G1-Renewal Process as Repairable System Model
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1 G-Reewl Process s Reprble Sysem Model M.P. Kmsky d V.V. Krvsov Uversy of Mryld College Prk USA Ford Moor Compy Derbor USA Absrc Ths pper cosders po process model wh moooclly decresg or cresg ROCOF d he uderlyg dsrbuos from he loco-scle fmly kow s he geomerc process Lm 988. I erms of reprble sysem relbly lyss he process s cpble of modelg vrous resoro ypes cludg beer h ew.e. he oe o covered by he populr G-Reewl model Kjm & Sum 986. The dscve propery of he process s h he mes bewee successve eves re obed from he uderlyg dsrbuos s he scle prmeer of ech s moooclly decresg or cresg. The pper dscusses properes d mxmum lkelhood esmo of he model for he cse of he Expoel d Webull uderlyg dsrbuos. Key words: gg rejuveo homogeey o-homogeey g-reewl geomerc process. Acroyms: CDF CIF GPR HPP IID MLE NHPP ORP PDF ROCOF cumulve dsrbuo fuco cumulve esy fuco geerlzed reewl process homogeeous Poso process depede d declly dsrbued mxmum lkelhood esmo o-homogeeous Poso process ordry reewl process probbly desy fuco re of occurrece of flures. Iroduco I reprble sysem relbly lyss f upo flure sysem s resored o s "good-sew" codo d he me bewee flures c be reed s depede d declly dsrbued IID rdom vrble he he flure occurrece c be modeled by he Ordry Reewl Process ORP. If upo flure he sysem s resored o he "sme-s-old" codo he he ppropre model o descrbe he flure occurrece s he No-Homogeeous Posso Process NHPP. The me bewee cosecuve flures hs cse s o IID rdom vrble. I sese he NHPP c be vewed s reewl process wh he "sme-s-old" repr ssumpo Krvsov 7. A mpor
2 prculr cse of boh ORP d NHPP s he Homogeeous Posso Process HPP whose uderlyg flure mes re dsrbued oelly. I s cler h eve hough rcve mhemclly he "good-s-ew" d "sme-s-old" repr ssumpos re ofe excepos rher h he rule from he sdpo of prccl relbly egeerg. Geerlly hey could be reed s he "lmg" codos o whch sysem could be resored. I rely fer he repr he sysem s lkely o fd self bewee he wo codos. Of gre eres herefore s modelg oher repr ssumpos such s he ermede "beer-hold-bu-worse-h-ew". A erly pproch o cover more h oe repr ssumpo wh he sme probblsc model s dscussed Brow d Prosch 98. Ths mehod ssumes h upo flure repr co resores he sysem o he "good-s-ew" codo wh probbly of p or he "sme-s-old" codo wh probbly of -p where s he ge of he sysem flure. A more geerl model s he so-clled G-Reewl process Kjm M d Sum 986 whch res ORP d NHPP s specl cses. The GRP s roduced usg he oo of vrul ge: A S where A d S s he sysem's vrul ge before d fer he -h repr respecvely d s he resoro or repr effecveess fcor. I s cler h for he ge of he sysem fer he repr s "re-se" o zero whch correspods o he "good-s ew" repr ssumpo d represes he ORP. Wh he sysem s resored o he "sme-s-old" codo whch s he cse of he NHPP. The cse of < < correspods o he ermede "beer-h-old-bu-worse-h-ew" repr ssumpo. Flly wh > he vrul ge s A > S so h he repr dmges he sysem o hgher degree h ws jus before he respecve flure whch correspods o he "worse-h-old" repr ssumpo. Oe lmo of he GRP model s s bly o model "beer h ew" resoro for whch he eed rses some prccl pplcos e.g. relbly growh modelg Crow 98. The cosdered below geomerc process Lm overcomes hs prculr drwbck.. Geomerc process: probblsc model The loco-scle fmly of uderlyg dsrbuos s cosdered. Afer ech -h flure... he sysem s resored dmged such wy h s scle prmeer s chged o
3 where s he resoro dmge prmeer < < so h for he me o he frs flure for he me bewee frs d secod flure d so o. Ths rsformo of he scle prmeer s smlr o he oe used he well-kow ccelered lfe me model Cox & Oks 984; Nelso 99. To exe he suggesed model mkes more physcl relbly sese h he respecve NHPP model erms of resoro ssumpo.e. "sme-s-old" ssumpo. If he process cocdes wh he ordry Reewl process. If > he roduced process s obvously mprovg oe d f < he process s gg deerorg. Tble shows mulpler o he scle prmeer of he uderlyg dsrbuos of he mes bewee cosecuve eves for some vlues of. Tble. Mulpler o he scle prmeer of he uderlyg Dsrbuos of he mes bewee successve eves for some vlues of. Eve I he gve coex we sugges cllg he cosdered geomerc po process s he G-Reewl Process due o cer smlry o he G-Reewl Process roduced erler by Kjm d Sum 986. Ag by logy wh G-Reewl Euo he euo for he cumulve esy fuco CIF of he G-Reewl Process wll be correspodgly clled he G-Reewl euo. I should be oed h he process does o hve esblshed me e.g. Wg d Phm 6 cll hs process us-reewl process... G-Reewl Euo The loco-scle dsrbuo for couous rdom vrble r. v. s defed s hvg he cumulve dsrbuo fuco CDF he followg form:
4 The respecve probbly desy fuco s The me o he h flure T s gve by u F F u f T X X X X where X... re depede r.v. whch he frmework of he G-Reewl Process re dsrbued ccordg o he followg cumulve dsrbuo fuco CDF F X u F X... 4 The dsrbuo of he me o he -h flure T s dffcul o fd s closed-form resso eve he cse of he ordry reewl process.e. whe excep for he oel d Gmm dsrbuo mog he populr lfeme dsrbuos. Noe h he process cosdered corry o he ordry reewl process he X s re o declly dsrbued. The euo for he cumulve esy fuco CIF lso kow s he g-reewl euo of he process c be foud s k k W F 5 where F k s k-fold covoluo of he cumulve dsrbuo fucos 4. Noe h F k PrT k <. s The respecve re of occurrece of flures ROCOF c be foud usg s defo dw k w f 6 d k.. G-Reewl Process wh Expoel Uderlyg Dsrbuo The process wh oel uderlyg dsrbuo s cosdered. The me o he frs flure hs he oel dsrbuo wh PDF 4
5 f 7 Accordg o 4 he me bewee he frs d he secod flures hs he followg PDF f 8 Correspodgly he me bewee he -h d he -h flures hs he followg PDF f The covoluo of f d f.e. f *f c be foud s f * f f x x dx x dx f x f x dx f x f x dx 9 I c be show h he Lplce rsform of he PDF of me o he h flure f s gve by s s * f Bsed o he Lplce rsform of he covoluo f k c be foud s k * f k s Π s The verse of s o vlble closed form whch s why we deful o obg he CIF for G-Reewl Process v Moe Crlo smulo smlr o he soluo of he G-Reewl euo we suggesed Kmsky & Krvsov 998. Fgures d show he CIF's of he G-Reewl Process wh uderlyg oel dsrbuo. I s eresg o oe h he coex of he G-Reewl he uderlyg oel dsrbuo provdes hgh flexbly modelg boh mprovg d deerorg processes corry o he HPP. 5
6 Cumulve Iesy Fuco Cumulve Iesy Fuco Tme Tme Fg.. CIF of he G-Reewl Process wh uderlyg oel dsrbuo scle prmeer of d vrous egve vlues of. Fg.. CIF of he G-Reewl Process wh uderlyg oel dsrbuo scle prmeer of d vrous posve vlues of... G-Reewl Process wh Webull Uderlyg Dsrbuo Fgures d 4 show he CIF's of he G-Reewl Process wh he posve resoro prmeer d he uderlyg Webull dsrbuo wh he scle prmeer of d he cresg d decresg hzrd fucos respecvely. The cocvy of he CIF for < ~.7 Fgure mgh be reled o he cresg hzrd fuco of he uderlyg dsrbuo. The subseue covexy of he CIF for >.7 mgh be led by he posve resoro prmeer whch correspods o he mprovg GR process. The overll covexy of he CIF Fgure 4 mgh be led by he decresg hzrd fuco of he uderlyg dsrbuo d he posve resoro prmeer whch correspods o he mprovg GR process. The cocvy of he CIF Fgure 5 mgh be led by he cresg hzrd fuco of he me-o-frs-flure dsrbuo d egve resoro prmeer whch correspods o he deerorg GR process. The relve "lery" of he CIF Fgure 6 mgh be led by he decresg hzrd fuco of he uderlyg dsrbuo whch s prlly "compesed" by he egve resoro prmeer of he GR process. 6
7 Cumulve Iesy Fuco Cumulve Iesy Fuco Tme Tme Fg.. CIF of he G-Reewl Process wh uderlyg Webull dsrbuo scle prmeer of shpe prmeer of.5 d resoro prmeer of. Fg. 4. CIF of he G-Reewl Process wh uderlyg Webull dsrbuo scle prmeer of shpe prmeer of.5 d resoro prmeer of Cumulve Iesy Fuco Cumulve Iesy Fuco Tme Tme Fg. 5. CIF of he G-Reewl Process wh uderlyg Webull dsrbuo scle prmeer of shpe prmeer of.5 d resoro prmeer of -.. Fg. 6. CIF of he G-Reewl Process wh uderlyg Webull dsrbuo scle prmeer of shpe prmeer of.5 d resoro prmeer of -... G-reewl process: mxmum lkelhood esmo.. D Le be me o he frs flure be he me bewee he frs flure d he secod flure so h s he me bewee he -h flure d he ls h flure. The es observo s ermed he me... G-Reewl Euo wh Expoel Uderlyg Dsrbuo For he uderlyg dsrbuo 7 he lkelhood fuco c be wre s follows: 7
8 8... Tkg logrhms of he fuco d dffereg wh respec o d oe ges d l d d l d Sysem of euos c be solved umerclly... G-Reewl Euo wh Webull Uderlyg Dsrbuo As he prevous cse he sme ype of flure-ermed d re cosdered. The PDF of he uderlyg me o he frs flure Webull dsrbuo s f > 4 For he bove uderlyg dsrbuo he lkelhood fuco s.... X 5 Tkg he logrhm of hs lkelhood fuco oe ges: l l... l l l l l l l 6 Dffereg hs fuco wh respec o d d eug he dervves o zero oe ges:... l d L d Thus he frs euo s
9 9 7- Tkg he dervve wh respec o oe ges l l l... l l l l l l d d l Accordgly he secod euo s l 7- Ad kg he dervve wh respec o oe ges he hrd euo d d l whch s. 7- Ag Euos 7. c be solved umerclly o ob MLE esmes of he GR Process wh he uderlyg Webull dsrbuo..4. Cse Sudy Cosder flure mes bewee cosecuve flures dscussed by Bsu & Rgdo : { }. The d re of he flure-ermed ype. The G- Reewl process wh he uderlyg oel dsrbuo s ssumed s probblsc model. Fgure 7 shows MLE of he CIF obed by solvg Sysem. I s eresg o oe h he CIF exhbs proouced covexy corry o lery whch mgh be uvely eced from po
10 process wh he uderlyg oel dsrbuo. The oel dsrbuo prmeer s esmed usg MLE o be 4.78 d G-R resoro prmeer s Cumulve Iesy Fuco CIF MLE D Tme Fg. 7. G-Reewl wh Expoel Uderlyg dsrbuo s Model o D Se of Bsu & Rgdo. REFERENCES [] Bsu A.P. d Rgdo S.E. Sscl Mehods for he Relbly of Reprble Sysems Wley. [] Brow M. d Prosch F. 98 "Imperfec Mece" I: Crowley J. d Johso R. ed Survvl Alyss. Vol 98 p [] Cox D.R. d Okes D. 984 Alyss of Survvl D Chpm & Hll CRC Moogrphs o Sscs & Appled Probbly. [4] Crow L. R. 98 Cofdece Iervl Procedures for he Webull Process wh Applco o Relbly Growh. Techomercs 4 pp [5] Kmsky M.P. d Krvsov V.V. 998 "A Moe Crlo Approch o Reprble Sysem Relbly Alyss". I: Probblsc Sfey Assessme d Mgeme New York: Sprger p [6] Krvsov V.V. 7 "Prccl Exesos o NHPP Applco Reprble Sysem Relbly Alyss" Relbly Egeerg & Sysem Sfey Vol. 9 # 5 pp [7] Kjm M. d Sum N. 986 "A Useful Geerlzo of Reewl Theory: Coug Process Govered by No-egve Mrkov Icremes" J. Appl. Prob. p [8] Lm Y. 988 Geomerc Process d Replceme Problem Ac Mh. Appll. Syc 4 p [9] Lm Y. 9 A geomerc process d-shock mece model IEEE Trscos o Relbly V. 58 No. p [] Wg H. d Phm H. 6 Relbly d Opml Mece Sprger Lodo.
11 Abou he uhors Mrk Kmsky s he Chef Ssc he Ceer of Techology d Sysems Mgeme of he Uversy of Mryld College Prk USA. Dr. Kmsky s resercher d cosul relbly egeerg lfe d lyss d rsk lyss. He hs coduced umerous reserch d cosulg projecs fuded by he goverme d dusrl compes such s Deprme of Trsporo Cos Gurds Army Corps of Egeers US Nvy Nucler Regulory Commsso Amerc Socey of Mechcl Egeers Ford Moor Compy Qulcomm Geerl Dymcs d severl oher egeerg compes. He ugh severl grdue courses o Relbly Egeerg he Uversy of Mryld. Dr. Kmsky s he uhor d co uhor of over 5 publcos jourls coferece proceedgs d repors. Vsly Krvsov s Seor Sff Techcl Specls relbly d sscl lyss wh Ford Moor Co. He holds M.S. d Ph.D. degrees EE from Khrkov Polyechc Isue Ukre d Ph.D. Relbly Egeerg from he Uversy of Mryld USA. Dr. Krvsov s he uhor d co uhor of 5 professol publcos cludg book o Relbly Egeerg d Rsk Alyss 9 peed veos d Ford corpore secre veos. He s edor of he Elsver's Relbly Egeerg d Sysem Sfey jourl d s member of he IEEE Relbly Socey. Pror o Ford Krvsov held he poso of Assoce Professor of Elecrcl Egeerg Ukre d h of Reserch Affle he Uversy of Mryld Ceer for Relbly Egeerg. Furher formo o Dr. Krvsov's professol cvy s vlble
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