NHPP and S-Shaped Models for Testing the Software Failure Process
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1 Irol Jourl of Ls Trds Copug (E-ISSN: Volu, Issu, Dcr NHPP d S-Shpd Modls for Tsg h Sofwr Flur Procss Dr. Kr Arr Asss Profssor K.J. Soy Isu of Mg Suds & Rsrch Vdy Ngr Vdy Vhr Mu. Id. dshuh_3@yhoo.co/rrr@ssr.soy.du Asrc: No-hoogous Posso procss plys por rol sofwr d hrdwr rlly grg. I y rlsc suos hr r wo or or chg pos NHPP odls. I h sofwr rlly, h ur of h flur d s ffcd y y fcors, such s sg vro, sg srgy, d rsourc lloco. Ths fcors r sl hrough h r procss of rlly lyss. I hs ppr, w s h dffr chg pos ccordg o hr xsc y usg so s sscs. Kywords: Chg pos, rlly d S -Shpd Modl.. Iroduco NHPP odls ply por rol sofwr d hrdwr rlly. Mus.l (987, X (99, Ph (999 d Sgpurwll d Wlso (999 og ohrs dvlopd h dffr sofwr rlly odls. Th frs NHPP sofwr rlly odl s proposd y Gol d Ouoo (979, hy ssud h h sofwr flur sy s proporol o h xpcd ur of udcd flurs. Mus f Ouoo (984 gv h logrhc Posso xcuo odl. Zho993, frs cosdrd h chg-po prol sofwr rlly. H odfd h Jls-Mord odl (97 o s h loco of chg po. Chg ( d Zou(3 uss so usful NHPP sofwr rlly odls wh chg po. Shyur(3 corpord oh prfc duggg d chg-po prol o NHPP odl. I h NHPP odl, hr s oly o chg-po d h uow chg po c sd y h xu llhood hod or h LS hod. Howvr, y rlsc suos, h chg-po s uow. Chg ( shows h h wo chg-pos oppos ch ohr. Ch d Gup ( cosdrd prol of ulpl chg pos. I h prs ppr, w cosdrd chgpo dcos. A frs, w gv h NHPP odls wh ulpl chg pos d xu llhood hod s usd o s h chg pos d ohr prrs of odl. To s h xsc of chgpo(s h s sscs s proposd.. NHPP Modls wh Chg Pos My NHPP odls r vry usful o dscr h sofwr flur procss. I h prs ppr h dlyd S Shpd odl wh o chg po s cosdrd, d fr h dlyd S-Shpd odl wh ulpl chg pos r cosdrd.. S-Shpd Modl Sofwr flur procsss r clld s ful coug procss. L N ( s cosdrd s h cuulv ur of sofwr flur y. Th N ( s clld s NHPP wh vlu fuco ( d flur sy (. Gol d Ouoo (979 ssu h h sofwr flur sy ( s proporol o h xpcd ur of udcd flur.., d ( ( ( d ( Whr, s l ur of fuls cod h sofwr d s clld s h ful dco r, h vlu fuco d sy fuco r, Ad ( ( ( R (3 Suppos h sofwr flurs r od d sofwr procss lsd T. L... T whch flurs r
2 Irol Jourl of Ls Trds Copug (E-ISSN: Volu, Issu, Dcr osrvd. Th log-llhood fucos of h osrvd d r, log L (, log λ ( (T (T log log T T T Th xu llhood sor of prrs, r od y solvg h followg wo quos, T (4 ( ( Th log-llhood fuco s, ( log L (,,, log N( log( log( ( N ( (8 T T T 3. S- Shpd Modl wh Chg-Pos (5 Whvr h sofwr sg procss s gog o, h ur of h flur d c ffcd y fcors such s sg vro, sg srgy, rsourcs lloco d so o. I hs cs, s good o us chg-po hod rlly lyss. Th ful dco r s o cos s ssud o hv chg-po. Thrfor, h ful dco r sg c dfd s, ( (6 Th s h chg pos d r h ful dco rs for d fr h chg pos. If h chg po odl s quvl o dlyd S-Shpd odl. Udr h ssupo, d ( ( ( ( d Th vlu fuco d sy fuco c xprssd s, Ad ( ( (7 By Ngug.l (984, ssus h log-llhood fuco ds o fy s h chg po ds o flur fro low. Hc, h s vlu of co od y xzg h log-llhood fuco ovr,t. Wg d Wg (5 rsrc h chg po h rvl,. Hr, w cosdrd h chg po lyg h rvl,. Thrfor h ss vlu of h prrs ˆ,â,ˆ, ˆ r, logl (ˆ,â,ˆ,ˆ x,,,, x x...,,3...,,,, logl,,, x logl,,, If,, h h ss of prr of,, r od y followg quos, ( For prr, ( For prr, (T (9 ( (T (3 For prr, T T (T Slrly, opl soluo of,,,. ( r od s,
3 Irol Jourl of Ls Trds Copug (E-ISSN: Volu, Issu, Dcr ( For prr, (T ( ( ( (... ( ( For prr, (3 For prr T (T (3 T (T (4 I ddo, f 3, h s of, d c od ccordgly, (T (5 T (T (6 T (T (7 Th opl soluo s dod y,, rspcvly. Th h ss ˆ,â,ˆ, ˆ c od y coprg h vlus of log L,,,,...,,,. d L,,, 3,..., log Hr, w prs S-Shpd odl wh chg po, ful dco r s gv y, (,,..., Whr,...,,..., s chg po d,,..., r ful dco rs. Th vlu fuco d sy fuco r s follows: Th log-llhood fuco s. logl (,...,,,,,..., N( log( log( log( log(... (... ( N( log (... ( Th chg pos r rsrcd h rvl,, h log-llhood fuco s oudd d ss of ˆ, ˆ,...,ˆ, ˆ,ˆ,..., ˆ c xu log llhood fuco s xprssd s, logl ˆ, ˆ,...., ˆ,ˆ,ˆ,...,ˆ logl,,...,,,,..., x x,.., (,,,,..., Th ss ˆ, ˆ,...,ˆ, ˆ,ˆ,..., ˆ c od slrly. 4. Ts Sscs of Sgl Chg Po Suppos h flur s,,..., r dsrud s h ordr sscs dpd d dcl rdo spl of sz fro h dsy, f ( (,, I T Whr ( ( (T (T T (T ( d d I(. r h dco T fucos? So w cosruc h llhood ro s sscs:
4 Irol Jourl of Ls Trds Copug (E-ISSN: Volu, Issu, Dcr S log x x f, f, x x L(,,,,,, log x, L (, ~ ~ logl â,ˆ,ˆ, ˆ logl (,,,,,, 5. Sps for Tsg Procdur (8 STEP: - For h S-Shpd dlyd odl h ss of ~, ~ r od y solvg quo 4 d 5. Sp: - For h S-Shpd dlyd odl wh chgpo, h ss ˆ,â,ˆ, ˆ c od y coprg h vlus of logl (,,, s dscussd rlr.,...,,d LogL ( Sp: 3- Clcul S y quo (8. Sp: 4- Th s -lvl s o rjc H f S (, Whr ( s h uppr po of h -dsruo wh o dgr of frdo. Ohrws w ccp h hypohss. 5. A Expl,, ss ˆ d ˆ s lrgr h h hrshold vlu, h hr s chg-po. Hr w c us ou s sscs o s f ˆ = 56 s chg-po. Th vlu of h s sscs s S = 8. > (.5 = 3.84, d w rjc h ull hypohss.. hr s chg-po h sg procss. Tl. Sofwr flur s d: sys T Sofwr Flur s (CPUs , Th d s show l. Ar collcd fro h sys T d Mus (979. Ths d s cluds 36 fuls h sg phs. Hr w us h hod proposd rlr o s h xsc of chg-pos. Th S-Shpd Dlyd odl f of hs d wh o chg-po rsuld prr ss of â 4.3 d 5 ˆ Th S-Shpd Dlyd odl wh o chg-po rsuld prr ss ˆ 4 ˆ 5 ˆ of , d ˆ 58 (.., = 6. Our so of chg-po grs wh h of Wg d Wg (5 d Zou (3. ˆ Alhough s clr h h ss ˆ d r sgfcly dffr. If h dffrc w h 6. Cocluso I sofwr rlly h prol of chg-po s cosdrd d so NHPP sofwr rlly odl wh chg-po hs proposd. Prcclly, h chg-po s uow, d s possl h hr s or h o chg-po. I hs rcl, w cosruc s sscs o s h xsc of chg-po y usg S-Shpd Modl. I h sg procss w fd h hr s xsc of chg-pos. I h sofwr sg phs, sos h flur co osrvd xcly d oly h ur of flurs up o gv s ow. Th d us of hs sg s groupd d. Bu, h lo of y sudy
5 Irol Jourl of Ls Trds Copug (E-ISSN: Volu, Issu, Dcr h y s sscs s o usd for h groupd d. Chg ( suggsd h sg for groupd yp of d. Rfrcs ( Chg, Y.P. ( : Eso of prrs for o-hoogous Posso procss: sofwr rlly wh chg po odl, Couco Sscs: Sulo d Copuo ( Ch, J., Gup, A. K. (: O chg-po dco d so Couco Sscs: Sulo d Copuo (3 Gol, A. L., Ouoo, K. (979 : T dpd rror dco r odl for sofwr rlly d ohr prforc surs, IEEE Trsco o Rlly86-. (4 Jls, Z., Mord, P.B. (97 : Sofwr rlly rsrch, I : Frrgr, W., d. Sscs copur Prforc Evluo. Nw Yor: Acdc Prss, pp (5 Nguy, H. T., Rogrs, G.S., Wlr, E. A. (984: Eso chg-po hzrd r odls Bor (6 Zho. (993: Chg-po prol sofwr d hrdwr rlly, Couco Sscs: Thory d Mhods (7 Zou, F. Z. (3 : chg-po prspcv o h sofwr flur procss Sofwr Tsg Vrfco d Rlly
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