Assessing Reliable Software using SPRT based on LPETM

Transcription

1 Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju Assssig Rliabl Sofwar usig SRT basd o LETM R. Saya rasad hd, Associa rofssor Dp. of CS &Egg. AcharyaNagarjua Uivrsiy D. Hariha Assisa rofssor Dp. of Elcroics & Compur Egg. K.L.Uivrsiy R. Sidhura Dp. of CSE K.L.Uivrsiy ABSTRACT Sofwar rliabiliy assssm is icrasigly impora i dvlopig ad sig w sofwar producs. Logarihmic oisso Excuio Tim Modl (LETM) is a sofwar rliabiliy modl which prdics h xpcd failurs ad hc rlad rliabiliy quaiis br ha xisig sofwar rliabiliy modls. I uss No-Homogous oisso rocss(nh) wih a ma valu fucio ha is dpd o xpoially fallig faul dcio ra. Th wll kow squial robabiliy Raio Ts(SRT) procdur of saisical scic is adopd for his modl i ordr o dcid upo h rliabiliy / urliabiliy of dvlopd sofwar. Th modl is valuad by usig 6 Ss. Gral Trms Dcisio Rul, Sofwar sig, Sofwar failur daa,qualiy Sofwar.. Kywords LETM, Maximum Liklihood Esimaio, Urliabl Sofwar, Ma valu fucio, Isiy fucio.. INTRODUCTION I h aalysis of sofwar failur daa w of dal wih ihr ir failur ims or umbr of rcordd failurs i a giv im irval. If i is furhr assumd ha h avrag umbr of rcordd failurs i a giv im irval is dircly proporioal o h lgh of h irval ad h radom umbr of failur occurrcs i h irval is xplaid by a oisso procss h w kow ha h probabiliy quaio of h sochasic procss rprsig h failur occurrcs is giv by a homogous oisso procss wih h xprssio N (.)! Sibr (7) obsrvs ha if classical sig sragis ar usd (o usag sig), h applicaio of sofwar rliabiliy growh modls may b difficul ad rliabiliy prdicios ca b misladig. Howvr, h obsrvs ha saisical mhods ca b succssfully applid o h failur daa. H dmosrad his obsrvaio by applyig h wllkow squial probabiliy raio s (SRT) of Wald (47) for a sofwar failur daa o dc urliabl sofwar compos ad compar h rliabiliy of diffr sofwar vrsios. I his papr w cosidr a popular SRGM proposd by Gol ad Okumoo(7) ad adop h pricipl of Sibr (7) i dcig urliabl sofwar compos i ordr o accp/rjc a dvlopd sofwar. For brviy w do h SRGM as GOM. Th failur isiy is liarly dcrasig i is ma valu fucio. Th hory proposd by Sibr (7) is prsd i Scio for a rady rfrc. Exsio of his hory o h LETM is prsd i Scio 3. Th procdur for paramr simaio is prsd i scio 4. Applicaio of h dcisio rul o dc urliabl sofwar compos wih rspc o h proposd SRGM is giv i Scio 5.. WALD'S SEQUENTIAL TEST FOR A OISSON ROCESS Th squial probabiliy raio s (SRT) was dvlopd by A.Wald a Columbia Uivrsiy i 43. Du o is usfulss i dvlopm work o miliary ad aval quipm i was classifid as Rsricd by h Espioag Ac ( Wald, 47). A big advaag of squial ss is ha hy rquir fwr obsrvaios (im) o h avrag ha fixd sampl siz ss. SRTs ar widly usd for saisical qualiy corol i maufacurig procsss. A SRT for homogous oisso procsss is dscribd blow. L {, } b a homogous oisso procss wih ra. I our cas, =umbr of failurs up o im ad is h failur ra (failurs pr ui im ). Suppos ha w pu a sysm o s (for xampl a sofwar sysm, whr sig is do accordig o a usag profil ad o fauls ar corrcd) ad ha w wa o sima is failur ra. W cao xpc o sima prcisly. Bu w wa o rjc h sysm wih a high probabiliy if our daa suggs ha h failur ra is largr ha ad accp i wih a high probabiliy, if i s smallr ha ( < < ). As always wih saisical ss, hr is som risk o g h wrog aswrs. So w hav o spcify wo (small) umbrs α ad β, whr α is h probabiliy of falsly rjcig h sysm. Tha is rjcig h sysm v if λ. This is h "producr s" risk. β is h probabiliy of falsly accpig h sysm.tha is accpig h sysm v if λ. This is h cosumr s risk. Wih spcifid choics of ad such ha < <, h probabiliy of fidig failurs i h im spa (, ) wih, as h failur ras ar rspcivly giv by N (.) (.) Th raio a ay im is cosidrd as a masur of dcidig h ruh owards or, giv a squc of im isas say N 3... K ad h 6

2 Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju corrspodig ralizaios. of. Simplificaio of givs Th dcisio rul of SRT is o dcid i favor of favor of, i or o coiu by obsrvig h umbr of failurs a a lar im ha '' accordig as is grar ha or qual o a cosa say A, lss ha or qual o a cosa say B or i bw h cosas A ad B. Tha is, w dcid h giv sofwar produc as urliabl, rliabl or coiu h s procss wih o mor obsrvaio i failur daa, accordig as (.3) (.4) (.5) Th approxima valus of h cosas A ad B ar ak as, B xp( ) A B B A Whr ad ar h risk probabiliis as dfid arlir. A simplifid vrsio of h abov dcisio procsss is o rjc h sysm as urliabl if falls for h firs im abov h li U. N a b o accp h sysm o b rliabl if im blow h li L. N a b N( ), N( ),... N( K ) (.6) falls for h firs (.7) To coiu h s wih o mor obsrvaio o (, ) as h radom graph of [, ] is bw h wo liar boudaris giv by quaios (.6) ad (.7) whr a log N (.8) log b log b log log (.) (.) Th paramrs,, ad ca b chos i svral ways. O way suggsd by Sibr (7) is.log q q whr q, If λ ad λ ar chos i his way, h slop of N U () ad N L () quals λ. Th ohr wo ways of choosig λ ad λ ar from pas projcs (for a compariso of h projcs) ad from par of h daa o compar h rliabiliy of diffr fucioal aras (compos). 3. SEQUENTIAL TEST FOR SOFTWARE RELIABILITY GROWTH MODEL I Scio, for h oisso procss w kow ha h xpcd valu of = λ calld h avrag umbr of failurs xpricd i im ''.This is also calld h ma valu fucio of h oisso procss. O h ohr had if w cosidr a oisso procss wih a gral fucio (o cssarily liar) m() as is ma valu fucio h probabiliy quaio of a such a procss is Dpdig o h forms of m() w g various oisso procsss calld NH for our modl h ma valu fucio is m()=a.log(+b) W may wri.log q q q y m () m () N( ) Y., y,,, y! m () m (). m ( ). m ( ) 7

3 Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju whr, ar valus of h ma valu fucio a spcifid ss of is paramrs idicaig rliabl sofwar ad urliabl sofwar rspcivly. For isac h modl w hav b cosidrig is m() fucio, coais a pair of paramrs a, b wih a as a muliplir. Also a, b ar posiiv. L, b valus of h NH a wo spcificaios of b say rspcivly. I ca b show ha for our modls m() a b is grar ha ha a b. Symbolically m ()<m ().Th h SRT procdur is as follows: Accp h sysm o b rliabl i.., i.., (3.) Dcid h sysm o b urliabl ad rjc if i.., (3.) m () m () b, b b m () m () Coiu h s procdur as log as (3.3) Subsiuig h appropria xprssios of h ma valu fucio m() of LETM w g h dcisio ruls ad ar giv i followigs lis m()=a.log(+b) Accpac rgio: Rjcio rgio: b. m ( ). m ( ) B B log m( ) m( ) N () log m ( ) log m ( ) log m( ) m( ) N () log m ( ) log m ( ) log m ( ) m ( ) log m ( ) m ( N () log m ( ) log m ( ) log m ( ) log m ( ) log a N () b log b b b (3.4) A log a N () b log b Coiuaio rgio: (3.5) (3.6) I may b od ha i h abov modl h dcisio ruls ar xclusivly basd o h srgh of h squial procdur (,) ad h valus of h ma valu fucios amly,,. If h ma valu fucio is liar i passig hrough origi, ha is, m() = λ h dcisio ruls bcom dcisio lis as dscribd by Sibr (7). I ha ss quaios (3.), (3.), (3.3) ca b rgardd as gralizaios o h dcisio procdur of Sibr (7).Th applicaios of hs rsuls for liv sofwar failur daa ar prsd wih aalysis i Scio ARAMETER ESTIMATION aramr simaio is of primary imporac i sofwar rliabiliy prdicio. Oc h aalyical soluio for m() is kow for a giv modl, paramr simaio is achivd by applyig a wll kow chiqu of Maximum Liklihood Esimaio (MLE). Dpdig o h forma i which s daa ar availabl, wo diffr approachs ar frquly usd. A s of failur daa is usually collcd i o of wo commo ways, im domai daa ad irval domai daa. Th ida bhid maximum liklihood paramr simaio is o drmi h paramrs ha maximiz h probabiliy (liklihood) of h sampl daa. Th mhod of maximum liklihood is cosidrd o b mor robus (wih som xcpios) ad yilds simaors wih good saisical propris. I ohr words, MLE mhods ar vrsail ad apply o mos modls ad o diffr yps of daa. Alhough h mhodology for maximum liklihood simaio is simpl, h implmaio is mahmaically is. Assumig ha h daa ar giv for h cumulaiv umbr of dcd rrors yi i a giv im-irval (, i ) whr i =,,,. ad < < < < h h log liklihood fucio (LLF) aks o h followig form. Likly hood fucio by usig λ() is:l = i= λ i.th logarihmic liklihood fucio for irval domai daa (pham, 6) is giv by: Log L = b b log a a N () b b log log b b m () m () b b b b log i=(y i y i ) log[m( i ) m( i- )] m( ) Th maximum liklihood simaors (MLE) ofѳ,ѳ,,ѳ k ar obaid by maximizig L or, whr is l L. By maximizig, which is much asir o work wih ha L, h maximum liklihood simaors (MLE) of Ѳ,Ѳ,,Ѳ k ar h simulaous soluios of k quaios such ha: 8

4 Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju (ᴧ ) Ѳ j = j=,,,k Th paramrs a ad b ar simad usig iraiv Nwo Raphso Mhod, which is giv as x To sima a ad b, for a sampl of uis, firs obai h liklihood fucio: L = i= ab b Tak h aural logarihm o boh sids, Th Log Liklihood fucio is giv as: Log L = log [ λ( i )] = log [ i= ab b ] i= = i=(y i y i )log [(a. log [ + b i ])-(a.log[+b i ])] a. log [ + b ] Th paramr a is simad by akig h parial drivaiv w.r. a ad quaig i o. (i. a = i=(y i y i ) log [+b ] log L a = ) Th paramr b is simad by iraiv Nwo Raphso Mhod usigb + = b g(b ). which is subsiud i g (b ) fidig a. whr g(b) & g (b) ar xprssd as follows. g(b)= i= y i y i [ i g (b)= y i y i [ i i= x f ( x ) f '( x ) i ] ( i i )+ i + i +b i i [log log L log L g( b) g' ( b) b b (+b i ) (+b i ) ; i ][ log +b i log [+b i] ] - a +b i [+b i ] +b +bi +b i ] 5. SRT ANALYSIS OF LIVE DATA SETS W s ha h dvlopd SRT mhodology is for a sofwar failur daa which is of h form [, ] whr is h obsrvd umbr of failurs of sofwar sysm or is sub sysm i uis of im. I his scio w valua h dcisio ruls basd o h cosidrd ma valu fucios for six diffr daa ss of h abov form, borrowd from Wood (6), ham (5). Basd o h simas of h paramr b i ach ma valu fucio, w hav chos h spcificaios of b, b quidisa o ihr sid of sima of b obaid hrough a S o apply SRT such ha b < b < b. Th choics ar giv i h followig abl. S ham (5) has ham (5) has Wood (6) Rlas Wood (6) Rlas Wood (6) Rlas 3 Wood (6) Rlas 4 Tabl 5.: Spcificaios of b, b Esima of a Esima of b b b Usig h slcd b,b ad subsquly h m (),m ()for ach modl w calculad h dcisio ruls giv by Equaios 3.4, 3.5, squially a ach of h daa ss akig h srgh ( α, β ) as (.5,.5). Ths ar prsd for h modl i Tabls i= [y i y i ] X log [+b ] (+b ) Tabl 5.: SRT for LETM S T ham(5) has R.H.S of quaio (3.4) Accpac rgio ( ) R.H.S of Equaio (3.5) Rjcio Rgio( ) ham (5) has

5 Wood (6) Rlas Wood (6) Rlas Wood (6) Rlas 3 Wood (6) Rlas Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju From h abov abl w s ha a dcisio ihr o accp or rjc h sysm is rachd much i advac of h las im isa of h daa(h sig im).th followig cosolidad abl rvals h iraios rquird o com o a dcisio abou h sofwar of ach S. S Tabl 5.3: Cosolidad Tabl of Dcisios ham (5) has ham (5) has Wood (6) Rlas Wood (6) Rlas Wood (6) Rlas 3 Wood (6) Rlas 4 LETM Modl Iraios 3 8 Dcisio.75 7 Accp.6 6 Accp.74 Coiuous.74 Coiuous. 3 Coiuous.67 Coiuous Th abov cosolidad abl shows ha LETM as xmplifid for 6 Ss idica ha h modl is prformig wll for Ss i arrivig a a dcisio. For h rmaiig 4 Ss LTEM is icoclusiv. Thrfor, w may coclud ha h modl LETM is mos appropria modl o dcid upo rliabiliy / urliabiliy of sofwar. Th auhors ar xplorig h possibiliy of prformac of a w SRGM grad o h basis of dpdc of ma valu fucio o h faul dcio ra i a xpoially dcrasig mar. 6. REFERENCES [] GOEL, A.L ad OKUMOTO, K. (7). A Tim Dpd Error Dcio Ra Modl For Sofwar Rliabiliy Ad Ohr rformac Masurs, IEEE Trasacios o Rliabiliy, vol.r-8, pp.6-, 7. [] MUSA, J.D., ad OKUMOTO, K. (84). A Logorihmic oisso Excuio Tim Modl For Sofwar Rliabiliy Masurm, rocdig Svh Iraioal Cofrc o Sofwar Egirig, Orlado, [3] HAM, H.(5). A Gralizd Logisic Sofwar Rliabiliy Growh Modl, OSEARCH, Vol.4, No.4, [4] ham. H., 6. Sysm sofwar rliabiliy, Sprigr. [5] STIEBER, H.A.(7). Saisical Qualiy Corol: How To Dc Urliabl Sofwar Compos, rocdigs of h 8h Iraioal Symposium o Sofwar Rliabiliy Egirig, 8-.

6 Iraioal Joural of Compur Applicaios (75 888) Volum 47 No., Ju [6] WALD (47). Squial Aalysis, Wily,Nw York. [7] WOOD, A.(6). rdicig Sofwar Rliabiliy,IEEE Compur, [8] R.Saya rasad ad G. Krisha Moha.(). Dcio Of Rliabl Sofwar Usig SRT O Tim Domai,Iraioal Joural of Compur Scic, Egirig ad Applicaios, Vol., No.4, pp.-. [] R. Saya rasad, N. Supriya ad G. Krisha Moha (). Dcio Of Rliabl Sofwar Usig SRT Iraioal Joural of Advacd Compur Scic ad Applicaios Vol., No: 8, pp [] R. Saya rasad ad D. Hariha (). Discovry of Rliabl Sofwar usig GOM o Irval Domai, Iraioal Joural of Compur Applicaios Volum 3 No.5, pp.7-. [] R. Saya rasad ad D. Hariha (). Dcio of Rliabl Sofwar usig HLSRGM, Iraioal Joural of Compur Iformaio Sysms,pp.4-53.