A DISCRETE PARAMETRIC MARKOV-CHAIN MODEL OF A TWO NON-IDENTICAL UNIT COLD STANDBY SYSTEM WITH PREVENTIVE-MAINTENANCE

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1 IJRRA 7 (3) Decembe 3 A DICRETE PARAMETRIC MARKOV-CHAIN MODEL OF A TWO NON-IDENTICAL UNIT COLD TANDBY YTEM WITH PREVENTIVE-MAINTENANCE Rakes Gua¹ * & Paul Badwa² ¹ ²Deamen of ascs C. Caan ng Unvesy Meeu-54 (Inda) ABTRACT Ts ae deals w e of analyss of a wo non-dencal uns (un- un-) cold sby sysem model assumng wo modes- nomal oal falue of eac un. Te un- s sen fo evenve manenance (PM) afe s wokng fo a om eod of me. A sngle eaman s avalable w e sysem fo PM of un- ea of bo uns. Te om vaables denong e me o PM PM me of un- falue mes ea mes of bo uns follow geomec dsbuons w dffeen aamees. Te vaous measues of sysem effecveness ae obaned by usng egeneave on ecnque. Keywods: Tanson obably mean sooun me egeneave on elably MTF avalably of sysem busy eod of eaman.. INTRODUTION All ecoveable uns/sysems wc ae used fo connuous o nemen sevce fo long eod of me ae subec o manenance. Manenance acon can be classfed n ee caegoes- evenve manenance (PM) coecve manenance ea manenance. Te PM s a so of ea a s done befoe e un acually fals. Usually a eodc olcy fo PM s adoed. Howeve s no always ossble o efom s manenance acon exacly a e me wen desed us one would exec e me a wc e PM s made o be a om vaable w eas small dseson abou e desed me. Dung suc manenance e un can be ee n oeave mode o can be swced off fo some me. I can be assumed a afe eac manenance oeaon sysem/un woks as good as new. A vey few auos ncludng [-7] ave analyzed sysem models w e conce of PM.e. afe wokng fo a om amoun eod of me a un goes fo s PM. In all ese sysem models s assumed a me o PM PM me as well as me o falue me o ea ae connuous om vaables. Te uose of e esen sudy s o analyze a wo non-dencal un cold sby sysem model w PM of a un unde dscee aamec Makov-Can.e. falue ea mes of a un me o PM PM me ae aken as dscee om vaables avng geomec dsbuons w dffeen aamees. Te followng economc elaed measues of sysem effecveness ae obaned by usng egeneave on ecnque- ) Tanson obables mean sooun mes n vaous saes. ) Relably mean me o sysem falue. ) Pon-wse seady-sae avalables of e sysem as well as execed u me of e sysem dung neval ( ). v) Execed busy eod of e eaman dung me neval ( ). v) Ne execed of eaned by e sysem dung a fne neval n seady-sae.. JUTIFICATION FOR CONIDERATION OF GEOMETRIC DITRIBUTION Te enomena of dscee falue ea me dsbuons may be obseved n e followng suaon. Le e connuous me eod s dvded as n of equal dsance on eal lne e obably of falue of a un dung me ( +); =.. s en e obably a e un wll fal dung ( +).e. afe assng successfully nevals of me s gven by ; = Ts s e.m.f of geomec dsbuon. mlaly f denoes e obably a a faled un s eaed dung ( +); =... en e obably a e un wll be eaed dung ( +) s gven by ; =. On e same way e om vaables denong me o PM PM me of a un may follow geomec 95

2 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model dsbuons. In vew of e dscee dsbuons e socasc model unde sudy leads o e dscee aamec Makov-Can w sae sace o 5 as sown n secon-4(b). 3. MODEL DECRIPTION AND AUMPTION ) Te sysem comses of wo non-dencal uns (un- un-). Inally un- s oeave un- s ke no cold sby.e. un- ges oy n oeaon ove e un-. ) Eac un of e sysem as wo modes- Nomal (N) Toal falue (F). ) Te un- s sen fo PM afe s wokng fo a om eod of me. Dung e PM acon e un- emans nacve sysem oeaes w sby (un-). Afe comleon of PM un- agan sas oeaon un- goes no sby. v) A sngle eaman s avalable w e sysem o lay e ole of PM of an oeave un- ea of bo uns. Te un- ges oy n PM ea ove e ea of un-. v) If un- fals a e same eoc wen PM of un- s due en un- enes o oal falue mode goes no ea. v) All e om vaables denong PM me me o PM of un- falue mes ea mes of bo uns ae ndeenden of dscee naue follow geomec dsbuons w dffeen aamees. v) Te sysem emans down wen e un- s unde PM un- s oally faled. v) Te sysem falue occus wen bo e uns ae n oal falue mode. x) Te eaed uns wok as good as new. 4. NOTATION AND TATE OF THE YTEM a) Noaons : q :.m.f. of falue me of un-; = q s :.m.f. of ea me of un-; = s. ab :.m.f. of me o PM of un- a b. cd :.m.f. of PM me of un- c d.. q Q :.m.f. c.d.f. of one se o dec anson me T fom sae : eady sae anson obably fom sae o Q. o Z : Pobably a e sysem soouns n sae a eocs uo (-). : Mean sooun me n sae. : ymbol dummy vaable used n geomec ansfom e. g. b) ymbols fo e saes of e sysems: GT q q q N / N : Un- (= ) n nomal (N) mode oeave/sby. o s F / F : Un- (= ) n oal falue (F) mode unde ea/wang fo ea. w N m : Un- n nomal (N) mode unde PM. W e el of above symbols e ossble saes of e sysem along w falue ea aes ae sown n e anson dagam (Fg. )-. 96

3 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model TRANITION DIAGRAM N N m 5 o N F m w c a c a c a 3 N N o s N F o a a F N o a 4 : U ae : Down sae : Faled ae : Regeneave Pon Fg. F F w 5. EXPLANATION OF TRANITION BETWEEN THE TATE ysem nally sas fom sae wee un- s oeave un- s ke no cold sby wc can fal n s sby sae. Fom s sae e sysem asses o sae n wo muually exclusve ways) Ee e oeave un- fals w ae so a e faled un- goes no ea e sby un- becomes oeave w a nsananeous efec swcng devce. ) Te un- fals a e same eoc wen PM of un- s due. In s case agan e suaon of sae ases wee e un- s unde ea un- s oeave. Te sysem may ans fom o f befoe e falue of un- e PM of un- s due so a un- enes no PM un- sas funconng. mlaly fom sae e followng ee muually exclusve suaons ase so as o ene e sysem no saes 4 3 esecvely. ) Befoe e falue of un- e ea of un- s comleed w ae. ) Befoe e comleon of ea of un- e un- s faled w ae. ) A e same eoc e ea of un- s comleed un- s faled w aes esecvely. On e same way e exs fom e oe saes can be obseved. 6. TRANITION PROBABILITIE Le Q be e obably a e sysem anss fom sae o anson me fom sae o en Q P T By usng smle obablsc agumens we ave dung me neval ( ).e. f T s e 97

4 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model a b Q bq bq q Q q s qs s Q q s 4 qs c Q 3 bq Q 3 bqs aq Q 3 bqs aq s Q 35 bqs 53 aq Q bq bq Q q s 3 qs cq Q d Q 5 a +b Q = bqs a s b s Q 34 bqs Q43 s Q d (-5) Te seady sae anson obables fom sae follows: a b bq s qs bq aqs 3 aq bq o cq a b can be obaned fom (-5) by akng as 3 3 q qs c aq qs d a s b s We obseve a e followng elaons old (6-) 7. MEAN OJOURN TIME Le be e sooun me n sae P T In acula bq bq s 4 5 (= 3 4 5) en mean sooun me n sae qs qs s gven by bqs 3 d (-6) c 98

5 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model 8. METHODOLOGY FOR DEVELOPING EQUATION In ode o oban vaous neesng measues of sysem effecveness we develo e ecuence elaons fo elably avalably busy eod of eaman as follows- a) Relably of e sysem- Hee we defne sas fom u sae R as e obably a e sysem does no fal u o eocs.(-) wen s nally. To deemne we egad e faled sae 4 as absobng sae. Now e exessons fo R ; = 3 5; we ave e followng se of convoluon equaons. mlaly R b q q u dur u q u dur u u u Z q R q R q35 R5 (7-3) R Z q R q R 3 3 R Z q R q R q R R Z q R q R q R R Z q R Z d q Z b q s Z q s 3 Z d b) Avalably of e sysem- Le A be e obably a e sysem s u a eoc (-) wen nally sas fom sae. By usng smle obablsc agumens as n case of elably e followng ecuence elaons can be easly develoed fo A ; = o 5. q34 A4 q35 A5 (3-37) A Z q A q A A Z q A q A q A A Z q A q A q A A Z q A q A q A A q A A q A Te values of Z c) Busy eod of eaman- ; = o 3 ae same as gven n secon 6(a). ) Due o PM of un-: Le be e obably a e eaman s busy a eoc (-) n e PM of an oeave un- wen m B sysem nally sas fom sae elaons fo m B. Usng smle obablsc agumens as n case of elably e ecuence ; =345 can be easly develoed as below. 5 99

6 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model Te values of m m m B q B q B m m m m B q B q B q B m m m m B Z q B q B q B m m m m B q B q B q B m m B q B m m m m q B q B B Z q B (38-43) Z Z 5 s same as gven n secon 6(a). ) Due o ea of un- un-: Le B be e esecve obables a e eaman s busy a eoc (-) n e ea of un- B un- wen sysem nally sas fom ecuence elaons fo B. Usng smle obablsc agumens as n case of elably e ; =345 can be easly develoed as below. Te dcoomous vaable akes values esecvely fo. B q B q B B Z q B q B q B B q B q B q B B Z q B q B q B B Z q B q B q B B q B (44-49) Z Z3 ave e same values as n secon 6(a) 9. ANALYI OF CHARACTERITIC a) Relably MTF- 4 Z s. Takng geomec ansfoms of (7-3) smlfyng e esulng se of algebac equaons fo R N (5) D * * 3 N q q q q q q Z q Z q q Z q q q R we ge q q Z q q q q q q q q Z q q q q q q Z q q q q q q q q q q q Z

7 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model 3 q 3 q q 3 q 3 q 5q 53 q q q 3q 3 q q 35q 53 q3q 3 q 3 qq3 q q3 q5q53 D q q q q q q q q q q Collecng e coeffcen of gven by- fom exesson (5) we can ge e elably of e sysem R N E T lm R (5) D N D b) Avalably Analyss- On akng geomec ansfoms of (3-37) smlfyng e esulng equaons we ge * N A D (5). Te MTF s * * q4 q43 Z qq3 q3 q5q53 Z q3 q4 q43 qq3 qq 3 N q q q q q q q q Z q Z q q q q q q q Z q q q q q q q q Z q qq3 q3 q5q53 q q3 q4q 43 qq3 qq 3 q q 34q 43 q 35q 53 q 3 q q 3 q 4q 43 q q 3 q 5q 53 D q q q q q q q q q q q q q q Te seady sae avalably of e sysem s gven by- As D N A lm A lm D a = s zeo eefoe by alyng L. Hosal ule we ge N A = - (53) D N

8 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model D Now e execed u me of e sysem uo eoc (-) s gven by so a - u x= c) Busy Peod Analyss- ) Due o PM of un-: Bu D D μ = A x u A (54) On akng geomec ansfoms of (38-43) smlfyng e esulng equaons we ge B m N3 D (55) N q q q q q q q q q q q q Z q Z q q q q q q q q q Z s same as n avalably analyss. In e long un e obably a e eaman s busy n PM of un- s gven by- m m 3 B lm B lm D a = s zeo eefoe by alyng L. Hosal ule we ge B m N N3 (56) D N D s same as n avalably analyss. Now e execed busy eod of e eaman n e ea of a oally faled un u o eoc (-) s gven by- o a m - m b x= μ = B x m b m B (57) 3

9 IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model ) Due o ea of un- un-: On akng geomec ansfoms of (44-49) smlfyng e esulng equaons fo we ge N 4 N 5 B B (58-59) D D D N q q q q q q q q q q q q Z q Z q q q q q q q q q Z N q q q q q q q q Z s same as n avalably analyss. In e long un e esecve obables a e eaman s busy n e ea of un- un- ae gven by- Bu D 4 B lm B lm D B lm B lm D N N 5 a = s zeo eefoe by alyng L. Hosal ule we ge B N4 D B N5 (6-6) D N N 5 3 D s same as n avalably analyss. Te execed busy eod of e eaman n e ea of un- un- u o eoc (-) ae esecvely gven by- o a B x B x b x B b b x B b (6-63). PROFIT FUNCTION ANALYI We ae now n e oson o oban e ne execed of ncued u o eoc (-) by consdeng e caacescs obaned n eale secon. Le us consde K = evenue e-un me by e sysem wen s oeave. K = cos e-un me wen eaman s busy n e PM of un-. K = cos e-un me wen eaman s busy n e ea of un-. 33

10 MTF IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model K 3 = cos e-un me wen eaman s busy n e ea of un-. Ten e ne execed of ncued u o eoc (-) gven by m P K K K K (64) u b b 3 b Te execed of e un me n seady sae s gven by- P P lm m A B B lm lm lm K K K m 3 3 K lm B K A K B K B K B (65) Beavo of MTF w esec o ₁ ₁ c 6 4 ₁=. c=.4 ₁=.5 c=.4 ₁=.3 c=.4 ₁=. c=.6 ₁=.5 c=.6 ₁=.3 c= Fg. Beavo of Pof (P) w esec o ₁ ₁ c 34

11 PROFIT IJRRA 7 (3) Decembe 3 Gua & Badwa A Dscee Paamec Makov-Can Model ₁=. c=.4 ₁=.5 c=.4 ₁=.3 c=.4 ₁=. c=.6 ₁=.5 c=.6 ₁=.3 c=.6 Fg. 3. GRAPHICAL REPREENTATION Te cuves fo MTF of funcon ave been dawn fo dffeen values of aamees c. Fg. decs e vaaons n MTF w esec o falue ae of an oeave un- fo ee dffeen values of ea ae..5.3 of a faled un- wo dffeen values of PM mes c.4.6 of an oeave un-. Fom e cuves we obseve a MTF deceases unfomly as e value of nceases. I also eveals a e MTF nceases w e ncease n deceases w e ncease n c. mlaly Fg. 3 eveals e vaaons n of (P) w esec o fo vayng values of c wen e values of oe aamees ae ke fxed as a. K K 5 K K3. Fom e cuves we obseve a of deceases unfomly as e value of nceases. I also eveals a e of nceases w e ncease n nceases w e ncease n c. Fom s fgue s clea fom e doed cuves a e sysem s ofable only f falue ae s geae an esecvely fo =..5.3 fo fxed value of c =.4. Fom smoo cuves we conclude a e sysem s ofable only f =..5.3 fo fxed value of c =.6. s geae an esecvely fo. REFERENCE []. L. R. Goel P. Gua Analyss of a wo-engne model w wo yes of falue evenve manenance Mcoelec. Relab (984). []. R. Gua Pobablsc analyss of a wo-un cold sby sysem w wo-ase ea evenve manenance Mcoelec. Relab. 6() 3-8 (986). [3]. R. Gua R. Ksan Damenda Kuma Cos benef analyss of a wo non-dencal un sby sysem w evenve manenance J. of Combnaocs Infomaon sysem scence 37() -3 (). [4]. G. Mokadds.. Elas E.A. olman A ee un sby edundan sysem w ea evenve manenance Mcoelec. Relab. 3() (99). [5]. G. Mokadds.. Elas Te elably funcon e avalably of a dulcaon edundan sysem w a sngle sevce facly fo evenve manenance ea Mcoelec. Relab (989). [6]. K. Mua A. Al-Al One un elably sysem subec o om socks evenve manenance Mcoelec. Relab. 8(3) (988). [7]. T. Nakagawa. Osak ocasc beavo of wo un aallel edundan sysem w evenve manenance Mcoelec. Relab (975). 35

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