DYNAMIC PROGRAMMING APPROACH TO TESTING RESOURCE ALLOCATION PROBLEM FOR MODULAR SOFTWARE

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1 DYAMIC PROGRAMMIG APPROACH TO TESTIG RESOURCE ALLOCATIO PROBLEM FOR MODULAR SOFTWARE P.K. Kpur P.C. Jh A.K. Brdh Astrct Tstg phs of softwr gs wth modul tstg. Durg ths prod moduls r tstd dpdtly to rmov mxmum possl umr of fults wth spcfd tm lmt or tstg rsourc udgt. Ths gvs rs to som trstg optmzto prolms, whch r dscussd ths ppr. Two Optmzto modls r proposd for optml llocto of tstg rsourcs mog th moduls of Softwr. I th frst modl, w mxmz th totl fult rmovl, sujct to udgtry Costrt. I th scod modl, ddtol costrt rprstg sprto lvl for fult rmovls for ch modul of th softwr s ddd. Ths modls r solvd usg dymc progrmmg tchqu. Th mthods hv llustrtd through umrcl xmpls. Ky words: Softwr Rllty, o Homogous Posso Procss, Rsourc Allocto, Dymc Progrmmg. Itroducto Growth softwr grg tchology hs ld to producto of softwr for hghly complx stutos occurrg dustry, sctfc rsrch, dfs d dy to dy lf. Cosqutly, th dpdc of mkd o computrs d computrsd systms s crsg dy y dy. Ay flur ths systms c cost hvly trms of moy d/or hum lvs. Though hgh rllty of hrdwr prt of ths systms c gurtd, th sm cot sd for softwr. Thrfor lot of mportc s ttchd to th tstg phs of th softwr dvlopmt procss, whr roud hlf th dvlopmtl rsourcs r usd [8]. Esstlly tstg s procss of xcutg progrm wth th xplct tto of fdg fults d t s ths phs, whch s mdl to mthmtcl modlg. It s lwys dsrl to rmov susttl umr of fults from th softwr. I fct th rllty of softwr s drctly proportol to th umr of fults rmovd. Hc th prolm of mxmzto of softwr rllty s dtcl to Dprtmt of Oprtol Rsrch, Fculty of Mthmtcl Sccs, Uvrsty of Dlh, Dlh 0007, IDIA 7

2 tht of mxmzto of fult rmovl. At th sm tm tstg rsourc r ot ulmtd, d thy d to judcously usd. I ths ppr w dscuss d solv such mgmt prolm of llocto of tstg rsourcs mog moduls, through Softwr Rllty Growth Modl SRGM. A Softwr Rllty Growth Modl SRGM s rltoshp tw th umr of fults rmovd from softwr d th xcuto tm/cpu tm/cldr tm. Svrl ttmpts hv md to rprst th ctul tstg vromt through SRGMs [,4,5,9]. Ths modls hv usd to prdct th fult cott, rllty d rls tm of softwr. SRGMs hv lso usd to mg th tstg phs. Ag lrg softwr cossts of moduls. Oft ths moduls r dvlopd dpdtly d ch modul my cot dffrt umr of fults d tht of dffrt svrty. Thrfor dstct SRGMs should usd to rprst th tstg procss of ch modul, s tstg for ths moduls r do dpdtly. A SRGM wth tstg ffort [9] hs chos to rprst th fult rmovl procss for th two optmzto prolms dscussd ths ppr. Th frst optmzto modl P mxmzs th totl umr of fults xpctd to rmovd, wh vll tstg rsourc s kow. Th mgmt ormlly sprs for som rllty lvl tht c trsltd trms of umr of fults rmovd. I th scod optmzto modl P w dd costrt P trms of mmum umr of fults sprd to rmovd from ch modul. Dymc progrmmg tchqu s usd to solv ths prolms. Ths s th frst tm tht ths hs do softwr grg, ccordg to our kowldg. Dymc progrmmg pproch, whch s sy to solv d udrstd provds glol optm for ths prolms. Th mthodology dscussd th ppr hs llustrtd through umrcl xmpls. ottos : umr of moduls th Softwr > : Expctd umr of fults th th modul,,, : Proportolty costt for th th modul x t : Currt tstg ffort xpdtur t tstg tm t t d t x w dw for th modul 0, : Th mout of tstg rsourc to lloctd to th th modul d totl tstg rsourc vll. m t : umr of fults rmovd 0,t] th th modul, m vlu fucto of HPP,,, T : Totl tstg tm * : Optml vlu of,,, f : Optml umr of fults rmovd upto th moduls.. corrspodg to th stg Dymc Progrmmg lgorthm 8

3 o : Asprto lvl of th modul.. umr of fults dsrd to rmovd from th modul p : Th mmum proporto of totl fults to rmovd from th modul.. Mthmtcl Modllg. Rsourc Allocto Prolm Cosdr softwr hvg moduls, whch r g tstd dpdtly for rmovg fults lyg dormt thm. Th durto of modul tstg s oft fxd wh schdulg s do for th whol tstg phs. Hc lmtd rsourcs r vll, tht d to lloctd judcously. If m fults r xpctd to rmovd from th th modul wth ffort, th rsultg tstg rsourc llocto prolm c sttd s follows [5,6]. mx m sujct to, 0,,, P Aov optmzto prolm s th smplst o s t cosdrs th rsourc costrt oly. Ltr ths ppr, w corport ddtol costrts to th sc modl. For solvg P fuctol rltoshp tw fult rmovl d rsourc cosumpto s rqurd, whch s dscussd th followg scto.. SRGM For Moduls A Softwr Rllty Growth Modl xpls th tm dpdt hvor of fult rmovl. As moduls r tstd dpdtly dstct SRGMs would rprst thr rllty growth. Th fluc of tstg ffort c lso cludd th SRGMs [9]. I ths ppr w dscuss th rsourc llocto prolm usg such SRGM for th moduls. Modl Assumptos. Softwr cosst of ft umr of moduls d tstg for ch modul s do dpdtly 9

4 . A modul s sujct to flurs t rdom tm cusd y fults rmg th softwr. 3. O flur, th fult cusg tht flur s mmdtly rmovd d o w fults r troducd. 4. Fult rmovl phomo s modlld y o Homogous Posso Procss HPP. 5. Th xpctd umr of fults rmovd t, t t to th currt tstg rsourc s proportol to th xpctd rmg umr of fults. Udr ssumpto 5, followg dffrtl quto my sly wrtt for th modul d m t dt m t,,,. x t Solvg quto wth th tl codto tht, t t 0, t 0, m t 0 w gt t m t,,, To dscr th hvour of tstg ffort, thr Expotl or Rylgh fucto hs usd [5,9]. Both c drvd form th ssumpto tht, " th tstg ffort rt s proportol to th tstg rsourc vll". d t c t [ α t ],,, 3 dt whr c t s th tm dpdt rt t whch tstg rsourcs r cosumd, wth rspct to th rmg vll rsourcs. Solvg quto 3 udr th tl codto 0 0 w gt t t α xp c k dk,,, 4 0 Wh c t β, costt β t t α,,, 5 If c t β. t, gvs Rylgh typ curv t β t α,,, 6 I ths ppr w hv chos xpotl fucto 5 to rprst tstg ffort th optmzto prolms. 30

5 .3 Estmto Of Prmtrs Th tstg ffort dt r gv th form of tstg ffort x k x < x <... < x cosumd tm 0, t ] ;,,..,. Th tstg ffort modl prmtrs α d β c stmtd y th mthod of lst squrs s follows. Mmz [ ˆ ] sujct to ˆ.. th stmtd vlu of tstg ffort s qul to th ctul vlu. Oc th stmts of α d β r kow, th prmtrs of th SRGMs for th moduls c stmtd through Mxmum Lklhood Estmto mthod usg th udrlyg Stochstc Procss, whch s dscrd y o Homogous Posso Procss. Durg stmto, stmtd vlus of α d β r kpt fxd. If th fult rmovl dt for modul s gv th form of cumultv umr of fults rmovd y j tm 0,t j ]. Th lklhood fucto for tht modul s gv s y y [ m ] t j m t j j j m t j m t j L, / y, W y j j y j! 3. Optml Allocto Of Rsourcs From th stmts of prmtrs of SRGMs for moduls, th totl fult cott th softwr s kow. Moduls tstg ms t rmovg mxmum umr of thm, wth vll rsourcs. Hc P c rsttd s Mxmz m Sujct to, 0,, PA PA c solvd usg Dymc Progrmmg Approch. From Bllm's prcpl of optmlty, w c wrt th followg rcursv quto []. 3

6 { } { f } f mx f mx,,, 7 0 To dx th moduls, thy c rrgd dscdg ordr of thr vlus of Through ths pproch rsourcs r lloctd to th moduls squtlly. But for som vlus of < r o or mor moduls wth hghr dx umr.. hvg lowr dtctlty my ot gt y llocto. W summrz ths rsult th followg smpl thorm. Thorm - If for y,,;, th vlus of,,... r zro d prolm rducs to - stg prolm wth r r r r log, r,,- r r rr 8 whr d / j j j,,, j j j Proof of th thorm s gv ppdx. As rsult of th ov llocto procdur, som moduls my ot tstd t ll. Ths stuto s ot dvsl. Ag mgmt oft sprs to chv crt mmum rllty lvl for th softwr d tht for ch modul of th Softwr.. crt prctgs of th fult cott ch modul of th Softwr s dsrd to rmovd. Hc P ds to sutly modfd to mxmz rmovl of fults th softwr udr rsourc costrt d mmum dsrd lvl of fults to rmovd from ch of th moduls th softwr. Th rsultg tstg rsourc llocto prolm c sttd s follows: mx m sujct to m p,,, 0 3

7 , 0,,, P P c solvd usg Dymc Progrmmg Approch thr y rducg th dmsolty of th prolm through Lgrg multplr or covrtg to P y susttuto. W frst cosdr th dmsolty rducto Dymc Progrmmg Approch [] s follows. [ α { }] 0 mx mφ, α α sujct to, α 0,, P3 Whr α,, s Lgrg multplr for th costrt corrspodg to th th modul. Th ov prolm c solvd y Dymc Progrmmg pproch whch Kuh-Tuckkr optmlty codtos r otd t ch stg []. At y stg α,, c zro or o-zro dpdg upo ffctvss or ffctvss of costrt rspctvly. Hc ch stg hs two posslts d corrspodg to ch posslty of prcdg stg prst stg hs two posslts. So t y stg, totl umr of css s -. Ifct, ov prolm rducs to tht of fdg optml pth y srchg for optml soluto t ch stg whch oly o opto could chos. Ths procdur dos ot provd closd form soluto. Hc wthout furthr lorto of th ov mthod, th susttuto mthod s doptd for covrtg th prolm P to th prolm P s follows: m mpls 0 0 Hc, 0 log sy,,, Thrfor P c rsttd s, Mxmz m sujct to,,, 0,,, P4 33

8 Lt Y,,, th P4 c wrtt s th prolm P gv low Y mx m mx sujct to Y sy Y 0,,,,,, P5 0 Th Prolm P5 s smlr to th prolm P d hc usg thorm- th prolm P5 c lso solvd. If for y,,,th Y, Y,..., Y r zros, th prolm P5 rducs to stg prolm d ts soluto s gv s Y log,,,- 9 f 0 Through quto 9 optml llocto of rsourcs to th moduls c clcultd. I th followg scto w umrclly llustrt ths rsults. 4. umrcl Exmpl It s ssumd tht prmtrs d for th th modul,... r lrdy stmtd usg th softwr flur dt. Cosdr softwr hvg 0 moduls whos prmtr stmts r s gv Tl-. Suppos th totl rsourc vll for tstg s Frst th prolm P s solvd d from th rcurso quto 7 optml llocto of rsourcs * for th moduls r computd. Ths r lstd Tl- log wth th corrspodg xpctd umr of fult rmovd, prctgs of fults rmovd d fults rmg for ch modul. Th totl umr of fults tht c rmovd through ths llocto s % of th fult cott s rmovd from th Softwr. It s osrvd 34

9 tht som moduls modul-9,0 th rmg fults ftr llocto s hgh. Ths c ld to frqut flur durg oprtol phs. Ovously ths wll ot stsfy th dvlopr d h my dsr tht t lst 50% of fult cott from ch of th moduls of th Softwr s rmovd.. p 0.5 for ch 0. Sc fults ch modul r tgrl vlus, rst tgr lrgr th 50% of th fult cott ch modul s tk s lowr lmt tht hs to rmovd. Th w llocto of rsourc log wth xpctd umr of fult rmovd, prctgs of fults rmovd d fults rmg for ch modul ftr solvg th prolm P through th prolm P5 s summrzd Tl-. Th totl umr of fults tht c rmovd through ths llocto s % of th fult cott s rmovd from th Softwr. I ddto to th ov f t s dsrd tht crt prctg of th totl fults r to rmovd th ddtol tstg rsourcs would rqurd. It s trstg to study ths trdoff d Tl-3 summrzs rsults, whr th rqurd prctg of fults rmovd s 60%. To chv ths, 3000 uts of ddtol tstg ffort s rqurd. Th totl umr of fults tht c rmovd through ths llocto s % of th fult cott s rmovd from th Softwr. Alyss gv Tls-, d 3 hlp provdg th dvlopr sght to th rsourc llocto d th corrspodg fult rmovl phomo d th ojctv c st ccordgly. Modul * m * % of fults rmovd Tl - % of fults rmg E E E E E E Totl

10 Modul o * Y * m Y m * % of fults rmovd Tl- % of fults rmg Totl Tl-3 Modul o * Y * m Y * m * % of fults rmovd % of fults rmg Totl

11 5. Cocluso I ths ppr w hv dscussd coupl of optmzto prolms occurrg durg modul tstg phs of softwr dvlopmt lf cycl. A dymc progrmmg pproch for fdg th optml soluto hs proposd. Usg smpl rcurso qutos t s show how fult rmovl for ch modul d tht of th softwr c mxmzd, y judcous llocto of rsourcs. It s osrvd tht ftr crt durto of tstg, fult rmovl coms dffcult th ss tht grtr ffort wll rqurd to rmov ch ddtol fult. As th rllty of softwr s of utmost mportc sctfc dcso mkg s rqurd whl dcdg th rsourc udgt. Th trdoff s show scto-4 c usful ths rgrd. Altrtvly f th dvlopr s ot too k o optml soluto ut s stsfd wth ffct soluto, Gol Progrmmg pproch my dsrl tht cs. W r furthr lookg to ths spct. Appdx: Proof of th thorm- W hv followg rcurso qutos gv 7: mx { f } f mx { f },,, 0 Th ov prolm c solvd through forwrd rcurso stgs s follows. Stg-: Lt th w hv mx { f } Stg-: Lt th w hv { } f f mx 0 Susttutg f ov w hv f mx 0 { } 37

12 38 ow lt, { } F th { } mx 0 F f Th mxm c foud through clculus. d Th suffccy codto c chckd through th scod drvtv codto: 0 d F d Th followg thr stutos c occur. 0 < d 0 d 0 > d If 0 < d, th 0. At 0 0 < d.. < Whch mpls >, othr words th dtctlty modul - s hghr th modul. Smlrly 0 > d mpls d w hv > Hc >, th tstg rsourcs would lloctd to modul - frst s th dtctlty s hghr thr. Flly f 0 d log,.. log,

13 39 d f.. f Whr,, ow procdg y ducto t c show for th stg, log d f for Th proof s complt. Rfrcs. Gol A.L., Softwr Rllty Modls: Assumptos, lmttos d pplclty, IEEE Trs. O softwr grg, SE-, pp. 4-43, Hdly, G., olr d Dymc Progrmmg, Addso-Wsly, Rdg Mss, Ichmor, T, Ymd, S. Ad shwk M., Optml llocto polcs for tstg-rsourc sd o Softwr Rllty Growth Modl, Procdgs of th Austrl Jp workshop o stochstc modls grg, tchology d mgmt, pp. 8-89, Kpur P.K. d Grg R.B.; Cost rllty optmum rls polcs for softwr systm wth tstg ffort, OPSEARCH, vol. 7, o., pp. 09-8, Kpur P.K., Grg R.B. d Kumr, S.; Cotrutos to Hrdwr d Softwr Rllty, World Sctfc, Sgpor, 999.

14 6. Kpur, P.K. d Brdh, A.K., Modllg, llocto d cotrol of rsourcs: trdscplry pproch softwr rllty d mrktg, Oprtos Rsrch, Eds. M. Agrwl d K. S, ros Pulshg Hous, w Dlh Kut P. d Koch H.S., Mgg tst procdurs to chv rll softwr, IEEE Trs. O Rllty, ol. R-3, pp , Mus J.D., Io A. Ad Okumoto K, Softwr Rllty- Msurmt, Prdcto d Applcto, Mc Grw Hll, Ymd S. Ad Ohtr H. Ad rhs H., Softwr Rllty Growth Modl wth tstg ffort, IEEE Trs. O Rllty, vol. R-35, pp. 9-3,