Stochastic Analysis of a Single Unit with a Protective Unit Discrete Parametric Markov Chain System Model

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1 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 I 50-5 ocasic Analysis of a ingle Uni wi a oecive Uni Discee aameic akov Cain ysem odel Rakes Gua Ai Tyagi Deamen of aisics C. Caan ing Univesiy eeu Absac- Te ae deals wi cos benefi analysis of a single uni (main uni) sysem model suoed by a eling uni. ain uni as ee ossible modes nomal () aial failue () oal failue (F). Weeas e eling uni (oecive uni) as wo ossible modes- nomal () oal failue (F). Te main uni can also wok wiou e oecive (eling) uni bu wi inceased failue ae. Te sysem is declaed as failed wen a leas main uni is ou of funcioning. A single eaiman is always available wi e sysem o eai e failed main oecive (eling) uni. Te failue eai imes of main uni oecive (eling) uni ae aken as indeenden om vaiables of discee naue aving geomeic disibuions wi diffeen aamees. Index Tems- Regeneaive oin eliabiliy TF availabiliy of sysem busy eiod of eaiman ne execed ofi. I I. ITRODUCTIO n e lieaue of eliabiliy vaious auos including [489] ave analyzed eaable sysem model wi one o moe unis by assuming diffeen conces suc as ee ossible modes slow swic imefec swic waiing ime of eaiman wo yes of eaimen insecion of a failed uni e-emive eea ioiy. Tey assumed coninuous disibuions of ime o failue ime o eai of a uni. Bu vey few auos Gua e al [56] ave obained e eliabiliy caaceisics of edundan sysem models by consideing discee disibuions of failue eai imes. ingle uni sysem models wi eling uni aving wo modes-omal Toal failue ave been sudies by few auos including [70] in e field of eliabiliy eoy. Te ole of eling (oecive) uni is o oec e failue of main uni. Te main uni may also wok wiou e oecive (eling) uni bu wi inceased failue ae. Vaious measues of sysem effeciveness ae obained by assuming e failue eai imes as coninuous om vaiables. Te uose of e esen ae is o analyze a single uni (main uni) sysem model wi eling (oecive) uni. Te main uni as ee ossible modes- nomal () aial failue () oal failue (F). Te sysem model is analyzed unde discee aameic akov-cain i.e. ime o failue eais ae aken as discee om vaiables. Te following economic elaed measues of sysem effeciveness ae obained by using egeneaive oin ecnique- i) Tansiion obabiliies mean sooun imes in vaious saes. ii) Reliabiliy mean ime o sysem failue. iii) oin-wise seady-sae availabiliy of e sysem as well as execed u ime of e sysem uo e eoc (-). iv) Execed busy eiod of e eaiman due o main uni eling uni uo e eoc (-). v) e execed ofi incued by e sysem uo ime eoc (-) in seady-sae is obained. II. ODEL DECRITIO AD AUTIO i) Te sysem comises of a single uni (main uni) is suoed by a eling (oecive) uni. Iniially bo e unis ae oeaive. ii) Te eling uni is used o oec e failue of main uni. ain uni as ee ossible modes nomal () aial failue () oal failue (F) weeas e eling uni as only wo ossible modes- nomal () oal failue (F). iii) Te main uni can ene ino F-mode wiou assing oug e -mode. iv) Wen e eling uni fails e main uni can also oeae wi inceased failue ae in bo -mode -mode. v) Te sysem sos funcioning wen main uni is failed eling uni is good because e oeaion of eling uni in is siuaion is woless. vi) A single eaiman is always available wi e sysem o eai e failed main uni eling uni. Te main uni is eaied only wen i enes ino F-mode. vii) Te ioiy in eai is being given o e main uni ove e eling uni. Teefoe if duing e eai of eling uni e main uni enes ino F-mode en main uni is aken u fo eai disconinuing e eai of eling uni. viii) Eac eaied uni (main o eling) woks as good as new. ix) Failue eai imes of e unis follow indeenden geomeic disibuions wi diffeen aamees.

2 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 I 50-5 III. OTATIO AD TATE OF TE YTE a) oaions : x q i i :.m.f. of failue ime of main uni fom -mode o -mode -mode o F-mode esecively fo i= wen eling uni is oeaive i qi. x i qi i >i :.m.f. of failue ime of main uni fom -mode o -mode -mode o F-mode esecively fo i= wen eling uni is failed i qi. x q :.m.f. of failue ime of eling uni q. x s i i : s.m.f. of eai ime of ain eling uni esecively fo i= i i. q i g i g :.m.f. C.d.f. of one se o diec ansiion ime fom sae i o. i i Zi i i : eady sae ansiion obabiliy fom sae i o. : obabiliy a e sysem sooun in sae i u o eocs 0 (-). : ean sooun ime in sae i. : ymbol dummy vaiable used in geomeic ansfom e. g. GT q q q i i i 0 b) ymbols fo e saes of e sysems: 0/ 0 : ain uni is in -mode/- mode oeaive. o / g : eling uni is in nomal () mode oeaive/good. F : ain uni is in oal failue (F) mode unde eai. F / Fw : eling uni is in oal failue (F) mode unde eai/wais fo eai. Wi e el of above symbols e ossible saes of e sysem ae as follows: 0 OO OF OO 0F 4 g F 5 F Fw Te ansiion diagam of e sysem model alongwi e ansiion aes is sown in Fig..

3 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 I 50-5 TRAITIO DIAGRA 0 o o o o o F F g o F F F w 4 5 : U ae : Failed ae : Regeneaive oin Fig. i Le be e obabiliy a e sysem ansis fom sae fom sae i o en i Ti By using simle obabilisic agumens we ave q q 0 qq 0 qq q q qq 0 qq q qq 0 q s q s qs s q s q s q s q qq q qq 4 q q qq 5 qq q qq q s q s 4 qs s qs 5 q s q s 40 s 5 s (-4) IV. TRAITIO ROBABILITIE i o T duing ime ineval (0 ) i.e. if i is e ansiion ime 0 Te seady sae ansiion obabiliies fom sae i o can be obained fom (-4) by aking as follows: q q q qq q q q q q s s q q 4 q s q s q q q q

4 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 4 I qq 40 5 q qs 4 qs 5 s qs We obseve a e following elaions old (5-9) V. EA OJOUR TIE T Le i be e sooun ime in sae i (i=0 4 5) en mean sooun ime E T T i i i In aicula qq 0 qq qs qs qs qs s 4 5 qq q q (say) (0-4) i in sae i is given by VI. ETODOLOGY FOR DEVELOIG EUATIO In ode o obain vaious ineesing measues of sysem effeciveness we develoed e ecuence elaions fo eliabiliy availabiliy busy eiod of eaiman as followsa) Reliabiliy of e sysem- Ri ee we define as e obabiliy a e sysem does no fail u o eocs 0.. (-) wen i is iniially saed fom u sae i. To deemine i we egad e failed saes 4 R 5 as absobing sae. ow e exession fo i ; i=0 ; we ave e following se of convoluion equaions. R q q q u R u q u R u q u R u u0 u0 u0 Z q R q R q R imilaly R Z q R q R q R 0 0 R Z q R R Z q R (5-8) Wee Z q s Z q q Z = q s b) Availabiliy of e sysem- Ai Ai Le be e esecive obabiliies a e sysem is u in nomal mode aial failue mode of main uni a eoc (-) wen sysem iniially sas fom sae i. Ten by using simle obabilisic agumens as in case of eliabiliy e Ai following ecuence elaions can be easily develoed fo ; i=0 o 5 =. A Z q A

5 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 5 I 50-5 A Z q A q A q A 0 0 A Z q A q A q A A Z q A q A q A A q A A q A 5 5 Wee (9-4) 0 esecively fo =. Te values of c) Busy eiod of eaiman- Bi Bi Zi ; i=0 o ae same as given in secion 6(a). Le be e esecive obabiliies a e eaiman is busy a eoc (-) in e eai of failed main uni eai of a failed eling uni wen sysem iniially sas fom sae i. Using simle obabilisic agumens as in case of eliabiliy k Bi availabiliy analysis e elaions fo ; i=0 o 5 k= can be easily develoed as below. k k B q q q k k k k B Z q B q B q B k k k k 0 0 B q B q B q B k k k k 4 4 B Z q B q B q B k k k k B Z q B k B Z q B k Wee (5-40) 0 esecively fo k=. Te values of Zi Z4 Z5 s ; i= ae same as given in secion 6(a). VII. AALYI OF RELIABILITY AD TF R Taking geomeic ansfoms of (5-8) simlifying e esuling se of algebaic equaions fo R0 D (4) Wee * * * * * * * * * * * * * * Z 0 qq Z0 q0z q0 q qq q0 q0q Z q * * * * * * * * 0 qq q q0q q 0 Z * * * * * * * * * * * D q q q q q q q q q q q R0 0 we ge Collecing e coefficien of fom exession (4) we can ge e eliabiliy of e sysem. Te TF is given by- ET lim R D (4) Wee D 0 0 VIII. AVAILABILITY AALYI On aking geomeic ansfoms of (9-4) simlifying e esuling equaions fo = we ge

6 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 6 I 50-5 A D * 0 Wee A * D 0 (4-44) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Z q q q q q q q q q q q q Z q q q q q q q q q q q q q Z q q q q q q q q q q q q q Z q q q q q q q q q q q q q D * * q q * * q q * * q q * q * * q q * q * * q q * * * * * * * * * * * * * * * * * * * * * * * * * * q 0 q0 qq q0q5 qq5 q5 q0q5 q5 qq 5 q q q q q q q q q q q q q q * * * * * * * * * * * * * * * * * 4 q5q q5q4 q4q5 q0 q4 qq4 q5q q4q5 q5q 4 Te seady sae availabiliies of e sysem due o oeaion of e sysem in -mode -mode of main uni ae given by- A0 lima0 lim D A0 lima0 lim D D ow since a = is zeo eefoe by alying L. osial ule we ge A0 D A0 D (45-46) Wee D ow e execed uime of e sysem in - mode -mode of main uni u o eoc (-) ae esecively given by u 0 x0 A x so a A u 0 u 0 x0 A x A u 0 (47-48)

7 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 7 I 50-5 IX. BUY ERIOD AALYI On aking geomeic ansfom of (5-40) simlifying e esuling equaions fo k= we ge * 4 * 5 B0 = B0 = D D (49-50) Wee * * * * * * * * * * * * * 4 Z 4 q0 q qq4 q4 q qq4 q4 q0 qq4 q4 q * * * * * * * * * * * * * * 5q q5q4 q4q5 q0 q4 qq4 q5q q4q5 * * * * * * q5q4 Z 5 q0 q qq5 q5 q q5 qq5 q0 qq5 q5 q0 q5 qq5 * * * * * * * * * * * * * 5 Z q0 qq q0q5 qq5 q5 q0q5 q5 qq5 * * * * * * * * * * * * * Z q0 q qq q0 q q5q5q q0 qq 5q5 D is same as in availabiliy analysis. In e long un e esecive obabiliies a e eaiman is busy in e eai of failed main uni failed eling uni ae given by- 4 B0 limbo lim D 5 B0 limbo lim D Bu D a = is zeo eefoe by alying L. osial ule we ge 4 B 0 D Wee B 0 5 D (5-5) D is same as in availabiliy analysis. ow e execed busy eiod of e eaiman in eai of failed main uni failed eling uni u o eoc (-) ae esecively given by- b 0 x0 B x o a B 0 b b 0 x0 b B x B0 (5-54) X. ROFIT FUCTIO AALYI We ae now in e osiion o obain e ne execed ofi incued u o eoc (-) by consideing e caaceisics obained in ealie secion. Le us conside K 0 =evenue e-uni ime by e sysem wen main uni is oeaive in nomal mode ().

8 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 8 I 50-5 K =evenue e-uni ime by e sysem wen main uni is oeaive in aial mode (). K =cos e-uni ime wen eaiman is busy in e eaiing failed main uni. K =cos e-uni ime wen eaiman is busy in e eaiing failed eling uni. Ten e ne execed ofi incued u o eoc (-) given by K0u Ku Kb Kb (55) Te execed ofi e uni ime in seady sae is given by- lim lim B0 A0 A0 B0 K0 lim K lim K lim K lim K A K A K B K B (56) XI. GRAICAL REREETATIO Te cuves fo TF ofi funcion ave been down fo diffeen values of aamees. Fig. deics e vaiaions in TF wi esec o eai ae (₂) of failed eling uni fo ee diffeen values of failue ae (₁) of main uni fom is nomal mode o aial failue mode wen eling uni is good wo diffeen values of failue ae (₃) of eling uni weeas e values of oe aamees ae ke fixed as ₁=0.6 ₂=0.8 ₂=0. ₁=0.6. Fom e cuves we obseve a execed life of e sysem inceases oo muc slowly wi almos linea end as e value of inceases i deceases wi e incease values of ₁ ₃. imilaly Fig. eveals e vaiaions in ofi funcion () wi esec o ₂ fo vaying values of ₁ ₃ wen e values of oe aamees ae ke fixed as ₁=0.6 ₂=0.8 ₂=0. ₁=0.6 K₀=0 K₁=5 K₂=40 K₃=50. Fom e figue i is clealy obseved fom e smoo cuves a e sysem is ofiable only if eai aamee ₂ is geae an fo ₁= esecively fo fixed values of ₃=0.0 fom doed cuves we conclude a e sysem is ofiable only if ₁ is geae an fo ₁= esecively fo fixed values of ₃=0.06. [] Goel LR umaz Z Analysis of a single uni sysem wi eling uni icoelecon Reliab. Vol.-5 o (99). [4] Goalan aidu R ocasic beavio of a wo-uni eaable sysem subec o insecion icoelecon. Reliab. (4) 77-7(98). [5] Gua R Vasney G A wo idenical uni aallel sysem wi Geomeic failue eai ime disibuions J. of Comb. Info. & ysem ciences Vol. o (007). [6] Gua R Vasney G A wo non idenical uni aallel sysem wi Geomeic failue eai ime disibuions IAR Tans. Vol. o. 7 9 (006). [7] Gua R Cauday Kuma K Cos-Benefi Analysis of a ingle- Uni ysem odel wi eling Uni ue Alied aemaika ciences Vol. LXVII; o. -ac 008. [8] Gua Jaiswal K Goel LR Analysis of wo-uni sby edundan sysem unde aial failue e-emive eai ioiy ysem ci (98). [9] Gua Jaiswal K Goel LR ocasic beavio of a sby edundan sysem wi ee modes icoelecon Reliab. () 9- (98). [0] Kuma Gua R Reliabiliy analysis of a single uni /G/ sysem model wi eling uni J. of Comb. Info. & ysem ciences Vol. o (007). [] uai K Goel V Reliabiliy of a sysem wi wo yes of eai faciliies icoelecon. Reliab (98). [] akagawa T Osaki ocasic beavio of a wo uni sby edundan sysem wi imefec swic ove IEEE Tans. Reliab. R (975). REFERECE [] Goel LR ama C ocasic analysis of wo uni sby sysem wi wo swicing devices icoelecon. Reliab. 6(5) (986). [] Goel LR Tyagi K Gua R Analysis of a ssndby sysem wi deenden eai ime slow swicing device icoelecon. Reliab. 4() 8-86(994). AUTOR Fis Auo Rakes Gua Deamen of aisics C. Caan ing Univesiy eeu econd Auo Ai Tyagi Deamen of aisics C. Caan ing Univesiy eeu ID: geadsas@yaoo.in

9 Inenaional Jounal of cienific Reseac ublicaions Volume 4 Issue 4 Ail 04 9 I 50-5 Beavio of TF wi esec o ₁ ₁ ₃ Fig. Beavio of ofi () wi esec o ₁ ₁ ₃ Fig.

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