Parameters Estimation in a General Failure Rate Semi-Markov Reliability Model

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1 Joura of Saca Theory ad Appcao Vo. No. (Sepember ) - Parameer Emao a Geera Faure Rae Sem-Marov Reaby Mode M. Fahzadeh ad K. Khorhda Deparme of Sac Facuy of Mahemaca Scece Va-e-Ar Uvery of Rafaja Rafaja Ira Deparme of Sac Shraz Uvery Shraz Ira m.fah@vru.ac.r horhda@uc.ac.r A em-marov proce wh four ae ha bee apped for modeg wo dmar u cod adby yem. A he mome ha operag u fa he adby u wched o operae by ug a wchg devce ha avaabe wh uow probaby. I ao aumed ha he faure rae of u ha he geera form h where are o-egave uow parameer. I favor of em-marov rucure of he yem maxmum ehood ad he Baye emaor of he uow parameer α are obaed whe are o-egave ow coa. Furhermore he emaor are obaed for yem wh mar u. Fay o compare he reu a muao udy doe. Keyword: Bayea emao od adby yem Maxmum ehood Sem-Marov proce. Iroduco Sem-Marov procee owaday have bee apped may area of cece uch a reaby heory. I fac may of he reaby yem ca be modeed by em-marov procee uch a wo dmar u cod adby yem whch ha wdey bee uded ad ued dury. Emao of parameer cuded reaby yem a commo job reaby aay. Recey h area emag he parameer of fe me drbuo have bee receved peca aeo. I 98 Sarmah ad Dharmadhar have obaed he mome emaor of he parameer cuded -ou-of-:g reparabe yem whe he faure ad repar me drbuo of he u are expoea wh uow parameer []. Afer wo decade Sarha ad E-Gohary () have emaed he parameer of h mode by maxmum ehood ad Bayea mehod []. They howed ha hee mehod perform beer ha mome emao mehod. Ao he parameer of he feme -ou-of-m cod adby yem wh mperfec wche have bee emaed ug wo dffere approache by A-Ruzaza ad Sarha []. I E-Gohary uded a peca cae of wo dmar u cod adby yem. He ued he Marov reewa heory o emae he uow parameer of wo mar u cod adby yem wh mperfec wche whe faure rae are eary deped o feme of he yem []. For deaed decrpo of em-marov ad Marov reewa procee ee []. The geera faure rae mode ha codered he pree paper deveop ear faure rae expoea Webu ad Rayegh drbuo mode a we a may oher reaby yem wh reac pecfed fe me drbuo. I h arce a em-marov proce wh four ae ha bee codered for modeg wo dmar u cod adby yem. A he mome ha operag u fa he adby u wched o operae by ug a wchg devce ha avaabe wh uow probaby. I ao aumed ha he faure rae of u ha he geera form h where are o-egave uow parameer. I favor of a em-marov rucure for he yem maxmum ehood ad he Baye emaor of he uow parameer Pubhed by Aa Pre opyrgh: he auhor

2 α have bee obaed whe are o-egave ow coa. By he fac ha he yem wh mar u a peca cae of he uded mode he emaor have ao bee obaed whe he yem ha mar u. Fay o compare he reu a muao udy ha bee doe. oder a em-marov proce wh fe ae pace ao e P p : be a probaby drbuo o : p p. A Marov reewa proce may be defed a foow: Le ξ T T deoe a wo-dmeoa ochac proce wh vaue he ξ T a Marov reewa proce f. P j T T T P j T. P T p. We ao aume ha he probabe (.) do o deped o ad deoe hem by P j T j j a he reewa ere ad j a he reewa ere marx. The aocaed coug j proce repreeg he oa umber of rao wh [ ] deoed by N : where N up : T. A ochac proce X : where X N caed he em-marov proce o geeraed by Marov reewa proce wh a drbuo P ad he ere.. Decrpo of he Mode We w coder a mechaca yem whch perform by he foowg eg:.. Noao ad aumpo The yem co of wo dmar u whch operae cod adby cofgurao a wch ad a repar facy. Whe operag u fa he adby u wched o operae by aco of a wchg devce. The eve ha wchg devce perform we whe requred deoed by A wh probaby P ( A ). The yem fa wheher operag u fa ad he repar job ha o bee fhed ye or boh he operag u ad he wch have bee faed. I h cae he whoe faed yem w be repaced by a ew deca oe. The faure rae of u ha he geera form h where are oegave uow parameer wherea are o-egave ow parameer. The fe me of operag u are o-egave radom varabe V wh drbuo fuco F.. The egh of repar perod of u a o-egave radom varabe wh drbuo fuco G.. Repacg me of he faed yem a o-egave radom varabe wh drbuo fuco M.. A above radom varabe are muuay depede... The em-marov mode I order o decrbe he em-marov reaby mode ad derve he aocaed reewa ere we w roduce he foowg ae:. he yem faed;. he u operag ad u uder repar;. he u operag ad u uder repar;. he u operag ad u adby mode. Le be he ucceve me a of he yem chage (ae rao). Ao e he proce X deoe he ae of he yem a me. Defe Z X wh he ae pace (.) Pubhed by Aa Pre opyrgh: he auhor 6

3 Z T a Marov reewa proce wh ae pace. The aocaed em-marov proce X ad he foowg reewa ere M F G xdf x G xdf x F G x df x G x df x F F. By aumg he geera faure rae h he dey fuco of he fe me are a:. Emao of he Parameer f exp[ ( )]. he I h eco we w oba he maxmum ehood ad Baye emaor of he uow vecor baed o a equece of obervao z from he radom vecor Z T T T a a rajecory of he em-marov proce... Maxmum ehood emao I favor of em-marov rucure of he yem he maxmum ehood fuco become: where ;α A exp[ ( )] G G L z By ag dervave of og L z ;α : j j j. A m G G w.r. uow parameer we have G G G G Pubhed by Aa Pre opyrgh: he auhor 7

4 . α o eay. So umerca echque are requred o cacuae he MLE' of hee parameer. I he foowg peca cae he MLE may be drecy derved from he above yem of equao: Uuay evauag expc expreo for he MLE' of ) G G a repecvey ad ;.e. feme of he u are expoea. I h cae he MLE become ˆ m ˆ m ˆ m where m ad m m are carda umber of he e repecvey. The varace-covarace marx of he maxmum ehood emaor V may be obaed a ) G m m V. m G a repecvey ad ;.e. feme of he u are Webu wh ow hape parameer. I h cae ˆ m ˆ m adˆ m ao V m. m m Sce he yem wh mar u are apped more ha dmar oe we ao chec he above reu for he mar cae. By combg ae ad he ae pace reduce o he foowg form;. The yem faed;. Oe u operag ad aoher uder repar;. Oe u operag ad aoher adby mode. By coderg f exp a he dey fuco for he fe me of u he maxmum ehood equao become Pubhed by Aa Pre opyrgh: he auhor 8

5 where G G G G G are obaed from ovg he yem of equao: ad. The foowg reu G a ad ;.e. feme of he u are expoea. I h cae ˆ m ad ˆ m where m. Ao ) Le m V. m Remar. The reu of E-Gohary [] w be obaed by coderg he above formua. G a ad ;.e. feme of he yem u are Webu wh ow hape parameer. I h cae ˆ m ad ˆ m. Smar o he precedg cae we have ) Le m V. m Remar. The formua whch have bee obaed for h cae exed he reu of E-Gohary [] o he Webu drbuo whch a more geera drbuo... The Baye emao I order o oba he Baye emaor for he vecor of uow parameer aumpo are adoped: A: A: ha pror dey fuco behave a depede radom varabe h. A: The o fuco whe he vecor α emaed by ˆα quadrac. α he foowg By a cacuao proce mar o [] we w arrve a our ma heorem whch gve he. marga poeror pdf of Theorem. The h r mome of he marga poeror pdf of r Φ r r r r r Φ are gve by h r mome of he r (.) where he Kroeer dea ad Pubhed by Aa Pre opyrgh: he auhor 9

6 where ad Θ Θ Φ u u u u u D 8 h d Θ u h m 7 8 m 6 u e d G j j j j j j 6 G j j 6 j j j 6 j j 8 D j 8 7 j 8 j. he doma of. Theorem. Uder aumpo A-A we have: () The Baye emaor for z Φ Φ ˆ E. () The mmum poeror r aocaed o he Baye emaor Φ Φ z Φ Φ var. Proof. By ag r (.) he dered reu foow mmedaey.... Smar u Uder he aumpo A-A we ca oba he foowg reu for he yem wh mar u. where () The Baye emaor for z Φ Φ ˆ E. () The mmum poeror r aocaed o he Baye emaor ˆ Φ Φ var z Φ Φ Pubhed by Aa Pre opyrgh: he auhor

7 u Φ u u u D h d Θ Θ h m u d e... Exampe Dmar cae: Le ha a Bea pror pdf h ao e a he parameer Φ u u u u u become The r r β r ha Gamma drbuo h r r exp Γ r r Γr m 7 8 m r u Φ u u u u u D β r Smar cae: Le Φ u u u become r r u Γ m m r u ha a Bea pror drbuo. Ao e a he parameer Γr m r u. m r u Φ u u u D r u β r Γ r ha Gamma drbuo.. (.) Remar. I (.) e he we w oba exacy he ame reu of E-Gohary [].. A Smuao Sudy I h eco we geerae hree ampe of ze of em-marov adby mode wh mar u o compare he reu. I aumed ha he exac vaue of he uow parameer ued o geerae he ampe are.7. ad. Tabe. Obervao Sampe Obervao Pubhed by Aa Pre opyrgh: he auhor

8 Tabe. Sojour me Sampe T I obag Bayea emaor of uow parameer we aume ha α α ad are radom varabe wh pror drbuo Bea( 6) Gamma( ) ad Gamma( ) repecvey. Tabe. Emao of parameer by dffere mehod Sampe Maxmum ehood mehod Bayea mehod α α α α α α Tabe gve he perceage of reave error for he emaor obaed by each mehod. Tabe. Reave error(%) Sampe Maxmum ehood mehod Bayea mehod α α α α α α Reu of abe ad how ha he Bayea procedure gve beer emae ha maxmum ehood mehod. Referece. A-Ruzaza A. S. ad Sarha A. M. Emaor for parameer cuded cod adby yem wh mperfec wche I. J. Reab. App. 6 () ar E. Iroduco o ochac procee. Prece-Ha Ic 97.. E-Gohary A. Bayea emao of parameer a hree ae reaby em-marov mode App. Mah. compu. () Sarha A. ad E-Gohary A. Parameer emao of -ou-of-: G reparabe yem App. Mah. ompu. () Sarmah. P. ad Dharmadhar. A. D. Emao of parameer of -ou-of :G reparabe yem ommu. Sa. Theor. Meh. () (98) Pubhed by Aa Pre opyrgh: he auhor