The Infinite NHPP Software Reliability Model based on Monotonic Intensity Function

Transcription

1 Id Jourl of Scece d Techology, Vol 8(4), DOI:.7485/js/25/v84/68342, July 25 ISSN (Pr) : ISSN (Ole) : The Ife Sofwre Relly Model sed o Moooc Iesy Fuco Te-Hyu Yoo * Deprme of Scece To, Nmseoul Uversy, Souh Kore; yh@su.c.kr Asrc Sofwre esg (deuggg) order o reduce coss erms of chges he sofwre relly d esg coss, eed o kow dvce s more effce. I hs pper, proposes he relly model wh moooc esy fucos (Power-Lw, Mus-Okumoo d Comperz model), whch mde ou effcecy pplco for sofwre relly. Algorhm o esme he prmeers used o mmum lkelhood esmo d seco mehod, model seleco sed o me squre error d coeffce of deermo, for he ske of effce model, were employed. Alyss of flure usg rel d se for he ske of proposg moooc esy fuco ws employed. Ths lyss of flure d compred wh moooc esy fuco. I order o surce for he relly of d, Lplce red es ws employed. I hs sudy, he proposed moooc esy fuco s more effce erms of relly hs re. Thus, moooc esy fuco c lso e used s lerve model. From hs pper, sofwre developers hve o cosder he growh model y pror kowledge of he sofwre o defy flure modes whch c e le o help. Keywords: Lplce Tred Tes, Moooc Iesy Fuco,, Sofwre Relly. Iroduco Sofwre flures cused y flure of compuer sysems our socey c led o huge losses. Thus, sofwre relly he sofwre developme process s mpor ssue. These ssues of he user requremes mee he cos of esg. Sofwre esg (deuggg) order o reduce coss erms of chges he sofwre relly d esg coss, eed o kow dvce s more effce. Thus, he relly, cos, d cosdero of relese me for sofwre developme process re essel. Eveully he sofwre o predc he coes of defec he produc developme model s eeded. Ul ow, my sofwre relly models hve ee proposed. No-Homogeous Posso Process () models rely o ecelle model,2 erms of he error dscovery process, d f ful occurs, mmedely remove he deuggg process d he ssumpo h o ew ful hs occurred. I hs feld, ehced o-homogeous Posso Process model ws preseed y Gokhle d Trved. Goel d Okumoo 2 proposed epoel sofwre relly models. I hs model, he ol umer of defecs hve S-shped or epoel-shped wh me vlue fuco ws used. The geerlzed model reles o hese models, delyed S-shped relly growh model d fleco S-shped relly growh model were proposed y Ymd d Oh 3. Zho 4 proposed sofwre relly prolems chge po d Shyur 5 usg he geerlzed relly growh models proposed. Phm d Zhg 6 esg mesured coverge, he sly of model, wh sofwre sly c e evlued preseed. Relvely recely, Hug 7, geerlzed logsc esg-effor fuco d he chge-po prmeer y corporg effce echques o predc sofwre relly, were prese. Kue-Che 8 c epl he lerg process h sofwre mgers o ecome fmlr wh he sofwre d es ools for S-ype model. I ddo, Km 9 suded ou he comprve sudy of delyed S-shped d ereme vlue dsruo sofwre relly model usg he perspecve of lerg effecs *Auhor for correspodece

2 The Ife Sofwre Relly Model sed o Moooc Iesy Fuco d Sh d Km suded ou he comprve sudy of sofwre opml relese me sed o sofwre relly model usg epoel d log shped ype for he perspecve of lerg effec. I hs pper, proposes he relly model wh moooc esy fucos (Power-Lw, Mus-Okumoo 3 d Comperz model), whch mde ou effcecy pplco for sofwre relly. Algorhm o esme he prmeers used o mmum lkelhood esmor d seco mehod, model seleco sed o me squre error d coeffce of deermo, for he ske of effce model, ws employed. Alyss of flure usg rel d se for he ske of proposg moooc esy fuco ws employed. Ths lyss of flure d compred wh moooc esy fuco. The purpose of hs sudy s o lyze sofwre relly model wh moooc esy fuco. 2. Reled Works 2. Model Ths s clss of me dom, sofwre relly models whch ssume h sofwre flures dsply he ehvor of No-Homogeeous Posso process (). The prmeer of he sochsc process, l()whch deoes he flure esy of he sofwre me, s me-depede. Le N() deoe he cumulve umer of fuls deeced me d m() deoe s epeco. The m() = E [N()] d he flure esy l() s reled s follows 9, : Ad, m () = l () sds dm() d () = l () (2) N() ws kow o hve Posso proly desy fuco(pdf) wh prmeer m(), h s: [ m ( )] m () pn ( ( ) = ) = e, =,,, (3)! These me dom models for he process c e descred y he proly of flure re possle. Ths model s he flure esy fuco (flure occurrece res per ful) l() epressed dfferely, lso me vlue he fuco m() wll e epressed dfferely. The models c e furher clssfed o fe flure d fe flure cegores. Fe flure models ssume h he epeced umer of fuls deeced gve fe mou of esg me wll e fe, wheres he fe flures models ssume h fe umer of fuls would e deeced fe esg me. Thus, usg Geerl Order Sscs (GOS) hs ecome flure 2. O he oher hd, usg Record Vlue Sscs (RVS) hs ecome fe. Typclly, he Geerlzed Order Sscs (GOS) model hs N defecs d y of he N defecs from he proly desy fuco (PDF); geered po ccordg o he order ssc s he me po of flure. I hs model, he me of ech repr, ew defec s ssumed o o occur. However, he cul suo he po of repr ew flure my occur. So s o dd for hs suo, he Record Vlue Sscs (RVS) model c e used model d me vlue fuco ws s follows: sy. m () = l( F ( )) (4) Where, ep ( m()) = F() d F() s cumulve he dsruo fuco d f() s he proly desy fuco. Therefore, from Equo (4) usg he reled equos of Equo (), esy fuco c e he hzrd fuco (h(),. I oher words, l( ) = m () = f()/( F()) = h( ) (5) Le q deoe he epeced umer of fuls h would e deeced gve fe flure models. The, he me vlue fuco of he fe flure models c lso e wre s 9, : m() = q F() (6) Noe h F() s Cumulve Dsruo Fuco (CDF). I Equo (4), he (seous) flure esy fuco l() cse of he fe flure models s gve y: Ths c e re-wre s: q() = q F'() (7) l() [ q m ( )] F = () [ q m ( )] h () F (8) Where h() s he flure occurrece re per ful of he sofwre, or he re whch he dvdul fuls 2 Vol 8 (4) July 25 Id Jourl of Scece d Techology

3 Te-Hyu Yoo mfes hemselves s flures course of esg process. The quy [q m()] deoes he epeced umer of fuls remg he sofwre me. Sce [q m()] s moooclly o-cresg fuco of me (cully [q m()] should decrese s more d more fuls re deeced d removed ), he ure of he overll flure esy, l() s govered y he ure of flure occurrece re per ful h(), from Equo (6). The flure occurrece re per ful h() c e cos or cresg, decresg. I hs seco, we descre some of he fe flure models log wh her hzrd fucos. Le {, =, 2, } deoe he sequece of mes ewee successve sofwre flures. The deoe he me ewee ( ) s d h flure. Le deoe flure me, so h: = (9) = The jo desy or he lkelhood fuco of, 2,, c e wre s, : m ( ) fx, X 2,, X(, 2,, ) = e l ( ) () = For gve sequece of sofwre flure mes (, 2,, ), h re relzos of he rdom vrles (X, X 2,, X ), he prmeers of he sofwre relly growh models re esmed usg he Mmum Lkelhood Mehod (MLE). As resul, he codol relly R ( d ) s kow s follows 3, : l ( ) d R ( ) d = e = ep[ { m ( d+ ) m ( )}] + d () Where d deoe msso me d s he ls flure me. 2.2 Model Comprso wh Rel Dse I order o vesge he effecveess of he model, he comprso crer 8, usg me squre errors d coeffce of deermo re descred s follows: 2.2. Me Squre Errors (MSE) m m MSE = ( ( ) ( )) 2 = k (2) Where m( ) s he ol cumuled umer of errors oserved wh me ( ), m ( ) s esmed cumuled umer of errors me oed from he fg me vlue fuco, s he umer of oservos d k s he umer of prmeers Coeffce of Deermo (R 2 ) R-Squre (R 2 ) c mesure how successful he f s eplg he vro of he d. I s defed s follows: R 2 = = [ m ( ) m ( )] [ m ( ) m ( )/ ] = j= j 2 2 (3) 3. Sofwre Relly Model usg Cosderg Moooc Iesy Fucos 3. Power Lw Iesy Fuco I s kow, s follows: Power lw esy fuco 2. l P L()= (4) Noe h (> ) scle prmeer d (> ) s shpe prmeer (, ). The me vlue fuco, usg he esy fuco Equo (4), s derved s follows: m ( Θ) = l ( s) ds= P L P L (5) Noe h Θ = (, ) s prmeer spce. Therefore, for fe flure model usg Equo (6), he lkelhood fuco s follows: L ( Θ ) = ep = (6) Noe: = (, 2,, ). The mmum lkelhood esmo mehod of he prmeer esmo mehod ws used. Log-lkelhood fuco of Equo (4) s he epresso derved s follows. l L ( Θ ) = l + l ( ) l Usg he epressos (7), ă MLE = d MLE (7) ssfy he followg equo for he mmum lkelhood esme of ech prmeer. Vol 8 (4) July 25 Id Jourl of Scece d Techology 3

4 The Ife Sofwre Relly Model sed o Moooc Iesy Fuco l L ( Θ ) = = (8) l L ( Θ ) = = l l = (9) Usg he epressos (), he relly s derved s follows: R ( d ) ep[ ( d ) = + + ] Noe h d s msso me. 3.2 Mus-Okumoo Iesy Fuco (2) Me vlue fuco of Mus-Okumoo model s kow s logrhmc fuco 3,4. The me vlue fuco s s follows: mm O( Θ) = l( + ) (2) Noe h (> ) s scle prmeer, (> ) s shpe prmeer d Θ = (, ) s prmeer spce. (, ). The esy fuco, usg me vlue fuco Equo (2), s derved s follows: l () = m () = ( + ) M O M O (22) Therefore, for fe flure model usg Equo (6), he lkelhood fuco s follows L ( Θ ) = ep[ l( + )] = ( + ) (23) Noe: = (, 2,, ). The mmum lkelhood esmo mehod of he prmeer esmo mehod ws used. Log-lkelhood fuco of Equo (22) s he epresso derved s follows. l L ( Θ ) = l+ l l( + ) l( + ) = (24) Usg he epressos (24), ă MLE d MLE ssfy he followg equo for he mmum lkelhood esme of ech prmeer. l L ( Θ ) = l( + ) = (25) l L ( Θ ) = = (26) = ( ) + + Smlrly, he relly s derved s follows: Rˆ( d ) = ep[ { m ( d+ ) m ( )}] (27) Noe h d s msso me. m( ) l( ), m( d ) = + + = l[ + ( + d)], m( d+ ) = l[ + ( + d)]. 3.3 Gomperz Iesy Fuco I proly d sscs, he Gomperz dsruo 5 s couous proly dsruo. The Gomperz dsruo s ofe ppled o descre he dsruo of dul lfesps y demogrphers d cures. Reled felds of scece such s ology d geroology lso cosdered he Gomperz dsruo for he lyss of survvl. More recely, compuer scess hve lso sred o model he flure res of compuer codes y he Gomperz dsruo. I mrkeg scece, hs ee used s dvdul-level model of cusomer lfeme. The proly desy fuco d he cumulve dsruo fuco for Gomperz dsruo usg vrous felds of dusry dsruo used re s follows: f (, ) = e e ep( e ) (28) F (, ) = ep( ( e )) (29) Noe h (> ) s shpe prmeer d (> ) s scle prmeer. (, ). I fe flure, hzrd fuco d esy re sme form. The hzrd fuco, usg Equo (28) d Equo (29), s derved s follows: F'( ) f () h () = = = e = lg P() F () F () (3) Noe h (> ) s scle prmeer, (> ) s shpe prmeer. The me vlue fuco, usg he esy fuco Equo (3), s derved s follows: mg P( ) G P( s) ds ( e Θ = l = ) (3) Smlrly, for he Gomperz model, he lkelhood fuco s follows: L ( Θ ) =( e ) ep[ ( e )] (32) = Noe: = (, 2,, ). The mmum lkelhood esmo mehod of he prmeer esmo mehod ws used. Log-lkelhood 4 Vol 8 (4) July 25 Id Jourl of Scece d Techology

5 Te-Hyu Yoo fuco of Equo (3) s he epresso derved s follows. = l L ( Θ ) = l+ l l ( e ) (33) Usg he epressos (33), ă MLE d MLE ssfy he followg equo for he mmum lkelhood esme of ech prmeer. l L ( Θ ) = ( e ) = l L ( Θ ) (34) = + e = (35) = Smlrly, he relly s derved s follows: Rˆ( d ) = ep[ { m ( d+ ) m ( )}] (36) Noe h d s msso me. m( ) = ( e ), m( d+ ) = e ( d ), m( ) e ( ) ( + d ) d+ = ( + ). I order o prese relly model d lyze he red for frs d should e preceded red es 7. I geerl, he Lplce red es lyss s used. As resul of hs es hs Fgure, s dced he Lplce fcor s ewee 2 d 2, relly growh shows he properes. Thus, usg hs d, s possle o esme he relly 9. I hs pper, umercl coverso d (Flure me (hours).) order o fcle he prmeer esmo ws used. The resul of prmeer esmo hs ee summrzed Tle 2. These clculos, solvg umerclly, he l vlues gve o. d 3 d olerce vlue for wdh of ervl ( 5 ) gve usg C-lguge checkg deque coverge, were performed ero of mes. The resul of Me Squre Error (MSE) d coeffce of deermo (R 2 ) re hs ee summrzed Tle 2. For sofwre model comprso Tle 2, MSE (whch mesures he dfferece ewee he cul vlue d he 4. Sofwre Flure Tme D I hs seco, w o lyze he proposed relly model usg sofwre flure me d 6. Ths flure d s lsed Tle. Tle. Flure Numer Flure me d Flure Tme (hours) Flure Tme Flure Numer Flure Tme (hours) Flure Tme Fgure. Tle 2. Model Lplce red es. Prmeer esmo, MLE d MSE MLE Power-Lw ˆ MLE = ˆ MLE = Mus-Okumoo ˆ MLE = ˆ MLE =.647 Gomperz ˆ MLE = ˆ MLE =.683 Noe. MLE: Mmum lkelhood esmo; MSE: Me squre error; R 2 : Coeffce of deermo Model Comprso MSE R Vol 8 (4) July 25 Id Jourl of Scece d Techology 5

6 The Ife Sofwre Relly Model sed o Moooc Iesy Fuco predced vlue) show h Gomperz model h Mus-Okumoo d power-lw hs smll vlue. Therefore, he Gomperz s pprecly eer h oher fe model. Also, R 2 (whch mes h he predcve power of he dfferece ewee predced vlues) show h Gomperz h oher model hs hgh vlue. Thus, Gomperz fe model s he uly model. Eveully, erms of MSE d R 2, Gomperz fe regrd s es model ecuse MSE s he smlles d R 2 s he hghes h oher models. I erms of comprso of relly Fgure 2, cse of relly for ssumed msso me, Mus-Okumoo model h oher models hs show hgh relly. Eveully, he relly hs sesve for he msso me. The resul of me vlue fucos re hs ee summrzed Fgure 3. I fgure, pers of me vlue fuco hve he edecy of o-decresg form. Also, hs fgure show h Gomperz model h y models s he uly model ecuse Gomperz fe model h y model esmes close o he Fgure 2. Relly of ech model. rue vlue from me vlue fucos. Ad, he resul of esy fucos re hs ee summrzed Fgure 4. I hs fgure, esy fucos of power-lw model hve he edecy of o-cresg d Mus-Okumoo model hve o-decresg form. The oher hd, Gomperz model hve he edecy of erly cos form. 5. Cocluso Sofwre relly growh model c esme he opml sofwre relese me d he cos of he esg effors. More ccure model s eeded o decrese he esg cos d crese he prof of relesg sofwre. The use of sofwre cos model c help predc he opml sofwre relese me ccurely. Compred wh prevous models, he proposed model kes o ccou he ol umer of fuls dscovered y users durg he sofwre opero perod or sofwre mece fer s relese, rher h smply ssume h resdul fuls h re o deeced wll e ll foud y he user. I c e see h he cos of cul ful deuggg s lower h he cos of removg ll remg fuls he opero phse. So he opml sofwre relese me s hed of me, d s more relsc. I furher sudes, we eed o check he vldy d effecveess of our proposed sofwre relly growh model d he sofwre cos model uder he modelg frmework y usg much cul flure d. Ad wll cosder oher prmeers whch c lso deerme he sofwre esg cos, whch s o cosdered hs pper. The uder hs codo, wll lyze he ew cos model o cosly mmze he cos of sofwre developme d ge more profs whe relesg he sofwre. Geerlly, whe lerg fcor s he hghes d uoomous errors-deeced fcor s he lowes, model ws effecve Fgure 3. Me vlue fuco of ech model. Fgure 4. Iesy fuco of ech model. 6 Vol 8 (4) July 25 Id Jourl of Scece d Techology

7 Te-Hyu Yoo depressg. Therefore, hs pper, he proposed model c e used s lerve model hs feld. As lerve o hs re feel h he coe s vlule reserch. 6. Ackowledgme Fudg for hs pper ws provded y Nmseoul Uversy. 7. Refereces. Gokhle SS, Trved KS. A me/srucure sed sofwre relly model. Als of Sofwre Egeerg. 998; 8: Goel AL, Okumoo K. Tme Depede Error - Deeco Re model for sofwre relly d oher performce mesure. IEEE Trs Relly. 979; R-28(3): Ymd S, Oh H. S-shped sofwre relly modelg for sofwre error deeco. IEEE Trs Relly. 983; 32: Zho M. Chge-po prolems sofwre d hrdwre relly. Commuco S Theory Mehods. 993; 22(3): Shyur H-J. A sochsc sofwre relly model wh mperfec deuggg d chge-po. J Sys Sofwre. 23; 66: Phm H, Zhg X. sofwre relly d cos models wh esg coverge. Eur. J Oper Res. 23; 45: Hug C-Y. Performce lyss of sofwre relly growh models wh esg-effor d chge-po. J Sys Sofwre. 25; 76: Kue-Che C, Yeu-Shg H, Tz-Zg L. A sudy of sofwre relly growh from he perspecve of lerg effecs. Rel Eg Sys Sf. 28; 93: Hee-Cheul K. The comprve sudy of delyed S-shped d ereme vlue dsruo sofwre relly model usg he perspecve of lerg effecs. Ierol Jourl of Advcemes Compug Techology (IJACT). 23; 5(9):2 8.. Hyu-D S, Hee-Cheul K. The comprve sudy of sofwre opml relese me sed o hpp sofwre relly model usg epoel d log shped ype for he perspecve of lerg effecs. Ierol Jourl of Advcemes Compug Techology (IJACT). 23; 5(2):2 9.. Kuo L, Yg TY. Byes compuo of sofwre relly. J Am S Assoc. 996; 9: Avlle from: hp:// pr/seco/pr72.hm 3. Mus JD, Okumoo K. A logrhmc Posso eecuo me model for sofwre relly mesureme. ICSE 84. Proceedgs of 7h Ierol Coferece o Sofwre Egeerg; 984. p Vek SRK, Rveedr Bu B. A log sed pproch for sofwre relly modelg. I J Adv Res Compu Sc Sofwre Eg. 24; 4(2): Avlle from: hp://e.wkped.org/wk/gomperz_ dsruo 6. Hykw Y, Telfr G. Med posso-ype processes wh pplco sofwre relly. Mh Compu Model. 2; 3: Kou K, Lpre JC. Hdook of sofwre relly egeerg. I: Lyu MR, edor. Chper Tred Alyss. New York, NY: McGrw-Hll; 996. p Vol 8 (4) July 25 Id Jourl of Scece d Techology 7