Software Development Cost Model based on NHPP Gompertz Distribution

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1 Idia Joural of Scic ad Tchology, Vol 8(12), DOI: /ijs/2015/v8i12/68332, Ju 2015 ISSN (Pri) : ISSN (Oli) : Sofwar Dvlopm Cos Modl basd o NHPP Gomprz Disribuio H-Chul Kim 1* ad Kyug-Soo Kim 2 1 Dparm of Idusrial ad Maagm Egirig, Namsoul Uivrsiy, Souh Kora; kim1458@su.ac.kr 2 Dparm of Ir Iformaio, Baksok Culur Uivrsiy, Souh Kora; kkskim@bscu.ac.kr Absrac Sofwar rliabiliy i h sofwar dvlopm procss is a impora issu. Sofwar procss improvm hlps i fiishig wih rliabl sofwar produc. I his sudy, rliabiliy sofwar cos modl, cosidrig shap paramr basd o lif disribuio from h procss of sofwar produc sig, was sudid. Th compariso problm of Gol-Okumoo modl ad Gomprz modl rliabiliy growh modl ha is widly usd i h fild of rliabiliy was prsd. Th sofwar failur modl was usd fii failur o-homogous Poisso procss modl ad h paramr simaio usig maximum liklihood simaio was coducd, for aalysis of sofwar cos modl cosidrig shap paramr. This rsarch hlpd sofwar dvloprs i ordr o idify sofwar dvlopm cos. Kywords: Dvlopm Cos Modl, Gomprz Modl, Laplac Trd Ts, NHPP 1. Iroducio Uil ow, may sofwar rliabiliy modls hav b proposd. No-homogous Poisso Procss (NHPP) modls rly o a xcll modl 1,2 i rms of h rror discovry procss, ad if a faul occurs, immdialy rmov h dbuggig procss ad h assumpio ha o w faul has occurrd. I his fild, hacd o-homogous Poisso Procss modl was prsd by Gokhal ad Trivdi 1. Gol ad Okumoo 2 wr proposd a xpoial sofwar rliabiliy modls. I his modl, h oal umbr of dfcs hav S-shapd or xpoial-shapd wih a ma valu fucio was usd. Th gralizd modl rlis o hs modls, dlayd S-shapd rliabiliy growh modl ad iflcio S-shapd rliabiliy growh modl wr proposd by Yamada ad Ohba 3. Zhao 4 proposd a sofwar rliabiliy problms i chag poi ad Shyur 5 usig h gralizd rliabiliy growh modls proposd. Pham ad Zhag 6 sig masurd covrag, h sabiliy of modl, wih sofwar sabiliy ca b valuad prsd. I his papr, h failur ra ovr im (hazard fucio) icrass h Gomprz disribuio, followig h disribuio shap paramr of 2 is applid o h lif disribuio modl so as o compar h cos of sofwar dvlopm, was proposd. 2. Rlad Works 2.1 NHPP Modl This is a class of im domai 1,3,7 sofwar rliabiliy modls which assum ha sofwar failurs display h bhavior of a No-homogous Poisso Procss (NHPP). Th paramr of h sochasic procss, λ() which dos h failur isiy of h sofwar a im, is im-dpd. *Auhor for corrspodc

2 Sofwar Dvlopm Cos Modl basd o NHPP Gomprz Disribuio L N() do h cumulaiv umbr of fauls dcd by im ad m() do is xpcaio. Th m() = E[N()] ad h failur isiy λ() is rlad as Ad, N() was kow o hav a Poisso PDF (probabiliy dsiy fucio) wih paramr m(), ha is: Various im domai modls hav appard i h liraur which dscribs h sochasic failur procss by NHPP. Ths modls diffr i hir failur isiy fucio λ() ad hc m(). L θ do h xpcd umbr of fauls ha would b dcd giv fii failur NHPP modls. Th, h ma valu fucio of h fii failur NHPP modls ca also b wri as: Whr F() is a CDF (cumulaiv disribuio fucio). From Equaio (4), h (isaaous) failur isiy λ() i cas of h fii failur NHPP modls is giv by: Th joi dsiy or h liklihood fucio of x 1, x 2,..., x ca b wri as 1,3,8 : 2.2 Sofwar Dvlopm Cos Modl Sofwar cos modl is dfid as follows 9. No. m () = λ () sds dm() d 0 = λ( ) [ m ( )] m () pn ( ( ) = ) =, = 01,,,! m () =θ F () λ() = θ F () m( x ) f X X ( x x x x 1 2 1,,,, 2,, ) = ( i 1 i ) λ = X E= E + E + E + E = E + C + C m() C [ m( + ) m ( )] 4 (1) (2) (3) (4) (5) (6) (7) E: Th xpcd oal cos of all h sofwar dvlopm cycl; E 1 : Th cos of sofwar dsig ad iiial sofwar dvlopm; lik aalyzig daa, h amou of ma powr, h CPU im ad so o. Is valu is cosa. E 2 : Th cos of sofwar sig; i ohr words, E 2 = C 2 whr C 2 dos h cos pr ui im ad dos im. E 3 : Th cos of rmovig a faul, ha cosiss of aciviis lik dcig h udrlyig faul ad rmovig h faul. I is rlad o h sofwar rliabiliy modlig; i ohr words, E 3 = C 3 m(), Whr C 3 dos h cos of rmovig a faul i h sig phas, m() dos h xpcd umbr of fauls dcd o h im. E 4 : Th cos of fixig a failur, i h opraioal phas ha is also rlad o h sofwar rliabiliy modlig; i ohr words, E 4 = C 4 [m( + ')- m() ], whr C 4 dos h cos of fixig a faul which is obsrvd by usrs i h sofwar opraioal phas afr h sofwar rlas ad ' dos h im of opraig ad maiaiig h sofwar afr rlasig h sofwar sysm. I raliy, h valu of C 4 is much largr ha ha of C 2 ad C 3. As a rsul, opimal sofwar rlas im ca b g accordig o h followig quaio: E = ( E + E + E + E ) = C + C m () C [ m ( + ) m ( )] = 3. Sofwar Rliabiliy Gol- Okumoo ad Gomprz Modl I his scio, Gol-Okumoo ad Gomprz modl wr proposd, 3.1 Gol-Okumoo Modl Th mos basic modl i h fild is Gol-Okumoo modl 2. Th Gol-Okumoo modl is a simpl No- Homogous Poisso Procss (NHPP) modl wih h ma valu fucio m( θ, β) = θ(1 - -β ). Whr h paramr θ is h umbr of iiial faul i h sofwar ad h paramr β is h faul dcio ra. Th corrspodig failur isiy fucio is λ( θ, β) = θβ -β. Th probabiliy dsiy fucio of Gol-Okumoo modl has h form is f ( β) = β -β. Th corrspodig cumulaiv disribuio fucio is F( β) = 1 - -β.... No ha β(>0) is shap paramr. (0, ]. Th maximum liklihood simaios of ach paramr ca b summarizd as β x θ =, = + θ x x i= 1 i (9) β x 1 β (8) 2 Vol 8 (12) Ju Idia Joural of Scic ad Tchology

3 H-Chul Kim ad Kyug-Soo Kim 3.2 Comprz Modl I probabiliy ad saisics, h Gomprz disribuio 10 is a coiuous probabiliy disribuio. Th Gomprz disribuio is of applid o dscrib h disribuio of adul lifspas by dmographrs ad acuaris. Rlad filds of scic such as biology ad groology also cosidrd h Gomprz disribuio for h aalysis of survival. Mor rcly, compur sciiss hav also sard o modl h failur ras of compur cods by h Gomprz disribuio. I markig scic, i has b usd as a idividual-lvl modl of cusomr lifim. Th probabiliy dsiy fucio ad h cumulaiv disribuio fucio for Gomprz disribuio usig various filds of idusry disribuio usd ar as b η f( b, η) = bη xp( η ), F ( b, η) = 1 xp( η( 1)) b No ha η(>0) is shap paramr ad b(>0) is scal paramr. (0, ]. Thus, h ma valu fucio ad h isiy fucio of h fii failur NHPP modl usig (4) ad (5) quaios ca b xprssd as b m () = θf ( b, η) = θ 1 xp( η( 1)), λ() = θ f( b, η) = θbη Usig h xprssios (15), for h maximum liklihood sima of ach paramr, θˆmle ad βˆmle saisfy h followig quaio. No. x_ = (x 1, x 2, x 3,..., x ), Θ is paramr spac. Th xprssio (12) ad (13) ca b summarizd as b b η xp( η ) b (10) (11) l LNHPP ( Θ x) = 1 xp( η ( 1)) θ θ = 0 (12) l LNHPP ( Θ x) = + b b x i i η θη x θ =, 1 xp( η ( 1)) ( 1 i x = 1 i= 1 i η ) = 0 i = + η + θη x = i x x i 1 i= 1 i b η( 1) (13) (14) 4. Illusraio Th daa s of sofwar fauls aalyzd hr was obaid from liraur 11. I his papr, w cosruc h corrspodig sofwar rliabiliy growh modl by usig h da i Tabl 1, ad o h basis of SRGM, w calcula h oal cos of sofwar sig by usig h proposd sofwar cos modl. Th accordig o modifyig h valu of diffr paramrs 12, w dircly illusra h ffc of various paramrs o h opimal sofwar rlas im. I ordr o prs a rus modl o aalyz h rd for firs d aa should b prcdd by rd s 13. I gral, h Laplac rd s aalysis is usd. As a rsul of his s i his Figur 1, as idicad i h Laplac facor is bw 2 ad -2, rliabiliy growh shows h propris. Thus, usig his daa, i is possibl o sima h rliabiliy 14,15. I his papr, umrical covrsio daa ( im 0.01) i ordr o facilia h paramr simaio was usd. Th rsul of paramr simaio has b summarizd i Tabl 2. Rfrrig o h paramrs i h liraur 12,14, w modify som valus ad g h followig codiios 13 : (A) Assumig, E 1 = 50, C 2 =30, C 3 =5, C 4 =50, '=200, w ca g: Tabl 1. umbr im daa im irval umbr im irval Vol 8 (12) Ju Idia Joural of Scic ad Tchology 3

4 Sofwar Dvlopm Cos Modl basd o NHPP Gomprz Disribuio Figur 1. Laplac rd s. Tabl 2. Modl Paramr Esimas Of Each Modl MLE (Maximum liklihood simaio) Gol-Okumoo θˆmle = βˆmle = Comprz disribuio θˆmle = bˆmle = Figur 2. Th curv udr h codiio of (A). As show i Figur 2, h growh curv of h proposd modl firsly dcrass ad h icrass. This shows ha h umbr of rsidual fauls i sofwar sysm is lss ad lss durig h procss of faul rmovig, ad h probabiliy of ha h rmaiig fauls ar obsrvd by usrs afr sofwar rlas is lowr ad lowr. I h arly phas of sig hr ar sill may fauls i sofwar which ar asily dcd ad rmovd, ad h cos of rmovig a faul i his phas is far lowr ha ha of rmovig a faul i h opraio phas, so h oal cos of sofwar dcrass wih h procss of fauls dbuggig. Bu i h lar phas h umbr of fauls rmaiig i sofwar is alrady lss, ad i his sig phas h im of dcig a faul is vry log ad h cos of rmovig a faul bcoms highr ha ha i h opraio phas, so h cos curv icrass cosaly wih im. From h rd of h cos curv w ca calcula h opimal sofwar rlas im ad i is also h mos ralisic siuaio. Mos cass ar cosis wih his i h procss of acual sofwar dvlopm 9,12. Wh comparig basic Gol-Okumoo modl ad Comprz disribuio modl, cas of sig cos pr ui of a dfc i h procss is largr ha h Gol-Okumoo modl, opimal sofwar rlas of Comprz disribuio modl ca b s ha h im dlay. Bu h cos is almos similar o apparig. (B) Assumig, E 1 = 50, C 2 =5, C 3 =30, C 4 =50, '=200, ca g h rsuls i Figur 3. Rsuls i Figur 3 od ha ohr assumpios ar h sam; cas of sig cos pr ui is smallr ha rmov a dfc i sig procss. I his cas, opimal Figur 3. Th curv udr h codiio of (B). sofwar rlas for h Comprz disribuio modl ca b s ha h im dlay. Bu, i rms of cos, bcaus of cos for Comprz disribuio modl ha basic Gol- Okumoo modl lss, Comprz disribuio modl ca b ffciv. I his papr, usig basic Gol-Okumoo modl ad Comprz disribuio modl, rsul of h compariso of sofwar cos modls, i rm of opimal sofwar rlas basic Gol-Okumoo is ffici modl ad i rms of cos Comprz disribuio modl ffici modl. Sofwar rliabiliy growh modl ca sima h opimal sofwar rlas im ad h cos of sig ffors. Mor accura modl is dd o dcras h sig cos ad icras h profi of rlasig sofwar. Th us of sofwar cos modl ca hlp prdic h opimal sofwar rlas im accuraly. Compard wih prvious modls, h proposd modl aks io accou h oal umbr of fauls discovrd by usrs durig h sofwar opraio priod or sofwar maiac afr is rlas, rahr ha simply assum ha rsidual fauls ha ar o dcd will b all foud by h usr. I ca b s ha h cos of acual faul dbuggig is lowr ha h cos of rmovig all rmaiig fauls i h opraio phas. So h opimal sofwar rlas im is ahad of im, ad i is mor ralisic. 4 Vol 8 (12) Ju Idia Joural of Scic ad Tchology

5 H-Chul Kim ad Kyug-Soo Kim 5. Coclusio Sofwar rliabiliy growh modl ca sima h opimal sofwar rlas im ad h cos of sig ffors. Mor accura modl is dd o dcras h sig cos ad icras h profi of rlasig sofwar. Th us of sofwar cos modl ca hlp prdic h opimal sofwar rlas im accuraly. Compard wih prvious modls, h proposd modl aks io accou h oal umbr of fauls discovrd by usrs durig h sofwar opraio priod or sofwar maiac afr is rlas, rahr ha simply assum ha rsidual fauls ha ar o dcd will b all foud by h usr. I ca b s ha h cos of acual faul dbuggig is lowr ha h cos of rmovig all rmaiig fauls i h opraio phas. So h opimal sofwar rlas im is ahad of im, ad i is mor ralisic. Thrfor, i his papr, h proposd Comprz disribuio modl ca b usd as a alraiv modl i his fild i rms of cos. As a alraiv o his ara fl ha h co is a valuabl rsarch. 6. Rfrcs 1. Gokhal SS, Trivdi KS. A im/srucur basd sofwar rliabiliy modl. Aals of Sofwar Egirig. 1998; 8(1-4): Gol AL, Okumoo K. Tim dpd rror - dcio ra modl for sofwar rliabiliy ad ohr prformac masur. IEEE Tras Rliabiliy. 1979; R-28(3): Yamada S, Ohba H. S-shapd sofwar rliabiliy modlig for sofwar rror dcio. IEEE Tras Rliabiliy. 1983; 32(5): Zhao M. Chag-poi problms i sofwar ad hardwar rliabiliy. Commuicaio Sa Thory Mhods. 1993; 22(3): Shyur H-J. A sochasic sofwar rliabiliy modl wih imprfc dbuggig ad chag-poi. J Sys Sofwar. 2003; 66(2): Pham H, Zhag X. NHPP sofwar rliabiliy ad cos modls wih sig covrag. Eur J Opr Rs. 2003; 145(2): Kui-Ch C, Yu-Shiag H, Tzai-Zag L. A sudy of sofwar rliabiliy growh from h prspciv of larig ffcs. Rliabiliy Egirig ad Sysm Safy. 2008; 93(10): H-Chul K. Th comparaiv sudy of NHPP dlayd S-shapd ad xrm valu disribuio sofwar rliabiliy modl usig h prspciv of larig ffcs. Iraioal Joural of Advacms i Compuig Tchology. 2013; 5(9): Zhag Y, Wu K. Sofwar cos modl cosidrig rliabiliy ad im of sofwar i us. Joural of Covrgc Iformaio Tchology. 2012; 7(13): Availabl from: hp://.wikipdia.org/wiki/gomprz_ disribuio 11. Saya Prasad R, Rao KRH, Kaha RRL. Sofwar rliabiliy masurig usig modifid maximum liklihood simaio ad SPC. Iraioal Joural of Compur Applicaios. 2011; 21(7): Zhag XM, Pham H. A sofwar cos modl wih rror rmoval ims ad risk coss. Sysm Scic. 1998; 21: Kaou K, Lapri JC. Hadbook of sofwar rliabiliy girig. I: Lyu MR, dior. Chapr: Trd Aalysis. McGraw-Hill, Nw York, NY. 1996; p H-Chul K, Hyoug-Ku P. Th comparaiv sudy of sofwar opimal rlas im basd o burr disribuio. Iraioal Joural of Advacms i Compuig Tchology. 2010; 2( 3): H-Chul K, Ja-Wook K. Trucad log shapd yp sofwar rliabiliy growh modl. Covrgc ad Hybrid Iformaio Tchology. Lcur Nos i Compur Scic. 2012; 7425: Vol 8 (12) Ju Idia Joural of Scic ad Tchology 5