Assessing Software Reliability Using SPC An Order Statistics Approach
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1 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Assssig Softw Rliility Usig SPC A Od Sttistics Appoch.Rmchd H Ro D. R.Sty Psd D. R.R..thm Dptmt of Comput Scic, A.S.N. Dg Collg, Tli, Idi [email protected] Dptmt of Comput Scic, Achy Ngju Uivsity, Gutu, Idi [email protected] 3 Dpt. of Sttistics, Achy Ngju Uivsity, Gutu, INDIA, [email protected] Atct Th my softw liility modls tht sd o th tims of occucs of os i th duggig of softw. It is show tht it is possil to do symptotic lilihood ifc fo softw liility modls sd o od sttistics o NoHomogous Poisso Pocsss (NHPP, with symptotic cofidc lvls fo itvl stimts of pmts. I pticul, itvl stimts fom ths modls otid fo th coditiol filu t of th softw, giv th dt fom th duggig pocss. Th dt c goupd o ugoupd. Fo somo mig dcisio out wh to mt softw, th coditiol filu t is impott pmt. Od sttistics usd i wid vity of pcticl situtios. Thi us i chctiztio polms, dtctio of outlis, li stimtio, study of systm liility, liftstig, suvivl lysis, dt compssio d my oth filds c s fom th my oos. Sttisticl Pocss Cotol (SPC c moito th focstig of softw filu d thy cotiut sigifictly to th impovmt of softw liility. Cotol chts widly usd fo softw pocss cotol i th softw idusty. I this pp w poposd cotol mchism sd o od sttistics of cumultiv qutity tw ovtios of tim domi filu dt usig m vlu fuctio of Hlf ogistics Distiutio (HD sd o NHPP. ywods Od Sttistics, Sttisticl Pocss Cotol (SPC, Hlf ogistics Distiutio (HD, NHPP. INTRODUCTION Th moitoig of Softw liility pocss is f fom simpl ctivity. I ct ys, svl uthos hv commdd th us of SPC fo softw pocss moitoig. A fw oths hv highlightd th pottil pitflls i its us[]. Th mi thust of th pp is to fomliz d pst y of guidlis i disciplid pocss with viw to hlpig th pctitio i puttig SPC to coct us duig softw pocss moitoig. Ov th ys, SPC hs com to widly usd mog oths, i mufctuig idustis fo th pupos of cotollig d impovig pocsss []. Ou ffot is to pply SPC tchiqus i th softw dvlopmt pocss so s to impov softw liility d qulity []. It is potd tht SPC c succssfully pplid to svl pocsss fo softw dvlopmt, icludig softw liility pocss. SPC is tditiolly so wll doptd i mufctuig idusty. I gl softw dvlopmt ctivitis mo pocss ctic th poduct ctic which ms it difficult to DOI :.5/ijcs..4
2 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August pply SPC i stight fowd m. Th utiliztio of SPC fo softw liility hs th sujct of study of svl schs. A fw of ths studis sd o liility pocss impovmt modls. Thy tu th sch light o SPC s ms of ccomplishig high pocss mtuitis. Som of th studis fuish guidlis i th us of SPC y modifyig gl SPC picipls to suit th spcil quimts of softw dvlopmt [] (Bu d Ow[3]; Flo d Clto[4]. It is spcilly otwothy tht Bu d Ow povid smil guidlis y dlitig th tchiqus cutly i vogu fo mgig d cotollig th liility of softw. Sigifictly, i doig so, thi focus is o cotol chts s fficit d ppopit SPC tools. It is ccptd o ll hds tht Sttisticl pocss cotol cts s powful tool fo igig out impovmt of qulity s wll s poductivity of y mufctuig pocdu d is pticully lvt to softw dvlopmt lso. Viwd i this light, SPC is mthod of pocss mgmt though pplictio of sttisticl lysis, which ivolvs d icluds th dfiig, msuig, cotollig, d impovig of th pocsss [5].. Odd Sttistics Od sttistics usd i wid vity of pcticl situtios. Thi us i chctiztio polms, dtctio of outlis, li stimtio, study of systm liility, liftstig, suvivl lysis, dt compssio d my oth filds c s fom th my oos [6]. Od sttistics dls with poptis d pplictios of odd dom vils d of fuctios of ths vils. Th us of od sttistics is sigifict wh filus fqut o it filu tim is lss. t X dot cotiuous dom vil with poility dsity fuctio f(x d cumultiv distiutio fuctio F(x, d lt (X, X,, X dot dom smpl of siz dw o X. Th oigil smpl ovtios my uodd with spct to mgitud. A tsfomtio is quid to poduc cospodig odd smpl. t (X (, X (,, X ( dot th odd dom smpl such tht X ( < X ( < < X ( ; th (X (, X (,, X ( collctivly ow s th od sttistics divd fom th pt X. Th vious distiutiol chctistics c ow fom Blish d Coh [7]. Th itfilu tim dt pst th tim lps tw vy two coscutiv filus. O th oth hd if sol witig tim fo filus is ot sious polm, w c goup th itfilu tim dt ito o ovlppig succssiv su goups of siz 4 o 5 d dd th filu tims with i ch su goup. Fo istc if dt of itfilu tims vill w c goup thm ito disjoit sugoups of siz 5. Th sum totl i ch sugoup would dot th tim lps tw vy 5 th od sttistics i smpl of siz 5.I gl fo itfilu dt of siz, if (y tul o lss th d pfly fcto, w c covitly divid th dt ito disjoit sugoups (/ d th cumultiv totl i ch sugoup idict th tim tw vy th filu. Th poility distiutio of such tim lps would tht of th th odd sttistics i sugoup of siz, which would qul to th pow of th distiutio fuctio of th oigil vil (m(t. Th whol pocss ivolvs th mthmticl modl of th m vlu fuctio d owldg out its pmts. If th pmts ow thy c t s thy fo th futh lysis, if th pmts ot ow thy hv to stimtd usig smpl dt y y dmissil, fficit mthod of distiutio. This is sstil cus th cotol limits dpd o m vlu fuctio, which it dpds o th pmts. If softw filus quit fqut pig tc of itfilu is tdcious. If filus mo fqut od sttistics pfl. 3. Modl Dsciptio To clcult th pmt vlus d cotol limits usig Od Sttistics ppoch, w cosidd Hlf ogistic Distiutio [8][].
![Som of th studis fuish guidlis i th us of SPC y modifyig gl SPC picipls to suit th spcil quimts of softw dvlopmt [] (Bu d Ow[3]; Flo d Clto[4].](/docs-images/53/11686854/images/page_2.jpg "It is spcilly otwothy tht Bu d Ow povid smil guidlis y dlitig th tchiqus cutly i vogu fo mgig d cotollig th liility of softw.")
3 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Th m vlu fuctio of HD [8] is m( t t ( t ( 3. To gt m(t vlu fo th Od Sttistics, t m(t to th pow [ m(t ] t t 3. [ m(s ] 3.3 Divtio with spct to t of qutio 4.3. m'(s. m'( s. ( ( ( ( ( m( s ' m (s ms ( ' og og m (s. ( ms ( og 3.6 [ log log log log ( log( ( log( ] 3.7 Divtio with spct to 3
![To gt m(t vlu fo th Od Sttistics, t m(t to th pow [ m(t ] t t 3. [ m(s ] 3.](/docs-images/53/11686854/images/page_3.jpg "3 Divtio with spct to t of qutio 4.3. m'(s. m'( s. ( ( ( ( ( 3.4 3.")
4 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August 4. ( ( s. s ( ( ( ( s ( s (
5 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August 5 s ( g( s ( g( s s s g( 3.8 ' s s. ( g( s. [ ] ( ( 3.9
6 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August 4. Moitoig th tim tw filus usig cotol cht Th slctio of pop SPC chts is sstil to ffctiv sttisticl pocss cotol implmttio d us. Th my chts which us sttisticl tchiqus. It is impott to us th t cht fo th giv dt, situtio d d [9].Th dvcs chts tht povid mo ffctiv sttisticl lysis. Th sic typs of dvcd chts, dpdig o th typ of dt th vil d ttiut chts. Vil cotol chts dsigd to cotol poduct o pocss pmts which msud o cotiuous msumt scl. X, R chts vil cotol chts. Attiuts chctistics of pocss which sttd i tms of good d, ccpt o jct, tc. Attiut chts ot ssitiv to vitio i th pocss s vils chts. Howv, wh dlig with ttiuts d usd poply, spcilly y icopotig l tim pto lysis, thy c ffctiv impovmt tools. Fo ttiut dt th : pchts, cchts, pchts, d uchts. W hv md th cotol cht s Filus Cotol Cht i this pp. Th sid cotol cht hlps to ssss th softw filu phom o th sis of th giv it filu tim dt 5. Estimtio of Pmts d Cotol imits Giv th dt ovtios d smpl siz d pmts usig qutios (3.,(3.8,(3.9, th d computd y usig th popul NR mthod. A pogm witt i C ws usd fo this pupos. Th qutio fo m vlu fuctio of Hlf ogistic Distiutio is giv y Th Cotol limits otid s follows: Dlt th tm fom th m vlu fuctio. Equt th miig fuctio succssivly to.99865,.35,.5 d solv fo t, fo hlf logistic distiutio, i od to gt th usul Six sigm cospodig cotol limits, ctl li. F(t log It givs t t U 4..7 t t 4. t t C 4.3 6
7 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Th cotol limits such tht th poit ov th m(t U (4.(UC is lm sigl. A poit low th m(t (4.3 (C is idictio of tt qulity of softw. A poit withi th cotol limits idicts stl pocss. 5. Dvlopig Filus Cht: Giv th itfilu dt th vlus of m(t t T c, T u, T d t th giv itfilu tims clcultd. Th succssiv diffcs of th m(t s t, which lds to vlus. Th gph with th sid itfilu tims to o Xxis, th vlus of succssiv diffcs m(t s o Yxis, d th 3 cotol lis plll to Xxis t m(t, m(t U, m(t C spctivly costituts filus cotol cht to ssss th softw filu phom o th sis of th giv itfilus tim dt. 6. Illusttio Th pocdu of filus cotol cht fo filu softw pocss is illusttd with xmpl h. Tl show th tim tw filus of softw poduct []. Tl: Softw filu dt potd y Mus(975 [] Fult Tim Fult Tim Fult Tim Fult Tim
8 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Tl: Pmt stimts d thi cotol limits of 4 d 5 od Dt St Od m t m t m t Tl ( U ( C ( Tl: 3 Succssiv diffcs of 4 od m(t s of Tl 4od Succssiv 4od Diffc s Fult m(t Of m(t s Fult Cumultivs Cumultivs m(t Succssiv Diffc s Of m(t s E E E E E E E E E E
9 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Filus Cotol Cht Succsiv Diffc UC C.7368 C Filu Num Fig : Filus Cotol Cht of Tl 3 Tl: 4 Succssiv diffcs of 5 od m(t s of Tl 5od Succssiv 5od Diffc s Fult m(t Of m(t s Fult Cumultivs Cumultivs m(t Succssiv Diffc s Of m(t s
10 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August Filus Cotol Cht Succsiv Diffc UC C C Filu Num Fig : Filus Cotol Cht of Tl 4 7. CONCUSION Th Filus Cotol Chts of Fig to hv show out of cotol sigls i.. low C. By ovig Filus Cotol Chts, w idtifid tht filus situtio is dtctd t ly stgs. Th ly dtctio of softw filu will impov th softw liility. Wh th cotol sigls low C, it is lily tht th ssigl cuss ldig to sigifict pocss dtiotio d it should ivstigtd. Hc, w coclud tht ou cotol mchism poposd i this chpt with od sttistics ppoch givig positiv commdtio fo its us to stimt whth th pocss is i cotol o out of cotol. ACNOWEDGMENTS Ou ths to Dptmt of Comput Scic d Egiig; Dptmt of Sttistics, Achy Ngju Uivsity; Dptmt of Comput Scic, Athui Styy Dg Collg, Tli, fo povidig cssy fcilitis to cyout th sch wo. REFERENCES [] N. Boffoli, G. Buo, D. Cvivo, G. Mstlloi; Sttisticl pocss cotol fo Softw: systmtic ppoch; 8 ACM /8/. []. U. Sgut, O. Dmios; Utiliztio of sttisticl pocss cotol (SPC i mgt softw ogiztios: Pitfllsd suggstios; Spig Scic Busiss mdi Ic. 6. [3] Bu,A. d Ow,M.996. Sttisticl Mthods fo Softw qulity. Thomso pulishig Compy. ISBN 8537X. [4] Clto, A.D. d Floc, A.W Sttisticlly cotollig th Softw pocss. Th 99 SEI Softw Egiig Symposim, Softw Egiig Istitut, Cgi Mllo Uivsity. [5] Mutsumi omuo; Expics of Applyig SPC Tchiqus to softw dvlopmt pocsss; 6 ACM x/6/5. [6] A M. Mthi ;Od Sttistics fom ogistic Dstiutio d Applictios to Suvivl d Rliility Alysis;IEEE Tsctios o Rliility, vol.5, No.; 3 [7] Blish.N., Cliffod Coh; Od Sttistics d Ifc; Acdmic Pss ic.;99. 3
11 Ittiol Joul of Comput Scic, Egiig d Applictios (IJCSEA Vol., No.4, August [8] R.StyPsd, Softw Rliility with SPC ; Ittiol Joul of Comput Scic d Emgig Tchologis; Vol, issu, Apil [9] Rold P.Ajd;SPC CHART slctio pocss;pgm 67(9959Elsvi scic ltd. [] Hog Phm; Systm Rliility; Spig;5;Pg No.8 [] M.Xi, T.N. Goh, P. Rj; Som ffctiv cotol cht pocdus fo liility moitoig; Elsvi scic td, Rliility Egiig d systm sfty 77( 43 5 [] R.stypsd, Hlf ogistic Softw Rliility Gowth Modl,Ph.D. Thsis,7 Authos.Rmchd H Ro, civd Mst s dg i Tchology with Comput Scic fom D. M.G.R Uivsity, Chi, Tmildu, Idi,. H is cutly woig s Associt Pofsso d Hd of th Dptmt, i th Dptmt of Comput Scic, A.S.N. Dg Collg, Tli, which is ffilitd to Achy Ngju Uivsity. H hs 8 ys tchig xpic d ys of Idusty xpic t Mog Stly, USA s Softw Alyst. H is cutly pusuig Ph.D., t Dptmt of Comput Scic d Egiig, Achy Ngju Uivsity, Gutu, Adh Pdsh, Idi. His sch is softw Egiig. H hs pulishd svl pps i Ntiol & Ittiol Jouls. D. R. Sty Psd civd Ph.D. dg i Comput Scic i th fculty of Egiig i 7 fom Achy Ngju Uivsity, Adh Pdsh. H civd gold mdl fom Achy Ngju Uivsity fo his out stdig pfomc i Msts Dg. H is cutly woig s Associt Pofsso d H.O.D, i th Dptmt of Comput Scic & Egiig, Achy Ngju Uivsity. His cut sch is focusd o Softw Egiig, Softw liility. H hs pulishd svl pps i Ntiol & Ittiol Jouls. R.R..tm is pofsso of sttistics t Achy Ngju Uivsity,Gutu Idi. H hs 3 ys of tchig xpic i sttistics t Ud Gdut d Post Gdut pogms. As sch i Sttistics, h hs succssfully guidd my studts fo M.Phil d Ph.D. i sttistics. H hs uthod mo th 6 sch pulictios ppd vious sttistics d comput scic jouls pulishd i Idi d oth coutis li US, U, Gmy, Pist, Sil d Bgldsh. H hs f fo Joul of Applid Sttistics (U., METRON (Itly, Pist Joul of Sttistics (Pist, IAPQR Tsctios (Idi, Assm Sttisticl Rviw (Idi d Gujt Sttisticl Rviw (Idi. H hs spcil sp i tchicl sssios of um of Smis d Cofcs. His s of sch itst Sttisticl Ifc, Rliility Studis, Qulity Cotol Mthods d Actuil Sttistics. As tch his pst tchig s Poility Thoy, Rliility, d Actuil Sttistics. His li tchig topics iclud Sttisticl Ifc, Mthmticl Alysis, Optios Rsch, Ecoomtics, Sttisticl Qulity cotol, Msu thoy. 3
![Rj; Som ffctiv cotol cht pocdus fo liility moitoig; Elsvi scic td, Rliility Egiig d systm sfty 77( 43 5 [] R.stypsd, Hlf ogistic Softw Rliility Gowth Modl,Ph.D. Thsis,7 Authos.](/docs-images/53/11686854/images/page_11.jpg "Rmchd H Ro, civd Mst s dg i Tchology with Comput Scic fom D. M.G.R Uivsity, Chi, Tmildu, Idi,. H is cutly woig s Associt Pofsso d Hd of th Dptmt, i th Dptmt of Comput Scic, A.S.N.")
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