Software Reliability Growth Models Incorporating Fault Dependency with Various Debugging Time Lags

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1 Sofwre Relbly Growh Models Incorporng Ful Dependency wh Vrous Debuggng Tme Lgs Chn-Yu Hung 1 Chu-T Ln 1 Sy-Yen Kuo Mchel R. Lyu 3 nd Chun-Chng Sue 4 1 Deprmen of Compuer Scence Nonl Tsng Hu Unversy Hsnchu Twn. Deprmen of Elecrcl Engneerng Nonl Twn Unversy Tpe Twn. 3 Compuer Scence nd Engneerng Deprmen The Chnese Unversy of Hong Kong Shn Hong Kong. 4 Deprmen of Compuer Scence nd Informon Engneerng Nonl Cheng-Kung Unversy Tnn Twn. Absrc Sofwre relbly s defned s he probbly of flure-free sofwre operon for specfed perod of me n specfed envronmen. Over he ps 30 yers mny sofwre relbly growh models (SRGMs) hve been proposed nd mos SRGMs ssume h deeced fuls re mmedely correced. Acully hs ssumpon my no be relsc n prcce. In hs pper we frs gve revew of ful deecon nd correcon processes n sofwre relbly modelng. Furhermore we wll show how severl esng SRGMs bsed on NHPP models cn be derved by pplyng he me-dependen dely funcon. On he oher hnd s generlly observed h muully ndependen sofwre fuls re on dfferen progrm phs. Somemes muully dependen fuls cn be removed f nd only f he ledng fuls were removed. Therefore here we ncorpore he des of ful dependency nd me-dependen dely funcon no sofwre relbly growh modelng. Some new SRGMs re proposed nd severl numercl emples re ncluded o llusre he resuls. Epermenl resuls show h he proposed frmework o ncorpore boh ful dependency nd me-dependen dely funcon for SRGMs hs frly ccure predcon cpbly. 1. Inroducon Drmc dvnces n sofwre echnologes hve grely promoed he growh of compuer pplcons. More nd more crcl pplcons such s bnkng pymen sysems cred crd nd shred ATM Sysems ec. re beng developed. The sofwre for hese pplcons s becomng ncresngly comple nd sophsced. Thus relbly wll become he mn gol for sofwre developers. Sofwre relbly s ofen defned s he probbly of flure-free sofwre operon for specfed perod of me n specfed envronmen [1]. Over he ps 30 yers mny Sofwre Relbly Growh Models (SRGMs) hve been proposed for esmon of relbly growh of producs durng sofwre developmen processes [-6]. From our sudes we fnd h mny ppers consder n NHPP s sochsc process o descrbe he ful process nd relbly growh of mos SRGMs s epressed s eponenl curve [7]. On he oher hnd Ohb [ 8-9] proposed n nfleced S-shped model o descrbe he sofwre flure-occurrence phenomenon wh muul dependency of deeced fuls. He hough h he eponenl SRGM ws somemes nsuffcen nd nccure o nlyze cul sofwre flure d for relbly ssessmen. Moreover Ymd e l. [ ] lso presened delyed S-shped SRGM ncorporng he me dely beween ful deecon nd ful correcon. Acully Ohb conceved h here were wo ypes of fuls n sofwre sysem: muully ndependen fuls nd muully dependen fuls [8]. The muully ndependen fuls re on dfferen progrm phs. Muully dependen fuls cn be removed f nd only f he ledng fuls re removed. Ler Kpur e l. [1-13] proposed n SRGM h ook cre of he underlyng ful dependency. They consdered h n sofwre sysem he ful removl depended on he prevously removed fuls nd h would resul n dely of he ful removl process. One common ssumpon of convenonl SRGMs s h deeced fuls re mmedely removed. In prcce hs ssumpon my no be relsc n sofwre developmen. We know h sofwre esng nd debuggng re very comple nd epensve processes. The me o remove ful depends on he compley of he deeced fuls he sklls of he debuggng em he vlble mnpower or he sofwre developmen envronmen ec. Therefore he me delyed by he deecon ndor correcon process should no be neglgble. There re some ppers h hve ddressed he problem of delyed ful correcon me [14-4]. For emple Schnedewnd [15-17] proposed n pproch o model he ful-correcon process by usng consn delyed fuldeecon process. He ssumed h he re of ful correcon ws proporonl o he re of flure deecon. However f hs ssumpon s no me n prcce he model wll underesme he remnng fuls n he code [0]. Ler Xe nd Zho [18 0] poned ou h hs

2 ssumpon ws oo resrcve. They eended he Schnedewnd model o connuous verson by subsung me-dependen dely funcon for he consn dely. Moreover Gošev-Popsojnov nd Trved [1] presened sofwre relbly modelng frmework bsed on Mrkov renewl process whch ncorpored he possble s- dependence mong successve sofwre runs number of runs beween flures nd occurrence me of flure. In hs pper we frs gve revew of ful deecon nd correcon processes n sofwre relbly growh models. Furhermore we show how severl esng SRGMs bsed on NHPP models cn be derved by pplyng he medependen dely funcon. On he oher hnd s probbly h muully ndependen sofwre fuls re on dfferen progrm phs nd muully dependen fuls cn be removed f nd only f he ledng fuls were removed. Thus we wll ncorpore he des of flure dependency nd me-dependen dely funcon no sofwre relbly growh modelng. The res of he pper s orgnzed s follows. Secon gves bref revew of chrcerscs of he NHPP models wh delyed correcon process nd shows how some esng NHPP models cn be renerpreed from vewpon of delyed correcon process. We consder flure dependency n sofwre relbly ssessmen n Secon 3. Furhermore we wll nroduce how o ncorpore he des of flure dependency nd me-dependen dely funcon no sofwre relbly growh modelng. The epermens nd numercl resuls re presened n Secon 4. Fnlly he concludng remrks re gven n Secon 5.. Revews of ful deecon nd correcon processes n sofwre relbly growh models Mos SRGMs hve some bsc ssumpons concernng he sofwre error-deecon process [ 4-5 7]: (1) The ful removl process follows he Nonhomogeneous Posson Process (NHPP). () The sofwre sysem s subjec o flures rndom mes cused by he mnfeson of remnng fuls n he sysem. (3) All fuls re ndependen nd eqully deecble. (4) Ech me flure occurs he ful h cused s mmedely nd perfecly removed. A deeced error s removed wh cerny nd correcon of errors kes only neglgble me. No new fuls re nroduced. I s noed h he ssumpon (4) ssumes h deeced fuls re mmedely removed. In fc hs ssumpon my no be relsc n prcce. In generl fndng ful durng esng s one hng nd fng s noher nd ofen here s consderble me dely beween he wo. Therefore he me delyed by he correcon process s no neglgble. Schnedewnd [15-17] ever modeled he fulcorrecon process by usng delyed ful-deecon process. He ssumes h he ful-deecon process follows he NHPP nd he re of chnge of he men vlue funcon (MVF) s eponenlly decresng. Under he bove ssumpon s shown h he ful deecon process cn be modeled by n NHPP wh eponenlly decresng nensy funcon!().e.!( ) $ & ep[ #% ] & " 0 % " 0 (1) where & nd %'re he prmeers of he model [18]. Therefore he MVF of ful deecon process s gven by m! ( ) $ (& % )(1 # ep[ #% ]). () Xe nd Zho [18 0] epln h Schnedewnd ssume he re of ful correcon s proporonl o he number of ful deeced nd lgs ful deecon process by consn dely (. Th s he MVF s depced s m ( # ( ) $ (& % )(1 # ep[ #% ( # ( )]) ) (. (3) Obvously he ful-deecon process n he Schnedewnd model s somorphc o he Goel-Okumoo model ecep he Goel-Okumoo model s vewed s connuous-me process [0]. Xe nd Zho poned ou h hs ssumpon s oo resrcve nd hey eended he Schnedewnd model o connuous verson by subsung me-dependen dely funcon for he consn dely (( ) [18 0]. Th s Eq. () nd Eq. (3) cn be chnged s m $ (& % )(1 # ep[ #% ]) (4) nd m ( # ( ) $ (& % )(1 # ep[ #% ( # ( )]) ) (. (5) In fc mos esng SRGMs cn be renerpreed s delyed ful-deecon models h cn model he sofwre ful deecon nd correcon processes. Therefore we cn remove he mprccl ssumpon h he ful-correcon process s perfec nd esblsh correspondng medependen dely funcon o f he ful-correcon process. Defnon 1: Gven ful-deecon nd ful-correcon process one defnes he dely-effec fcor *( o be me-dependen funcon h mesures he epeced dely n correcng deeced ful ny me. Defnon : An SRGM s clled delyed-me NHPP model f obeys he followng ssumpons: (1) The ful deecon process follows he NHPP. () The sofwre sysem s subjec o flures rndom mes cused by he mnfeson of remnng fuls n he sysem. (3) All fuls re ndependen nd eqully deecble. (4) The re of chnge of he MVF s eponenlly decresng. (5) The deeced fuls re no mmedely removed nd lgs he ful deecon process by dely-effec fcor *(. Bsed on he bove ssumpons (1)-(4) he orgnl MVF of NHPP model s m orgnl # ep[ # r]) " 0 r " 0 (6)

3 where s he epeced number of nl fuls nd r s he ful deecon re. From he ssumpon (5) n defnon nd Eq. (6) he new MVF cn be depced s m $ morgnl ( # *( ) $ ( 1 # ep[ # r]ep[ r* ]) " 0 r " 0. (7) We hus derve he followng heorem. Theorem 1: Gven dely-effec fcor *( we hve [19]: () The ful-deecon nensy of he delyed-me NHPP SRGM s + ( ) $ dm( d* $ r ep[ # r]ep[ r* ] (1 # ) " 0 r " 0. (8) (b) d* - 1. In he followng we wll revew hree convenonl SRGMs h cn be drecly derved from Defnon 1 Defnon nd Theorem 1. We cn derve he fuldeecon nensy from Eq. (8) nd check he condon of Theorem 1. Goel-Okumoo Model: Ths model frs proposed by Goel nd Okumoo [ 4] s one of he mos populr NHPP model n he feld of sofwre relbly modelng. If * $ 0 hen we hve d* $ 0-1 (9) nd m # ep[ # r]) " 0 r " 0. Ymd Delyed S-Shped Model: The Ymd Delyed S-Shped model s modfcon of he NHPP o obn n S-shped curve for he cumulve number of flures deeced such h he flure re nlly ncreses nd ler decys [ ]. If *( $ (ln( 1. r )) r hen we hve d* $ 1(1. r - 1 (10) nd m # (1. r ep[ # r]). Ymd Webull-Type Tesng-Effor Funcon Model: Ymd e l. [ 7] proposed sofwre relbly model ncorporng he moun of es-effor epended durng he sofwre esng phse. The esng-effor cn be represened s he mn power number of CPU hours or he number of eecued es cses ec. In generl he esng-effor durng he esng phse nd he medependen behvor of developmen effor n he sofwre developmen process cn be descrbed by Webull curve. If *( $. & ep[ #% ] #& hen we hve # 1 d *( $ 1 # &% ep[ #% ] - 1 (11) nd m $ {1 # ep[ # r& (1 # ep[ #% ])]}. Inuvely he correcon process cn be vewed s lernng process snce he sofwre esng ems wll fmlr wh he debuggng envronmens nd ools s me proceeds. These ems' sklls cn be grdully mproved nd hus he moun of me lg wll be lesser. In oher words he dely-effec fcor s non-ncresng n he crcumsnces. 3. Consderng flure dependency n sofwre ful modelng Assumpons [1-13]: (1) The ful deecon process follows he NHPP. () The sofwre sysem s subjec o flures rndom mes cused by he mnfeson of remnng fuls n he sysem. (3) The ll deeced fuls cn be cegorzed s ledng fuls nd dependen fuls. Besdes he ol number of fuls s fne. (4) The men number of ledng fuls deeced n he me nervl ( +0] s proporonl o he men number of remnng ledng fuls n he sysem. Besdes he proporonly s consn over me. (5) The men number of dependen fuls deeced n he me nervl ( +0 s proporonl o he men number of remnng dependen fuls n he sysem nd o he ro of ledng fuls removed me nd he ol number of fuls. Besdes he proporonly s consn over me. (6) The deeced dependen ful my no be mmedely removed nd lgs he ful deecon process by dely-effec fcor *(. Th s *( s he me dely beween he removl of he ledng ful nd he removl of he dependen ful(s). (7) No new fuls re nroduced durng he ful removl process. Le denoes he epeced number of nl fuls. Besdes 1 s he ol number of ledng fuls nd s he ol number of dependen fuls deeced n he sofwre produc. Therefore from ssumpons (3) & (4) we hve = 1 +. For he ske of convenence n he followng prgrph we wll le m( be he MVF of he epeced number of fuls deeced n me (0 ]. Therefore m( s n ncresng funcon of nd m(0)=0. Here we ssume m( = m 1 ( + m ( (1) where m 1 ( s he MVF of he epeced number of ledng fuls deeced n me (0 ] nd m ( s he MVF of he epeced number of dependen fuls deeced n me (0 ]. Consequenly f he number of deeced ledng fuls s proporonl o he number of remnng ledng fuls hen we obn he followng dfferenl equon: dm 1 $ r [ 1 # m1( )] (13) where s he epeced number of nl fuls nd r s he ful deecon re. Solvng he bove dfferenl equon under he boundry condon m 1 (=0 we hve

4 m 1 $ 1(1 # ep[ # r]). Smlrly from ssumpons (6) & (7) we hve dm m1( # *( ) $ 1 [ # m ]. (14) Plese noe h he dependen fuls cn be removed only when he ledng ful s perfecly removed. In he followng we wll gve deled descrpon of possble behvor of *(. (Cse 1) If *(=0 Eq. (14) becomes dm 1(1 # ep[ r]) $1 [ # m( ] #. (15) Assumng he nl condon m (0)=0 we obn 11 (1 # ep[ # r]) # r11 m $ (1 # ep[ ]) (16) r where 1 s he dependen ful removl re. Here we le 1 = P & =(1P) (where P s he proporon of he ledng fuls). From Eq. (1) we obn he MVF m( s follows [1-13] : m $ m1(. m # Pep[ # r] # (1 # P) P1 ep[ (1 # ep[ # r]) # P1 ]). (17) r (Cse ) If *( ) $ (ln(1. r) r Eq. (14) becomes dm 1(1 # (1. rep[ # r]) $1 [ # m( ]. (18) By solvng he bove equon under he boundry condon m (0)=0 he MVF s gven by # 11 ( r. ep[ # r]. rep[ # r] # ) m( # ep[ ]) (19) r nd m # P(1. rep[ # r] # (1 # P) P1 ep[ 1 # ep[ # r] 3# P1 1. ep[ # r] 3]). (0) r (Cse 3) If *( ) $. & ep( #% ) # & Eq. (14) becomes dm 1{1 # ep[ # r& (1 # ep[ #% ])]} $ 1 [ # m( ]. (1) When γ=1 or γ= for Ymd s Webull-ype esng-effor funcon model we obn he eponenl or he Rylegh curve respecvely. Acully hey re specl cses of he Webull esng-effor funcon [1-13]. For emple f γ=1 Eq. (1) cn be solved nd s gven by m $ ( 1 # 11 ep[ # r& ] ep[ r& ] % # 6 4r& ep[ # % ] r& 53 ep[ # ]) % () where 6 [ z] $ # 7 8 ep[ # ]. #z Therefore m # Pep[ # r& 1 # ep[ #% ] 3 ] # (1 # P) P 1 ep[ # r& ] ep[ r& ]% # 6 [ r& ]. 6 [ r& ep[ #% ]] 3 ep[ # ]). % (3) On he oher hnd f γ= we hve 0 % m $ 1 # ep[ # 7 P1 # 1. ep[( # 1. ep[ # y ]) r& 3dy ] nd. 7 % P 1 # 1. ep[( # 1. ep[ # y ]) r& ]) dy (4) ( % m( $ 1# P ep[ # r& (1 # ep[ # ])] #(1 # ) P 0 % P1 # 1. ep[( # 1. ep[ y r d ]) &] 7 # 3. # 1. ep[( # 1. ep[ y ]) r& ] 3dy] 3 ep[ # y % 7 P 1 #. (5) 4. Numercl emples 4.1. D descrpon We choose wo rel d ses s llusrons. The frs d se (DS1) ws from sudy by Ohb [9]. The sysem ws PLI dbse pplcon sofwre conssng of ppromely lnes of code. Durng nneeen weeks CPU hours were consumed nd bou 38 sofwre fuls were removed. The second d se (DS) n hs pper ws from he echncl repor for he projec of Recor Vessel Level Indcon Sysem (RVLIS deecon sysem used o monor he level of wer whn he recor vessel) [5]. The codng lnguge s VersPro.03 nd he developmen plform s GE FANUC PLC I ook 5 weeks o complee he es. Durng he es phse 30 sofwre fuls were removed. The complee flure d s gven n Tble 1. Tble 1: Rel sofwre flure d se (RVLIS). Week CNF Week CNF Week CNF Week CNF CNF: Cumulve number of flures 4.. Crer for model s comprson The comprson crer we use o compre vrous models performnce re descrbed s follows: (1) The Nose s defned s [6]: n 9 $ 1 # 1 ) r # 1 ( r # r (6) where r s he predced flure re. () The Men Squre of Fng Error (MSE) s defned s [13]: k 1 4m( ) m 5 k 9 # (7) $ where m s he observed number of fuls by me. ]3

5 (3) The Men Error of Predcon (MEOP) s defned s [7]: n : 9 $ n # m ;( n # k.1) (8) k where n s he observed cumulve number of flures me s nd m s he predced cumulve number of flures me s =k k+1 n Performnce nlyss In hs secon we wll evlue he proposed models nd severl esng NHPP models. Due o he lmon of spce here we only consder Eq. (0) s llusron Cse DS1. Frsly ll prmeers of he proposed models re esmed by usng he mehod of les squres esmon (LSE) or mmum lkelhood esmon (MLE) [ ]. Tble shows he esmed prmeers of Eq. (0) nd he performnce comprsons of dfferen SRGMs for DS1. I s noed h he proposed model (.e. Eq. (0)) esmes P=0.7 for hs d se. The resul suggess h he sofwre my conn wo cegores of fuls 7% re ledng fuls nd 8% re dependen fuls. Moreover he possble vlues of''p re lso dscussed nd lsed n Tble. As seen from Tble he proposed model lmos provdes he lowes MEOP f compred o he Goel-Okumoo model nd he Ymd dely S-shped model. Overll he MVF of proposed model provdes good f o hs d Cse DS. Smlrly prmeers of ll seleced models re esmed nd he reled MVFs re obned. All seleced models re compred wh ech oher bsed on objecve crer. Tble 3 shows he esmed prmeers of Eq. (0) nd he performnce comprsons of dfferen SRGMs for DS. The proposed model esmes P=0.77 nd ndces h he sofwre conns wo cegores of fuls 77% re ledng fuls nd 3% re dependen fuls. Moreover he possble vlues of'p re lso lsed n Tble 3. On he oher hnd we know h he nflecon S-shped model s bsed on he dependency of fuls by posulng he ssumpon: some of he fuls re no deecble before some oher fuls re removed [5]. Therefore my provde us some nformon for reference. Afer he smulon we fnd h he esmed vlue of nflecon re (whch ndces he ro of he number of deecble fuls o he ol number of fuls n he sofwre) s for DS. I ndces h he growh curve s slghly S-shped [1-13]. On he verge he proposed model performs well n hs cul d. 5. Conclusons In hs pper we ncorpore boh flure dependency nd me-dependen dely funcon no sofwre relbly ssessmen. Specfclly ll deeced fuls cn be cegorzed s ledng fuls nd dependen fuls. Moreover he ful-correcon process cn be modeled s delyed ful-deecon process nd lgs he deecon process by me-dependen dely. Thus he proposed dely-effec fcor cn be used o mesure he epeced me-lg n correcng he deeced fuls durng sofwre developmen. Some new SRGMs re proposed nd severl numercl llusrons bsed on wo rel d ses re presened. Epermenl resuls show h he proposed frmework o ncorpore boh flure dependency nd me-dependen dely funcon for SRGM hs frly ccure predcon cpbly. 6. Acknowledgmens Ths reserch ws suppored by he Nonl Scence Councl Twn under Grn NSC E nd ws lso subsnlly suppored by grn from he Reserch Grn Councl of he Hong Kong Specl Admnsrve Regon Chn (Projec No.CUHK43600E). Moreover we re hnkful o Shn-Shng Shyu Chung-Ln Lee nd Ch-Yun Chng Insue of Nucler Energy Reserch Aomc Energy Councl Eecuve Yun Twn for provdng he second d se. The uhors lso hnk severl nonymous referees for her consrucve revews nd commens. Tble : Comprson resuls of dfferen SRGMs for DS1. Model r θ P MSE MEOP Nose Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Goel-Okumoo model Ymd Dely S-shped model

6 Tble 3: Comprson resuls of dfferen SRGMs for DS. Model r θ P MSE MEOP Nose Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Eq. (0) Goel-Okumoo model Ymd Dely S-shped model References [1] Amercn Insue of Aeronucs nd Asronucs Recommended Prcce for Sofwre Relbly ANSIAIAA R Februry [] M. Xe Sofwre Relbly Modelng World Scenfc Publshng Compny [3] J. D. Mus Sofwre Relbly Engneerng: More Relble Sofwre Fser Developmen nd Tesng McGrw-Hll [4] M. R. Lyu Hndbook of Sofwre Relbly Engneerng McGrw Hll [5] C. Y. Hung M. R. Lyu nd S. Y. Kuo A Unfed Scheme of Some Non-Homogenous Posson Process Models for Sofwre Relbly Esmon IEEE Trns. on Sofwre Engneerng Vol. 9 No. 3 pp Mrch 003. [6] J. D. Mus A. Innno nd K. Okumoo Sofwre Relbly Mesuremen Predcon nd Applcon McGrw Hll [7] S. Ymd Sofwre Relbly Models nd Ther Applcons: A Survey Proceedngs of he Inernonl Semnr on Sofwre Relbly of Mn-Mchne Sysems pp Aug. 000 Kyoo Unversy Kyoo Jpn. [8] M. Ohb Infecon S-Shped Sofwre Relbly Growh Model Sochsc Models n Relbly Theory Sprnger- Verlg Berln pp [9] M. Ohb Sofwre Relbly Anlyss Models IBM Journl of Reserch nd Developmen Vol. 8 No. 4 pp [10] S. Ymd M. Ohb nd S. Osk S-Shped Relbly Growh Modelng for Sofwre Error Deecon IEEE Trns. Relbly Vol. R-3 No. 5 pp [11] S. Ymd M. Ohb nd S.Osk S-Shped Sofwre Relbly Growh Models nd Ther Applcons IEEE Trns. Relbly Vol. R-33 No. 4 pp [1] P. K. Kpur nd S. Younes Sofwre Relbly Growh Model wh Error Dependency Mcroelecroncs nd Relbly Vol. 35 No. pp [13] P. K. Kpur R. B. Grg nd S. Kumr Conrbuons o Hrdwre nd Sofwre Relbly World Scenfc Publshng Compny [14] S. S. Gokhle P. N. Mrnos M. R. Lyu nd K. S. Trved Effec of Repr Polces on Sofwre Relbly Proceedngs of Compuer Assurnce pp June 1997 Ghersburg Mrylnd. [15] N. F. Schnedewnd Modelng he Ful Correcon Process Proceedngs of he 1h Inernonl Symposum on Sofwre Relbly Engneerng pp Nov. 001 Hong Kong Chn. [16] N. F. Schnedewnd An Inegred Flure Deecon nd Ful Correcon Model Proceedngs of 18h Inernonl Conference on Sofwre Mnennce pp Oc. 00 Monrel Quebec Cnd. [17] N. F. Schnedewnd Ful Correcon Profles Proceedngs of he 14h Inernonl Symposum on Sofwre Relbly Engneerng pp Nov. 003 Denver Colordo. [18] M. Xe nd M. Zho The Schnedewnd Sofwre Relbly Model Revsed Proceedngs of he 3rd Inernonl Symposum on Sofwre Relbly Engneerng pp Oc. 199 Reserch Trngle Prk Norh Croln. [19] J. H. Lo S. Y. Kuo M. R. Lyu nd C. Y. Hung Modelng Ful Deecon nd Correcon Processes n Sofwre Relbly Anlyss IEEE Trns. on Relbly n Revson. [0] D. Wllce nd C. Colemn Applcon nd Improvemen of Sofwre Relbly Models Techncl Repor Sofwre Assurnce Technology Cener Oc [1] K. Gošev-Popsojnov nd K. S. Trved Flure Correlon n Sofwre Relbly Models IEEE Trns. Relbly Vol. 49 No. 1 pp Mrch 000. [] L. A. Tomek J. K. Muppl nd K. S. Trved Modelng Correlon n Sofwre Recovery Blocks IEEE Trns. Sofwre Engneerng Vol. 19 pp Nov [3] J. A. Morgn G. J. Knfl nd W. E. Wong Predcng Ful Deecon Effecveness Proceedngs of he 4h Inernonl Sofwre Mercs Symposum pp Nov Albuquerque New Meco. [4] T. Doh N. Ko nd S. Osk Opml Sofwre Relese Polces wh Debuggng Tme Lg Inernonl Journl of Relbly Quly nd Sfey Engneerng Vol. 4 No. 3 pp [5] C. Y. Hung C. T. Ln H. K. Lo Y. S. Su nd B. T. Ln Inroducon o Sofwre Relbly nd Is Applcons Techncl Repor NTHU EECS Indusrl Affles Progrm (EECSIAP) Jn [6] M. R. Lyu nd A. Nkor Applyng Sofwre Relbly Models More Effecvely IEEE Sofwre pp July 199. [7] M. Zho nd M. Xe On he Log-Power NHPP Sofwre Relbly Model Proceedngs of he 3rd Inernonl Symposum on Sofwre Relbly Engneerng pp.14- Oc. 199 Reserch Trngle Prk Norh Croln.

Supporting information How to concatenate the local attractors of subnetworks in the HPFP

Supporting information How to concatenate the local attractors of subnetworks in the HPFP n Effcen lgorh for Idenfyng Prry Phenoype rcors of Lrge-Scle Boolen Newor Sng-Mo Choo nd Kwng-Hyun Cho Depren of Mhecs Unversy of Ulsn Ulsn 446 Republc of Kore Depren of Bo nd Brn Engneerng Kore dvnced