Study of Reliability Measures of a Two Units Standby System Under the Concept of Switch Failure Using Copula Distribution

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1 Amrican Journal of Computational and Applid Matmatic 24, 4(4): 8-29 DOI:.5923/j.ajcam Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution Ciwa Mua Dala, Vijay Vir Sing * Dpartmnt of Matmatic & Statitic, Yob Stat Univrity, PMB 44 Gujba Road Damaturu, Yob Stat, Nigria Abtract Ti papr dal wit t tudy of tandby complx ytm wic conit of a main unit and two tandby unit. T tandby unit ar connctd to t main unit via an automatic witc and t ytm i opratd by a uman oprator. T tandby unit ar connctd wit t main via a witc in uc a way tat ty can prform t tak immdiatly aftr failur of main unit. Wn t tandby unit prform t tak, t main unit go for rpair and a oon a it i rpaird, t load of tandby unit again go to t main unit. In ca main unit i not rpaird and t firt tandby unit fail, t cond tandby unit tart and prform t tak. A uman oprator oprat t ytm and trfor t uman failur can appar in tat wr ytm i in oprational mod. T failur rat ar contant and aum to follow xponntial ditribution but rpair follow two typ of ditribution (gnral and Gumbl-Hougaard family copula) ditribution. T ytm i tudid by upplmntary variabl tcniqu. Som important maur of rliability uc a availability, MTTF, nitivity and profit function av bn dicud. Som particular ca av bn dicud for diffrnt valu of diffrnt variabl. Kyword Rliability, Availability, Human failur, Snitivity analyi, MTTF and profit function. Introduction Earlir, t rarcr and cintit, wo tudid t rliability caractritic of rpairabl complx ytm, proclaimd tir validation of t rult in t fild of rliability by taking diffrnt failur rat and on rpair. Mot of t rarcr aumd tat t faild unit can b rpaird by a ingl rpairman. Tu, wnvr t ytm / ubytm fail, on typ of rpair i mployd to rpair t ytm, wic tak mor tim for rpair t faild unit, rulting t indutry /organization uffrd wit a grat lo. T autor in [2, 2, 3, 22] tudid t rliability caractritic of a complx ytm undr and prmptiv rum rpair policy uing copula ditribution. T rarcr in [4, 5,, 5] tudid availability of complx ytm wit common cau failur and rliability of duplx tandby ytm by upplmntary variabl tcniqu [6] and Laplac tranform. In ca, an ordinary rpair facility i mployd to rpair a faild unit, it will av a grat impact on functioning of indutry / organization, rulting t organization may uffrd a grat lo and t manufacturr will lo own markt rputation. Human failur ari in t ytm du to wrongly opration by untraind or * Corrponding autor: ingvijayvir68@gmail.com (Vijay Vir Sing) Publid onlin at ttp://journal.apub.org/ajcam Copyrigt 24 Scintific & Acadmic Publiing. All Rigt Rrvd inxprinc oprator. Apparanc of uman failur compltly brakdown t ytm and can damag many important part of t ytm. In ti contxt t autor in [3] tudid t automation cnario analyi of uman and ytm rliability undr diffrnt typ failur and gnral rpair policy. Availability of ytm dfin t rliability tat t ytm will prform it intndd work ovr a priod of tim wn t rpair i availabl. Tu unavailability i t probability tat t ytm i not abl to prform intndd tak. T autor in [8] tudid unavailability analyi of afty ytm undr aging upplmntary variabl tcniqu and Laplac tranform. T autor in [8] tudid a complx ytm by conidring intrting modling of ytm wic aving tr unit at upr priority, priority and ordinary undr prmptiv rum rpair policy mploying upplmntary variabl tcniqu and Laplac tranform. W obrvd, in many ituation in ral lif, wr mor tan on rpair i poibl btwn two adjacnt tranition tat, t ytm i rpaird by copula ditribution [, 6], wic coupl t gnral and xponntial ditribution. If t main unit of ytm fail, t tandby unit tart prforming t tak of t faild unit. T ytm could b rpaird by mploying gnral rpair, by t rpairman wit i own capability but wnvr t ytm i in complt failur mod, it ould b rpaird by mploying Copula (Gumbl-Hougaard family copula) ditribution. Uing ti tratgy, t autor in [7, 4, 7, 9] tudid t rliability

2 Amrican Journal of Computational and Applid Matmatic 24, 4(4): caractritic of complx ytm, wic conit of two ubytm wit controllr and tandby complx ytm wit waiting rpair policy uing Gumbl-Hougaard family copula ditribution. T automatic controlld witc play an important rol in improvmnt t rliability of a rpairabl ytm. T prim aim in any indutry or organization i to gain mor profit wit lat xpnditur. Trfor, wn t ytm i in oprational mod via tandby unit, it ould b rpaird by ordinary rpair facility and wnvr t ytm i in compltly brakdown mod, it ould b rpaird by copula. Rcntly t autor in [2, 2] tudid t rliability maur of a tandby complx ytm wit diffrnt typ of failur undr waiting rpair diciplin uing Gumbl-Hougaard family Copula ditribution by conidring two typ of failur i.. partial failur (minor and major) and complt failur. Eitr arlir t rarcr ignord t concpt of witc failur or t concpt of gnral rpair policy ad mployd to rpair t faild unit / ubytm. Trfor, in ti papr, w av paid our attntion on t tudy of complx ytm, wic conit of two tandby rdundant unit undr t concpt of an auto-cut witc and uman failur. W av analyzd t ytm for diffrnt ituation wr t ytm av on tandby unit, ignoring t witc failur and t uman failur. Initially, t main unit tart functioning and on failur of main unit, t firt tandby unit tak it load and t availabl rpairr tart rpairing t faild unit. In ca t faild unit rpaird bfor failur of tandby unit, it tak load of tandby unit and t tandby unit go for tandby mod. If t firt tandby unit fail bfor rpair of main unit tn cond tandby unit tak load of firt tandby unit and rpair in continuou will b givn to main faild unit. If main unit i rpaird bfor t failur of cond tandby unit, it tart functioning du to auto cut dvic, rpair i aignd to firt faild unit and tn to cond unit. Tu t main unit will nvr will b in non-oprational mod, it will b in itr oprational mod or undr rpairing. T ytm will b in Tabl. Stat Dcription for Matmatical Modl of Fig. 2 compltly failur mod in t following ituation: ) bot tandby unit fail bfor rpairing of t main unit, 2) Switc fail at any intant during oprational mod, 3) T uman failur occur at any tag wn t ytm i in oprational mod. Human oprator oprat t ytm; t uman rror can ari at any tag wn t ytm i in oprational mod. All failur rat ar aumd to b contant and follow xponntial ditribution, but t rpair follow two typ of ditribution namly: Gnral ditribution and Gumbl-Hougaard family copula ditribution. Wnvr t ytm i in partially faild tat [S, S 2, S 3, S 4, S 5, S 6, S 7 ] of Fig. 2, t gnral rpair i mployd but wnvr t ytm i in compltly faild tat [ S 8, S 9 ] of Fig. 2, it i rpaird by Gumbl-Hougaard family copula ditribution. T rarcr mainly paid tir attntion for t tudy of om traditional maur including rliability, availability, MTTF and cot analyi wic av bn tudid arlir by vari invtigator, uing variou mtod. T tudy of rliability nginring in not only ufficint up to t maur, but till tr ar many maur, wo tudy i alo rquird in t fild of rliability tory. Ti nd tudy of nitivity analyi for variou maur a dmandd by variou indutri & organization. Hnc for in t prnt papr, w av dicud t nitivity of rliability and MTTF, including t traditional maur. T papr i organizd a follow: Sction 2 xplain t dcription of matmatical modl. Sction 3 dcrib t formulation and olution of matmatical modl. T ction 4 dcrib t MTTF of t ytm. Sction 5 nitivity analyi of rliability and MTTF of dcribd modl. Finally, ction 6 of t papr dcrib t intrprtation and concluion of t papr. 2. Matmatical Modl Dcription 2.. Stat Dcription for Matmatical Modl Stat S S S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 Dcription Main unit i in oprational mod and bot rdundant unit ar in good working condition. T tat rprnt t main unit of ytm a faild and t firt rdundant unit i in oprational mod, faild unit i bing rpaird. T ytm i in oprational mod. T tat rprnt tat t main unit av takn t load of rdundant unit aftr gtting rpair, rdundant unit i bing rpaird. T ytm i in oprational mod. In ti tat firt rdundant unit of t ytm av faild, main unit of t ytm av not yt bn rpaird. T ytm i in oprational mod du to cond rdundant unit. T ytm a compltly faild du to failur of cond rdundant unit bfor rpair of main unit, main unit i undr rpair. T tat rprnt tat main unit av tart opration aftr bing rpaird, firt faild rdundant unit i running undr rpair. T ytm i in oprational mod. T tat rprnt tat t aftr gtting rpaird t firt rdundant unit tart functioning and main unit i in rpairing procding, ytm i in oprational mod wit l fficincy. In tat S 7 t firt rdundant unit of t ytm av faild and t cond rdundant unit i in oprational mod wit l fficincy. In ti tat t ytm i in a compltly brakdown tat du to failur of auto cut witc. T ytm a compltly faild du to uman failur by t oprator.

3 2 Ciwa Mua Dala t al.: Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution T dcription of tat conclud tat in tat S t ytm i in prfct tat wr main unit and bot rdundant unit and auto cut witc i in good working condition. S, S 2, S 3, S 5, S 6, S 7, ar t tat wr t ytm i in dgradd mod and t rpair i bing mployd, tat S 4, S 8, and S 9 ar t tat wr t ytm i in compltly failur mod Nomnclatur Tabl 2. Paramtr and Symbol ud in Analyi of Modl T Tim cal variabl. Laplac tranform variabl. λ / λ Failur rat of du to uman failur/ witc failur rat of ytm. λm / λ / λ 2 Failur rat of main unit/ firt rdundant unit/ cond rdundant unit of t ytm. φ(x)/ (y)/ξ(z) Rpair rat for main unit/firt rdundant unit / cond rdundant unit. P i (t) T probability tat t ytm i in S i tat at intant for i = to 9. P () Laplac tranformation of P(t). P i (x, t) T probability tat a ytm i in tat S i for i= to 9; t ytm i running undr rpair and lapd rpair tim i x, t. E p (t) K, K 2 Expctd profit during t intrval [, t). Rvnu and rvic cot pr unit tim rpctivly. µ (x)= C θ (u (x), u 2 (x)) T xprion of joint probability (faild tat S i to good tat S ) according to Gumbl-Hougaard family copula i givn / a Cθ ( u( x), u2( x)) = xp[ x θ + {log φ( x)} θ ] θ, wr, u = φ(x), and u 2 = x, wr θ i a paramtr Aumption T following aumption ar takn trougout t dicuion of t modl: Initially in S tat and all unit a wll a witc of ytm i in good working condition. T ytm work uccfully till rdundant unit ar i in working condition. T ytm fail if bot rdundant unit fail bfor rpair of main unit. T witc i intalou and auto cut it did not tak tim for witcing to rdundant unit. T ytm can b rpaird wn it i in dgradd tat or compltly faild tat. All failur rat ar contant and ty follow an xponntial ditribution. Human failur /complt failur& witc failur of t ytm nd immdiat rpairing, tr for it i rpaird by Gumbl-Hougaard family copula. It i aum tat rpaird ytm work lik a nw ytm and tr will b no damag don du to rpair. A oon a t faild unit gt rpair it rady to prform t tak wit full fficincy Sytm Configuration Human oprator Auto-cut witc Main unit Firt tandby unit Scond tandby unit Figur. Sytm Configuration of Modl

4 Amrican Journal of Computational and Applid Matmatic 24, 4(4): Stat Tranition Diagram of Modl Figur 2. Tranition Stat Diagram of Modl 3. Formulation and Solution of Matmatical Modl 3.. Matmatical Formulation of Modl By t probability of conidration and continuity of argumnt, w can obtain t t of diffrnc diffrntial quation govrning t prnt matmatical modl a own in appndix. Appndix. ould b r: Solving (2)-(29), wit lp of quation (3) to (38) on may gt,

5 22 Ciwa Mua Dala t al.: Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution P () = D () (39) λ ( S ( )) () m ϕ + λ + λ + λ P = D () ( + λ + λ + λ ) λλ ( S ( )) m 2() m ϕ + λ + λ P = DA () ( + λ + λ ) λλ ( ( 2 )) 3() m C Sϕ + λ + λ + λ P = DA () ( + λ + λ + λ ) λλ ( ( )) 5() m C Sϕ + λ + λ P = DA () ( + λ + λ ) m λλ ( ) ( ( 2 )) 6() mc Sϕ + λm + λ Sϕ + λ + λ + λ P = D () B ( + λ + λ + λ ) 2 λ λ ( ) ( ( 2 )) 7() mc Sϕ + λm + λ Sϕ + λ + λ + λ P = D () B ( + λ + λ + λ ) ( S ( )) 2 µ 4( ) = 4(, ) (46) P P () ( S ( )) µ 8( ) = 8(, ) (47) P P () ( S ( )) PH( ) P (, ) () µ = H (48) D ( ) = + λ + λ [ P(, S ) ( + λ + λ + λ )) + P(, S ) µ ( ) + P(, S ) ( + λ + λ ) m ϕ 8 2 ψ m + P(, S ) ( + λ + λ + λ ) + P (, S ) µ ( )] 6 ξ H A = ( λ S ( + λ + λ + λ )) B = ( λsϕ ( + λ2 + λ + λ)) λ and C m = λ2 + A Wr, m ϕ 2 T Laplac tranformation of t probabiliti tat t ytm i in up (i.. itr good or dgradd tat) and faild tat at any tim i a follow: 3.2. Particular Ca Pup () = P() + P() + P2() + P3() + P5() + P6() + P7() (49) 2 2 (4) (4) (42) (43) (44) (45) Pdown() = Pup () (5) A. Availability Analyi: For particular ca t tudy of availability i focu on following ca, A. Availability (Availability of ytm), A2. Availability (Switc i ignor) & A3. Availability (Human failur i ignor); Wn rpair follow xponntial ditribution, tting θ θ / θ xp[ x + {log ϕ( x)} ] ϕ Sµ () =, S θ θ / θ ϕ () =, taking t valu of diffrnt paramtr a λ m =.3, + xp[ x + {log ϕ( x)} ] + ϕ λ =.2, λ 2 =.5, λ =.2, λ =.25, φ =, θ =, x =, in (49), tn taking t invr Laplac tranform, on can

6 Amrican Journal of Computational and Applid Matmatic 24, 4(4): obtain, (.42 t) 6 (.5886 t) (.5886 t) A = x t (.858 t) (.426 t) (.223 t) (.8899 t) (. 53 t) (5A) (.42 t) (.995 t) (.32 t) A2 = ( t) 4 ( t) ( t) x (.566 t) 4 (.426 t) 3 (.9973 t) x.9747x ( t) (.546 t) (.3 t) 3 (.9627 t) 5 (.9627 t) A3 = x.3688x 3 ( t) 2 (.74 t) 4 (.36 t) +.27x x +.233x (.3 t) ( t) (.3962 t) (5A2) (5A3) For, t=,, 2, 3, 4, 5, 6, 7, 8, 9; unit of tim, on may gt diffrnt valu of P up (t) wit t lp of (5 A), (5 A2) & (5 A3) a own in Fig. 3. Tim(t) P up (t)/availability A A2 A Availability for ytm A3 Availability for ytm A Figur 3. Availability a function of tim Availability for ytm A3 B. Rliability Analyi: Taking all rpair qual to zro in quation (47) and taking invr Laplac tranform, on can av xprion for t rliability of ytm and for givn valu of failur rat λ m =.3, λ =.2, λ 2 =.5, λ =.2, λ =.25, in (49), w gt (52)a and (52)b; Rt () = m m 2 + m t m m m 2 2 ( λ ) ) ( 2 )) m+ λ t λλ m + λλ m + λ λ λ λ ( ) ( 2 ) ( ) ( )( 2 ) mλ + λm λ+ λ + λ t λm λ+ λ + λ t λm λ λ λm λ λ + + λm λ λ ( λm λ λ) ( λ λ ( λ λ )( λ λ λ )( λ λ λ ) ( λ ( λ λ λ λ λ ) λ ( λ λ λ λ (52a) (.42 t) (.57 t) (.52 t) Rt ( ) = (.627t+ 3.68) (52b)

7 24 Ciwa Mua Dala t al.: Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution Tim (t) Rliability Figur 4. Rliability a function of tim 4. Man Tim to Failur (MTTF) Taking all rpair zro and t limit a tnd to zro in (49) for t xponntial ditribution; on can obtain t MTTF a: λλ ( 2 ) ( ) m + λ + λm λλm + λm λm MTTF = ( λm + λ) λm + λ λm + λ2 + λ λ + λ+ λ Stting λ m =.3, λ =.2, λ 2 =.5, λ =.25, λ =.2 and varying λ m, λ, λ 2, λ, λ on by on rpctivly a.,.2,.3,.4,.5,.6,.7,.8,.9 in (53), on may obtain t variation of M.T.T.F. wit rpct to failur rat a own in Fig.4. Failur Rat MTTF λ m MTTF λ m MTTF λ m MTTF λ m MTTF λ m (53) MTTF λm MTTF λm MTTF λm MTTF λm MTTF λm Figur 5. MTTF a function of Failur rat

8 Amrican Journal of Computational and Applid Matmatic 24, 4(4): Tim T Tabl 6. Snitivity of rliability a a function of tim (tt) λm (tt) λ (tt) λ2 (tt) λ (tt) λ 系列 Sri -.83 系列 2 Sri 系列 Sri3 3 系列 Sri4 4 系列 Sri5 5-2 Figur 6. For nitivity of rliability a a function failur rat Tabl 7. Snitivity of MTTF a a function of failur rat Variation in λ m, λ, λ 2, λ, λ (MMMMMMMM) λm (MMMMMMMM) λ (MMMMMMMM) λ2 (MMMMMMMM) λ (MMMMMMMM) λ Snitivity Analyi 5.. Snitivity of Rliability T nitivity of t rliability can b caractriz a t rat of cang of rliability wit rpct to input factor, mot rgularly dfind a t partial drivativ of rliability wit rpct to failur rat. Tu, t nitivity of rliability and MTTF can b dfind a t rat of variation of outcom maur wit rpct to input factor. Tr for t nitivity of rliability can b obtain by diffrntiating t (52a) and (52 b) wit rpct to λ m, λ, λ 2, λ, λ., and tting λ m =.3, λ =.2, λ 2 =.5, λ =.2, λ =.25 on can obtain Tabl 6 and Figur 6, rpctivly for nitivity of rliability. Snitivity of MTTF can b obtaind by making partial drivativ of MTTF wit rpct to failur rat. Rult ar igligtd a in Tabl and corrponding Figur.7 rpctivly.

9 26 Ciwa Mua Dala t al.: Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution 系列 Sri 系列 Sri2 2 系列 Sri3 3 系列 Sri4 4 系列 Sri5 5 系列 Sri6 6 Figur 7. Snitivity of MTTF a a function of failur rat 5.2. Snitivity of MTTF Snitivity analyi for cang in MTTF rulting from cang in t ytm paramtr i.. ytm failur rat λ m, λ, λ 2, λ, λ. By diffrntiating Equation (5) wit rpct to failur rat λ m, λ, λ 2, λ, λ rpctivly on gt valu of (tt) (tt) λ2, (tt) λ, (tt) λ. 6. Cot Analyi Lt t rvic facility b alway availabl, tn xpctd profit during t intrval [, t) i; For t am t of paramtr of (47), on can obtain (52). Trfor, (tt), λm λ t Ep () t = K Pup () t dt K2t (54) (.42 t) 6 (.5876 t) ( t) Ep ( t) / A = K( x +.36 ( 2,7232 t) (.858 t) (.426 t) (.223 t) (.889 t) (.53 t) ) K2 t (54A) (.42 t) (.995 t) (.32 t) (.969 t) Ep ( t) / A2 = K( (.36 t) ( t) (.566 t) 4 (.426 t).838x x (54A2) (.9973 t) (.8383 t) (.546 t) ) K2 t (.3 t) (.9627 t) 5 (.5229 t) Ep ( t) / A3 = K( x 4 ( 2786 t) (.74 t) (.36 t).3754x (.3 t) ( t) (.396 t) ) K2 t (54A3) Stting K = and K 2 =.5,.25,.5,. and.5 rpctivly and varying t =,, 2, 3, 4, 5, 6, 7, 8, 9, unit of tim, t rult for xpctd profit can b obtain a own in Fig.5.

10 Amrican Journal of Computational and Applid Matmatic 24, 4(4): Tabl 8. For xpctd profit Tim (t) E p (t)/a; K 2 =.5 E p (t) A E p (t)/a; K 2 =.25 E p (t)/a2; K 2 =.5 E p (t) A2 E p (t)/a2; K 2 =.25 E p (t)/a3; K 2 =.5 E p (t) A3 E p (t)/a3; K 2 = Ep(t) A Ep(t)/A; K2=.5 Ep(t) A Ep(t)/A; K2=.25 Ep(t) A2 Ep(t)/A2; K2=.5 Ep(t) A2 Ep(t)/A2; K2=.25 Ep(t) A3 Ep(t)/A3; K2=.5 Ep(t) A3 Ep(t)/A3; K2= Intrprtation of Rult and Concluion Fig.3 provid information ow t availability of t complx rpairabl ytm cang wit rpct to t tim wn failur rat ar fixd at diffrnt valu. Wn failur rat ar fixd at lowr valu λ m =.3, λ =.2, λ 2 =.5, λ =.25, λ =.2, availability of t ytm dcra and ultimatly bcom tady to t valu zro aftr a ufficint long intrval of tim. Hnc, on can afly prdict t futur bavior of a complx ytm at any tim for any givn t of paramtric valu, a i vidnt by t grapical conidration of t modl. Availability of ytm for wic uman failur i ignord i dcra up to tim t=6, but it again tart incraing again. In figur.4 provid t variation in rliability of non-rpairabl ytm. Fig. 5, yild t man-tim-to-failur (M.T.T.F.) of t ytm Figur 8. Expctd profit a function of tim wit rpct to variation in λ m, λ, λ 2, λ, and λ rpctivly wn t otr paramtr av bn takn a contant. T variation in MTTF corrponding to failur rat λ, λ p ar almot i vry clour but corrponding to λ B, λ t variation i vry ig wic indicat tat t bot ar mor rponibl to propr opration of t ytm. T maur of nitivity of rliability and MTTF a dicud in ction C of papr wic rat of cang of t valu of output variabl wit cang of output variabl. Wn rvnu cot pr unit tim K i fixd at, rvic cot K 2 =.5,.25,.5,.,.5, profit a bn calculatd and rult ar dmontratd by grap in Fig.8. A critical xamination from Fig.8 rval tat xpctd profit incra wit rpct to t tim wn t rvic cot K 2 fixd at minimum valu.5. Finally, on can obrv tat a rvic cot incra, profit dcra. In gnral, for low rvic cot, t xpctd profit i ig in comparion to ig rvic cot.

11 28 Ciwa Mua Dala t al.: Study of Rliability Maur of a Two Unit Standby Sytm Undr t Concpt of Switc Failur Uing Copula Ditribution Appndix + λm + λ P() t = ϕ() x P(,) x t dx + ψ( y) P2( y,) t dy t () + ξ () z P6(,) z t dz + µ () x P8(,) x t dx + µ () x PH (,) x t dx + + λ + λ+ λ + ϕ() x P(,) xt = t x + + λm + λ + ψ( y) P2 ( yt,) = t y + + λ + λ2 + λ + ϕ() x P3(,) xt = t x + + ϕ() x P4 (,) xt = t x + + λm + λ + ϕ( y) P5 ( yt,) = t y + + λ+ λ + λ + ξ() z P6(,) zt = t z + + λ + λ2 + λ + ϕ() x P7(,) xt = t x + + µ () x P8(,) xt = t x + + µ () x PH (,) xt = t x Boundary condition (2) (3) (4) (5) (6) (7) (8) (9) () P(, t) = λmp( t) () P2(,) t = ϕ() x P3(,) x t dx (2) P3(, t) = λp(, t) + λmp2(, t) (3) P4(, t) = λ2p3(, t) + λmp5(, t) + λ2p7(, t) (4) P5(,) t = ϕ() x P4(,) x t dx (5) P6(,) t = ψ( y) P5( y,) t dy + ϕ() x P5(,) x t dx (6) P7(, t) = λp6(, t) (7) P8(, t) = λ ( P(, t) + P3(, t) + P6(, t) + P7(, t)) (8) P (, t) = λ ( P ( t) + P(, t) + P (, t) + P (, t) H P (, t) + P (, t) + P (, t)) B. Solution of t modl (9) Taking Laplac tranformation of quation ()-(9) and uing quation wit lp of initial condition, P (t)= and otr tat probabiliti ar zro at t [ + λm + λ] = + ϕ P () () x P (,) x dx + µ ( x) P ( x, ) dx + µ ( x) P ( x, ) dx 8 + ϕ( y) P ( y, ) dy + ξ( z) P ( z, ) dx λ + λ+ λ + ϕ( x) P( x, ) = x + + λm + λ + ψ( y) P2 ( y, ) = y H (2) (2) (22) + + λ + λ2 + λ + ϕ( y) P3( x, ) = x (23) ( x) P4 ( x, ) x η + + = + + λm + λ + ψ( y) P5 ( y, ) = y + + λ+ λ + λ + ξ() z P6(,) z = z (24) (25) (26) + + λ + λ2 + λ + ϕ( x) P7( x, ) = x (27) + + µ ( x) P8( x, ) = x ( ) + + µ x PH ( x x, ) = Laplac tranform of boundary condition (28) (29) (, ) = λm ( ) (3) P P P2(, ) = ϕ( x) P3( x, ) dx (3)

Amrican Journal of Computational and Applid Matmatic 24, 4(4): 8-29 29 P (, ) = λ P(, ) + λ P (, ) (32) 3 m 2 P (, ) = λ P (, ) + λ P (, ) + λ P (, ) (33) 4 2 3 m 5 2 7 P5(, ) = ϕ( x) P4( x, ) dx

12 Amrican Journal of Computational and Applid Matmatic 24, 4(4): P (, ) = λ P(, ) + λ P (, ) (32) 3 m 2 P (, ) = λ P (, ) + λ P (, ) + λ P (, ) (33) m P5(, ) = ϕ( x) P4( x, ) dx (34) P6(, ) = ψ( y) P5( y, ) dy + ϕ( x) P7( x, ) dx P P (35) 7(, ) = λ 6(, ) (36) P8(, ) = λ ( P(, ) + P3(, ) + P6(, ) + P7(, )) (37) PH(, ) = λ( P( ) + P(, ) + P2(, ) + P3(, ) (38) + P (, ) + P (, ) + P (, )) REFERENCES [] A. Gami; S. Yacout and M.-S. Ouali (2); Evaluating t Rliability Function and t Man Ridual Lif for Equipmnt wit Unobrvabl Stat, IEEE, Tranition on Rliability., Vol. 59, Iu, pp [2] A. K. Govil (974); Oprational baviour of a complx ytm aving lf-lif of t componnt undr prmptiv-rum rpair diciplin, Microlctronic Rliability, Vol. 3, pp [3] Alitair G. Sutcliff (27); Automating Scnario Analyi of Human and Sytm Rliability, IEEE Tranaction on Sytm, Man and Cybrntic-Part A: Sytm and Human, Vol.37 (2), pp [4] B. S. Dillon, and N. Yang (992); tocatic analyi of tandby ytm wit common cau failur and uman rror. Microlctronic Rliability, Vol.32 (2), pp [5] B. S. Dillon, and N. Yang (993); Availability of a man-macin ytm wit critical and non-critical uman rror. Microlctronic Rliability, Vol.33 (), pp [6] D. R. Cox (995); T analyi of non-markov tocatic proc by t incluion of upplmntary variabl, Proc. Camb. Pil. Soc. (Mat. Py. Sci.), Vol. 5, pp [7] D. K. Rawal, M. Ram & V.V. Sing (24); Modling and availability analyi of intrnt data cntr wit variou maintnanc polici. IJE Tranaction A: Baic Vol. 27, No. 4. pp [8] E. A. Olivira, A.C.M. Alvim and P. F. Frutuoo Mlo (25); Unavailability analyi of afty ytm undr aging by upplmntary variabl wit imprfct rpair. Annal of Nuclar Enrgy, Vol.32, p p [9] E. J. Vandrprr (99); Rliability analyi of a two-unit paralll ytm wit diimilar unit and gnral ditribution. Microlctronic Rliability, Vol.3, pp [] H. Gnnimr (22); Modl Rik in Copula Bad Dfault Pricing Modl, Working Papr Sri, Working Papr No. 9, Swi Banking Intitut, Univrity of Zuric and NCCR FINRISK. [] M. Ram, and S. B. Sing (2); Analyi of a Complx Sytm wit common cau failur and two typ of rpair faciliti wit diffrnt ditribution in failur Intrnational Journal of Rliability and Safty, Vol. 4(4), pp [2] M. Ram, and S. B. Sing (28); Availability and Cot Analyi of a paralll rdundant complx ytm wit two typ of failur undr prmptiv-rum rpair diciplin uing Gumbl-Hougaard family copula in rpair, Intrnational Journal of Rliability, Quality & Safty Enginring, Vol.5 (4), pp [3] M. Ram, and S. B. Sing (2); Availability, MTTF and cot analyi of complx ytm undr prmptiv-rpat rpair diciplin uing Gumbl-Hougaard family copula. Intrnational Journal of Quality & Rliability Managmnt, Vol. 27(5), pp [4] M. Ram, S. B. Sing& V.V. Sing (23); Stocatic analyi of a tandby complx ytm wit waiting rpair tratgy IEEE Tranaction on Sytm, Man, and Cybrntic Part A: Sytm and uman Vol.43, NO. 3. pp, [5] P. P. Gupta, and M. K. Sarma (993); Rliability and M.T.T.F valuation of a two duplx-unit tandby ytm wit two typ of rpair. Microlctronic Rliability, Vol. 33(3), pp [6] R. B. Nln (26); An Introduction to Copula (2nd dn.) (Nw York, Springr). [7] V. V. Sing, S B Sing, M. Ram & C.K. Gol (22); Availability, MTTF and Cot Analyi of a ytm aving Two Unit in Sri Configuration wit Controllr. Intrnational Journal of Sytm Auranc Enginring and Managmnt, Volum 4, Iu 4, pp [8] V. V. Sing, S. B. Sing, M. Ram and C. K. Gol (2); Availability analyi of a ytm aving tr unit upr priority, priority and ordinary undr prmptiv rum rpair policy, Intrnational Journal of Rliability and Application, Vol. (), pp [9] V.V. Sing, M Ram & Dilip Rawal (23); Cot Analyi of an Enginring Sytm involving ubytm in Sri Configuration. IEEE Tranaction on Automation Scinc and Enginring. Vol;, Iu; 4.pp [2] V.V. Sing, Jyoti Gulati (24); Availability and cot analyi of a tandby complx ytm wit diffrnt typ of failur, undr waiting rpair diciplin uing Gumbl-Hougaard copula. IEEE, www. ixplor.org. DOI.9/ICICICT PP (8-86). [2] V. V. Sing, Dilip Kumar Rawal (2); Availability analyi of a ytm aving two unit in ri configuration wit controllr and uman failur undr diffrnt rpair polici, Intr. J. of Scint. Eng. R. Vol 2 (), pp. -9. [22] X. Cai, X. Wu and X. Zou (25); Dynamically Optimal Polici for Stocatic Scduling Subjct to Prmptiv- Rpat Macin Brakdown, IEEE Tranaction On Automation Scinc and Enginring, Vol. 2, No.2, pp

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC