JOSE L. HURTADO, FRANCISCO JOGLAR 1, MOHAMMAD MODARRES 2

Transcription

1 Inrnaonal Journal of Prformably Engnrng, Vol., No., July 25, pp RAMS Consulans Prnd n Inda. Inroducon JOSE L. HURTADO, FRANCISCO JOGLAR, MOHAMMAD MODARRES 2 Dparmn of Mchancal Engnrng, Rlably Engnrng Program, Unvrsy of Maryland, 2 Mar Moun Hall, Collg Park, MD , USA Rcvd on Fb. 28, 25 Absrac: Kma and Suma hav proposd a sochasc modl calld h gnralzd rnwal procss GRP o dscrb h avalably characrscs of rparabl sysms by nroducng h noon of h vrual ag of h sysm. Yanz al. offr maxmum lklhood smaon MLE approach for smang paramrs of h GRP modls. Du o h complxy of h quaons, a clos soluon s no avalabl, and numrcal soluons ar proposd wh lmd succss. Ths papr dscrbs an alrnav for calculang h paramrs of GRP modls usng a Gnc Algorhm GA approach o solv complx MLE quaons. Th rsuls usng hs approach confrm and xnd conclusons of h Kma and Suma, and Yanz al. works. Exampls of applcaons of GA hav bn prsnd. Th papr also concluds ha undr cran condons, applcaon of h mnmal rpar assumpon provd a rasonabl answr for h avalably of rparabl uns. Ky Words: gnralzd rnwal procss, rparabl sysms, gnc algorhm, vrual ag, war ou. Engnrng sysms can b cagorzd no wo basc yps: non-rparabl, and rparabl []. Mos of h complx sysms ar rparabl. Analyzng rnds n h ra of occurrnc of falurs ROCOF s h mhod of choc o assss avalably of rparabl sysms. If h ROCOF shows h xsnc of a rnd, a falur procss known as h non-homognous Posson procss NHPP may b usd. On h ohr hand, f hr s no rnd.., ROCOF s consan, h falur procss s known as h homognous Posson procss HPP or prfc rnwal procss. Falur daa n h lar cas nds o b ndpndn and dncally dsrbud IID bcaus hs s h basc assumpon of h prfc rnwal procss. Scnc Applcaons Inrnaonal Corporaon, Rson, VA, USA. 2 Corrspondng auhor: modarrs@glu.umd.du 37

2 38 J. L. Hurado, F. Joglar and M. Modarrs Howvr, n pracc, nhr h NHPP nor HPP sm ralsc for rparabl sysms. Ths s bcaus usng h HPP mans ha h rpard uns mus hav bn brough o a lk nw sa afr rpar, whch s ovrly opmsc. On h ohr hand, h NHPP assums ha rpard uns rman n xacly h sam condon as h on us bfor h falur occurrd, or as bad as old condon, whch s somwha pssmsc. Dffrnly sad, HPP assums prfc rpar and NHPP assums mnmal rpar. Ralscally, a rpard sysm would b n a br condon han was us bfor h falur, bu cranly no lk-nw condon. [2] Som rsarchrs hav proposd probablsc modls o dal wh h sas bwn hs wo modls. For xampl, Laky and Rgdon [3] nroducd h modulad powr law procss, whch s a spcal cas of h nhomognous and modulad gamma procss nroducd by Brman [4]. Dorado al. [5] nroducd a gnral rpar modl, consdrd a br han mnmal rpar modl. Kma and Suma s GRP modl [6] may b consdrd as spcal cass of Dorado al. modl. Th GRP modl s a sochasc modl o dscrb opraon of a rparabl sysm as a funcon of m. Such a sysm s manand hrough nspcon and rpar acons. Th GRP usd n modlng prformanc rparabl sysms has bn h bass of rcn suds by Kamnsky and Krsov [7], Krsov [8] and Yanz, al. [9]. Th purpos of hs papr s o prsn an ovrvw of h mhodology dvlopd by Kma and Suma and dscuss, xpand and mprov Yanz, al [9] applcaons of hs mhodology. Th lar work offrd a mahmacal approach o assss h MLE of h paramrs of a Wbull dsrbuon rprsnng h random varabl, vrual-m-ofalur. Vrual ag s vwd as h ag ha h un s physcally xprncng fnss ag rahr han h lapsd m n opraon acual ag. Th nroducon of h vrual ag maks h MLE calculaon proposd by Yanz al. complx, and n som cass only local soluons ar possbl ha may dffr from h corrc global soluons. In hs papr, h GA approach has bn proposd o rmdy lmaons of Yanz, al. soluon. Whl Yanz, al. us a ROCOF modl n form of a powr law, n hs papr h scop of h applcaon has bn xpandd o nclud boh h log-lnar and powr law ROCOF modls. In hs cas, a GA s proposd as an opmzaon ool o sma h paramrs of h on vrual-m-o-falur dsrbuon and h ROCOF modls. Morovr, h GA approach allows smaon of h paramrs of h on modls ha maxmz hr lklhood funcons. In hs papr, a cursory rvw of h GRP modl proposd by Kma and Suma s provdd, followd by h lmnary aspcs of h GA chnqu, ncludng mplcaons of hr us n hs rsarch. Th papr prsns a compur-basd formulaon of h GA approach for paramr smaon. Th rsuls oband by applyng h proposd approach o four xampls of acual falur daass rpord and modld by ohrs has bn prsnd, compard and dscussd. Fnally, h papr prsns som conclusons and rcommndaons for furhr rsarch. 2. Ovrvw of GRP Accordng o Kma [6], vry ofn rpar acvs ar assumd o b mnmal. Tha s, rpar rsors a sysm o h oprang condon us pror o h occurrnc of falur. Ths assumpon sms rasonabl for sysms conssng of many componns ach havng hr own falur mod, snc h rpar of h fald componn wll no

3 Gnralzd Rnwal Procss 39 apprcably nhanc h ovrall sysm s oprang nvronmn. For xampl an auomobl ha nds a rpar, say rplacmn of h xhaus ppng, would mnmally xprnc a chang n s ovrall condon afr h rpar. On h ohr hand, for sysms wh only a fw maor componns, sms mor rasonabl o hnk ha rpar can brng h sa of a fald sysm o a lvl whch s somwhr bwn complly nw and ha of us pror o falur. To addrss sysms of h lar cas, Kma nroducd h noon of vrual ag of h sysm as follows: L V n b h vrual ag of h sysm mmdaly afr h nh rpar, h sysm xprncs s n+h rpar a an acual m T n+, accordng o: Pr[T n V n F y F y y ] F y Whr F. s h cumulav dnsy funcon of m o falur of h sysm, and T n+ s h m bwn h nh and n+h falur.., h nrarrval of m bwn n succssv falurs. Dfn a paral sum Sn k T k, wh S =, rprsnng h ral ag of h sysm a h m of h nh falur or h lapsd m snc h sysm has bn placd no opraon. L a n rprsn h rpar qualy ndx of h nh rpar. I s assumd ha h nh rpar can rmov h damags ncurrd durng T n. In ohr words, s slcd such ha rducs T n o a n T n, and h vrual ag afr h nh rpar bcoms V n = V n- + a n X n. A smplfcaon assumpon s o us a consan a n =. For hs cas, h vrual ag s gvn by V n S n n T k k 2 L Z y y b a random varabl dsrbud rprsnng h cumulav dsrbuon funcon n Equaon. Snc a nw sysm has h vrual ag V =, h frs nrarrval of falur T has h sam dsrbuon as Z. Afr h frs rpar, h vrual ag of h sysm bcoms V = T, and h scond lfm T 2 s hn dsrbud as T 2 = Z V, and V 2 = S 2. In gnral, h nh lfm s dsrbud as T n = Z Vn-, and V n = S n. I s mporan o no ha f hn V n S n, manng ha h rpar ruvnas h sysm. On h ohr hand, f hn V n S n, manng ha h sysm s n a wors condon afr h rpar. If = hn V n = S n, whch s h mnmal rpar cas, and f = hn, snc V n =, h sysm s rnwd by ach rpar and h rsulng falur procss s an ordnary rnwal procss. Suppos ha h nh falur occurs a m. Ths mans ha S n = and V n =. Th condonal lf dsrbuon GT V n = of T n+ s gvn, from, by G T F T F 3 F Kma appld hs mhodology n som xampls, concludng ha h mnmal rpar assumpon s no bad f h qualy of rpar s no prfc, say.6.

4 4 J. L. Hurado, F. Joglar and M. Modarrs 3. Ovrvw of Gnc Algorhm A rvoluon n bologcal hough, and ndd n human phlosophy, bgan whn Charls Darwn and Alfrd Russl Wallac prsnd hr vdnc for h hory of voluon bfor h Lnnan Socy of London on July, 858. Classcal Darwnan voluonary hory, combnd wh h slconsm of Wsmann and h gncs of Mndl, has now bcom a rahr unvrsally accpd s of argumns known as h no-darwnan paradgm[]. No-Darwnan assrs ha h hsory of h vas maory of lf s fully accound for by only a vry fw sascal procsss acng on and whn populaons and spcs. Ths procsss ar rproducon, muaon, compon, and slcon. Rproducon s an obvous propry of all lf, muaon s guarnd n any sysm ha connuously rproducs slf n a posvly nropc unvrs. Compon and slcon bcom h nscapabl consquncs of any xpandng populaon consrand o a fn arna. Evoluon s hn h rsul of hs fundamnal nracng sochasc procsss as hy ac on populaons, gnraon afr gnraon []. Th da of usng voluonary compuaon as a problm solvng chnqu xss snc h 95s. Snc hn, four maor approachs hav volvd: Evoluonary Programmng, Evoluonary Srags, Gnc Algorhms and Gnc Programmng. All of hs ar algorhms ha hav bn nsprd by h noons of voluon and survval n naur []. In hs work, w ar nrsd n h applcaon of GA o assss paramrs of a GRP modl by opmzaon of h rsuls ha h modl gvs whn compard o h undrlyng daa oband from acual vdncs of falur and rpar acons. Th GA wr frs nroducd by Holland [2] wh a wd array of applcaons n opmzaon. Ths mhodology s dsgnd o mmc h naural gnc bhavors by ncorporang spcfc mahmacal opraors o rplca procsss such as crossovr, muaon, and rcombnaon. Th gnrc approach o GA s mplmnd as follows:. Th problm o b addrssd s dfnd n an obcv funcon ha ndcas h fnss of any ponal soluon. For xampl how wll h GRP modl wh spcfc paramr valus prdcs h daa obsrvd. 2. A populaon of candda soluons n our cas paramrs of h GRP modl s nalzd subc o cran consrans. Typcally, ach ral s codd as a vcor x, calld chromosom, wh lmns dscrbd as gns and varyng valus a spcfc posons calld allls. 3. Each chromosom n h populaon s dcodd no a form appropra for valuaon and s hn assgnd a fnss scor, accordng o h obcv. 4. A probably of rproducon s assgnd o ach chromosom. Th lklhood of bng slcd s proporonal o s fnss rlav o h ohr chromosoms n h populaon. Tchnqus such as h roul whl, ournamn and sochasc unvrsal samplng ar usd o slc h parns. a. Roul whl: In ordr o rproduc offsprng, parns nd o b slcd. Th mos commonly usd mhods ar roul whl slcon and rank slcon. Th ky for roul whl slcon s fnss. Th fr h chromosoms ar, h mor chancs hy wll hav o b

5 Gnralzd Rnwal Procss 4 slcd. Imagn roul whr all chromosoms ar placd, ach has s own scon accordng o s fnss funcon. Th fs on gs h largs ara, whl h wors gs h smalls ara. Thn a marbl s hrown hr and a chromosom s slcd [3]. b. Tournamn: In ournamn, h slcon a group of ndvduals s randomly chosn from h populaon. Thy may b drawn from h populaon wh or whou rplacmn. Ths group aks par n a ournamn; ha s, a wnnng ndvdual s drmnd dpndng on s fnss valu. Th ndvdual havng h hghs fnss valu s usually chosn drmnscally hough occasonally a sochasc slcon may b mad. In boh cass only h wnnr s nsrd no h nx populaon and h procss s rpad n ms o oban a nw populaon [4]. c. Sochasc unvrsal samplng SUS: I s a smpl, sngl-phas samplng algorhm. I s zro basd, has a mnmum sprad and wll achv all N sampls n a sngl ravrsal. On a roul whl, hr s a sngl ponr, whch ndcas h wnnr. Th SUS s analogous o a roul whl wh N qually spacd ponrs. Hnc, a sngl spn rsuls n N wnnrs. Snc h sum of h populaon s xpcd valus s N, h ponrs ar xacly. apar. Thus an ndvdual s guarand [xpcd valu] n sampls and no mor han [xpcd valu]. Hnc, SUS has mnmal sprad. Furhrmor, n a randomly ordrd populaon, an ndvdual s slcon probably s basd solly on h nal spn and h magnud of hs xpcd valu. Hnc, SUS has zro bas [5]. 5. A nw populaon of chromosoms s gnrad accordng o h assgnd probabls of rproducon. Opraors such as dscr rcombnaon, nrmda rcombnaon, Exndd ln rcombnaon and muaon ar usd o gnra h nw chromosom or offsprng. Ths opraors works accordng o h followng: a. Dscr rcombnaon: Ths funcon prforms an xchang of varabl valus bwn h ndvduals [6]. For ach varabl, h parn who conrbus s varabl o h offsprng s chosn randomly wh qual probably. Typcal valus for h probabls of rcombnaon ar.6 [7],.75 o.95 [8] and.95 [9]. Dscr rcombnaon can b usd wh any knd of varabls bnary, ral or symbols. b. Inrmda rcombnaon: s a mhod only applcabl o ral varabls. Hr h varabl valus of h offsprng ar chosn somwhr around and bwn h varabl valus of h parns [6]. Offsprng ar producd accordng o h rul: Offsprng = parn +αparn 2 - parn,whr α s a scalng facor chosn unformly a random ovr an nrval [-d, + d]. In nrmda rcombnaon d =, for xndd nrmda rcombnaon d >. A good choc s d =.25. Each varabl n h offsprng s h rsul of combnng h varabls accordng o h abov xprsson wh a nw α chosn for ach varabl.

6 42 J. L. Hurado, F. Joglar and M. Modarrs c. Exndd ln rcombnaon: gnras offsprng n a drcon dfnd by h parns ln rcombnaon. I ss mor ofn ousd h ara dfnd by h parns and n h drcon of parn. Th pon for h offsprng s dfnd by faurs of h muaon opraor of h Brdr GA [2]. Exndd ln rcombnaon s only applcabl o ral varabls. d. Muaon: hs opraor s appld afr rcombnaon, and plays an mporan rol n gnc algorhm, bcaus hlps o prvn h problms of prmaur convrgnc assocad wh h rpad us of crossovr. Offsprng varabls ar muad by h addon of small random valus sz of h muaon sp, wh low probably.[2]. Typcal valus for h probabls of muaon ar. [7],.5 o. [8] and. [9]. I s consdrd a background opraor, assurng ha h crossovr has a full rang of allls and h adapv plan s no rappd on local opma. Typcally, a pon s locally opmal f no mprovmn can b mad by sarchng n a nonmpy nghborhood around ha pon. Ths may no b h cas for gnc algorhms rlyng solly on crossovr. Th squnc of rals may sagna anywhr, a any homognous collcon of pons. Undr such condons, h pon s locally opmal only bcaus h sarch algorhm s ncapabl of procdng furhr []. 6. Th procss sops f a suabl soluon has bn achvd or f h compur m has xprd. Ohrws, h procss procds o sp hr, whr h nw chromosoms ar scord. 4. Applcaon of Gnc Algorhm o GRP A numrcal funcon calculang h paramrs of h log-lnar and powr law modls was dvlopd usng MATLAB, and h Gnc and Evoluonary Algorhm Toolbox crad for us wh MATLAB [2]. Fgur shows h srucur of h algorhm. S a r Gnra nal po pulao n Evalua Obcv funco n Gnra nw populaon No Opm zao n crra m? Ys Slco n Rco m bnao n M uao n Bs ndvdual R suls Fgur.Algorhm for h smaon of h paramrs Fg. : Algorhm for h Esmaon of h Paramrs Th assumpons,valus and funcons usd n h program ar xpland n h followng: Codng: A ral varabl codng s usd n our algorhms. Th us of bnary srngs s no unvrsally accpd n h GA lraur. Mchalwcz [22] ndcad ha for ral-valud numrcal opmzaon problms, floang-pon rprsnaons ouprform bnary rprsnaon bcaus hy ar mor conssn, mor prcs, and lad o fasr xcuon.

7 Gnralzd Rnwal Procss 43 Gnraons and Populaon sz: Ths wo varabls ar ghly nrwnd n h GA soluons. A bg populaon sz wll ncras h compuaonal m for ach gnraon. As par of hs rsarch, dffrn populaon szs and gnraons numbrs wr sd. A sz populaon of 6 and gnraon numbr of 5 for h powr law modl, and a populaon sz of 8 and gnraon numbr of 5 for h LogLnar modl showd h bs rsuls. Parn Slcon funcons: In hs rsarch w usd Roul whl, Tournamn, and Sochasc Unvrsal Samplng funcons. Rcombnaon opraor: A probably of.6 was usd n h cod for h opraors: Dscr, Inrmda and Exndd ln rcombnaon. Muaon opraor: In hs cas, w us h muaon funcon for ral valus. Obcv funcon: In hs cas, h obcv funcon s h lklhood funcon of h log-lnar and powr law modls. Two falur daa ss wr analyzd usng all parn slcon chnqus, and gnc opraors o fnd h bs combnaon. From hs analyss, was obsrvd ha h hr parn slcon and h dscr rcombnaon opraor showd h bs f wh h daa n boh xampls. Thrfor, hs parn slcon and dscr rcombnaon wr usd o analyz h ohr xampls. 5. GRP-Log-lnar Modl Ths modl s basd on h m dpndn Posson procss dvlopd by Cox and Lws [23], n whch h ROCOF aks h form of: = δ δ 4 from hs quaon, and usng Eq. 3, w hav h condonal cdf funcon: F 5 and akng s drvav, h probably dnsy funcon for s f 6 Clarly, w can s ha whn =, = -, and h f bcoms: f 7

8 J. L. Hurado, F. Joglar and M. Modarrs 44 and, f =, hn w hav ha = and h f bcoms: f 8 Bcaus h frs falur dos no oby h condonaly mnond prvously, h lklhood funcon L, mplmnd n h GA roun wh h purpos of obanng h paramrs,, and ha maxmz L, s gvn by L = n n f f As a numrcal chck of h GA roun, rsuls oband wh sng =, and usng daa from wo xampls n h lraur ar compard wh h rpord rsuls n Exampls 6. and 6.2, Mkr and Escobar, pp [24] and found ha hy ar dncal. 6. GRP-Powr Law Modl Ths s a mor popular modl n h analyss of rparabl sysms. I s a Posson procss, whch has h sam funconal form as h hazard ra funcon of a Wbull dsrbuon. Ths rmnology has ofn rsuld n confuson bwn h powr law procss and h Wbull dsrbuon, whch ar dffrn concps as nod by Aschr [25]. Th ROCOF for hs modl has h followng xprsson: = β α α β from hs quaon, and usng q. 3, w hav h condonal cdf funcon: F and akng s drvav, h probably dnsy funcon for : f 2

9 Gnralzd Rnwal Procss 45 If =, hn = -, and h f bcoms f 3 If =, hn, and h f bcoms, f 4 Smlar o h log-lnar modl, lklhood quaon of h powr law modl s oband hrough Equaon 9. I s mplmnd n h GA roun o oban h paramrs, and ha maxmz L. L = n 5 7. Exampls of h GA approach o GRP-Powr Law and GRP-Log-lnar modls Th applcaons conss of four daa s found n h lraur [2,23,24]. Th followng abls ls h daa and s sourcs: Tabl : Exampl -Tm Bwn Succssv Falurs In Hours of Ar Condonng Equpmn-Cas. Proschan, 963 [23] Tabl 2: Exampl 2-Tm Bwn Succssv Falurs In Hours of Ar Condonng Equpmn Cas 2. Proschan, 963 [23] Tabl 3: Exampl 3-Tm Bwn Succssv Falurs In Hou rs for h U.S.S. Halfbak Nº 4 Man Propulson Dsl Engn [24]

10 46 J. L. Hurado, F. Joglar and M. Modarrs Tabl 4: Exampl 4 -Tm Bwn Succssv Falurs In Hours for Arcraf Gnraor Rpord by Duan 964 [2] Tabls 5 and 6, show h paramr smas oband usng h proposd GA roun for h four xampls abov, and h acual rnd n ROCOF. Tabl 5: GRP-Loglnar Rsuls Exampl U Tabl 6: GRP- Powr Law Rsuls Exampl U Th rsuls for boh modls for h xampl 3 confrm h Kma concluson, whch sas ha a mnmal rpar assumpon s a good on for h valus of grar han.6 and lowr han. Basd on hs rsuls, h valu can b xndd a las o.39, whch s h paramr ha maxmzs h lklhood funcon for hs daa s. Ths s USS Halfbak Nº 4 Man Propulson Log Lnar Modl Daa GR NHPP HPP Tm Fg. 2: Expcd Numbr of Falurs for Exampl 3. Log-Lnar Modl N falurs USS Halfbak Nº 4 Man Propulson Powr Law Modl Tm Daa GR NHPP HPP Fg. 3: Expcd Numbr of Falurs for Exampl 3. Powr Law Modl an nrsng obsrvaon, bcaus h valu of mans ha h vrual ag of h sysm s ruvnang alhough h cnrod rnd s U and shap paramr show an ncrasng falur ra. Accordng o hs, w can say ha h sysm droraon s bcaus of h war ou of s pars, and no bcaus h rpar acon. On h ohr hand, rsuls from xampl 2 Fgurs 4 and 5 shows ha h rnwal procss s a good assumpon whn h valu of s lowr han.. Th valu of h rnds n h ra of occurrnc of falurs usng cnrod s U also confrms hs samn. Thrfor, can b concludd ha f h rnd show no rnd, h analys can drcly modl h daa usng h rnwal procss, and do no spnd m usng complx GRP modls.

11 Gnralzd Rnwal Procss 47 3 Plan #4 Proschan Log Lnar Modl 25 Plan #4 Proschan Powr Law Modl Daa GR NHPP HPP Tm Fg. 4: Expcd Numbr of Falurs for Exampl 2. Log-Lnar Modl Daa GR NHPP HPP Tm Fg. 5: Expcd Numbr of Falur for Exampl 2. Powr Law Modl Rsuls from xampl and 4 show som of h bnfs of h Kma approach. Th valus of n hs daass ar grar han, whch mans ha h vrual ag of h sysm s no ruvnang. Morovr, h curvs of h GRP approach for boh modls show a br f wh h daa han h mnmal rpar procss. In hs cas, w hav an xclln xampl ha shows complxy of modlng rparabl sysms, bcaus boh xampls show ha h rpar ffcvnss s grar han, bu h valu of h rnd and shap facor ar dffrn. In xampl Fgurs 6 and 7 h rnd and shap facor ndca an ncrasng falur ra, bu n h xampl 4 Fgurs 8 and 9, hs valus ndca a dcrasng falur ra. Thrfor, w can say ha alhough h rpar acon s dfcn, h sysm can b mprovng or dgradng s prformanc Plan #2 Proschan Log Lnar Modl Daa GR NHPP HPP Tm Fg. 6: Expcd Numbr of Falurs for Exampl. Log-Lnar Modl Plan #2 Proschan Powr Law Modl Daa GR NHPP HPP Tm Fg. 7: Expcd Numbr of Falurs for Exampl. Powr Law Modl 3 Arcraf gnraor Log Lnar Modl 3 Arcraf gnraor Powr Law Modl N falurs 2 5 N falurs Daa GR NHPP HPP Tm Daa GR NHPP HPP Tm Fg. 8: Expcd Numbr of Falurs for Exampl 4. Log-Lnar Modl Fg. 9: Expcd Numbr of Falurs for Exampl 4. Powr Law Modl

12 48 J. L. Hurado, F. Joglar and M. Modarrs Comparng h rsuls from xampl 3 and 4, on may obsrv ha h GRP-Powr law modl fs h daa br han h GRP-Log-lnar modl. Howvr, Fgurs and suggs ha hs dffrnc s mnmal. Thrfor, falurs can b prdcd usng any of h wo modls. 9 USS Halfbak Nº 4 Man Propulson 8 6 Arcraf gnraor Daa GR PLaw GR LLnar Tm Fg. : Powr Law and Log-Lnar Modls Comparson Expcd Numbr of Falurs for Exampl 3 GRP Approach Daa GR PLaw GR LLnar Tm Fg. : Powr Law and Log-Lnar Modls Comparson Expcd Numbr of Falurs for Exampl 4 GRP Approach Th daa s also usful o vrfy h capabls of h modl prdcng falurs. Usng only a fw daa pons, Fgur 2 llusra ha dffrn rsuls ar oband whn h numbr of daa pons ar vard. For xampl, a 25, hours w hav abou 3 and 6 falurs basd on calculaons mad wh 2 and 4 falur pons, rspcvly. Accordng o h ral daa, a 25, hours, h numbr of falurs s 7. Thrfor, h bs smas ar oband whn h analyss s carrd ou wh h maxmum numbr h daa avalabl. 2 USS Halfbak Nº 4 Man Propulson P o w r L a w M o d l N fa lu r s Fg. 2: Expcd Numbr of Falurs Consdrng Dffrn Daa Pons for Exampl Conclusons Tm Daa 7 Fal 6 Fal 5 Fal 4 Fal 3 Fal 2 Fal Fal Th rsuls oband suggs ha h GA paramrs assssmn chnqu dscussd n hs papr s an xclln approach o modl prformanc of rparabl sysms. Th quaons wr valdad comparng our rsuls wh h valus oband for Mkr and Escobar [24] for h USS Halfbak, and USS grampus daa s hrough h us of h MLE approach. Morovr, h us of GA for h paramr smaon has an advanag ovr h classcal MLE, bcaus GA s abl o calcula paramrs for all h cass. In h cas of h MLE, h soluon for som cass canno b oband du o h numrcal complxy of h sysm of quaons. Spcfcally, h MLE formulaon could no b conssnly solvd numrcally n xampls and 4.

13 Gnralzd Rnwal Procss 49 Alhough h GRP-Powr Law modl shows a br f han h GRP-Log-lnar modl, h dffrnc s mnmal, whch confrms why hs modls ar rounly usd n h analyss of rparabl sysms. Th rnd analyss s a good frs sp n h analyss of rparabl sysms, bcaus suggss wha procss o us, rducng h nd of mor sophscad modls. Wh h applcaon of h GRP mhodology, s possbl o oban mor nformaon abou h rparabl sysm. Th nroducon of h vrual ag paramr allows h analys o undrsand h bhavor of h sysm br. Howvr, h rsuls show ha hs approach s usful for sysms wh grar han, bcaus h mnmal rpar procss s a good assumpon for bwn. and, and h rnwal procss works xclln for lowr han.. Usng h proposd approach, s possbl o drmn how h rpar acon s affcng h prformanc of h qupmn or sysm. Rfrncs [] Crow L.H. Rlably analyss for complx, rparabl sysms. In: Proschan f., Srflng R.J., dors. Rlably and Bomry. Phladlpha: SIAM; 974. [2] Black S.E., Rgdon S.E. Sascal nfrnc for a modulad powr law procss. Journal of Qualy Tchnology 996; 28.:8-9. [3] Laky M.J., Rgdon S.E. Th modulad powr law procss. Procdngs of h 45 h Annual Qualy congrss. ASQC, Mlwauk, WI.pp [4] Brman M. Inhomognous and modulad gamma procss. Bomrka 68, pp [5] Dorado C., Hollandr M., Shuraman J. Nonparamrc smaon for a gnral rpar modl. Th annals of Sascs 997, 25.3.:4-6. [6] Kma M., Mormura H., Suzuk Y. Prodcal rplacmn problm whou assumng mnmal rpar. Europan Journal of Opraonal Rsarch. 988; 37: [7] Kamnsky M. Krsov V. A Mon Carlo approach o rparabl sysm rlably analyss. Probablsc safy assssmn and managmn, Nw York: Sprngr; 998. pp [8] Krsov V. A Mon Carlo approach o modlng and smaon of h gnralzd rnwal procss n rparabl sysm rlably analyss. Dssraon for h Dgr of Docor of Phlosophy, Unvrsy of Maryland; 2. [9] Yanz M., Joglar F., Modarrs M. Gnralzd rnwal procss for analyss of rparabl sysms wh lmd falur xprnc. Rlably Engnrng & Sysm Safy 22, 77:67-8. [] Fogl D. Evoluonary Compuaon. Nw York. IEEE Prss [] Nural and voluonary chnqus for naural languag procssng. hp://cns.ua.ac.b/~kool/nnga.hml. [Accssd on May 3, 23]. [2] Holland J. H. Adapaon n naural and arfcal sysms. Ann Arbor: Th Unvrsy of Mchgan Prss, 975 [3] Gnc Algorhm. hp:// fangmn/ chapr4.hm. [Accssd on May 3, 23]. [4] Blckl. T. Tournamn slcon. hp:// [Accssd on May 3, 23].

14 5 J. L. Hurado, F. Joglar and M. Modarrs [5] Bakr J.E. Rducng bas and nffcncy n h slcon algorhm. Proc. Of h scond Inrnaonal Confrnc on Gnc Algorhms. Ed. J.J. Grfns. Massachuss Insu of Tchnology, Cambrdg, MA. pp [6] Mühlnbn H., Schlrkamp-Voosn D. Prdcv Modls for h Brdr Gnc Algorhm: I. Connuous Paramr Opmzaon. Evoluonary Compuaon,. pp [7] D Jong K.A. An analyss of h bhavor of a class of gnc adapv sysms. Ph.D hss, Unvrsy of Mchgan. Dss. Absr. In. 36, 54B, Unvrsy Mcroflms No [8] Shaffr J. D., Caruana R.A., Eshlman L.J., Das R. A sudy of conrol paramrs affcng onln prformanc of gnc algorhms for funcon opmzaon. In Shaffr, pp [9] Grfns J.J. Opmzaon of conrol paramrs for gnc algorhms. IEEE ransacons of Sysms, Man and Cybrncs SMC 6, [2] Mühlnbn H.: Th Brdr Gnc Algorhm - a provabl opmal sarch algorhm and s applcaon. Colloquum on Applcaons of Gnc Algorhms, IEE 94/67. London [2] Gnc and Evoluonary Algorhm Toolbox for us wh Malab GEATbx. Copyrgh 996 Harmu Pohlhm. Grmany. [22] Mchalwcz Z. Gnc Algorhms + Daa Srucurs = Evoluon Programs. Nw York. Sprngr-Vrlag [23] Cox D.R., Lws P.A.W. Th Sascal Analyss of Srs of Evns. Mhun & CO LTD, London, 978. [24] Mkr W., Escobar L. Sascal Mhods for Rlably Daa. Nw York. John Wly & Sons, INC [25] Aschr H., Fngold H. Rparabl Sysm Rlably modlng, nfrnc. Msconcpons and hr causs. Nw York. Marcl Dkkr, Inc Jos L. Hurado s a Faculy Rsarch Asssan and PhD sudn a h Unvrsy of Maryland. H rcvd a bachlor s dgr n Mchancal Engnrng from h Unvrsdad Smon Bolvar-Vnzula, and a Masr dgr from h Unvrsy of Maryland. H has mor han 2 yars of xprnc n h aras of nspcon, mannanc, and rlably n ol rfnrs. Francsco Joglar compld hs PhD a h Unvrsy of Maryland workng for h Cnr for Tchnology Rsk Suds. Hs rsarch nrs ncluds rsk, uncrany and rlably analyss. Mr. Joglar s currnly workng for SAIC. Mohammad Modarrs s a Profssor of Nuclar Engnrng and Rlably Engnrng and Drcor of h Cnr for Tchnology Rsk Suds a h Unvrsy of Maryland, Collg Park. Hs rsarch aras ar probablsc rsk assssmn, uncrany analyss, and Physcs of Falur Modlng. In h pas 23 yars ha h has bn wh h Unvrsy of Maryland, h has srvd as a consulan o svral govrnmnal agncs, prva organzaons and naonal laboraors n aras rlad o rsk analyss, spcally applcaons o h nuclar powr plans. Profssor Modarrs has ovr 2 paprs n archval ournals and procdngs of confrncs and hr books n varous aras of rsk and rlably ngnrng. H s a Unvrsy of Maryland Dsngushd Scholar- Tachr. Dr. Modarrs rcvd hs Ph.D. n Nuclar Engnrng from Massachuss Insu of Tchnology n 98, hs M.S. n Mchancal Engnrng from Massachuss Insu of Tchnology n 977.