) Th raio N ( )! a any im is considrd as a masur of dciding h ruh owards or, givn a squnc of im insans say K and h corrsponding ralizaions N( ), N( 2),... N( K ) of N(). Simplificaion of givs xp( ) Th dcision rul of SPRT is o dcid in favor of, in favor of or o coninu by obsrving h numbr of failurs a a lar im han '' according as N is grar han or qual o a consan say A, lss han or qual o a consan say B or in bwn h consans A and B. Tha is, w dcid h givn sofwar produc as unrliabl, rliabl or coninu h s procss wih on mor obsrvaion in failur daa, according as A (2.3) B (2.4) B A (2.5) Th approxima valus of h consans A and B ar akn as A, B Whr and ar h risk probabiliis as dfind arlir. A simplifid vrsion of h abov dcision procsss is o rjc h sysm as unrliabl if N() falls for h firs im abov h lin U. 2 N a b (2.6) o accp h sysm o b rliabl if N() falls for h firs im blow h lin Volum 2, Issu 6, Jun 23 Pag 35

3 Volum 2, Issu 6, Jun 23 ISSN L. N a b (2.7) To coninu h s wih on mor obsrvaion on (, N()) as h random graph of [, N()] is bwn h wo linar boundaris givn by quaions (2.6) and (2.7) whr a (2.8) log log b log b 2 log log (2.9) (2.) Th paramrs,, and can b chosn in svral ways. On way suggsd by Sibr (997) is.log q q,.log q q q whr q If λ and λ ar chosn in his way, h slop of N U () and N L () quals λ. Th ohr wo ways of choosing λ and λ ar from pas projcs (for a comparison of h projcs) and from par of h daa o compar h rliabiliy of diffrn funcional aras. 3. INFLECTION S-SHAPED MODEL Sofwar rliabiliy growh modls (SRGM s) ar usful o assss h rliabiliy for qualiy managmn and singprogrss conrol of sofwar dvlopmn. Thy hav bn groupd ino wo classs of modls concav and S-shapd. Th mos imporan hing abou boh modls is ha hy hav h sam asympoic bhavior, i.., h dfc dcion ra dcrass as h numbr of dfcs dcd (and rpaird) incrass, and h oal numbr of dfcs dcd asympoically approachs a fini valu. Th inflcion S-shapd modl was proposd by Ohba in 984. This modl assums ha h faul dcion ra incrass hroughou a s priod. Th modl has a paramr, calld h inflcion ra, ha indicas h raio of dcabl fauls o h oal numbr of fauls in h arg sofwar. Tru, susaind xponnial growh canno xis in h ral world. Evnually all xponnial, amplifying procsss will uncovr undrlying sabilizing procsss ha ac as limis o growh. Th shif from xponnial o asympoic growh is known as sigmoidal, or S-shapd, growh. Ohba modls h dpndncy of fauls by posulaing h following assumpions: Som of h fauls ar no dcabl bfor som ohr fauls ar rmovd. Th dcion ra is proporional o h numbr of dcabl fauls in h program. Failur ra of ach dcabl faul is consan and idnical. All fauls can b rmovd. b Assuming [Ohba 984b]: b( ) b c a c b This modl is characrizd by h following man valu funcion: m( ) Whr b is h failur dcion ra, and c is h inflcion facor. Th failur innsiy funcion is givn as: b Volum 2, Issu 6, Jun 23 Pag 35

4 Volum 2, Issu 6, Jun 23 ISSN ab ( ) b c b c SEUENTIAL TEST FOR SOFTWARE RELIABILITY GROWTH MODELS In Scion 2, for h Poisson procss w know ha h xpcd valu of N() = λ calld h avrag numbr of failurs xprincd in im ''.This is also calld h man valu funcion of h Poisson procss. On h ohr hand if w considr a Poisson procss wih a gnral funcion (no ncssarily linar) m() as is man valu funcion h probabiliy quaion of a such a procss is y m( ) m( ) P N( ) Y., y,,2, y! Dpnding on h forms of m() w g various Poisson procsss calld NHPP. For h Inflcion S-shapd modl h a b man valu funcion is givn as m( ) b whr a, b c W may wri m ( ) m ( ). m ( ) N ( )!. m ( ) N( )! N ( ) N ( ) Whr, m ( ), m ( ) ar valus of h man valu funcion a spcifid ss of is paramrs indicaing rliabl sofwar and unrliabl sofwar rspcivly. L, b valus of h NHPP a wo spcificaions of b say b, b whr b b rspcivly. I can b shown ha for our modls m a b is grar han ha a b. Symbolically m m. Thn h SPRT procdur is as follows: Accp h sysm o b rliabl B i.., log m ( ) m ( ) N( ) log m ( ) log m ( ) Dcid h sysm o b unrliabl and rjc if A i.., log m ( ) m ( ) N( ) log m ( ) log m ( ) Coninu h s procdur as long as log m ( ) m ( ) log m ( ) m ( ) N( ) log m ( ) log m ( ) log m ( ) log m ( ) Subsiuing h appropria xprssions of h rspciv man valu funcion m() of Inflcion S-shapd modl w g h rspciv dcision ruls and ar givn in followings lins Accpanc rgion: (4.) (4.3) (4.2) Volum 2, Issu 6, Jun 23 Pag 352

5 Volum 2, Issu 6, Jun 23 ISSN b b a c b b c c b b c b b c log N( ) log Rjcion rgion: b b a c b b c c b b c b b c log N( ) log Coninuaion rgion: b b b b b b b b b b b b b b b b a c a c log log c c c c N c c log log c c I may b nod ha in h abov modl h dcision ruls ar xclusivly basd on h srngh of h squnial procdur (, ) and h valus of h rspciv man valu funcions namly, m ( ), m ( ). If h man valu Volum 2, Issu 6, Jun 23 Pag 353 (4.4) (4.5) (4.6) funcion is linar in passing hrough origin, ha is, m() = λ h dcision ruls bcom dcision lins as dscribd by Sibr (997). In ha sns quaions (4.), (4.2), (4.3) can b rgardd as gnralizaions o h dcision procdur of Sibr (997). Th applicaions of hs rsuls for liv sofwar failur daa ar prsnd wih analysis in Scion SPRT ANALYSIS OF LIVE DATA SETS Th dvlopd SPRT mhodology is for a sofwar failur daa which is of h form [, N()]. Whr, N() is h failur numbr of sofwar sysm or is sub sysm in unis of im. In his scion w valua h dcision ruls basd on h considrd man valu funcion for Fiv diffrn daa ss of h abov form, borrowd from Pham (26) and Lyu. Basd on h simas of h paramr b in ach man valu funcion, w hav chosn h spcificaions of b b, b b quidisan on ihr sid of sima of b obaind hrough a Daa S o apply SPRT such ha b < b < b. Assuming h valu of. 2 5 and c. 5 h choics ar givn in h following abl. Tabl 5.: Esimas of a, b & Spcificaions of b, b for Tim domain Daa S Esima of a Esima of b b b DS DS DS DS DS m ( ), m ( ) for h modl, w calculad h dcision ruls givn by Using h slcd b, b and subsqunly h Equaions and 5.3.5, squnially a ach of h daa ss aking h srngh ( α, β ) as (.5,.2). Ths ar prsnd for h modl in Tabl 5.2. Th following consolidad abl rvals h iraions rquird o com o a dcision abou h sofwar of ach Daa S.

6 Volum 2, Issu 6, Jun 23 ISSN Daa S DS Tabl 5.2: SPRT analysis for 5 daa ss of Tim domain daa T 3. 2 N( ) Accpanc rgion ( ) Rjcion Rgion ( ) Dcision Accpanc DS2 DS3 DS Accpanc Rjcion Rjcion DS Coninu From h abov abl, a dcision of ihr o accp, rjc h sysm or coninu is rachd much in advanc of h las im insan of h daa. Volum 2, Issu 6, Jun 23 Pag 354

7 Volum 2, Issu 6, Jun 23 ISSN CONCLUSION. Th abov consolidad abl shows ha Inflcion S-shapd modl as xmplifid for 5 Daa Ss indica ha h modl is prforming wll in arriving a a dcision. Th modl has givn a dcision of accpanc for 2 Daa Ss i. DS & DS2, a dcision of rjcion for 2 Daa Ss i. DS3 & DS4 and Coninu for Daa s i. DS5. Thrfor, w may conclud ha, applying SPRT on daa ss w can com o an arly conclusion of rliabiliy / unrliabiliy of sofwar. REFERENCES [] Dr. R.Saya Prasad, K. Prasada Rao and G.Krishna Mohan. (22). Conrol Char Procdur for Sofwar Rliabiliy: Inflcion S-Shapd Modl, Inrnaional Journal of Elcronics Communicaion and Compur Enginring (IJECCE). Volum 3, Issu 6, Nov-Dc, pp.[ ]. [2] GOEL, A.L and OKUMOTO, K. (979). A Tim Dpndn Error Dcion Ra Modl For Sofwar Rliabiliy And Ohr Prformanc Masurs, IEEE Transacions on Rliabiliy, vol.r-28, pp.26-2, 979. [3] Michal. R. Lyu, Th hand book of sofwar rliabiliy nginring, McGrawHill & IEEE Compur Sociy prss. [4] Pham. H., (26). Sysm sofwar rliabiliy, Springr. [5] Saya Prasad (27). Half logisic Sofwar rliabiliy growh modl Ph.D Thsis of ANU, India. [6] STIEBER, H.A. (997). Saisical ualiy Conrol: How To Dc Unrliabl Sofwar Componns, Procdings h 8h Inrnaional Symposium on Sofwar Rliabiliy Enginring, 8-2. [7] Wald. A., 947. Squnial Analysis, John Wily and Son, Inc, Nw York. [8] Wood, A. (996). Prdicing Sofwar Rliabiliy, IEEE Compur, [9] Xi, M., Goh. T.N., Ranjan.P., Som ffciv conrol char procdurs for rliabiliy monioring -Rliabiliy nginring and Sysm Safy Auhors: Dr. R. Saya Prasad Rcivd Ph.D. dgr in Compur Scinc in h faculy of Enginring in 27 from Acharya Nagarjuna Univrsiy, Andhra Pradsh. H rcivd gold mdal from Acharya Nagarjuna Univrsiy for his ousanding prformanc in a firs rank in Masrs Dgr. H is currnly working as Associaiv Profssor and H.O.D, in h Dparmn of Compur Scinc & Enginring, Acharya Nagarjuna Univrsiy. His currn rsarch is focusd on Sofwar Enginring. H publishd 5 rsarch paprs in Naional & Inrnaional Journals. Mr. K.Prasad Rao, Working as a Profssor and Dircor, Dp. of M.C.A, CMR Insiu of Tchnology. H is having 22 yars of xprinc as a Had & Lcurr in Compur Scinc fild. H Publishd paprs in 3 Naional and Inrnaional journals. H is pursuing Ph.D a Acharya Nagarjuna Univrsiy. His rsarch inrss lis in Sofwar Enginring. Mr. G. Krishna Mohan, working as a Radr in h Dparmn of Compur Scinc, P.B.Siddharha Collg, Vijayawada. H obaind his M.C.A dgr from Acharya Nagarjuna Univrsiy, M.Tch from JNTU, Kakinada, M.Phil from Madurai Kamaraj Univrsiy and pursuing Ph.D from Acharya Nagarjuna Univrsiy. H qualifid, AP Sa Lvl Eligibiliy Ts. His rsarch inrss lis in Daa Mining and Sofwar Enginring. H publishd 4 rsarch paprs in various Naional and Inrnaional journals. Volum 2, Issu 6, Jun 23 Pag 355