Optimal Testing Resource Allocation, and Sensitivity Analysis in Software Development

Transcription

1 Opal Tesng Resource Allocaon, and Sensvy Analyss n Sofware Developen Chn-Yu Huang, Meber, IEEE Mchael R. Lyu 2, Fellow, IEEE Keywords: Sofware relably, esng resource allocaon, non-hoogeneous Posson processes, sensvy analyss. Absrac We consder wo knds of sofware esng-resource allocaon probles. The frs proble s o nze he nuber of reanng fauls gven a fxed aoun of esng-effor, and a relably objecve. The second proble s o nze he aoun of esng-effor gven he nuber of reanng fauls, and a relably objecve. We have proposed several sraeges for odule esng o help sofware projec anagers solve hese probles, and ake he bes decsons. We provde several syseac soluons based on a non-hoogeneous Posson process odel, allowng syseac allocaon of a specfed aoun of esng-resource expendures for each sofware odule under soe consrans. We descrbe several nuercal exaples on he opal esng-resource allocaon probles o show applcaons & pacs of he proposed sraeges durng odule esng. Experenal resuls ndcae he advanages of he approaches we proposed n gudng sofware engneers & projec anagers oward bes esng resource allocaon n pracce. Fnally, an exensve sensvy analyss s presened o nvesgae he effecs of varous prncpal paraeers on he opzaon proble of esng-resource allocaon. The resuls can help us know whch paraeers have he os sgnfcan nfluence, and he changes of opal esng-effor expendures affeced by he varaons of faul deecon rae & expeced nal fauls. C. Y. Huang s wh he Deparen of Copuer Scence, aonal Tsng Hua Unversy, Hsnchu, Tawan e-al: cyhuang@cs.nhu.edu.w). 2 M. R. Lyu s wh he Copuer Scence and Engneerng Deparen, The Chnese Unversy of Hong Kong, Shan, Hong Kong e-al: lyu@cse.cuhk.edu.hk).

2 Acronys 3 HPP SRGM TEF TE MLE LSE oaon ) λ) w κ ) W κ ) a r α A κ β v Rx ) non-hoogeneous Posson process sofware relably growh odel esng-effor funcon esng-effor axu lkelhood esaon leas squares esaon expeced ean nuber of fauls deeced n e, ], ean value funcon falure nensy for ), d)/d curren esng-effor consupon a e cuulave esng-effor consupon a e expeced nuber of nal fauls faul deecon rae per un esng-effor oal aoun of esng-effor evenually consued consupon rae of esng-effor expendures n he generalzed logsc esng-effor funcon consan paraeer n he generalzed logsc esng-effor funcon srucurng ndex whose value s larger for beer srucured sofware developen effors consan paraeer weghng facor o easure he relave porance of a faul reoval fro odule condonal sofware relably. ITRODUCTIO A copuer syse coprses wo ajor coponens: hardware, and sofware. Wh he seadly growng power & relably of hardware, sofware has been denfed as a ajor sublng block n achevng desred levels of syse dependably. We need qualy sofware o produce, anage, acqure, dsplay, and rans nforaon anywhere n he world. Sofware producers us ensure he adequae relably of he delvered sofware, he e of delvery, and s cos. Accordng o he ASI defnon: Sofware relably s defned as he probably of falure-free sofware operaon for a specfed perod of e n a specfed envronen []. Alernavely, ay be vewed fro he perspecve of general use on a varey of dfferen npus, n whch case s he probably ha wll correcly process a randoly chosen npu. Many Sofware Relably Growh Models SRGM) were developed n he 97s-2s []-[2]. SRGM descrbe falures as a rando process, whch s characerzed n eher es of falures, or he nuber of falures a fxed es. 3 The sngular and plural of an acrony are always spelled he sae. 2

3 In addon o sofware relably easureen, SRGM can help us predc he faul deecon coverage n he esng phase, and esae he nuber of fauls reanng n he sofware syses. Fro our sudes, here are soe SRGM ha descrbe he relaonshp aong he calendar esng, he aoun of esng-effor, and he nuber of sofware fauls deeced by esng. The esng-effor TE) can be represened as he nuber of CPU hours, he nuber of execued es cases, or huan power, ec [2]. Musa e al. [2] showed ha he effor ndex, or he execuon e s a beer e doan for sofware relably odelng han he calendar e because he observed relably growh curve depends srongly on he e dsrbuon of he TE. In he sofware developen phase, esng begns a he coponen level, and dfferen esng echnques are approprae a dfferen pons n e. Tesng s conduced by he developer of he sofware, as well as an ndependen es group [3]. One ajor sofware developen challenge s ha esng s oo expensve & lenghy, ye he projec schedule has o ee a delvery deadlne. Mos popular coercal sofware producs are coplex syses coposed of a nuber of odules. As soon as he odules are developed, hey have o be esed n a varey of ways, and ess are derved fro he developer s experence. Praccally, odule esng s he os dealed for of esng o be perfored. Thus, projec anagers should know how o allocae he specfed esng resources aong all he odules & develop qualy sofware wh hgh relably. Fro our sudes [4]-[23], here are any papers ha have addressed he probles of opal resource allocaon. In hs paper, we frs consder wo knds of sofware esng-resource allocaon probles, and hen propose several sraeges for odule esng. aely, we provde syseac ehods for he sofware projec anagers o allocae a specfc aoun of TE expendures for each odule under soe consrans, such as ) nzng he nuber of reanng fauls wh a relably objecve, or 2) nzng he aoun of esng-effor wh a relably objecve. Here we eploy a SRGM wh generalzed logsc esng-effor funcon o descrbe he e-dependency behavors of deeced sofware fauls, and he esng-resource expendures spen durng odule esng. The proposed odel s based on on-hoogeneous Posson processes HPP). The reanng conens of hs paper conss of four secons. Secon 2 descrbes an SRGM wh generalzed logsc TEF. In Secon 3, he ehods for esng resource allocaon & opzaon for odular sofware esng are nroduced. uercal exaples for he opu TE allocaon probles are deonsraed n Secon 4. In Secon 5, we analyze he sensvy of paraeers of proposed SRGM. 3

4 2. SRGM WITH GEERALIZED LOGISTIC TESTIG-EFFORT FUCTIO 2. Sofware Relably Modelng A nuber of SRGM have been proposed on he subjec of sofware relably []. Tradonal SRGM, such as he well-known Goel-Okuoo odel, and he Delayed S-shaped odel, have been shown o be very useful n fng sofware falure daa. Yaada e al. [6]-[8] odfed he G-O odel, and ncorporaed he concep of TE n an HPP odel o ge a beer descrpon of he sofware faul phenoenon. Laer, Huang e al. [24], [25] proposed a new SRGM wh he logsc TEF o predc he behavor of falure occurrences, and he faul conen of a sofware produc. Based on our pas experenal resuls [26], [27], hs approach s suable for esang he relably of sofware applcaon durng he developen process. The followng are he odelng assupons: ) The faul reoval process s odeled as a HPP, and he sofware applcaon s subjec o falures a rando es caused by he reanng fauls n he syse. 2) The ean nuber of fauls deeced n he e nerval, + ) by he curren TE s proporonal o he ean nuber of reanng fauls n he syse a e, and he proporonaly s a consan over e. 3) TE expendures are descrbed by a generalzed logsc TEF. 4) Each e a falure occurs, he correspondng faul s edaely reoved, and no new fauls are nroduced. Le ) be he ean value funcon of he expeced nuber of fauls deeced n e, ]. Because he expeced nuber of deeced fauls s fne a any e, ) s an ncreasng funcon of, and ). Accordng o hese assupons, we ge + ) ) r [ a )]. ) wκ ) Tha s, + ) ) r l [ a )] w ). 2) κ Consequenly, f he nuber of deeced fauls due o he curren TE expendures s proporonal o he nuber of reanng fauls, we oban he dfferenal equaon d ) r [ a )]. 3) d w ) κ 4

5 Solvng he above dfferenal equaon under he boundary condon ), we have exp[ r W ))]) ) a exp[ r Wκ ) Wκ ))]) a. 4) oe ha paraeer a s he nuber of nal fauls, and hs nuber s usually a represenave easure of sofware relably. I can also provde an esae of he nuber of falures ha wll evenually be encounered by he cusoers. Besdes, paraeer r s he faul deecon rae, or he rae of dscoverng new fauls n sofware durng he esng phase. In general, a he begnnng of he esng phase, any fauls can be dscovered by nspecon, and he faul deecon rae depends on he faul dscovery effcency, he faul densy, he esng-effor, and he nspecon rae [3]. In he ddle sage of he esng phase, he faul deecon rae norally depends on oher paraeers, such as he execuon rae of CPU nsrucon, he falure-o-faul relaonshp, he code expanson facor, and he scheduled CPU execuon hours per calendar day [2]. We can use hs rae o rack he progress of checkng acves, o evaluae he effecveness of es plannng, and o assess he checkng ehods we adoped [25]. In fac, ) s non-decreasng wh respec o esng e. Knowng s value can help us deerne wheher he sofware s ready for release, and f no, how uch ore of he esng resources are requred. I can also provde an esae of he nuber of falures ha wll evenually be encounered by he cusoers. Yaada e al. [], [26] repored ha he TE could be descrbed by a Webull-ype curve, and he Webull curve s one of he hree known exree-value dsrbuons. Alhough a Webull-ype curve can f he daa well under he general sofware developen envronen, wll show he apparen peak phenoenon when he value of he shape paraeer s greaer han 3 [26]. Fro our pas sudes [27], [28], a logsc TEF wh a srucurng ndex was proposed, whch can be used o consder & evaluae he effecs of possble proveens on sofware developen ehodology. The dea of a logsc TEF was proposed by F.. Parr [29]; predcs essenally he sae behavor as he Raylegh curve, excep durng he early par of he projec. For a saple of soe wo dozen projecs suded n he Yourdon projec survey, he logsc TEF was farly accurae n descrbng expended TE [3]. In [28], we exended he logsc TEF o a generalzed for, and he generalzed logsc TEF s forulaed as Wκ ). 5) κ + ακ Ae Coen [JWR]: Do you nsead ean never repored? The curren TE consupon s dwκ ) wκ ). 6) d 5

6 The TE reaches s axu value a e A ln κ ax. 7) ακ The condonal relably funcon afer he las falure occurs a e s obaned by [], [2] R cond ) R + ) exp[ + ) ))]. 8) Takng he logarh on boh sdes of he above equaon, we oban ln R cond cond ) + ) )). 9) Here we wll defne anoher easure of relably,.e., he rao of he cuulave nuber of deeced fauls a e o he expeced nuber of nal fauls. ) R ). ) a oe ha R) s an ncreasng funcon n. Usng R), we can oban he requred esng e needed o reach he relably objecve R, or decde wheher he relably objecve can be sasfed a a specfed e. If we know ha he value of R) has acheved an accepable level, hen we can deerne he rgh e o release hs sofware. 2.2 Mehods of Model s Paraeer Esaon To valdae he proposed odel, experens on real sofware falure daa wll be perfored. Two os popular esaon echnques are Maxu Lkelhood Esaon MLE), and Leas Squares Esaon LSE) [], [2], [26]. For exaple, usng he ehod of LSE, he evaluaon forula S, A, α) of Equaon 5) wh κ s depced as Mnze n 2 S, A, α ) [ W W )], ) k where W s he cuulave esng-effor acually consued n e, ], and W k ) s he cuulave TE esaed by Equaon 5). Dfferenang S wh respec o, A, and α, seng he paral dervaves o zero, and rearrangng hese ers, we can solve hs ype of nonlnear leas square probles. We oban n S 2 W + Aexp[ α] + Aexp[ α] Thus, he leas squares esaor s gven by solvng he above equaon o yeld 2) 6

7 7 + + n n A A W 2 ] exp[ ] exp[ 2 2 α α. 3) ex, we have 2 2 ]) exp[ ] exp[ ] exp[ + + n A A W A S α α α, 4) and 2 2 ]) exp[ ] exp[ ] exp[ + + n A A A W S α α α α. 5) The oher paraeers A & α can also be obaned by subsung he leas squares esaor no Equaons 4) & 5). Slarly, f he ean value funcon s descrbed n Equaon 4), hen he evaluaon forula S2a, r) can be obaned as Mnze n r a S 2 )] [ ), 2, 6) where s he cuulave nuber of deeced fauls n a gven e nerval, ], and ) s he expeced nuber of sofware fauls esaed by Equaon 4). Dfferenang S2 wh respec o a & r, seng he paral dervaves o zero, and rearrangng hese ers, we can solve hs ype of nonlnear leas square probles. On he oher hand, he lkelhood funcon for he paraeers a & r n he HPP odel wh ) n Equaon 4) s gven by L Pr{ ), 2 ) 2,..., n ) n } ))] ) exp[ )! )} ) { n,7) where for. Therefore, akng he logarh of he lkelhood funcon n Equaon 7), we have )!] ln[ )) ) )] ) ) ln[ ln n n n L 8) Fro Equaon 3), we know ha ) - ) aexp[rw - )] exp[rw )]). Thus, )])] exp[ )] ) ln[exp[ ) ln ln n n rw rw a L + )!] ln[ )]) exp[ n n rw a 9) Consequenly, he axu lkelhood esaes a & r can be obaned by solvng

8 ln L ln L a r 2) 3. TESTIG-RESOURCE ALLOCATIO FOR MODULE TESTIG In hs secon, we wll consder several resource allocaon probles based on an SRGM wh generalzed logsc TEF durng sofware esng phase. Assupons [4], [5], [7], []-[4], [27]: ) The sofware syse s coposed of odules, and he sofware odules are esed ndvdually. The nuber of sofware fauls reanng n each odule can be esaed by an SRGM wh generalzed logsc TEF. 2) For each odule, he falure daa have been colleced, and he paraeers of each odule, ncludng he faul deecon rae and he odule faul weghng facor, can be esaed. 3) The oal aoun of esng resource expendures avalable for he odule esng processes s fxed, and denoed by W. 4) If any of he sofware odules fals upon execuon, he whole sofware syse s n falure. 5) The syse anager has o allocae he oal esng resources W o each sofware odule, and nze he nuber of fauls reanng n he syse durng he esng perod. The desred sofware relably afer he esng phase should acheve he relably objecve R. Fro Secon 2., he ean value funcon of a sofware syse wh odules can be forulaed as M) v ) v a exp[ rw )]) 2) If v for all, 2,,, he objecve s o nze he oal nuber of fauls reanng n he sofware syse afer hs esng phase. Ths ndcaes ha he nuber of reanng fauls n he syse can be esaed by v a exp[ rw )] v a exp[ rw ] 22) We can furher forulae wo opzaon probles as follows. 8

9 3. Mnzng he nuber of reanng fauls wh a gven fxed aoun of TE, and a relably objecve A successful es s one ha uncovers an as-ye-undscovered faul. We should know ha ess show he presence, no he absence, of defecs [3]. I s possble o execue every cobnaon of pahs durng esng. The Pareo prncple ples ha 8 percen of all fauls uncovered durng esng wll lkely be raceable o 2 percen of all progra coponens [3]. Thus he queson of how uch o es s an poran econoc queson. In pracce, a fxed aoun of TE s generally spen n esng a progra. Therefore, he frs opzaon proble n hs paper s ha he oal aoun of TE s fxed, and we wan o allocae hese effors o each odule n order o nze he nuber of reanng fauls n he sofware syses. Suppose he oal aoun of TE s W, and odule s allocaed W esng effors; hen he opzaon proble can be represened as []-[4], [7]-[8], [27] Mnze: v a exp[ rw ] 23) Subjec o: W W, W 24) R ) exp[ rw ] R, 2,..., ). 25) Fro Equaon 25), we can oban W ln[ R ],, 2,..., 26) r Le D ln[ R ],, 2,...,. Thus, we have r W W, W,, 2,...,, and W C, where C ax, D, D2, D3,..., D ). Tha s, he opal esng resource allocaon can be specfed as below [7]-[9] Mnze: v a exp[ rw ] 27) Subjec o: W W, W and W C. Le X W C, hen we can ransfor he above equaons o Mnze: va exp[ r C ]exp[ r X ] 28) 9

10 Subjec o: X W C, X,, 2,...,. 29) oe ha he paraeers v, a, and r should already be esaed by he proposed odel. To solve he above proble, he Lagrange ulpler ehod can be appled. The Lagrange ulpler ehod ransfors he consraned opzaon proble no he unconsraned proble by nroducng he Lagrange ulplers [22], [27], [3], [32]. Consequenly, Equaons 28) & 29) can be splfed as 2 Mnze: L X, X,..., X, λ ) v a exp[ r C ]exp[ r X ] + λ X W + C ) 3) Based on he Kuhn-Tucker condons KTC), he necessary condons for a nu value of Equaon 3) exs, and can be saed as [2]-[5], [3], [32] L X, X 2,..., X, λ) A:,, 2,...,. 3a) X L X, X 2,..., X, λ) A2: X,, 2,...,. 3b) X A3: λ { X W C )},, 2,...,. 3c) Theore. A feasble soluon X, 2,..., ) of Equaon 3) s opal ff a) λ v a r exp[ r C ] exp[ r X ], b) X { λ v a r exp[ r C ] exp[ r X ])} Proof: a) Fro Equaon 3), we have L X 2, X,..., X X, λ) v a r exp[ r C ] exp[ r X we know ha λ v a r exp[ r C ] exp[ r X ]. ] + λ. Therefore, fro Equaon 3a), b) Fro Equaons 3) & 3b), we have X { λ v a r exp[ r C ] exp[ r X ])}. Corollary. Le X be a feasble soluon of Equaon 3) a) X ff λ v a r exp[ r C ]. b) If X >, hen X {ln var exp[ rc ]) ln λ} / r. Proof:

11 a) If X, hen Theore par a) ples ha λ v a r exp[ r C ]. Besdes, f λ v a r exp[ r C ], hen fro Theore par b), we know ha X { v a r exp[ r C ] v a r exp[ r C ] exp[ r X ]} or X v a r exp[ r C ] { exp[ r X ]}. Because v, a, and r, we have X or exp[ r ],.e., X. Tha s, X. If λ > v a r exp[ r C ], hen X λ > v a r exp[ r C ] v a r exp[ r C ] exp[ r X ] because exp[ r ] ) or λ v a r exp[ r C ] exp[ r X ]. Therefore, fro Theore par b), we have X. Q.E.D. b) Fro Theore par b), we know ha f X >, hen λ v a r exp[ r C ] exp[ r X ]. Therefore, X {ln v a r exp[ r C ]) ln λ} / r. Q.E.D. Fro Equaon 3), we have L X X, X,..., X, λ) var exp[ r C ] exp[ r X ] + λ X 2 L X, X 2,..., X λ Thus, he soluon X s The soluon λ s, λ) X W + C X ln v a r exp[ r C ]) ln λ ) / r,, 2,...,. 32) / r )ln var exp[ r C ]) W + C λ exp 33) / r ) 2 3 X Hence, we ge X X, X, X,..., ) as an opal soluon o Equaon 3). However, he above X ay have soe negave coponens f v exp[ r C ] < λ, akng ar X nfeasble for Equaons 28) & 29). If hs s he case, he soluon X can be correced by he followng seps [4], [5], []. Algorh Sep : Se l. Sep 2: Calculae he equaons X [ln var exp[ r C ]) ln λ],, 2,..., l. r

12 l / r )ln var + exp[ r C ]) W C λ exp / r ) Sep 3: Rearrange he ndex such ha X X 2... X l. Sep 4: If X l, hen sop.e., he soluon s opal) Else, X l ; ll+. End If. Sep 5: Go o Sep 2. The opal soluon has he for X ln v a r ) exp[ rc ] lnλ)/ r, X,,2,..., l, l / r )lnv ar exp[ rc ]) W + C where λ exp l / r ) oherwse Algorh always converges n, a wors, seps. Thus, he value of he objecve 2 funcon gven by Equaon 28) a he opal soluon X, X,..., X ) as v a ] 34) exp[ r C ]exp[ r X 35) 3.2 Mnzng he aoun of TE gven he nuber of reanng fauls, and a relably objecve ow suppose Z specfes he nuber of reanng fauls n he syse, and we have o allocae an aoun of TE o each sofware odule o nze he oal TE. The opzaon proble can hen be represened as Mnze: Subjec o: W, 36) v a exp[ rw ] Z, W. 37) R ) exp[ rw ] R 38) Slarly, fro Equaon 38), we can oban W W ln R ),,2,...,. 39) r Followng slar seps descrbed n Secon 3., and leng X 2 W C, where

13 C ax, D, D2, D3,..., D ), we can ransfor he above equaons o Mnze: X + C ), 4) Subjec o: v a exp[ r C ]exp[ r X ] Z, X + C 4) To solve hs proble, he Lagrange ulpler ehod can agan be used. Equaons 4) & 4) are cobned o he equaon 2 ) Mnze: L X, X,..., X, λ ) X + C ) + λ v a exp[ r C ]exp[ r X ] Z 42) Theore 2. A feasble soluon X, 2,..., ) of Equaon 42) s opal ff a) λ exp[ r C + X )] / var, b) X { λ v a r exp[ r C ]exp[ r X ]} Proof: L X, X 2,..., X, λ ) a) Fro Equaon 42), we know ha X λ var exp[ r C ]exp[ r X ]. Besdes, fro Equaon 3a), we have λ v a r exp[ r C ]exp[ r X ],.e., λ v a r exp[ r C ]exp[ r X ]. Therefore, λ exp[ r C + X )] / v a r. b) Fro Equaons 42) & 3b), we have X { λ v a r exp[ r C ]exp[ r X ]}. Corollary 2. Le X be a feasble soluon of Equaon 42) a) X ff λ exp[ r C )]/ var. b) If X >, hen X ln λ var exp[ rc ]) / r. Proof: a) If X, hen Theore 2 par a) ples ha λ exp[ r C )] / var. Besdes, f λ exp[ r C )] / v a r, hen fro Theore 2 par b), we know ha X { exp[ r X ]}. Thus, we have X or exp[ r X ],.e., X. Tha s, X. If λ < exp[ r C )] / var, ha s, λ v a r < exp[ r C )] or λ v a r exp[ r C )] <. Hence, λ v a r exp[ r C ]exp[ r X ] < exp[ r X ]. Because exp[ r ], hen we have λ v a r exp[ r C ]exp[ r X ] or λ v a r exp[ rc ]exp[ r X ]. Therefore, fro X 3

14 Theore 2 par b), we have X. Q.E.D. b) Fro Theore 2 par b), we know ha f X >, λ v a r exp[ r C ]exp[ r X ]. Therefore, X ln λ v a r exp[ rc ]) / r. Q.E.D. Fro Equaon 42), we have L X, X,..., X, λ ) λvar exp[ r C ]exp[ r X ] + X 2 L X, X 2,..., X, ) λ va exp[ r C ]exp[ r X ] Z λ Thus, he soluon X s 43) 44) X ln λ v a r exp[ rc ]) / r,, 2,...,. 45) The soluon λ s Tha s, v a r / r ) λ 46) Z ln exp[ rc ] )) Z r / X, X 2, X 3,..., X X Hence, we ge ) r,, 2,...,. 47) X as an opal soluon o Equaon 42). However, he above X ay have soe negave coponens f Z var exp[ rc ] <, r akng X nfeasble for Equaons 4) & 4). In hs case, he soluon X can be correced by he followng seps. Slarly, we propose a sple algorh o deerne he opal soluon for he TE allocaon proble. Algorh 2 Sep : Se l. Sep 2: Calculae v l a r X [ln exp[ r C ] ) ],, 2,..., l. r Z r 2 X l Sep 3: Rearrange he ndex such ha X X.... Sep 4: If X l hen sop. Else updae X l ; ll+. End If. 4

15 Sep 5: Go o Sep 2. The opal soluon has he for X v a r l [ln exp[ rc ] )], r Z r Algorh 2 always converges n, a wors, seps.,2,..., l. 48) 4. EXPERIMETAL STUDIES AD RESULTS In hs secon, hree cases for he opal TE allocaon probles are deonsraed. Here we assue ha he esaed paraeers a & r n Equaon 2), for a sofware syse conssng of odules, are suarzed n Table I. Moreover, he weghng vecors v n Equaon 2) are also lsed. In he followng, we llusrae several exaples o show how he opal allocaon of TE expendures o each sofware odule s deerned. Suppose ha he oal aoun of TE expendures W s 5, an-hours, and R.9. Besdes, all he paraeers a & r of Equaon 2) for each sofware odule have been esaed by usng he ehod of MLE or LSE n Secon 2.2. We apply he proposed odel o sofware falure daa se [2], [5], [27], [33]. Here we have o allocae he expendures o each odule, and nze he nuber of reanng fauls. Fro Table I & Algorh n Secon 3., he opal TE expendures for he sofware syses are esaed, and shown n Table II. For exaple, usng he esaed paraeers a, r, he weghng facor v n Table I, and he opal TE expendures n Table II, he value of he esaed nuber of reanng fauls s 72 for Exaple. Tha s, he oal nuber of reanng fauls s nended o decrease fro 54 o 72 by usng esng-resource expendures of 5, an-hours, and abou a 33.6% reducon n he nuber of reanng fauls. Conversely, f we wan o decrease ore reanng fauls, and ge a beer reducon rae, hen we have o re-plan & consder he allocaon of esng-resource expendures;.e., usng he sae values of a, r, κ, and v, he opal TE expendures should be re-esaed. Therefore, we can know how uch exra aoun of esng-resource expendures s expeced [2], [5]. The nubers of nal fauls, he esaed reanng fauls, and he reducon n he nuber of reanng fauls for he oher exaples are shown n Table III. Fnally, suppose he oal nuber of reanng fauls Z s. We have o allocae he expendures o each odule, and nze he oal aoun of TE expendures. Slarly, usng Algorh 2 n Secon 3.2 & Table I, he opal soluons of TE expendures are 5

16 derved & shown n Table IV. Furherore, he relaonshp beween he oal aoun of esng-effor expendures, and he reducon rae of he reanng fauls, are also depced n Fgure. Table I: The esaed values of a, r, v, and κ. Module a r κ v n Exaple v n Exaple 2 v n Exaple Table II: The opal TE expendures usng Algorh. Module X for Exaple X for Exaple 2 X for Exaple Table III: The reducon n he nuber of reanng fauls. Exaple Inal fauls Reanng fauls Reducon %)

17 Table IV: The opal TE expendures usng Algorh 2. Module X for Exaple X for Exaple 2 X for Exaple Exaple Exaple3 Exaple2 Fgure : The Reducon Rae of Reanng Fauls vs. he Toal TE Expendures. 5. SESITIVITY AALYSIS In hs secon, sensvy analyss of he proposed odel s conduced o sudy he effec of he prncpal paraeers, such as he expeced nal fauls, and he faul deecon rae. In Equaon 4), we know ha here are soe paraeers affecng he ean value funcon, such as he expeced oal nuber of nal fauls, he faul deecon rae, he oal aoun of TE, he consupon rae of TE expendures, and he srucurng ndex, ec. Consequenly, we have o esae all hese paraeers for each sofware odule very carefully because hey play an poran role for he opal resource allocaon probles. In general, each paraeer s esaed based on he avalable daa, whch s ofen sparse. Thus, we analyze 7

18 he sensvy of soe prncpal paraeers, bu no all paraeers due o he laon of space. everheless, we sll can evaluae he opal resource allocaon probles for varous condons by exanng he behavor of soe paraeers wh he os sgnfcan nfluence. We perfor he sensvy analyss of opal resource allocaon probles wh respec o he esaed paraeers so ha aenon can be pad o hose paraeers deeed crcal [34]-[4]. In hs paper, we defne Relave Change RC) MTEE OTEE, 49) OTEE where OTEE s he orgnal opal TE expendures, and MTEE s he odfed opal TE expendures. 5. Effec of varaons on expeced nal fauls & faul deecon rae Algorh). Assung we have obaned he opal TE expendures o each sofware odule ha nze he expeced cos of sofware, hen we can calculae he MTEE concernng he changes of expeced nuber of nal fauls a for he specfc odule. The procedure can be repeaed for varous values of a. For nsance, for he daa se used n Secon 4 here we only use Exaple as llusraon), f he expeced nuber of nal fauls a of odule s ncreased or decreased by 4%, 3%, 2%, or %, hen he odfed TE expendures for each sofware odule can be obaned by followng he slar procedures. Table V shows soe nuercal values of he opal TE expendures for he case of 4%, 3%, 2%, and % ncrease o a. The resul ndcaes ha he esaed values of opal TE expendures wll be changed when a changes. Tha s, f a s ncreased by 4%, hen he esaed value of opal TE expendure for odule s changed fro 6254 o 7, and s RC s.2 abou 2% ncreen). Bu for odules 2, 3, 4, 5, 7, and 8, he esaed values of opal TE expendures are abou.2%,.2%,.2%,.%,.69%, and 2.32% decreen, respecvely. Therefore, he varaon n a has he os sgnfcan nfluence on he opal allocaon of TE expendures. Fro Table V, we can also know ha, f he change of a s sall, he sensvy of he opal esng-resources allocaon wh respec o he value of a s low. ex, fro Table VI, s shown ha, f a s decreased by 3%, he esaed value of opal TE expendure for odule s changed fro 6254 o 5452, and s RC s -.28 abou 2.8% decreen). I s noed ha for odules 2, 3, 4, 5, 7, and 8, he esaed values of opal TE expendures are abou.%,.3%, 3.29%,.7%,.79%, and 2.47% ncreen, respecvely. We have perfored an exensve sensvy analyss for he expeced nal fauls as 8

19 shown above. Bu each a s consdered n solaon. ow we ry o sudy he effecs of sulaneous changes of a & a j j ). If we le a & a 2 boh be ncreased by 4%, hen he esaed values of opal TE expendure for odules, and 2 are changed fro 6254 o 6972 abou.48% ncreen), and 3826 o 445 abou 5.39% ncreen), respecvely. Bu for odules 3, 4, 5, 7, and 8, he esaed values of opal TE expendures are abou 2.2%, 5.66%,.83%, 3.9%, and 4.23% decreen, respecvely. Therefore, he varaon n a & a 2 has he sgnfcan nfluence on he opal allocaon of TE expendures. Slarly, fro Table VII, we can also know ha f he changes of a & a 2 are less, he sensvy of he opal esng-resources allocaon wh respec o he values of a & a 2 s low. Fro Table VIII, s also shown ha f a & a 2 are boh decreased by 3%, he esaed values of opal TE expendure for odules, and 2 are changed fro 6254 o 5494 abou 2.2% decreen), and 3826 o 32 abou 6.3% decreen), respecvely. I s also noed ha for odule 3, 4, 5, 7, and 8, he esaed values of opal TE expendures are, respecvely, abou 2.38%, 6.2%,.94%, 3.27%, and 4.49% ncreen. Based on hese observaons, we can conclude ha f a s changed, wll have uch nfluence on he esaed values of opal TE expendure for odule. In fac, we can nvesgae he sensvy of faul deecon rae followng he slar seps descrbed above. Table IX shows nuercal values of he opal TE expendures for he case of 4%, 3%, 2%, and % ncrease o r. Table X shows nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r. uercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r & r 2 are shown n Table XI. Fnally, Table XII shows nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r & r 2. Table V: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n a Algorh ). Module X a.4) X a.3) X a.2) X a.)

20 Table VI: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n a Algorh ). Module X a.6) X a.7) X a.8) X a.9) Table VII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n a & a 2 Algorh ). Module X X X X a.4 & a 2.4) a.3 & a 2.3) a.2 & a 2.2) a. & a 2.) Table VIII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n a & a 2 Algorh ). Module X X X X a.6 & a 2.6) a.7 & a 2.7) a.8 & a 2.8) a.9 & a 2.9)

21 Table IX: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r Algorh ). Module X r.4) X r.3) X r.2) X r.) Table X: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r Algorh ). Module X r.6) X r.7) X r.8) X r.9) Table XI: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r & r 2 Algorh ). Module X X X X r.4 & r 2.4) r.3 & r 2.3) r.2 & r 2.2) r. & r 2.)

22 Table XII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r & r 2 Algorh ). Module X X X X r.6 & r 2.6) r.7 & r 2.7) r.8 & r 2.8) r.9 & r 2.9) Effec of varaons on expeced nal fauls & faul deecon rae Algorh 2). Assung we have obaned he opal TE expendures o each sofware odule, hen we can calculae he MTEE concernng he changes of expeced nuber of nal fauls a for he specfc odule. The procedure can be repeaed for varous values of a. Slarly, we nvesgae he possble change of opal TE expendures when he expeced nuber of nal fauls a s changed. For he daa se Exaple ) used n Secon 4, f he expeced nuber of nal fauls a of odule s ncreased or decreased by 4%, 3%, 2%, or %, hen he odfed TE expendures for each sofware odule can be re-esaed fro he algorhs n Secon 3. Frs, Table XIII shows soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n a. The resul ndcaes ha he esaed values of opal TE expendures wll be changed when a changes. For exaple, f a s ncreased by 4%, hen he esaed value of opal TE expendure for odule s changed fro 77 o 854, and s RC s.4 abou % ncreen). Besdes, for odules 3, 4, 6, 7, and 8, he esaed values of opal TE expendures are abou.86%, 4.32%, 5.55%,.39%, and.798% decreen, respecvely. Bu for odules 2, 5, and 9, he esaed values of opal TE expendures are abou.74%,.42%, and 6.79% ncreen, respecvely. Therefore, fro Table XIII, we can know ha, f he change of a s sall, he sensvy of he opal esng-resources allocaon wh respec o he value of a s low. ex, we show he sae coparson resuls n case ha a s decreased. Fro Table XIV, s shown ha, f a s decreased by 3%, he esaed value of opal TE expendure for odule s changed fro 77 o 6847, and s RC s -. 22

23 abou.7% decreen). So far, we have perfored an exensve sensvy analyss for he expeced nal fauls as shown above. However, each a s consdered n solaon. Agan we sudy he effecs of sulaneous changes of a & a j j ). If we le a & a 2 boh be ncreased by 4%, hen he esaed values of opal TE expendure for odules, and 2 are changed fro 77 o 854 abou.4% ncreen), and 4976 o 5674 abou 4.% ncreen), respecvely. Fro Table XV, we can fnd ha he varaon n a & a 2 ay have he os sgnfcan nfluence on he opal allocaon of TE expendures. Slarly, we can also know ha, f he changes of a & a 2 are less, he sensvy of he opal esng-resources allocaon wh respec o he values of a & a 2 s low. Fro Table XVI, we can see ha, f a & a 2 are boh decreased by 3%, he esaed values of opal TE expendure for odules, and 2 are changed fro 77 o 6847 abou.% decreen), and 4976 o 433 abou 3.3% decreen), respecvely. Slarly, we can nvesgae he sensvy of faul deecon rae for Algorh 2 followng he slar seps descrbed above. Table XVII shows nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r. Table XVIII shows nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r. Moreover, nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r & r 2 are shown n Table XIX. Fnally, Table XX shows nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r & r 2. Table XIII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n a Algorh 2). Module X a.4) X a.3) X a.2) X a.)

24 Table XIV: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n a Algorh 2). Module X a.6) X a.7) X a.8) X a.9) Table XV: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n a & a 2 Algorh 2). Module X X X X a.4 & a 2.4) a.3 & a 2.3) a.2 & a 2.2) a. & a 2.) Table XVI: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n a & a 2 Algorh 2). Module X X X X a.6 & a 2.6) a.7 & a 2.7) a.8 & a 2.8) a.9 & a 2.9)

25 Table XVII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r Algorh 2). Module X r.4) X r.3) X r.2) X r.) Table XVIII: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r Algorh 2). Module X r.6) X r.7) X r.8) X r.9) Table XIX: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % ncrease n r & r 2 Algorh 2). Module X X X X r.4 & r 2.4) r.3 & r 2.3) r.2 & r 2.2) r. & r 2.)

26 Table XX: Soe nuercal values of he opal TE expendures for he cases of 4%, 3%, 2%, and % decrease n r & r 2 Algorh 2). Module X X X X r.6 & r 2.6) r.7 & r 2.7) r.8 & r 2.8) r.9 & r 2.9) ACKOWLEDGEMET Ths research was suppored by he aonal Scence Councl, Tawan, under Gran SC E-7-87 and also parally suppored by a gran fro he Research Gran Councl of he Hong Kong Specal Adnsrave Regon, Chna Projec o. CUHK425/4E). REFERECES [] M. R. Lyu, Handbook of Sofware Relably Engneerng, McGraw Hll, 996. [2] J. D. Musa, Sofware Relably Engneerng: More Relable Sofware, Faser Developen and Tesng, McGraw-Hll, 999. [3] R. S. Pressan, Sofware Engneerng: A Praconer's Approach, 6/e, McGraw-Hll, 25. [4] P. Kuba, and H. S. Koch, Managng Tes-Procedure o Acheve Relable Sofware, IEEE Trans. on Relably, Vol. 32, o. 3, pp , 983. [5] P. Kuba, Assessng relably of Modular Sofware, Operaon Research Leers, Vol. 8, o., pp. 35-4, 989. [6] B. Llewood, Sofware Relably Model for Modular Progra Srucure, IEEE Trans. on Relably, Vol. 28, o. 3, pp , 979. [7] Y. W. Leung, Sofware Relably Growh Model wh Debuggng Effors, Mcroelecroncs and Relably, Vol. 32, o. 5, pp , 992. [8] Y. W. Leung, Dynac Resource Allocaon for Sofware Module Tesng, The Journal of Syses and Sofware, Vol. 37, o. 2, pp , May 997. [9] Y. W. Leung, Sofware Relably Allocaon under Unceran Operaonal Profles, Journal of he Operaonal Research Socey, Vol. 48, o. 4, pp. 4-4, Aprl

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28 Engneerng, Vol. 29, o. 3, pp , March 23. [25] S. Y. Kuo, C. Y. Huang, and M. R. Lyu, Fraework for Modelng Sofware Relably, Usng Varous Tesng-Effors and Faul-Deecon Raes, IEEE Trans. on Relably, Vol. 5, o. 3, pp. 3-32, Sep. 2. [26] C. Y. Huang, and S. Y. Kuo, Analyss and Assessen of Incorporang Logsc Tesng Effor Funcon no Sofware Relably Modelng, IEEE Trans. on Relably, Vol. 5, o. 3, pp , Sep. 22. [27] C. Y. Huang, J. H. Lo, S. Y. Kuo, and M. R. Lyu, Opal Allocaon of Tesng Resources for Modular Sofware Syses, Proceedngs of he IEEE 3h Inernaonal Syposu on Sofware Relably Engneerng ISSRE 22), pp.29-38, ov. 22, Annapols, Maryland. [28] C. Y. Huang, J. H. Lo, S. Y. Kuo, and M. R. Lyu, Sofware Relably Modelng and Cos Esaon Incorporang Tesng-Effor and Effcency, Proceedngs of he IEEE h Inernaonal Syposu on Sofware Relably Engneerng ISSRE'99), pp , ov. 999, Boca Raon, Florda. [29] F.. Parr, An Alernave o he Raylegh Curve for Sofware Developen Effor, IEEE Trans. on Sofware Engneerng, SE-6, pp , 98. [3] T. DeMarco, Conrollng Sofware Projecs: Manageen, Measureen and Esaon. Prence-Hall, 982. [3] G. L. ehauser, A. H. G. Rnnooy Kan, M. J. Todd, Opzaon: Handbooks n Operaons Research and Manageen Scence; v., orh-holland, 994. [32] M. S. Bazaraa, H. D. Sheral, and C. M. Shey, onlnear Prograng: Theory and Algorhs, 2nd Ed., John Wley & Sons, 993. [33] J. H. Lo, S. Y. Kuo, M. R. Lyu, and C. Y. Huang, Opal Resource Allocaon and Relably Analyss for Coponen-Based Sofware Applcaons, Proceedngs of he 26h Annual Inernaonal Copuer Sofware and Applcaons Conference COMPSAC 22), pp. 7-2, Aug. 22, Oxford, England. [34] M. Xe, and G. Y. Hong, A Sudy of he Sensvy of Sofware Release Te, Journal of Syses and Sofware, Vol. 44, Issue 2, pp , 998. [35] P. S. F. Yp, X. Lqun, D. Y. T. Fong, and Y. Hayakawa, Sensvy-Analyss and Esang uber-of-fauls n Reoval Debuggng, IEEE Trans. on Relably, Vol. 48, o. 3, pp. 3-35, 999. [36] S. S. Gokhale, and K. S. Trved, Relably Predcon and Sensvy Analyss Based on Sofware Archecure, Proceedngs of he IEEE 3h Inernaonal Syposu on Sofware Relably Engneerng ISSRE 22), pp , ov. 22, Annapols, Maryland. [37] A. Pasqun, A.. Crespo, and P. Marella, Sensvy of Relably-Growh Model o Operaonal Profle Errors vs. Tesng Accuracy, IEEE Trans. on Relably, Vol. 45, o. 4, pp , 996. [38] M. H. Chen, A. P. Mahur, and V. J. Rego, A Case Sudy o Invesgae Sensvy of Relably Esaes o Errors n Operaonal Profle, Proceedngs of he IEEE 5h 28

29 Inernaonal Syposu on Sofware Relably Engneerng ISSRE'94), pp , Oc. 994, Monerey, Calforna. [39] C. Y. Huang, J. H. Lo, J. W. Ln, C. C. Sue, and C. T. Ln, Opal Resource Allocaon and Sensvy Analyss for Sofware Modular Tesng, Proceedngs of he IEEE 5h Inernaonal Syposu on Muleda Sofware Engneerng ISMSE 23), pp , Dec. 23, Tachung, Tawan. [4] J. H. Lo, C. Y. Huang, S. Y. Kuo, and M. R. Lyu, Sensvy Analyss of Sofware Relably for Dsrbued Coponen-Based Sofware Syses, Proceedngs of he 27h Annual Inernaonal Copuer Sofware and Applcaons Conference COMPSAC 23), pp. 5-55, ov. 23, Dallas, Texas. ABOUT THE AUTHORS Dr. Chn-Yu Huang; Dep' of Copuer Scence; aonal Tsng Hua Unversy; Hsnchu, TAIWA. Inerne e-al): Chn-Yu Huang Meber IEEE) s currenly an Asssan Professor n he Deparen of Copuer Scence a aonal Tsng Hua Unversy, Hsnchu, Tawan. He receved he MS 994), and he Ph.D. 2) n Elecrcal Engneerng fro aonal Tawan Unversy, Tape. He was wh he Bank of Tawan fro 994 o 999, and was a senor sofware engneer a Tawan Seconducor Manufacurng Copany fro 999 o 2. Before jonng THU n 23, he was a dvson chef of he Cenral Bank of Chna, Tape. Hs research neress are sofware relably engneerng, sofware esng, sofware ercs, sofware esably, faul ree analyss, and syse safey assessen. He s a eber of IEEE. Dr. Mchael R. Lyu; Copuer Scence & Engneerng Dep ; The Chnese Unv. of Hong Kong; Shan, HOG KOG. Inerne e-al): lyu@cse.cuhk.edu.hk Mchael R. Lyu receved he B.S. 98) n elecrcal engneerng fro aonal Tawan Unversy; he M.S. 985) n copuer engneerng fro Unversy of Calforna, Sana Barbara; and he Ph.D. 988) n copuer scence fro Unversy of Calforna, Los Angeles. He s a Professor n he Copuer Scence and Engneerng Deparen of he Chnese Unversy of Hong Kong. He worked a he Je Propulson Laboraory, Bellcore, and Bell Labs; and augh a he Unversy of Iowa. Hs research neress nclude sofware relably engneerng, sofware faul olerance, dsrbued syses, age & vdeo processng, uleda echnologes, and oble neworks. He has publshed over 2 papers n hese areas. He has parcpaed n ore han 3 ndusral projecs, and helped o develop 29

30 any coercal syses & sofware ools. Professor Lyu was frequenly nved as a keynoe or uoral speaker o conferences & workshops n U.S., Europe, and Asa. He naed he Inernaonal Syposu on Sofware Relably Engneerng ISSRE), and was Progra Char for ISSRE'996, Progra Co-Char for WWW & SRDS 25, and General Char for ISSRE'2 & PRDC 25. He also receved Bes Paper Awards n ISSRE'98 and n ISSRE'23. He s he edor-n-chef for wo book volues: Sofware Faul Tolerance Wley, 995), and he Handbook of Sofware Relably Engneerng IEEE and McGraw-Hll, 996). He has been an Assocae Edor of IEEE Transacons on Relably, IEEE Transacons on Knowledge and Daa Engneerng, and Journal of Inforaon Scence and Engneerng. Professor Lyu s an IEEE Fellow. 3