ARITHMETICO GEOMETRIC PROCESS MAINTENANCE MODEL FOR DETERIORATING SYSTEM UNDER RANDOM ENVIRONMENT

Transcription

1 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST ARITHMTIO GOMTRI PROSS MAITA MODL FOR DTRIORATIG SYSTM UDR RADOM VIROMT D..Mahesaa Reddy ad D. B.Vekaa Ramudu2 Picipa, Si Baaji P.G. oege, Aaapu-55.A.P Idia. mai:mamadha2@gmai.com 2 Assisa Pofesso, Dep. of Saisics SSB Degee & P.G. oege, Aaapu-55.A.P Idia. mai:vekaaamudussb@gmai.com Absac I his pape, e sudied a aihmeico geomeic pocess maieace mode fo deeioaig sysem ude adom eviome. Assume ha he umbe of adom shocks up o ime, poduced by he adom eviome foms a couig pocess. Wheeve a adom shock aives, he sysem opeaig ime is educed. The successive educios i he sysem opeaig ime ae saisicay idepede ad ideicay disibued adom vaiabes. Assume ha he cosecuive epai imes of he sysem afe faiues, fom a iceasig aihmeico geomeic pocess. Ude he codiio ha he sysem suffes o adom shock, he successive opeaig imes of he sysem afe epais cosiue a deceasig aihmeico- geomeic pocess. A epaceme poicy, by hich he sysem is epaced a he ime of he faiue, is adoped. A expici expessio fo he aveage cos ae is deived. The, a opima epaceme poicy is deemied empiicay aso. ey Wods: ovouio, Geomeic pocess, Moooe Pocesses, Repai epaceme poicy, Reea Pocess.. Ioducio oside a sysem hich is subjeced o shocks. As shocks occu, a sysem has faiues. Faiues make a sysem moe expesive o u. I such siuaios oe impoa aea of iees i eiabiiy heoy is he sudy of maieace poicies i ode o educe he opeaig cos ad he isk of caasophic beakdos; hece i is desiabe o deemie a opima epaceme ime ad opima umbe of faiues fo he sysem. The faiue ae of he sysem is a fucio of age bu ca aso deped o he vaues of cocomia vaiabes descibig he effec of he eviome i hich i opeaes. The shock mode povides a eaisic fomuaio fo modeig ceai eiabiiy sysems siuaed i a adom eviome. A sysem is subjeced o shocks hich cause he sysem deeioae. I may cases, he deeioaio of a sysem is due o a iea cause such as agig ad accumuaed ea of he sysem ad a exea cause such as a eviomea faco migh be aohe easo fo sysem deeioaio. I pacice, if a compue is ivaded by some vius o aacked ih a aide, he opeaig ime of he compue is dimiished, o he compue ca beak do. This exampe shos ha he sysem is deeioaig due o exea causes. The effec of a iea cause o he sysem opeaig ime ca be a coiuous pocess, hie he effec of a exea cause such as a Radom shock migh fom a jump pocess. Theefoe i sudyig a maieace pobem fo a epaiabe sysem, oe shoud o oy coside he iea cause bu coside he effec of a adom shock poduced by he eviome. As a esu, oe shoud sudy a maieace mode ih adom shock ha is aso a impoa mode i eiabiiy heoy. The effec of a adom eviome o he sysem is hough a sequece of adom shock hich shoe he opeaig ime. I pacice, may exampes sho ha he effec of a adom shock is a educio ahe ha a peceage educio i esidua opeaig ime. I ohe ods, assume ha W acs addiivey ahe ha muipiciy. Fo exampe, a peso suffeig fom secod had smokig is vey seious, he effec is measued by a educio i ife ime. Simiay, ca damaged by affic accides educes is opeaig ime addiivey.wheeve he oa educio, i sysem opeaig ime i -, - + is gae ha he esidua opeaig ime, he he sysem fais : he chace ha a shock poduces a immediae faiue depeds o he compaaive disibuios of, ad,. To see he easoabeess of his poi, ISS : Vo. 5 o.3 Mach 23 46

2 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST coside he fooig exampes. I a affic accide, a he passeges i he bus suffe he same shock, so ha he educios i hei ife imes ae moe o ess he same, bu he effecs o diffee passeges migh be quie diffee. A od passege is moe fagie because of havig ess esidua ife ime ha a youge passege has: hus he ode passege ca be ijued moe seiousy ha a youge passege. The ode passege migh eve die, bu he youge passege migh oy suffe ad a igh-ijuy. This siuaio aso happes i egieeig. Suppose may machies ae isaed i I okshop, a of hem suffe he same shock poduced by a adom eviome, bu he effecs migh be diffee: as od machie coud be desoyed hee as a e machie migh be sighy damaged. This meas ha he effec of a adom shock depeds o he esidua ife ime of a sysem, if he educio i he esidua ife ime is geae ha he esidua ime, he he sysem fais. These o exampes aso sho hy W acs addiivey, ad if W acs muipicaivey, he sysem coud o fai afe suffeig a adom shock. I iceasig faiue ae shock mode a exea shock bigs o he some damage, ad such damage is o ehace he sysem faiue ae of some amou. Besides exea shocks, he sysem faiue ae is aso iceasig ove ime. So, he sysem faiue ae ca be deoed as a fucio of boh is age ad he umbe of shocks. I maieace pobems, besides, poicy T is aso appied, hee i he affeced sysem is epaced by a e ad S ideica oe a a soppig ime T. Fo he og u aveage cos Lam ad Zuckema8 sho ha ude some mid codiios, a opima * is a eas as good as a opima T*. Thee foe, ihou oss of geeaiy, he poicy ca be sudied. Impemeig poicy is moe coveie ha impemeig poicy T. This is a addiioa advaage of usig poicy. I his chape, e sudy a maieace mode ih adom shock hece; i is desiabe o deemie opima epaceme poicy fo such sysems. say ad say e a [5] sudied he heoy of poisso shock mode. Lae o, Bao ad Poscha [4] cosideed his pobem i hei moogaph. Ross [5] peseed a geeaized poisso shock mode. Shai uma ad Sumia [3] exeded poisso shock mode o geea shock mode, ad sudied such a shock mode i hich a sysem fais he he shock magiude of sige shock ousips a give posiive vaue. A he same ime, Fedma [6], Zuckema [2], Goieb [7] ad Abde- Hameed [] deemied especivey he opima epaceme poicy fo he diffee shock modes. Shooma[8], Sheu ad Lio [2] cosideed a -ou-of sysem subjec o shocks. I maieace pobems, mos eseach ok so fo assumed ha a faied sysem afe epai i be as good as e, his is he pefec epai mode. I pacice, i is o aays ue. Bao ad Hue [4] poposed he miima epai mode by assumig ha a faied sysem afe epai i fucio agai, bu ih he same faiue ae ad he same effecive age as a he ime of faiue. Lae o, Bo ad Poscha [2] ioduced he impefec epai mode, i hich ih pobabiiy p epai is a pefec epai ad ih pobabiiy -p epai is a miima epai. May eseach oks have bee caied ou by Pak [4], Pheps [2], Bock ea 3] ad ohes aog hese o diecios. Hoeve, a moe easoabe mode is he geomeic pocess epai mode, fis sudied by Lam [,], hich i his he successive suviva imes ae sochasicay deceasig ad he cosecuive epai ime ae sochasicay iceasig. Ude his assumpio, Lam [9] sudied o kids of epaceme poicy, oe based o he okig age T of he sysem ad he ohe based o he faiue umbe of he sysem. The objec is o choose opima epaceme poicies T* ad * especivey such ha he ogu aveage oss pe ui ime is miimized. The expici expessios fo he og-u aveage oss pe ui ime ude each epaceme poicy ca be evauaed, ad he coespodig opima epaceme poicies T* ad * ca be foud umeicay o aayicay. Zhag [23] geeaized Lam s []ok by a bivaiae epaceme poicy T, ude hich he sysem is epaced a he okig age T o a he ime of h faiue, hich eve occus fis. Ad ude some mid codiios, Zhag [23] shoed he opima poicy T, * is bee ha he opima poicy *. Ohe epaceme poicies ude geomeic pocess epai mode ae epoed by Lam [9], Sadje ad Zuckema [9], Saey [7] Leug ad Lee [8], Zhag e a [22], ad ohes aog his diecio. I his chape, e sudied a aihmeico geomeic pocess maieace mode fo deeioaig sysem ude adom eviome. Assume ha he umbe of adom shocks up o ime, poduced by he adom eviome foms a couig pocess. Wheeve a adom shock aives, he sysem opeaig ime is educed. The successive educios i he sysem opeaig ime ae saisicay idepede ad ideicay disibued adom vaiabes. Assume ha he cosecuive epai imes of he sysem afe faiues, fom a iceasig aihmeico geomeic pocess. Ude he codiio ha he sysem suffes o adom shock, he successive opeaig imes of he sysem afe epais cosiue a deceasig aihmeico- geomeic pocess. A epaceme poicy, by hich he sysem is epaced a he ime of he faiue, is adoped. A expici expessio fo he aveage cos ae is deived. The, a opima epaceme poicy is deemied empiicay aso. I modeig of hese deeioaig epaiabe sysems e sae he defiiios eaig o sochasic ode ad aihmeico geomeic pocess hich ae give i Lam [, ] ISS : Vo. 5 o.3 Mach 23 46

3 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST 2.Mode I his secio, e deveop a mode fo epaceme poicy fo a deeioaig sysem ude adom eviome speciaizig o aihmeico- geomeic pocess by maximizig og-u expeced ead pe ui ime ih he fooig assumpios. ASSUMPTIOS:. A e sysem is isaed a he begiig. I is epaced by a e ad s-ideica oe some ime ae. 2. Give ha hee is o adom shock, he {, =,2,..} fom AGP ad =>. Hoeve, o mae hehe hee is a adom shock o o. Le {Y, =,2,..} cosiues a AGP ih <a, b< ad Y = >. Le he cdf of ad Y be F ad G especivey ad he Pdf be f ad g especivey. F x= G y= F d ad a G d b 2 3. is he umbe of Radom shocks up o ime poduced by he adom eviome. {, } foms a couig pocess havig saioay ad s-idepede iceme. Wheeve a shock occus, he sysem opeaig ime is educed. {W, =,2,..} ae i i d v ; W is he educio i he sysem opeaig ime afe adom shock. The successive educios i he sysem opeaig ime ae addiive. If a sysem fais, i is cosed so ha he adom eviome has o effec o a faied sysem. 4. The pocesses {, =,2,..}, {Y, =,2,..} adom vaiabes, z ae s-idepede. The pocesses {, =,2,..}, {, } ad {W, =,2,..} ae aso s-idepede. 5. The epaceme poicy is appied. 6. The epai cos ae of he sysem is, he epaceme cos is R ad he ead ae of he sysem is. 7. The compeio ime of epai - deoed by - :he umbe of Radom shock i -, - +] poduced by he eviome is -, - + = ad - + ae especivey he umbe of Radom shock poduced i, - ] ad, - + ] ; he oa educio i he opeaig ime i -, - + is, -, - + = ] W i cosequey, ude he adom eviome, he i esidua ime a - + is S = - -, subjec o S heefoe, If S fo his mode, a mehodoogy fo obaiig opima umbe of faiues, hich maximizes og expeced ead pe ui ime is discussed beo. 3. Opima souio I his mode a cyce is compeed if a epaceme is compeed. Because a cyce is acuay a ime ieva beee o successive epacemes, he he successive cyces fom a Reea pocess. The successive cyces ogehe ih he coss icued i each cyce make a eea ead pocess. The sadad esu i eea ead pocess shos ha he aveage cos ae is: ISS : Vo. 5 o.3 Mach

4 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST he expeced cos icued i a cyce = 3. he expeced egh of a cyce To begi,sudy he disibuio of fo his pupose. Le -, - + ] of Radom shock hich occu i -, - + ] be. The fo >, sudy he codiioa pobabiiy: P { > / -, - + ] = } = P { = If { / S } > / -, - + ] = } = P { S >, [, ]/ -, - + ] = } = P { - -, - +]>, [, ]/ -, - + ] = } = P { - -, - + ] > / -, - + ] = } = P { -, - + ] > / -, - + ] = } = P { i } i = f x. h dxd D = x, x h k = Pdf k i,, x. i ad is he k-fod covouio of h ih isef. h = pdf[ i ], ad H = cdf[ i ]. P { > / -, - + ]=} = f x dx h F = d dh F d H = H = df P { > } i i hus = P { /, ] }.P{-, - ] ISS : Vo. 5 o.3 Mach

5 = } ], {. P dh F = - } {. P dh F The above equaio is due o he fac ha {, } has a saioay iceme popey. Theefoe, by oig ha F x = Fa -.x, he cdf, I, of is I x = P { x } = } {.. x P dh x a F By usig epaceme poicy, og-u expeced ead pe ui ime is R Z Y Y =. z y Y R = R Whee x xdi 3.3 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST ISS : Vo. 5 o.3 Mach

6 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST Hee 2 a b d d 2 ad 4. mpiica Resus ad ocusios Fo give fixed vaues of,,,,,, he opima epaceme poicy * is cacuaed as foos: Le, =5, a=.69, b=.4, =3, =8, =4, T=4, =2, d =.5, d 2 =.9, =5, =3, =5 Tabe : Vs ISS : Vo. 5 o.3 Mach

7 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST ocusio: Fom he abe & gaph, e see ha 3 = is he maximum of he og-u expeced ead cos pe ui ime of he sysem i.e he opima poicy is * = 3 ad e shoud epace he sysem a he ime of 3 d faiue. 2 By examiig fo vaious vaues of paamees, e ca comme ha he opima umbe of faiues does o affec much fo a cosideabe chage i a & b bu hee is subsaia chage i og-u expeced ead cos pe ui ime of he sysem. Refeeces: [] Abde-Hameed,M., Opimum Repaceme of sysem subjec o shocks, Joua of Appied pobabiiy, Vo.o.23, 986, pp 7-4. [2] Bo, M., ad Poscha, F., Impefec Repai, Joua of Appied Pobabiiy, Vo.2, 983, PP [3] Bock,H.W.,Boges,W.S., Savis,T.H., A geea age epaceme mode ih miima epai, ava Reseach Logisics, Vo.o.35, 988, pp [4] Bao, R.., ad Poscha, F., Saisica heoy of Reiabiiy ad ife esig, Ho. Rieha, Wiso, 975. [5] say, J.D., Masha, A.W., ad Poscha, F., Shock Modes ad ea pocess, Aas of pobabiiy, Vo., 973, pp [6] Fedma,R.M., Opima epaceme ih semi-makov shock modes, Joua of Appied pobabiiy, Vo. o.3, 976, pp 8-7. [7] Gay Goieb., Opima Repaceme mode fo shock Modes ih Geea faiue Rae, opeaios Reseach, Vo.3, o., 982, pp [8] Leug,..F.,ad Lee, Y.M., Usig geomeic pocesses o sudy maieace pobems fo egies, Ieaioa joua of Idusia gieeig, Vo.5, 998, pp [9] Lam Yeh., A opima Repaiabe Repaceme mode fo Deeioaig sysems, Joua of Appied pobabiiy, Vo o. 28, 99, pp [] Lam Yeh., Geomeic Pocesses ad Repaceme Pobems, Aca Mahemaicae Appicaae Siica, Vo.4, 988 a, pp [] Lam Yeh., A oe o he Opima Repaceme Pobem, Advaced Appied Pobabiiy, Vo.2, 988 b, pp [2] Pheps, R.I., Repaceme poicies ude miima epai, Joua of opeaioa Reseach sociey, Vo.32, 98, pp [3] Shahi uma, J.G., ad Sumia, U., Geea shock modes associaed ih coeaed Reea sequeces, Joua of Appied pobabiiy, Vo.2, 983, pp [4] Pak,.S., Opima umbe of Miima Repais befoe Repaceme, I Tasacios o Reiabiiy, vo. R-28, o.2, Ju 979, pp [5] Ross, S.M., Appied Pobabiiy Modes ih opimizaio Appicaios, Sa Facisco, Hode-Day, 97. [6] Ravichada uma T. ad Y.isha Reddy, opima Repai epaceme poicy fo o ype faiues mode ih α seies pocess, poceedigs of A.P. Akademi of scieces, Vo.23, 28, pp [7] Saey, A.D.J., O Geomeic Pocesses ad Repai Repaceme Pobems, Micoeecoics Reiabiiy, Vo.33, 993, pp [8] Shooma, M.L., Pobabiisic Reiabiiy-A gieeig Appoach, McGa Hi, e Yok, 968. [9] Sadje, W., ad Zuckema, D., Opima Saegies fo some Repai Repaceme Modes, Advaced Appied Pobabiiy, Vo.22, 99, pp [2] Shey-Hue Sheu., ad Tie Hsiag hag., A opima epaceme peiod fo a k-ou-of-: F sysem subjec o shocks, Ieaioa Joua of sysems sciece, 2, Vo.32, o.5, pp ISS : Vo. 5 o.3 Mach

8 D..Mahesaa Reddy e a. / Ieaioa Joua of gieeig Sciece ad Techoogy IJST [2] Zuckema, D., Opima soppig i semi-makov shock mode, Joua of Appied pobabiiy, Vo.5, 978, pp [22] Zhag, Y.L., Yam, R..M., ad Zuo, M.J., Opima Repaceme Poicy fo a Muisae Repaiabe Sysem, Joua of he Opeaioa Reseach Sociey, Vo. o.53, 22, pp [23] Zhag, Y.L., A Bivaiae Opima Repaceme Poicy fo a Repaiabe Sysem, Joua of Appied Pobabiiy, Vo.3, 994 pp ISS : Vo. 5 o.3 Mach