Reliability Equivalence of Independent Non-identical Parallel and Series Systems.

Transcription

1 Lf Scc Jua 0;9(3) h://wwwfccc aby Euvac f Idd N-dca Paa ad S Sy Yuy Abdad 3 ; A I Shawy ad M I A-Ohay D f Mah acuy f Scc Uvy f Daa KSA D f Sac acuy f Scc Kg Abduazz Uvy PO Bx 8003 Jddah 589 Saud Aaba 3 Mah D acuy f Scc Axada Uvy Egy yabdad@yahc Abac: h aby uvac fac f aa ad y wh dd -dca c a bad h dff hd a ud v uch y: () vg h uay f va c by dcag h fau a () addg a h c h y ad () addg a cd duda c h y h uvva fuc ud a a fac au f h y aby ca dff y dg h h fau w b dvd aa ad y Wbu Dbu [Yuy Abdad; A I Shawy ad M I A-Ohay aby Euvac f Idd N-dca Paa ad S Sy Lf Sc J 0;9(3): ] (ISSN: ) h://wwwfccc 8 Ky wd: aby; S ad Paa Sy; duc hd; duca; d duca; Wbu Dbu Iduc aby vaua a a ad ga fau f h ag dg ad a f a gg y h fau f chaca dvc uch a h a ad ca a ay way h f dah f bgca ga A aby uvac fac a fac by whch a chaacc f c f a y dg ha b ud d ach uay f a chaacc f h dg ad dff dg gadd a a adad Euva f dff y dg wh c a aby chaacc uch a a fau (M) uvva fuc ca f a dd åd (989) ha ducd h cc f aby uvac åd ( a 993b) ha ad uch cc ad aa y f w dd ad dca c åd (993a ad 993b) ha ud h dff hd v h aby f a y I h hd aud ha h aby f a y ca b vd by: () vg h uay f va c by dcag h fau a; () addg a h c h y; () addg a cd duda c h y Saha (000) ha ducd ga vg hd f a y I uch hd h aby f a y ca b vd by: () vg h uay f c by ducg h fau a by a fac 0< <; () aug h duca f c; () aug cd duca f c; (v) aug cd duda adby c ccd wh c ( f ach) by ad wch åd (993a 993b) ad Saha (000) hav ud h uvva fuc a h fac au f h aby y åd (993a 993b) ha cacuad h aby uvac fac f ad aa y whch c f dd ad dca c wh ca fau a Saha (000) ha ad h cc f aby uvac a y c f dd bu dca c aud ha h f f ach c fw h xa dbu Saha a (004) hav udd h uvac f dff dg f a fu dd ad dca c -aa y hy hav d dduc h aby uvac fac f a aa y I bag h fac h aby fuc ad a fau (M) f h y a ud a fac au ca dff y dg f ga y ad h vd y Saha (005) bad h aby uvac fac f a aa y wh dd ad dca c ug h ga hd whch abhd (000) wv Saha (000) bad h M h ca f dd ad -dca c y ad ca ca udd aa y wh h fau a f h y' c a ca Saha ad Muafa (006) ducd dff vc f aby uvac fac f a y cg f dd ad -dca c hy aud ha h fau a f h y c a fw h xa dbu ad ud h aby fuc ad h M a fac dv h aby uvac f h y h w gazd h u d Saha (000) Saha a (008) dcud h uvac fac f du (uby) ccd wh ach du cg f c aa h://wwwfccc 577 d@ccub

2 Lf Scc Jua 0;9(3) h://wwwfccc hy aud ha h y c a dd ad dca xay dbud E- Dac ad Khfa (008) hav cdd h uvac fac f du ccd aa wh uby cg f c hy aud ha h y c a dd ad dca Wbu dbu Saha (009) ha udd h uvac f dff dg f a ga -aa y bad h y aby fuc ad h M aud ha h y c v a dd ad fw h xa dbu Gay ag ba h M h f f d ac ud g Ad a (99) ad Aad (006) ha cu µ : E(X : ) f aa y ad µ : E(X : ) f h y h h f d ac ag f dd dcay dbud ad vaab hav b udd by ay auh g Baaa ad Abdad ( ) Abdad (004 a ad 004 b) Abdad ad Abahu (006) ad Abdad (00 0) Sc h fau f a a ad vaab auad ad øyad (004) a ud va au f h c f a f dbu h a h da ad h d a ud au h fau f a hf a hgh uch a h vaac ca a b ud h vaac f h cy ud au f aayzg I h a w ha u h uvva fuc ad h h fau () cacua h aby uvac fac f ad aa y cg f dd ad -dca c ha w cu h h fau μ : E X : ) f h aa y ad μ : E X : f h y h c a aud b fw h Wbu dbu h ca f dd -dcay dbud h u d h gaz h gv h au h f h a gazd a fw h aby fuc ad f h aa ad y a ducd Sc I Sc 3 w dvd h aby fuc ad f h aa ad y wh h y c v fw h Wbu dbu w dff hd f vg h y dg a ducd Sc 4 I Sc 5 h aby uvac fac ad γ -fac f h ga ad vd dg a d cudg a a daw Sc 6 h aby fuc ad Au a y cd f dd bu dca c Each c ha a f ( ) h aby fuc f h y whch c f c ccd aa ad a dfd cvy a ad () () () wh () h aby fuc f ach c ( ) h dfd by ( ) d (3) L ad b h fau f h aa ad y hy a cvy dfd by Π[ ] d (4) ad Π d (5) 3 h Wbu dbu L b ad vaab havg Wbu dbu wh aby fuc a 0 a > 0 (6) ( ) () () () h Wbu dbu ga a fay f dbu a chag vau ac wh ad h Wbu dbu duc xa ad aygh dbu wh aa cvy h Wbu dbu ha a bad vay f cag fau a wh ha h fau a () a fuc f (7) h fau f h aa ad y w b dvd h fwg h h h ad f h Wbu dbu a gv cvy by + + ) ( ) D (8) ad + ) D (9) h://wwwfccc 578 d@ccub

3 Lf Scc Jua 0;9(3) h://wwwfccc D Wh ( ) ( ) < < a h ua f ( ) f whch < < < ad ) h gaa fuc Pf I ay ha Π( ) d + ( ( + ) < < ( ) ) d U ug Γ θ d θ > 0 (0) θ w g E(8) h f f E(9) fw f h df Π d d ad ug E(0) ay h ad f h xa dbu a gv cvy by ( ) () Γ + D ad Γ ( + ) D () Wh D < < ( + ) + + Pf Sg h w g h u 4 Dg f vd h ad Paa y I h c w ha duc w dff hd v h y dg h f h duc hd ad h cd cad dudacy hd whch cd f h ad cd duca hd I duc hd aud ha h fau a f f h y c a ducd by a fac 0<< I h h duca hd aud ha f h y c a ducad aa wh f h y c a ducad aa by a fc wch h cd duca hd hf ca a uvac bw h duc hd ad h duca hd bad aby au I h wd h dg f h y ha vd accdg h duc hd hud b uva h dg f h y accdg f h dudacy hd h ca f h dg duc h -cad aby uvac fac 4 h duc hd f ad Paa y I h hd aud ha h y ca b vd by vg f c ha h fau a f c a ducd f 0<< L ad d h aby fuc f h ad aa y vd by ducg h fau a f f c by h fac h h f h ad aa y ud h duc hd dd by ( ) ( ) ad a gv cvy by + ) (3) + + ad ( ) + Γ ( + )( ( ) Z ( ) + + ( ) + Z ( ) Z Y Z Wh + ( ) + + < < + + < < ( ) Y ( )) (4 ) < < < + < < + + h://wwwfccc 579 d@ccub

4 Lf Scc Jua 0;9(3) h://wwwfccc h ubc f h a ua a w hav w ua O u f ad h h u f Pf h aby fuc f h y gv by + ( ) + + (5) h aby fuc f h aa y gv by ( ) ( ) ( ) ( ) Z + ( ) Z ( ) Z Z( + + wh Z Z < < + < < ) (6) h df f h whch a gv by E (4) ad (5) ad ug h ga E (0) w g h u 4 h h duca hd I h ca aud ha h y ca b vd by vg f c wh La h aby fuc f h aa ad y dd by ad ud h h duca hd a gv by ( ) I wh + I (7) (8) v v I < < ad + Pf h aby fuc f h y gv by () ( ) < < 3 ( + + ( + ) 3 ) ( ) + W ca w h a ua h f f E(7) h f f E(8) fw f h df I ad hc h f h 3 h fau f h aa ad y ad a gv by + ) ( ) + + ) D (0) wh D < < D (9) Pf By acg by h w g h u 43 h cd duca hd Ud h cd duca hd h y w b vd by vg f c L ad b h aby fuc f h ad aa y h + ad () ( ) ( ( )) () + wh d h aby f a y' c af wa vd accdg cd duca hd S w ca w () + (3) B ad Aa (983) h 4 h f h ad aa y w a ad cd duca hd a gv cvy by h://wwwfccc 580 d@ccub

5 Lf Scc Jua 0;9(3) h://wwwfccc ad + a ( Γ ( a a ) a ( ) + + ( + ( ) ( ) a ) ) (5) Wh a ( ) < < a 0 a 0 ( ) + < < + a ) ( + ) ( ) a ) ( ) + + a ) ( + ) ( + ) a + Pf E() w ca w + + a + ( ) a 0 + () () ( ) wh a a () a 0 a 0 hf () d (4) U ug h ga f E (0) w g E (4) h aby fuc f h aa y ca b w h f ( ) + ( ) ( ) + + Sc w ca w ( ) (6 ) + ( + ) ( ) + + () + + < < + ( ( + ) ) ( ) a a < < a 0 a 0 hf E(6) ca b bad c d U ug h ga f E (0) w g E (5) ad hc h f 5 h aby uvac fac ad γ - ac I h c h uvva ad h aby uvac fac f h ad aa y w b ducd h γ -ac f h ga ad vd y a a d 5 -d ad aa aby uvac fac h uvva aby uvac fac K w G ( γ ) wh G (f ad aa y); () f h(cd); K c ad 0 < γ < dfd a ha fac by whch h fau a f f h y' c hud b ducd d ach h aby f h y whch vd by vg f h ga y c accdg h(cd) duca hd Saha (009) h fwg f -a ua gv h u f f h ad aa y: ( ) G G γ hf f E (5) - (8) w ca w f h h aby uvac fac ( ad aa y) ha ( ) ( ) ( ) ( ) ( + ) + γ (7 ) h://wwwfccc 58 d@ccub

6 ( ) ( ) ( ) ( ) Lf Scc Jua 0;9(3) ( ) ( ) + ( ) I γ (8 ) wh I dfd La Sa ua f h cd aby uvac fac ( ad aa y) ca b w ug h E (5) () ad (6) () ha γ (9) ad γ (30) h u f h y f E (7) - (30) hav cd f S a uca chu ca b ud g h u wh c O h h had h aby uvac fac ca b ud ach h uay ( a) f h fau h duc hd ad h fau f h dg bad f h h(cd) duca hd f h ad aa y Saha (009) ha (3) (3) hf ug E (3) (0) (4) ad (4) (9) (5) w g -a y f ua whch ca b v ucay 5 γ -ac L L ( γ ) b h γ -ac f h ga y ad L G ( γ ) b h γ -ac f h vg y bad by vg f h y' c accdg G() f ad aa y ad () duca hd h γ -ac f h y havg aby fuc G () ad L G ( γ ) ca b bad by vg h ua L ( ) ( G γ ) γ (33) ad f h h ad cd duca hd w hav v h fwg ua ( ) L ( γ ) ( ) G G ( ) γ (34 ) wh c L h abv ua hav cd f u L S h uca chu a h b u h ca 6 cudg a h ca f dd -dca a ay ua w ua w h f w a h://wwwfccc guaa ha a c f h y a w wh h a ffccy v hy a ducd f h a facy h cd w ay au ha h c a ca f dff fac wh dff ffccy h u f h a xd h avaab h au g Saha ( ) Ma ad Saha (008) ad Saha a ( ) h y aby fuc ad h y fau () h ca f dd -dca c a ud udy h uvac f dff dg f h ad aa y Mv h d a h fc aud ha h fau a f h y c a fw h xa dbu (ca fau a) ad cud h a fau Wh h a h ad au h fau a f h y c a fuc f ad fw h Wbu dbu I bvu f h fgg h ha h fwg a: () f E (9) ad (3) ( ) h ga ha ua af ducg h fau a f f h c by h fac Sc h da f E (9) ga ha h da f E (3); () ay ( ) h ga ha whch a gv by E (8) ad (4) f h a a; () a ay dduc ha > ad c > ha h fau h cd duca hd ga ha h fau h h duca hd I aby uvac fac cudg h uvva ad h f h u abhd h au hwd ha < < ad < < < < ( ) ( ) < < aa f h c a dd dca -dca g Saha ( ) Saha ad Muafa (006) Saha a ( ) Xa ad Zhag (007) E-Dac ad Khafa (008) ad Ma ad Saha (008) Mv h u h d fc hwd ha h au f cag f h γ -fac cd duca hd ga ha h γ -fac bad by ug h h duca hd h://wwwfccc 58 d@ccub

7 Lf Scc Jua 0;9(3) h://wwwfccc I h fuu w ay udy h aby uvac fac f h fay f dbu h ca f dd -dca c ag wh hgh uay fwa ga f h acad cacua Acwdg h a ca ud ad ach u w ud by Dah f Scfc ach f Uvy f Daa (Pc N 0079) dg auh Yuy Abdad D f Mah acuy f Scc Uvy f Daa KSA Mah D acuy f Scc Axada Uvy Egy yabdad@yahc fc Abdad Y 004a ug h f d ac f -dcay dbud Eag vaab Saca Pa 45: Abdad Y 004b ug h f d ac f -dcay dbud Ga vaab wh aca I J Mah Ga hy ad Agba 4: -8 3 Abdad Y 00 ug h f d ac f dd dcay dbud Ba ad vaab Saca Pa 5: Abdad Y 0 A Laac af hd f d ac f dca ad vaab ad aca Pha-y dbu Sac ad Pbaby L 8: Abdad Y ad A W Abahu 006 Axag h f d ac f -dcay dbud Gaa vaab wh ga ha aa I J MahGa hy ad Agba 6(): Ad B Baaha N Nagaa N 99 A u Od Sac Jh Wy Nw Y 7 Aad M 006 O h a a f f h c f a aa y Jua f Saca Pag ad Ifc 36: Baaa M ad Abdad Y 000 ug h f d ac f dcay dbud Wbu vaab Jua f uaa ad Ad Mahac 7; Baaa M ad Abdad Y 004 ug h f d ac f dcay ad vaab Saca Mhd & Aca 3 :5-6 0 B ad Aa 983 aby vaua f gg y: cc ad chu Pu P Nw Y ad Ld E-Dac MA ad Khafa MM 008 aby uvac fac f a -aa y Wbu dbu Iaa Jua f aby ad Aca 9(): Ma M Saha MA 008 aby uvac fac f a aa y wh dca c Iaa Mahaca u 3(34): åd L 989 aby uvac: ud aca uay c ad aby Mahaca Sac ha Uvy f chgy 4 åd L 990 aby y f 3-a c Sud aca uay c ad aby Mahaca Sac ha Uvy f chgy 5 åd L 99 Pfac au f aby y wh a cd adby wh a ad wch Sud aca uay c ad aby Mahaca Sac ha Uvy f chgy 6 åd L 993a aby uvac Mccc ad aby 33: åd L993b aby uvva uvac Mccc ad aby 33: auad M øyad A 004 Sy aby hy: Md Saca Mhd ad Aca Scd Ed Jh Wy Nw Y 9 Saha AM 000 aby uvac f dd ad -dca c y aby Egg ad Sy Safy 67: Saha AM 00 aby uvac wh a bac /aa y Ad Mahac ad ua 3: 5-3 Saha MA A-uzaza AS Awa IA E- Ghay A 004 aby uvac f a aa y Ad Mahac ad ua 54: Saha A M 005 aby uvac fac f aa y aby Egg ad Sy Safy 87: Saha A M ad Muafa A 006 aby uvac f a y c f $$ dd ad -dca c Iaa Jua f aby ad Aca 7: Saha A M ad L A-hdha A ad Muafa A 008 Euvac fac f a aa- y Ad Scc 0: Saha A M 009 aby uvac fac f a ga -aa y aby Egg ad Sy Safy 94: Xa Y Zhag G 007 aby uvac fac Gaa dbu Ad Mahac ad ua 87: /7/0 h://wwwfccc 583 d@ccub