DETECTION OF RELIABLE SOFTWARE USING SPRT ON TIME DOMAIN DATA

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1 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August DETECTION OF RELIABLE SOFTWARE USING SRT ON TIME DOMAIN DATA G.Krisha Moha ad Dr. Satya rasad Ravi Radr, Dpt. of Computr Scic,.B.Siddhartha collg Vijayawada, Adhrapradsh, Idia. Associat rofssor, Dpt. of Computr Scic & Egg., Acharya Nagrjua Uivrsity, Nagarjua Nagar, Gutur, Adhrapradsh, Idia ABSTRACT I Classical Hypothsis tstig volums of data is to b collctd ad th th coclusios ar draw which may tak mor tim. But, Squtial Aalysis of statistical scic could b adoptd i ordr to dcid upo th rliabl / urliabl of th dvlopd softwar vry quickly. Th procdur adoptd for this is, Squtial robability Ratio Tst (SRT. I th prst papr w proposd th prformac of SRT o Tim domai data usig Wibull modl ad aalyzd th rsults by applyig o 5 data sts. Th paramtrs ar stimatd usig Maximum Liklihood Estimatio. KEYWORDS Wibull modl, Squtial robability Ratio Tst, Maximum Liklihood Estimatio, Dcisio lis, Softwar Rliability, Tim domai data.. INTRODUCTION Wald's procdur is particularly rlvat if th data is collctd squtially. Squtial Aalysis is diffrt from Classical Hypothsis Tstig wr th umbr of cass tstd or collctd is fixd at th bgiig of th xprimt. I Classical Hypothsis Tstig th data collctio is xcutd without aalysis ad cosidratio of th data. Aftr all data is collctd th aalysis is do ad coclusios ar draw. Howvr, i Squtial Aalysis vry cas is aalysd dirctly aftr big collctd, th data collctd upto that momt is th compard with crtai thrshold valus, icorporatig th w iformatio obtaid from th frshly collctd cas. This approach allows o to draw coclusios durig th data collctio, ad a fial coclusio ca possibly b rachd at a much arlir stag as is th cas i Classical Hypothsis Tstig. Th advatags of Squtial Aalysis ar asy to s. As data collctio ca b trmiatd aftr fwr cass ad dcisios tak arlir, th savigs i trms of huma lif ad misry, ad fiacial savigs, might b cosidrabl. I th aalysis of softwar failur data w oft dal with ithr Tim Btw Failurs or failur cout i a giv tim itrval. If it is furthr assumd that th avrag umbr of rcordd failurs i a giv tim itrval is dirctly proportioal to th lgth of th itrval ad th radom umbr of failur occurrcs i th itrval is xplaid by a oisso procss th w kow that th probability quatio of th stochastic procss rprstig th failur occurrcs is giv by a homogous poisso procss with th xprssio DOI :.5/ijcsa

2 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August ( t t N ( t (.! Stibr[5] obsrvs that if classical tstig stratgis ar usd, th applicatio of softwar rliability growth modls may b difficult ad rliability prdictios ca b misladig. Howvr, h obsrvs that statistical mthods ca b succssfully applid to th failur data. H dmostratd his obsrvatio by applyig th wll-kow squtial probability ratio tst of Wald [4] for a softwar failur data to dtct urliabl softwar compots ad compar th rliability of diffrt softwar vrsios. I this papr w cosidr popular SRGM Expotial imprfct dbuggig modl ad adopt th pricipl of Stibr i dtctig urliabl softwar compots i ordr to accpt or rjct th dvlopd softwar. Th thory proposd by Stibr is prstd i Sctio for a rady rfrc. Extsio of this thory to th SRGM Wibull is prstd i Sctio 3. Maximum Liklihood paramtr stimatio mthod is prstd i Sctio 4. Applicatio of th dcisio rul to dtct urliabl softwar compots with rspct to th proposd SRGM is giv i Sctio 5.. WALD'S SEQUENTIAL TEST FOR A OISSON ROCESS Th squtial probability ratio tst was dvlopd by A.Wald at Columbia Uivrsity i 943. Du to its usfulss i dvlopmt work o military ad aval quipmt it was classifid as Rstrictd by th Espioag Act (Wald, 947. A big advatag of squtial tsts is that thy rquir fwr obsrvatios (tim o th avrag tha fixd sampl siz tsts. SRTs ar widly usd for statistical quality cotrol i maufacturig procsss. A SRT for homogous oisso procsss is dscribd blow. Lt {N(t,t } b a homogous oisso procss with rat. I our cas, N(t umbr of failurs up to tim t ad is th failur rat (failurs pr uit tim. Suppos that w put a systm o tst (for xampl a softwar systm, whr tstig is do accordig to a usag profil ad o faults ar corrctd ad that w wat to stimat its failur rat. W ca ot xpct to stimat prcisly. But w wat to rjct th systm with a high probability if our data suggst that th failur rat is largr tha ad accpt it with a high probability, if it s smallr tha. As always with statistical tsts, thr is som risk to gt th wrog aswrs. So w hav to spcify two (small umbrs α ad β, whr α is th probability of falsly rjctig th systm. That is rjctig th systm v if. This is th "producr s" risk. β is th probability of falsly accptig th systm.that is accptig th systm v if. This is th cosumr s risk. With spcifid choics of ad such that < <, th probability of fidig N(t failurs i th tim spa (,t with, as th failur rats ar rspctivly giv by N [ ] ( t t t (. N! N [ ] ( t t t (. N! 74

3 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August Th ratio at ay tim t is cosidrd as a masur of dcidig th truth towards or, giv a squc of tim istats say t < t < t3 <... < tk ad th corrspodig ralizatios N, N,... N( t K of N(t. Simplificatio of givs xp( t + ( N t Th dcisio rul of SRT is to dcid i favor of, i favor of or to cotiu by obsrvig th umbr of failurs at a latr tim tha 't' accordig as is gratr tha or qual to a costat say A, lss tha or qual to a costat say B or i btw th costats A ad B. That is, w dcid th giv softwar product as urliabl, rliabl or cotiu [3] th tst procss with o mor obsrvatio i failur data, accordig as B A (.3 B (.4 < < (.5 A Th approximat valus of th costats A ad B ar tak as β A, α β B α Whr α ad β ar th risk probabilitis as dfid arlir. A simplifid vrsio of th abov dcisio procsss is to rjct th systm as urliabl if N(t falls for th first tim abov N t a. t + b (.6 th li U ( to accpt th systm to b rliabl if N(t falls for th first tim blow th li L (. N t a t b (.7 To cotiu th tst with o mor obsrvatio o (t, N(t as th radom graph of [t, N(t] is btw th two liar boudaris giv by quatios (.6 ad (.7 whr a (.8 log 75

4 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August α log β b log b β log α log (.9 (. Th paramtrs α, β, ad ca b chos i svral ways. O way suggstd by Stibr is.log q ( q,.log q q ( q whr q If ad ar chos i this way, th slop of N U (t ad N L (t quals. Th othr two ways of choosig ad ar from past projcts (for a compariso of th projcts ad from part of th data to compar th rliability of diffrt fuctioal aras (compots. 3. SEQUENTIAL TEST FOR SOFTWARE RELIABILITY GROWTH MODELS I Sctio, for th oisso procss w kow that th xpctd valu of N(t t calld th avrag umbr of failurs xpricd i tim 't'.this is also calld th ma valu fuctio of th oisso procss. O th othr had if w cosidr a oisso procss with a gral fuctio (ot cssarily liar m(t as its ma valu fuctio th probability quatio of a such a procss is [ ] y [ m ] ( m t N Y., y,,, y! Dpdig o th forms of m(t w gt various oisso procsss calld NH. For our Wibull ( bt modl th ma valu fuctio is giv as m a whr a >, b > W may writ m ( t [ m t ]. ( N! [ m t ] N ( t N ( t ( m ( t. ( N! Whr, m ( t, m ( t ar valus of th ma valu fuctio at spcifid sts of its paramtrs idicatig rliabl softwar ad urliabl softwar rspctivly. Lt, b valus of th NH at two spcificatios of b say, b b whr ( b b < rspctivly. It ca b show that for our modls m at b is gratr tha that at b. Symbolically m m procdur is as follows: Accpt th systm to b rliabl <. Th th SRT 76

5 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August B i.., m m [ m t ] [ m t ]. (. ( N N B log β + m m α i.., N log m log m Dcid th systm to b urliabl ad rjct if A β log + m m α i.., N (3. log m log m Cotiu th tst procdur as log as β β log + m m log + m m α α < N < (3.3 log m log m log m log m Substitutig th appropriat xprssios of th rspctiv ma valu fuctio m(t of Rayligh w gt th rspctiv dcisio ruls ad ar giv i followigs lis Accptac rgio: β ( bt ( b t log + a ( α N (3.4 ( b t log ( bt Rjctio rgio: β ( bt ( b t log + a( α N (3.5 ( b t log ( bt Cotiuatio rgio: β ( bt ( bt β ( bt ( b t log + a ( log + a ( α α < N < (3.6 ( b t ( b t log log ( bt ( bt It may b otd that i th abov modl th dcisio ruls ar xclusivly basd o th strgth of th squtial procdur (α,β ad th valus of th rspctiv ma valu fuctios amly, m ( t, m ( t. If th ma valu fuctio is liar i t passig through origi, that is, m(t t th dcisio ruls bcom dcisio lis as dscribd by Stibr (997. I that ss quatios (3. 77

6 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August (3., (3., (3.3 ca b rgardd as gralizatios to th dcisio procdur of Stibr (997. Th applicatios of ths rsults for liv softwar failur data ar prstd with aalysis i Sctio ML (MAXIMUM LIKELIHOOD ARAMETER ESTIMATION Th ida bhid maximum liklihood paramtr stimatio is to dtrmi th paramtrs that maximiz th probability (liklihood of th sampl data. Th mthod of maximum liklihood is cosidrd to b mor robust (with som xcptios ad yilds stimators with good statistical proprtis. I othr words, MLE mthods ar vrsatil ad apply to may modls ad to diffrt typs of data. Although th mthodology for maximum liklihood stimatio is simpl, th implmtatio is mathmatically its. Usig today's computr powr, howvr, mathmatical complxity is ot a big obstacl. If w coduct a xprimt ad obtai N idpdt obsrvatios, t, t, K, t N. Th th liklihood fuctio is giv by[] th followig product: L ( t, t, K, t θ, θ, K, θ L f ( t ; θ, θ, K, θ N k i k i Likly hood fuctio by usig (t is: L N i ( t Th logarithmic liklihood fuctio is giv by: Log L log ( ( t i i which ca b writt as [ ti ] i lo g ( m ( t i N Λ l L l f ( t ; θ, θ, K, θ Th maximum liklihood stimators (MLE of θ, θ, K, θ ar obtaid by maximizig L or Λ, k whr Λ is l L. By maximizig, which is much asir to work with tha L, th maximum liklihood stimators (MLE of θ, θ, K, θ ar th simultaous solutios of k quatios such k that: (, j,,,k Λ θ j Th paramtrs a ad b ar stimatd usig itrativ Nwto Raphso Mthod, which is giv as g ( x x + x g '( x For th prst modl of Wibull, th paramtrs ar stimatd from [9]. 5. SRT ANALYSIS OF LIVE DATA SETS W s that th dvlopd SRT mthodology is for a softwar failur data which is of th form [t, N(t] whr N(t is th failur umbr of softwar systm or its sub systm i t uits of tim. I this sctio w valuat th dcisio ruls basd o th cosidrd ma valu fuctio for Fiv diffrt data sts of th abov form, borrowd from [][7][8] ad SONATA softwar srvics. Basd o th stimats of th paramtr b i ach ma valu fuctio, w hav chos th spcificatios of b b δ, b b + δ quidistat o ithr sid of stimat of b obtaid through a data st to apply SRT such that b < b < b. Assumig th valu of δ.5, th choics ar giv i th followig tabl. Tabl : Estimats of a,b & Spcificatios of b, b Data St Estimat of Estimat of a b b b Xi AT&T i i k 78

7 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August IBM Lyu NTDS SONATA TROICO-R m, m ( t for th modl, w calculatd th Usig th slctd b, b ad subsqutly th dcisio ruls giv by Equatios 3., 3., squtially at ach t of th data sts takig th strgth ( α, β as (.5,.. Ths ar prstd for th modl i Tabl. Tabl : SRT aalysis for 7 data sts Data St T N(t R.H.S of quatio (5.3. Accptac rgio ( R.H.S of Equatio (5.3. Rjctio Rgio( Dcisio Xi Rjctio AT & T Rjctio IBM Rjctio Lyu Rjctio NTDS SONATA Rjctio Accptac 79

Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August TROICO-R 5 7 -.666864.776399 8 -.6493669.

CONCLUSION Th tabl shows that Wibull modl as xmplifid for 7 Data Sts idicat that th modl is prformig wll i arrivig at a dcisio.

8 Itratioal Joural of Computr Scic, Egirig ad Applicatios (IJCSEA Vol., No.4, August TROICO-R Rjctio From th abov tabl w s that a dcisio ithr to accpt or rjct th systm is rachd much i advac of th last tim istat of th data(th tstig tim. 6. CONCLUSION Th tabl shows that Wibull modl as xmplifid for 7 Data Sts idicat that th modl is prformig wll i arrivig at a dcisio. Out of 7 Data Sts th procdur applid o th modl has giv a dcisio of rjctio for 6, accptac for ad cotiu for o at various tim istat of th data as follows. DS, DS, DS3, DS4, DS5, DS7 ar rjctd at st, d, 3 rd,3 th, th ad d istat of tim rspctivly. DS6 is accptd at d istat of tim. Thrfor, w may coclud that, applyig SRT o data sts w ca com to a arly coclusio of rliabl / urliabl of softwar. 7. REFERENCES [] GOEL, A.L ad OKUMOTO, K. A Tim Dpdt Error Dtctio Rat Modl For Softwar Rliability Ad Othr rformac Masurs, IEEE Trasactios o Rliability, vol.r-8, pp.6-, 979. [] ham. H., Systm softwar rliability, Sprigr. 6. [3] Satya rasad, R., Half logistic Softwar rliability growth modl, 7, h.d Thsis of ANU, Idia. [4] Wald. A., Squtial Aalysis, Joh Wily ad So, Ic, Nw York [5] STIEBER, H.A. Statistical Quality Cotrol: How To Dtct Urliabl Softwar Compots, rocdigs th 8 th Itratioal Symposium o Softwar Rliability Egirig, [6] WOOD, A. rdictig Softwar Rliability, IEEE Computr, [7] Xi, M., Goh. T.N., Raja.., Som ffctiv cotrol chart procdurs for rliability moitorig -Rliability girig ad Systm Safty [8] Michal. R. Lyu, Th had book of softwar rliability girig, McGrawHill & IEEE Computr Socity prss. [9] Dr R.Satya rasad, G.Krisha Moha ad rof R R L Katham. Articl: Tim Domai basd Softwar rocss Cotrol usig Wibull Ma Valu Fuctio. Itratioal Joural of Computr Applicatios 8(3:8-, March.. Author rofil: First Author: Mr. G. Krisha Moha is workig as a Radr i th Dpartmt of Computr Scic,.B.Siddhartha Collg, Vijayawada. H obtaid his M.C.A dgr from Acharya Nagarjua Uivrsity i, M.Tch from JNTU, Kakiada, M.hil from Madurai Kamaraj Uivrsity ad pursuig h.d at Acharya Nagarjua Uivrsity. His rsarch itrsts lis i Data Miig ad Softwar Egirig. Scod Author: Dr. R. Satya rasad rcivd h.d. dgr i Computr Scic i th faculty of Egirig i 7 from Acharya Nagarjua Uivrsity, Adhra radsh. H rcivd gold mdal from Acharya Nagarjua Uivrsity for his outstadig prformac i Mastrs Dgr. H is currtly workig as Associat rofssor ad H.O.D, i th Dpartmt of Computr Scic & Egirig, Acharya Nagarjua Uivrsity. His currt rsarch is focusd o Softwar Egirig. H has publishd svral paprs i Natioal & Itratioal Jourals. 8

Statistics 3858 : Likelihood Ratio for Exponential Distribution

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