The Mean Residual Lifetime of (n k + 1)-out-of-n Systems in Discrete Setting

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1 Appled Mahemacs Publshed Ole February 4 (hp//wwwscrporg/oural/am hp//dxdoorg/436/am45346 The Mea Resdual Lfeme of ( + -ou-of- Sysems Dscree Seg Maryam Torab Sahboom Deparme of Sascs Scece ad Research brach Islamc Azad Uversy Tehra Ira Emal morab@mauacr Receved November 3; revsed December 3; acceped December 9 3 Copyrgh 4 Maryam Torab Sahboom Ths s a ope access arcle dsrbued uder he Creave Commos Arbuo Lcese whch perms uresrced use dsrbuo ad reproduco ay medum provded he orgal wor s properly ced I accordace of he Creave Commos Arbuo Lcese all Copyrghs 4 are reserved for SCIRP ad he ower of he ellecual propery Maryam Torab Sahboom All Copyrgh 4 are guarded by law ad by SCIRP as a guarda ABSTRACT I real lfe here are suaos whch he lfeme of he compoes of a echcal sysem (ad hece he lfeme of he sysem s dscree I hs paper we sudy he resdual lfe a ( + -ou-of- sysem uder he assumpos ha he compoes of he sysem are depede decally dsrbued wh commo dscree dsrbuo fuco F We defe he mea resdual lfeme (MRL of he sysem ad uder dffere scearos vesgae several agg ad sochasc properes of MRL KEYWORDS ( + -ou-of- Sysem; Dscree Lfeme; Falure Rae; Relably Iroduco I rece years researchers relably heory have show esfed eres he sudy of sochasc ad relably properes of echcal sysems The ( + -ou-of- sysem srucure s a very popular ype of redudacy echcal sysems A ( + -ou-of- sysem s a sysem cossg of compoes (usually he same ad fucos f ad oly f a leas + ou of compoes are operag ( Hece such sysem fals f or more of s compoes fal Le T T T deoe he compoe lfemes of he sysem ad assume ha T T T represe he ordered lfemes of he compoes The s easy o argue ha he lfeme of he sysem s T where T deoes he he order sascs correspodg o T 's Uder he assumpo ha T 's are couous radom varables several auhors have suded he resdual lfeme ad he mea resdual lfeme (MRL of he sysem uder dffere codos Assumg ha a me a leas r+ compoes of he sysem are worg he resdual lfeme of he sysem ca be defed as follows r T T Tr > r ( Amog he researchers who vesgaed he relably ad agg properes of he codoal radom r varable T uder varous codos ad for dffere values of ad r we ca refer o Baramov e al [] Asad ad Baramov [3] Asad ad Golforusha [4] L ad Zhao [5] ad Zhag ad Yag [6] The exeso o cohere sysems has also bee cosdered by several auhors; see amog ohers L ad Zhag [7] Navarro e al [8] Zhag [9] Zhag ad L [] Asad ad Kel Nama [] ad refereces here Recely M [3] cosdered he suao whch he compoes of he sysem had dscree lfemes ad vesgaed some of agg properes of he sysem The am of he prese paper s o sudy he MRL of ( + -ou-of- sysem uder dscree seg For hs purpose we assume ha T T T are o + -ou-of- egave eger valued radom varables deog he lfemes of he compoes of a

2 M TORABI SIAHBOOMI 467 sysem Furhermore we assume ha T are depede ad have a commo probably mass fuco ad survval fuco p PT S PT p The hazard rae of he compoes deoed by h ad r( s defed as follows h ( ( PT p PT S Oe ca easly show ha he survval ad probably mass fucos ca be recovered from he hazard rae respecvely as follows ( ( + ( S h H H ( p h h The MRL fuco of he compoes deoed by m plays a mpora role relably egeerg ad survval aalyss Assumg each compoe of he sysem has survved up o mes he MRL fuco m of each compoe s defed as m E TT I s o dffcul o show ha he survval fuco S( ca be represeed erms of L as below S S + ( S m + m + The rese of he paper s orgazed as follows We frs assume ha a me all compoes of he sysem are worg ad obag he fucoal form of he mea of T Ths s fac he MRL of he sysem deoed by H ( uder he codo ha all compoes of he sysem are operag a me I s show ha whe he compoes of he sysem have geomerc dsrbuo H ( s free of me The we prove ha f he compoes of he sysem have creased falure rae H ( s a decreasg fuco of I s also show ha whe he compoes of wo depedes are ordered erms of hazard rae orderg uder he codo ha all compoes of he wo sysems are alve her correspodg MRLs are also ordered The resuls are he exeded o he case where a leas ( r+ compoes of he sysem are operag I hs case we oba he fucoal form of he MRL of he sysem deoed by r H ( I s show ha H r ( ca be represeed as he mxure of H where he mxg fuco s ( + P PT < < T Tr r s decreasg me However s show usg a couer example ha whe he compoes of he sysem have decreased hazard rae s o ecessarly rue geeral ha H r ( s creasg me The fuco P ( meoed above has s ow eresg erpreao I shows he probably ha here are exacly faled compoes he sysem r gve ha a leas ( r+ compoes P are also vesgaed We prove ha he case where he compoes of he sysem have creased hazard rae he H r are worg a me Several properes of

3 468 M TORABI SIAHBOOMI The Mea Resdual Lfe Fuco of Sysem a he Compoe Level I hs seco we cosder a ( r depede dscree lfemes T T T wh commo probably mass fuco p PT ( vval fuco S( where T T + -ou-of- sysem ad assume ha he compoes of he sysem have ad sur Le also be he order sascs correspodg o T 's I wha follows frs we assume ha a me > all he compoes of he sysem are worg e T The resdual lfeme of he sysem uder he codo ha all compoes of he sysem are worg a me s T T (see Asad ad Baramoglu [3] Usg he sadard echques oe ca easly show ha S + x+ S + x+ P( T > + xt ( S S Hece he MRL fuco of he sysem deoed by H where ca be obaed as follows x ( ( > + H E T T P T x T S + x+ S + x+ x S S ( M ( ( + x+ S + x+ S + x+ ( x S ( x S M + ( + ( + x+ + + S x deoes he MRL fuco of a seres sysem cossg of + compoes Example Le he compoes of he sysem have geomerc dsrbuo wh probably mass fuco ad survval fuco We have M H ( θ( θ p PT ( ( ( S PT θ θ θ + x + + ( θ ( x+ ( + ( θ θ x ( θ x ( θ + + ( θ ( θ + ( ( + Noe ha he MRL of a sysem havg depede geomerc compoes does o deped o (3 (4

4 M TORABI SIAHBOOMI 469 The dsrbuo fuco of he order sascs T r ca be represeed erms of complee bea fuco as follows (see Davd ad Nagaraga [4] where F( x r r PT ( r x F ( x ( F( x u ( u du r Br ( r+ ( a+ b! B( ab ab!! Hece he MRL fuco of he sysem ca be represeed as H u u u S( + x+ ( d x B ( + (5 S Ths represeao s useful o prove he followg wo heorems Theorem If he compoes of he ( + -ou-of- sysem have a creasg (decreasg hazard H s decreasg (creasg Proof p( If h deoes he hazard rae of he compoes he h s creasg (decreasg f ad oly S rae he f for o-egave eger valued x ( + x S s decreasg (creasg Now he resul follows easly by represeao (5 The followg example gves a applcao of hs heorem Example 3 Le he compoes of he sysem have dscree Webull dsrbuo wh survval fuco The he MRL ( β S α H of he sysem s decreasg for α > ad creasg for α < Theorem 4 Le ad be wo ( + -ou-of- sysems wh depede compoes Le he compoes of ad have he probably mass fuco p( ad q( survval fucos S ( h respecvely If for h h he ad S ( ; ad hazard raes h ( ad H ( H ( where H ( ad Proof Noe ha for h ( h ( H deoe he mea resdual lfe of S ad S respecvely f ad oly f S + x+ S + x+ x S S The requred resul s mmedae ow from (5 Khorashadzadeh e al [5] suded dscree varace resdual lfe fuco for oe compoe Usg he fac ha oe ca easly prove he followg lemma Lemma 5 ( ( ( + PT T PT T PT T ( E T T + ( PT ( P T T (6 + ( ( + ( PT ( P T T E T T (7 +

5 47 M TORABI SIAHBOOMI Usg hs he varace of he resdual lfe fuco of ( all compoes are worg ca be derved erms of H ( Theorem 6 If E( T < he varace resdual lfe fuco ( H ( are relaed as Proof We have σ ( + + -ou-of- sysem uder he codo ha σ ad mea resdual lfe fuco P( T ( T + H H PT Var σ T T E T T E T T Usg Lemma 5 we ge he requred resul ( ( E T T E T T E T T H Now we sudy he MRL of ( + -ou-of- sysem uder he codo ha a leas ( r poes of he sysem are worg Tha s we cocerae o ( Frs oe ha where ad ( r r u r u ( > + r PT ( P T xt P T > xt r + com- r r H E T r ( u ( PT> + x PT> + x P ( T < P ( T u PT PT r P ( T P ( T < S + x+ S + x+ S u r S u u r S + x+ S + x+ P ( u u S S ( P P Z Z r r ( S ( ( S ( Z s a bomal radom varable wh parameers ( r r x ( > H P T xt r ( u u u r S + x+ S + x+ P ( u x u S S P H Equao (8 shows ha r H ( s a covex combao of H ( r u Noe ha H ( u (8 s

6 M TORABI SIAHBOOMI 47 gve by ( Example 7 Le T T deoe he lfemes of depede compoes whch are coeced a ( + -ou-of- sysem Le T be dsrbued as dscree Webull ( αβ wh ad The ad α ( + p PT β β < β < α > H ( ( β S PT P r α α α (( β α ( ( β u α α α + x+ + x+ u α ( β β u x Hece he MRL H r ( s gve by (8 Fgures ad show he graphs of r H example 7 3 whe α β 7 for dffere values of r ad r Remar 8 Le us cosder aga he codo radom varable T T Tr for whch he survval fuco s gve by ( The represeao ( shows ha T s fac he he order sascs r Fgure The MRL r 345 from he op respecvely 3 H of he sysem for he dscree webull dsrbuo wh 7 5 ad

7 47 M TORABI SIAHBOOMI r Fgure The MRL 3456 from he op respecvely 3 H of he sysem for he dscree webull dsrbuo wh 7 r 3 ad form of a dsrbuo wh survval fuco oe ca wre ( + x+ S Hece usg he resul of Davd ad Nagarae [8] ( x PT ( + x+ + + S > + S Hece ad H H + + ( r r H P H + + ( (9 Ths dcaes he MRL r H ( ca be expressed erms of smpler MRL H MRL of seres sysems The followg heorem gves bouds for r Theorem 9 I s always rue ha H + r H r+ H H Proof The proof s smlar o he proof of Theorem 3 of [4] ad hece s omed The ex heorem proves ha whe he pare dsrbuo has creased hazard rae H r erms of me whch s fac he creases

8 Theorem If h s creasg he r M TORABI SIAHBOOMI 473 H s decreasg Proof I order o prove he resul we eed o show ha for r ad fxed We have from (8 afer some algebra r H H + r r r r r ( + ( + ( + H H P H P H r ( ( ( ( r P H H + + H + P P + Bu he frs erm he above equaly s posve by Theorem Hece we us eed o prove ha he secod erm he above equaly s posve Assume ha φ ( ad oe ha φ ( s a creasg S fuco of The φ ( φ ( + φ ( φ ( r r φ ( φ ( + φ φ + H + H + ( r H ( + r r r r + Afer some algebrac mapulaos oe ca show ha he umeraor of he expresso s equal o r r ( φ ( φ ( + φ ( φ ( + ( H( + H ( + ( + I ca be easly show ha for > H( + H ( + > (see [3] O he oher had as φ ( s a creasg fuco of we have φ ( φ ( + φ ( φ ( + Ths mples ha he expresso ( s o-egave ad hece he proof s complee Remar As was already meoed for a sysem wh decreasg falure rae compoes H creasg me Ths resul however s o geerally rue for MRL H r graphs of h ad H r ( Example 7 As he graphs show ha however H r ( s a creasg fuco of for a perod of me ad he sars o decrease Remar I he followg we show ha s Fgures 3 ad 4 show he h s a decreasg fuco of me P has s ow eresg erpreao I fac uder he codo ha he sysem s worg a me P ( shows he probably ha here s exacly compoe falure he sysem The meoed codoal probably ca be wre as ( < < + r ( < r ( + < r PT ( < Tr PT ( + < Tr PT ( PT ( PT T T PT T PT T r r r r ( S ( S S + r r ( S ( S S r ( S ( φ ( r ( S ( φ P r ( (

9 474 M TORABI SIAHBOOMI r Fgure 3 The MRL r 5 H of he sysem for he dscree webull dsrbuo wh 7 5 ad Fgure 4 The falure rae of he sysem for he dscree webull dsrbuo

10 M TORABI SIAHBOOMI 475 where φ ( for such ha > less ha Also he followg we sudy some properes of P ( Theorem 3 For Also for r Proof We have S shows he odds of he eve ha a compoe has a lfeme P s decreasg fuco of ad for r s creasg fuco of lmp lmp P r r r φ whch s obvously a decreasg fuco of ( φ ( s a creasg fuco sce lm φ ( φ ( From ( we easly coclude ha P ( ad P ( lm P ( lm r r ( r φ r+ ( lm ( ad I hs case s easly see ha Pr ( s a creasg fuco of lm Pr ( Pr ( Theorem 4 The survval fuco S( ca be uquely deermed by P( ad P ( lm r as follows S ( P ( ( P( + ( + P ( Proof The resul easly follows from he fac ha for r whch gves ( P S + P + S Cosder he vecor P ( ( P( Pr ( Obvously ( he followg heorem Theorem 5 For all ( ( P P s Proof I order o prove he resul we eed o show ha for r or equvalely r r + P P r ad + ( P s a probably vecor we ca he prove

11 476 M TORABI SIAHBOOMI or Ths s equvale o show ha Bu as ( r r φ ( φ ( r r φ ( φ ( φ ( φ ( r r φ ( φ ( r l l ( φ ( φ ( φ ( φ ( (3 l l φ s creasg he brace he summaos for < l s always egave Hece he equaly (3 s vald Ths complees he proof of he heorem Theorem 6 Cosder wo ( + -ou-of- sysems Assume ha he compoes of he sysems have S ad S ( respecvely ad odds fucos φ ( ad φ respecvely If for all S( he P( s P ( r Proof Asad & Berred [6] proved ha ηr ( for fxed ad s a creasg fuco r of S S φ φ he depede lfemes wh survval fuco The assumpo ha mples ηr ( φ ( ηr ( φ( whch s equvale o say ha for all r ad all r r φ ( φ r r φ ( ( φ ( REFERENCES ( P ( s P ( [] I Baramov M Ahsaullah ad I Ahudov A Resdual Lfe Fuco of a Sysem Havg Parallel ad Seres Srucures Joural of Sascal Theory ad Applcaos Vol No pp 9-3 [] M Asad ad I Baramoglu A Noe o he Mea Resdual Lfe Fuco of he Parallel Sysems Commucaos Sascs Theoy ad Mehods Vol 34 No 5 pp - [3] M Asad ad I Baramoglu O he Mea Resdual Lfe Fuco of he -ou-of- Sysems a Sysem Level IEEE Trasacos o Relably Vol 55 No 6 pp hp//dxdoorg/9/tr [4] M Asad ad S Golforusha O he Mea Resdual Lfe Fuco of Cohere Sysems IEEE Trasacos o Relably Vol 57 No 4 8 pp hp//dxdoorg/9/tr876 [5] X L ad P Zhao Sochasc Comparso o Geeral Iacvy Tme ad Geeral Resdual Lfe of -ou-of- Sysems Commucaos Sascs Smulao ad Compuao Vol 37 No 5 8 pp 5-9 hp//dxdoorg/8/ [6] Z Zhag ad Y Yag Ordered Properes of he Resdual Lfeme ad Iacvy Tme of ( + -ou-of- Sysems uder Double Moorg Sascs & Probably Leers Vol 8 No 7-8 pp 7-77

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