International Journal Of Engineering And Computer Science ISSN: Volume 5 Issue 12 Dec. 2016, Page No.

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1 Ieraoal Joural Of Egeerg Ad Compuer Scece ISSN: Volume 5 Issue 1 Dec. 16, Page No Sofware Relably Model whe mulple errors occur a a me cludg a faul correco process K. Harshchadra 1, K. M. Majuaha ad Balakrsha 1. Deparme of Sascs, Bagalore Uversy, Begaluru. Bo-Sasca, Daa Corol Cedu Ida Servces (P) Ld. Begaluru. Deparme of Sascs, Vjaya College, Begaluru * Correspodece o: Dr. K. Harshchadra, Deparme of Sascs, Bagalore Uversy, Bagalore 56 56, Ida. E-mal d: harsh.jbc@gmal.com Absrac Mos of he sofware relably models are based o relably growh models whch deal wh oly falure deeco process. I hese models s assumed ha sofware fauls occur radomly a dffere me pos ad faul correco mes are eher gored or cosdered sgfca. I s also assumed ha oly oe faul s deeced a ay gve me po. I hs paper we propose a sofware relably model whch a radom umber of fauls are deeced wheever a falure occurs. The model also akes o accou he correco mes for he fauls deeced. The sofware falure mes ad he correco mes are assumed o follow expoeal dsrbuos. The umber of sofware fauls deeced a ay me po s assumed o follow a geomerc dsrbuo. The dsrbuo of he oal correco me s derved ad he model s formulaed as a alerag reewal process. The properes of he relably model are suded hrough he reewal process. We oba he maxmum lkelhood esmaors ad also asympoc erval esmaors of he sysem parameers ad her properes are dscussed. We also propose some large sample ess for he sysem parameers. Some umercal sudes have bee made o evaluae he power of he ess. Keywords: Sofware relably; Falure deeco process; Faul correco process; Maxmum lkelhood esmao; Reewal process; Large sample es; Power of es sasc. 1. INTRODUCTION Compuer sysems have become more ad more mpora moder socey. The problem of esmag relably of he sofware ad predcg he fuure behavor of compuer sofware falures has receved a grea deal of aeo over he las hree decades. Sofware relably s defed as he probably of falure free sofware operao for a specfed perod of me a specfed evrome. Durg he las years, umerous sofware relably models have bee developed by he researchers o provde useful formao abou how o mprove sofware relably. For a dealed revew of sofware relably models see [7, 9]. I mos of he sofware relably models dscussed he leraure s assumed ha wheever a sofware falures are deeced, hese fauls are mmedaely removed or correced he sese ha deeced K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 196

2 DOI: /jecs/v51.17 fauls are correced before coug he sofware esg process. Goel ad Okumoo [] ad Pham [9] have dscussed some sofware relably models wh oly faul deeco processes wh he assumpo of perfec ad mmedae faul correco. Huag e. al. [4], Huag ad L [5] ad Yamada e. al. [14] have dscussed some exesos of hese models. Igorg faul correco may o be realsc pracce, because fdg a sofware faul s oe hg ad fxg s aoher ad bewee hese wo here s a cosderable me delay, we call hs as correco me ad hs correco me depeds o he avalably of ma power, skll of a programmer ad experece of debuggg eam. Few models are developed akg o accou he faul correco processes FCPs for some work o hese les see [, 1, 11, 1, 1]. Schedewd [7] was he frs oe o corporae a cosa faul correco me a model havg o-homogeeous Posso process for faul deeco process ad Xe e. al. [1] have assumed radom me delay for he faul correco processes for modelg faul deeco ad correco processes, wh he emphass o FCPs descrbed by a delayed deeco process wh radom or deermsc delay. The sofware falure mes as well as correco mes are assumed o be expoeal. Furher, 1 see [] we proposed a sofware relably model akg o accou he faul correco me. The sofware falure mes as well as correco mes are assumed as expoeal. The model s formulaed as a alerag reewal process. The properes of he relably model are suded hrough he reewal process. The sysem parameers are esmaed by meas of he maxmum lkelhood esmaor. I almos all sofware relably models dscussed he leraure s assumed ha oly oe faul s deeced a a me. I hs paper, we propose a sofware relably model whch a radom umber of fauls are deeced wheever a falure occurs. The model also akes o accou he correco mes for he fauls deeced. The sofware falure mes ad he correco mes are assumed o follow expoeal dsrbuos. The umber of sofware fauls deeced a ay me po s assumed o follow a geomerc dsrbuo. The dsrbuo of he oal correco me s derved ad he model s formulaed as a alerag reewal process. The properes of he relably model are suded hrough he reewal process. We oba he maxmum lkelhood esmaors ad also asympoc erval esmaors of he sysem parameers ad her properes are dscussed. We also propose some large sample ess for he sysem parameers. Some umercal sudes have bee made o evaluae he power of he ess.. MODELING OF FAULT DETECTION AND CORRECTION PROCESSES I hs seco we develop a sofware relably model whch he me bewee successve sofware falures are assumed o a have commo expoeal dsrbuo ad he correco me have a geomerc compoudg of expoeal dsrbuo. K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1964

3 DOI: /jecs/v Model assumpos The sofware relably model s defed o followg assumpos:. All he fauls a program are muually depede from he falure deeco po of vew.. The me bewee successve falures s assumed o have a expoeal dsrbuo wh mea 1/.. Wheever a falure occurs, a radom umber of sofware fauls are deeced ad hs sze s assumed o follow a geomerc dsrbuo wh parameer p. v. The correco me for each sofware faul s assumed o have a expoeal dsrbuo wh mea 1/. v. The falure mes ad correco mes are muually depede... Developme of he model Suppose ha parcular sofware udergoes a esg process ad he sofware falure mes are observed. Le deoe he me a whch he deoe he sofware falure mes adx h sofware falure occurs ( 1,,... ). Le X, 1,,... 1 forms a sequece of decally ad depedely dsrbued.. d. radom varable havg commo expoeal dsrbuo wh mea1/. Assume ha wheever a falure occurs, he falure caused by fauls occur radom sze. Le M deoe he umber of sofware fauls deeced a he h me po 1,,.... Le Y M Y deoe he oal correco me for he M j1 j fauls deeced a he h me po 1,,.... Ad where Y j deoe he correco me for he h j faul deeced a he h me po, j 1,..., M ad 1,,.... Theorem.1 Le for j 1,,... M ad 1,,.... Y ' s be.. d. radom varables havg commo expoeal dsrbuo wh mea 1/ j LeY M Y, where M s a radom varable depede of Y ' s. Le he dsrbuo of j1 j j M be geomerc wh parameer p wh p.m.f. m1 P( M m) g m pq, m 1,,... ad p 1 (1) The { Y } forms a sequece of..d. radom varables havg a expoeal dsrbuo wh mea 1/ p. K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1965

4 DOI: /jecs/v51.17 Proof: Defe Y M Y. Sce Yj ' s are..d. radom varables ad M s are depede of Yj ' s, Y ' s are j1 j also..d. radom varables ad s desy fuco s gve by df( y) P y Y y dy m1 P M m ad y Y y dy Each Yj m1 g m P y Y y dy () ' s be..d. radom varables havg commo expoeal dsrbuo wh mea 1/ he Y M Y has a gamma dsrbuo wh parameers m ad ad M has geomerc dsrbuo wh j1 j parameer p he desy fuco of Y s gve by m m1 m1 y df( y) pq y e dy m1 m O smplfcao he expresso reduces o () py, 1, f y pe y ad p (4) Equao 4 s he desy fuco of radom geomerc sum of expoeal r.v s or commo expoeal dsrbuo wh mea 1/ p. Ths proves he heorem.. Reewal process model Le { X } deoe he sequece of falure mes ad { Y } deoe he sequece of correco mes. The sequece { X } ad { Y } cosues a alerag reewal process. If we deoe by Z X Y, 1,,..., he { Z } s also a reewal process. Sequece { Z } s a sequece of..d. radom varables wh desy, f Z z z f x f z x X y dx The desy fuco of Z reduces o λμp fz z e e z p ad λ λ μp μpz λz,, 1, Ths s he Hypo-expoeal dsrbuo. The mea ad varace of hs dsrbuo are gve by mea p p p p m ad varace (6) (5) K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1966

5 Le ad S Z ad N Sup S 1 : DOI: /jecs/v51.17, he N, s a reewal process duced by sequece Z N deoe he oal umber of reewals me erval,. Le M deoe he reewal fuco whch s equal o expeced umber of reewals,. The Laplace rasform of he reewal fuco s defed as.e. Where f * M s L M s * * f s 1 f s M * s p p s p s p p p * s s s p s he Laplace rasform of dsrbuo fuco Z. The reewal fuco of he faul deeco ad correco processes s defed as verse of he above Laplace rasform gve 7, ha s M p p 1e p The correspodg reewal desy fuco s dervave of he reewal fuco gve 8, ha s pe p (9) Reewal fuco M s also called as mea value fuco fauls deeced ad correced up o me fuco., ad reewal desy fuco (7) (8) MVF s defed as expeced umber of s also called as esy For hs model he falure rae s cosa ad s equal o ad s depede of me sce he las falure. Tha s, r S ad he relably fuco s Le x 1 exp R S P X S (1) M be he reewal fuco of, x dx. If 1 avalably fuco Z, he x dx s he probably of oe or more correcos x, ha s, f he sofware s operag codo a he al me po, he he A s gve by Pr Pr A T x T x dx K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1967

6 O smplfcao we ge, A DOI: /jecs/v51.17 p e p p p (11). MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS Suppose ha he sofware program has bee pu for esg ad assume ha he daa are gve for me bewee successve reewals,.e. he realzaos of he radom varables Z for 1,.... Gve he daa o successve reewals s obaed usg he desy fuco gve 5 as follows z he lkelhood fuco for he radom varable Z deoed by L, ; z p L z e e 1 p pz z, ; (1) Maxmum lkelhood esmao of ad : Takg log o boh sdes of 1, we ge log-lkelhood fuco ha s defed as l L, ; z pz z l L, ; z l l l p l p l e e (1) 1. Tha s, Take paral dervave o 1 w.r.. ukow model parameers ad, ad equag o zero we ge lkelhood equaos, hose are z ze pz z p 1 e e pz p pze pz z p 1 e e We do o ge closed form expressos for he maxmum lkelhood esmaes m.l.e.'s of ad. However, he m.l.e.'s ca be obaed by erave procedure. Le ˆ ad ˆ be he m.l.e.'s of parameers (14) ad, respecvely. We ca he oba he m.l.e.'s of he reewal fuco, reewal desy, relably fuco ad avalably fucos by replacg 8, 9, 1 ad 11 respecvely. ad by s m.l.e.'s ˆ ad ˆ expressos Asympoc cofdece ervals: If we deoe he m.l.e. of (, ) by ˆ ˆ, ˆ, he observed formao marx s he gve by K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1968

7 I l l l l E E l l l l E E DOI: /jecs/v51.17 ˆ, ˆ Ad hece he varace covarace marx would be I ervals for he parameers ad are ˆ V ˆ 1 (15). The approxmae ad ˆ V ˆ 1 1% cofdece respecvely, where ˆ V ad V ˆ are he varaces of ˆ 1 ad, ˆ whch are gve by he frs ad he secod, dagoal eleme of I, ad s he upper percele of sadard ormal dsrbuo. 4. HYPOTHESIS TESTING FOR ad Here we cosder a hypohess esg problem for he model parameers ad. These hypoheses esg problems are bascally of eres o compare wo alerave sofware falure deeco or correco processes. Cosder he problem of esg he hypohess H :, agas he alerave ha he equaly does o hold for a leas for oe parameer. (16) Here we propose hree large sample es procedures for 16. We sae below hree basc ceral lm heorems relaed o reewal processes whou proof. Lemma 1: Le N, be he reewal process geeraed by F ad, 1,,... Z be reewal mes wh dsrbuo fuco F, wh mea m E Z ad varace E Z m exs ad are fe. Le Z reewal process, we have where, 1,,... be reewal mes wh dsrbuo fuco F. The from he ceral lm heorem o N lm P m s s m s s he d.f. of sadard ormal varae. Based o 17 we propose he es sasc be Z N m (17) 1 (18) m K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1969

8 DOI: /jecs/v51.17 For esg he hypohess saed 16 he rule s o rejec H f Z1 Z. Power fuco of he above es s deoed by gve by 1, B B P Z z C P Z z C, 1 H1 H A 1 A1 p p p p Where A1, B1 p p p ad C p p p p p p 1 (19) Z Lemma : Le, 1,,... be reewals wh dsrbuo fuco F, for whch he mea m E Z ad varace E Z m exs ad are fe. Le from he ceral lm heorem o reewal process, we have N, be he reewal process geeraed by F. The Where, SN lm P s s m N S N Z 1 ha s me of he las eve a me. Based o we propose he es sasc Z For he hypohess saed 16 based o Power fuco of he above es s deoed by Where, S N p p p 1 he rule s o rejec H f Z, ha s,, 1 P H Z 1 z A P H Z 1 z A A p p p p p p () (1). Z () Lemma : Le dsrbuo of he legh of me he sofware sysem wll be correco process has bee gve by Barlow ad Huer see [1]. Tha s D N Y 1 correco process. The for large values of, he asympoc dsrbuo of be he legh of me ha he sofware uder D wll be K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 197

9 DOI: /jecs/v51.17 D p lm P s s p p Based o we defe he sasc Z D p p p I hs case for he hypohess 16 he rule s o rejec H f Z. Power fuco of he above es s deoed by, Where, H1 H1 Z ad s gve by, B B, 1- P Z z C P Z - z C A A A p p, B p p ad C p p p p () (4) (5) 5. NUMERICAL COMPUTATION To llusrae he esmao procedure ad applcao of he Sofware relably model (exsg as well as proposed), we have carred ou he daa aalyss of a real sofware daa se. The daa se had bee colleced durg 19 weeks of esg a real me commad ad corol sysem, ad 8 fauls were deeced durg esg. These daa are ced from Ohba e al. [8]. We make use of Akake s formao crera (AIC) for comparg he performaces of dffere models. AIC defed as AIC = -log lkelhood fucoa s maxmummvalue +N (6) Where, N represes he umber of parameers he model. Aalyzg daa se, we oba he esmao resuls ad AIC value for exsg models (Here we call Goel ad Okumoo [] model as SRM-1, Delayed S-shaped NHPP [14] model as SRM- ad Harshchadra ad Majuaha [] model as SRM-) s show Table 5.1 ad for dffere values of p proposed model summarzed Tables SIMULATION STUDY I Seco, we have dscussed he maxmum lkelhood mehod of esmao of he parameers of he falure me ad faul correco me dsrbuos. I was observed ha o closed form soluos are K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1971

10 DOI: /jecs/v51.17 avalable for he m.l.e s. I hs seco we provde he maxmum lkelhood esmaes of he parameers for some choce of he parameers ad sample szes hrough a smulao sudy. A sequece of observaos { x, 1,,... } s geeraed from a expoeal dsrbuos wh parameer wh sample sze ad aoher sequece of observaos{ y, 1,,... } geeraed from expoeal dsrbuos wh parameer of p wh sample sze. For he values of 8,1,16 ad,4,8 ad p.5,.8 ad m.l.e's of model parameers ad are esmaed usg expresso (14) for sample szes 5, 1. These resuls are show Table 6.1. From Table 6.1, may be observed ha he esmaed values are farly close o assumed values for sample sze 5 ad he performace of hese esmaes are furher beer for large sample sze 1. I may be oed ha for small sample sze such as 5 he echque provdes esmaes of he parameers very close o assumed value. I s also observed ha he sadard error reduces o zero as sample sze creases. 7. POWER OF THE TEST PROCEDURES I hs seco, we evaluae he power of he hree es sascs proposed seco 4. Cosder he hypohess H :, agas he alerave hypohess H :,. Le 4, 4 ad p.9, 5 ( s me us) ad.5 (level of sgfcace). The power curves for hree es sascs proposed seco 4 are show Fg I hs case may be observed ha he es based o he sasc Z does o perform well for values of 4 ad Z also does o perform sasfacorly for 6. The power curves for esg power curves for esg H : 4, 4 agas he alerave hypohess H : 4, 4. Le 4, 4 ad p.9, 5 ( s me us) ad.5. The hree proposed ess are show Fg. 7.. I hs case we ca see ha boh sascs Z 1 ad Z performace are good ad also may be observed ha he es based o he sasc Z does o perform well for values of CONCLUSIONS AND REMARKS I hs paper we have developed a sofware relably model whch akes o accou besdes he falure occurrece mes also he faul correco mes. The model s developed assumg expoeal dsrbuo for falure mes ad wheever falure occurs a radom umber of sofware fauls are deeced. Assumg faul correco mes o be expoeal dsrbuo he compoud dsrbuo of faul correco me s derved. The model s formulaed as a reewal process ad he properes of he sofware relably model K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 197

11 DOI: /jecs/v51.17 are suded hrough he properes of he reewal process. The maxmum lkelhood esmaes ad asympoc erval esmaes of he model parameers have bee obaed. Three dffere large sample ess have bee proposed for he model parameers ad her power fucos are evaluaed from umercal sudes. I s observed ha oe es perform well for wo sded aleraves for boh deeco ad correco alerave hypohess ad oe es performs well for oly oe wo sded aleraves of correco ype ad he oher oe es perform oly for oe sded aleraves for boh ype of deeco ad correco aleraves. These ess are bascally of mporace from he po of vew of comparg he performaces of dffere faul correco processes. The model ca be exeded o he cases of oher forms of falure mes ad faul correco mes whose rage of dsrbuo akg couous o-egave radom varables. Refereces [1]. Barlow R E, Huer L C. Relably aalyss of a oe u sysem. Operao Research 1961; 9(); - 8. []. Goel AL, Okumoo K. Tme-depede error-deeco rae model for sofware ad oher performace measures. IEEE Trasacos o Relably 1979; 8; []. Harshchadra K, Majuaha K M. Sascal ferece o sofware relably assumg expoeal faul correco me. Ecoomc Qualy Corol 1; 5; [4]. Huag C Y, Lyu M R, Kuo S Y. A ufed scheme of some o-homogeeous Posso process models for sofware relably esmao. IEEE Trasacos o Sofware Egeerg ; 9(); [5]. Huag C Y, L C T. Sofware relably aalyss by cosderg faul depedecy ad debuggg me lag. IEEE Trasacos o Relably 6; 55(); [6]. Lo J H, Huag C Y. A egrao of sofware falure deeco ad faul correco processes sofware relably aalyss. Joural of Sysems ad Sofware 6; 76(); [7]. Lyu M R. Hadbook of Sofware Relably Egeerg. McGraw-Hll, New York [8]. Obha M. Sofware relably aalyss models. IBM Joural of Research ad Developme 1984; 8; [9]. Pham H. Sysem Sofware Relably. Sprger-Verlag, 6. [1]. Schedewd N F. Aalyss of error processes compuer sofware. I: Proc. of he I. Cof. o Relable Sofware. IEEE compuer Socey Press, Loss Alamos, CA 1975; [11]. Schedewd N F. Modelg he faul correco. I: Proc. of he 1 h I. Symp. O Sofware Relably Egeerg. IEEE compuer Socey Press, Loss Alamos, CA 1; [1]. Wu Y P, Hu Q P, Xe M, Ng S H. Modelg ad aalyss of sofware faul deeco ad correco process by cosderg me depedecy. IEEE Trasacos o Relably 7; 56(4); K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 197

12 DOI: /jecs/v51.17 [1]. Xe M, Hu Q P, Wu Y P, Ng S H. A sudy of he modelg ad aalyss of sofware faul- deeco ad faul-correco process. Qualy ad Relably Egeerg Ieraoal 7; (4); [14]. Yamada S, Obha M, Osak S. S-shaped sofware relably growh models ad her applcaos. IEEE Trasacos o Relably 1984; ; K. Harshchadra, IJECS Volume 5 Issue 1 Dec., 16 Page No Page 1974