International Journal of Advanced Scientific Research and Management, Volume 3 Issue 11, Nov

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1 199 Algothm ad Matlab Pogam fo Softwa Rlablty Gowth Modl Basd o Wbull Od Statstcs Dstbuto Akladswa Svasa Vswaatha 1 ad Saavth Rama 2 1 Mathmatcs, Saaatha Collg of Egg, Tchy, Taml Nadu, Ida Abstact I ths pap, a w softwa lablty gowth modl basd o Wbull Od Statstcs dstbuto s dvlopd basd o No-Homogous Posso Pocss. A st of data s allowd to follow Wbull Od Statstcs dstbuto ad t tstd wth th modl. Algothm ad Matlab pogam a also gv ths pap to cut ths modl. Kywods: Wbull Od Statstcs dstbuto, Softwa Rlablty Gowth Modl (SRGM), No- Homogous Posso Pocss (NHPP), Ucostad Optmzato Tchqu, Statstcs Pocss Cotol. 1. Itoducto Rlablty of a Softwa [6] s dfd as th pobablty that a softwa systm woks wthout falu occug o spcfd opatg codtos fo a spcfd amout of tm. Assssg th lablty of a softwa ad thby matag softwa qualty dug softwa dvlopmt ad softwa usag s most madatoy. Softwa Rlablty Gowth Modls (SRGM) ca b appld to aalyz lablty of softwa. Ths modls dtct th softwa falu whch ca b adcatd ad thfo th lf tm of th softwa ascds whch hc cass th lablty of th softwa too. If a adom vaabl X s allowd to follow Wbull dstbuto, t s dsty fucto s gv by / f ( ) wh [0, ), 0 s th shap paamt, 0 s th scal paamt ad whos cospodg cumulatv dstbuto fucto s ( / ) F( ) (1 ). Lt us suppos that (X 1,..., X ) a jotly dstbutd adom vaabls. Th X s a aagd casg od s ts cospodg od statstcs. Thus X 1: X 2:... 2 Mathmatcs, Nhu Mmoal Collg, Tchy, Taml Nadu, Ida X :. A dpdt ad dtcally dstbutd sampl fom a absolutly cotuous dstbuto wth dsty f() has th jot dsty fucto of th od statstcs [5] as f (,,..., )! f ( ),... X1:,..., X : No-Homogous Posso Pocss (NHPP) modls o fault coutg modls ca b catgozd as ft ad ft falu modls dpdg o th spcfcato. Th umb of falus ths modl follows NHPP dstbuto. Th tsty fucto of falu ( ) s dfd as ( ) af ( ) wh a s th umb of falus pctd ad f() s th pobablty dsty fucto of X. Basd o NHPP assumptos, Ma valu fucto s m() = af() wh F() s th cumulatv dstbuto fucto of X ad a F. Motog th falu occuc pocss usg th tm chat s staghtfowad [7]. Th act pobablty lmts a usd to calculat th cotol lmts. Th upp cotol lmt, UCL, th ctal l, CL ad low cotol lmt, LCL ca b asly calculatd usg F( UCL,, ) 1 / 2 F( CL,, ) 0.5 F( LCL,, ) / 2 wh s th accptd fals alam sk, f th adom vaabl s tak as pstg t falu tm of a dvc, a cotol chat fo such a data would b basd o pobablty lmts of th falu tms. Ths lmts ad th ctal l a spctvly th solutos of

2 F( UCL,, ) F( CL,, ) 0.5 F( LCL,, ) If th plottd pot falls blow th LCL, t dcats that th pocss avag o th falu occuc that may hav casd whch sults a dcas th falu tm. Ths mas that pocss may hav dtoatd ad thus actos should b tak to dtfy th causs, whch may b movd. Gol ad Okumoto [9] pstd Gol ad Okumoto mpfct dbuggg modl whch s a stochastc modl basd o Posso Pocss wth No- Homogty (NHPP). Akladswa V.S., Pooma R. ad Saavth V. dvlopd a lablty gowth modl of a softwa basd o Lhma-typ Laplac dstbuto-i[2]. To tst Rlablty of a Softwa Akladswa V.S., Pooma R. ad Saavth V. usd Lhma-Typ Laplac dstbuto Typ II (LLD-II)[1] SRGM whch had a btt ft fo softwa falu data tha Gol-okumoto, Wbull, Epotal Gomtc, Pato III, Lhma-Typ Laplac dstbuto Typ I (LLD-I)dstbutos. Thy also dvlopd Lhma-Typ Laplac dstbutos Typ I ad Typ II[3,4] softwa lablty gowth modls too. I ths pap, lablty gowth modl of a softwa s dvlopd scto 2 wth ts paamt stmato. Algothm ad Matlab pogam a also gv ths scto. I scto 3, softwa falu data aalyss s pfomd ad pap s cocludd scto 4. 2.Wbull Od Statstcs Gowth Modl Lt X 1, X 2,..., X b th adom vaabls pstg a sampl of cumulatv tm btw falus. Lt X 1:, X 2:,..., X : b th ogal adom vaabls so that X 1: X 2:..., X : calld th od statstcs. Th pobablty dsty fucto of Wbull th od statstcs s gv by 1 ( 1) f : ( )... (2.1) wh [0, ), 0, 0, 1 Th cumulatv dstbuto fucto s F : ( ) 1 ( )( / ) ( / )... (2.2) 2.1. Paamt stmato Mthod of Mamum lklhood s usd to stmat ad. Th lklhood fucto of Wbull od statstcs s 1 ( 1) l (2.3) Th log-lklhood fucto s 1 ( 1) logl log (2.4) Usg ucostad optmzato tchqu, th mmum of log l s foud. 2.2 NHPP modl fo Wbull od statstcs SRGM Th ma valu fucto fo ths SRGM, usg (2.2), s m( ) a 1 ( )( / ) ( / )... (2.5) Th tsty valu fucto, usg (2.1), s 1 ( ) a... (2.6) a, th pctd umb of falus ( )( / ) ( / ) 1 ( / ) ( 1)( / )... (2.7) 2.3 Algothm fo Wbull Od Statstcs SRGM Stp 1: Fd th cumulatv data of th tm btw falus Stp 2: Choos th valu of Stp 3: Usg mmzato tchqus of o-la ucostad objctv fucto, fd - log l (2.4). Stp 4: Ma f ( z) M( f ( z)), usg ths, fd th mamum of log l multplyg th valu by (-1). Th valus of ad that gvs th mamum of 200

3 log l a th optmum valus of ad. Stp 5: Calculat th pctd umb of falus (2.7) usg ths paamts Stp 6: Fd th cotol lmts UCL, LCL ad CL Stp 7: Estmat th ma valu fucto (2.5) at all falu umbs. Stp 8: Th, fd th succssv dffcs of ma valu fuctos Stp 9: Plot th ma valu chat takg falu umbs alog X-as ad succssv dffcs alog Y-as Stp 10: Th falu umbs at whch th ma valu fucto s blow LCL, dtcts th falu of th softwa. 2.4 MATLAB Pogam fo Wbull Od Statstcs SRGM global =valu of y0=[tal valu fo paamts]; d=0;sd=0; optos=optmoptos(@fmuc,'algothm','quaswto'); [y,fval,tflag,output] = fmuc(@wbull_os,y0,optos); F=1-p(-(()/y(2))^y(1)); sum1=0; fo =: sum1=sum1+choosk(,)*(f^)*((1-f)^(-)); d a=/sum1; fo j=1: osf=0; F=1-p(-((j)/y(2))^y(1)); fo =: osf=osf+(choosk(,)*(f^)*((1-f)^())); d m=a*osf; d=[d m]; dummy=d(j+1)-d(j);sd=[sd dummy]; d UCL= *a;LCL= *a;CL=0.5*a; hold o plot(sd(:,3:+1)); lcl=lcl;flag1=lcl;ucl=ucl;flag2=ucl;cl=cl;flag 3=cl; fo =1: lcl=[lcl flag1]; ucl=[ucl flag2]; cl=[cl flag3]; d plot(lcl);plot(ucl);plot(cl); fucto fu=wbull_os(y) global u=[data] =lgth(u); 201 cum=u(1);sum1=u(1); fo =2: sum1=sum1+u(); cum=[cum sum1]; d =cum; =lgth(); pod=1; fo =1: F=1-p(-(()/y(2))^y(1)); f=(y(1)/y(2))*((()/y(2))^(y(1)-1))*p(- (()/y(2))^y(1)); os=(factoal()/(factoal(-1)*factoal()))*(f^(-1))*((1-f)^(-))*f; pod=pod*os; d fu=-log(pod); 3. Wbull od statstcs SRGM Datast 3.1 Cumulatv Tm btw Falus Falu Numb Tm btw falu tms CPU uts Cumulatv tm btw falus Th followg sult was obtad fo th datast wh tstd usg Wbull od statstcs SRGM. Tabl 3.2 gvs th mamum lklhood valus of datast wh pogam 2.4 s u fo all possbl valus of fo th abov datast.

4 SUCCESSIVE DIFFERENCES OFm() Tabl 3.2 Mamum lklhood valus fo Wbull Od Statstcs dstbuto at all possbl valus of Mamum Lklhood Valus Fom Tabl 3.2, t s foud that mamum lklhood valu fo datast1 s whch s obtad at I od statstcs of Wbull. Thus th SRGM pogam 2.4 s u fo = 1 to tst th falu dtcto. Tabl 3.1 gvs th cumulatv data btw falus. Paamts, Epctd umb of falus, a Tabl 3.3 gvs th ma valu fucto ad ts succssv dffcs. Cotol lmts UCL LCL CL Tabl 3.3 Succssv dffcs of ma valu fucto Succssv Falu Ma valu dffcs of Numb fucto m() m() MEAN VALUE CHART FOR WEIBULL ORDER STATISTICS AT =1 UCL= CL= LCL= FAILURE NUMBER Fgu 3.4 Fgu 3.4 gvs th ma valu chat at I od statstcs of Wbull SRGM. It s foud fom th 202

5 gaph that th falu umbs a dtctd at falu pots 15 ad Cocluso H lablty gowth modl of softwa s dvlopd basd o Od Statstcs of Wbull dstbuto ad t s tstd fo a st of data. Usg ucostad optmzato tchqu, th paamts a stmatd ad valuato of mamum lklhood valus at all ods s do. At th I od, th mamum out of ths s foud ad hc th softwa falu dtcto s do fo I od of Wbull od statstcs dstbuto ad t s dtctd at two falu pots 15 ad 18. Rfc [1] Akladswa V.S., Pooma R. ad Saavth V, Softwa Rlablty Gowth Modl usg Lhma-Typ Laplac Dstbuto-Typ II, Itatoal Joual of Appld Egg Rsach, Vol 10(Issu 13), ,(2015). [2] Akladswa. V. S ad Saavth. V, Lhma- Typ Laplac dstbuto Typ-I Softwa Rlablty Gowth Modl, OPSEARCH, Vol 54 (Issu 2), , (2017). [3] Akladswa. V. S. ad Saavth. V., Softwa Rlablty Gowth Modl basd o Od Statstcs of Lhma-Typ Laplac Dstbuto-Typ II, IJAER, Vol 10(Issu 82), 73-78, (2015). [4] Akladswa. V. S. ad Saavth. V., Algothm fo Softwa Rlablty Gowth Modl basd o Od Statstcs of Lhma- Typ Laplac Dstbuto-Typ I, Rs. J. Mathmatcal ad Statstcal Sc., 4(8), 1-6, (2016). [5] Balaksha. N. ad Coh,A.C., Od Statstcs ad Ifc: Estmato mthods, Acadmc, Bost, (1991). [6] Pham H, Systm Softwa Rlablty, Spg, Lodo(2006). [7] X M., Goh T.N. ad Raja P. Som ffctv cotol chat pocdus fo Rlablty Motog, Rlablty gg ad Systm Safty, Vol 77(Issu 1), [8] Gol. A. L. ad Okumoto. K., Tm dpdt o-dtcto at modl fo softwa lablty ad oth pfomac masus, IEEE Tas. Rlab R-28, Vol 28 (Issu 3), , (1979). 203