RENEWAL AND AVAILABILITY FUNCTIONS

Transcription

1 Advaes ad Appliatios i Statistis 14 Pushpa Publishig House, Allahabad, Idia Available olie at Volume 43, Number 1, 14, Pages RENEWAL AND AVAILABILITY FUNCTIONS Dilu B. Helvai ad Saeed Maghsoodloo Depamet of Idustrial ad Systems Egieerig Aubur Uiversity AL 36849, U. S. A. Abstrat The reewal ad availability futios for some ommo failure ad repair uderlyig distributios are explored. Exat results for the reewal futios ad availability of ormal time to failure ad time to repair, ad gamma time to failure ad expoetial time to repair are provided. Beause obtaiig the -fold ovolutios of time betwee yles for geeral lasses of failure ad repair distributios is itratable, we obtaied the availability futios for some ommolyeoutered failure distributios but at a ostat repair-rate. A MATLAB program was devised to perform all alulatios. 1. Itrodutio This aile geeralizes the wor i [1] by the same authors, where we ow assume the MTTR (mea time to repair) is ot egligible ad that TTR has a pdf (probability desity futio) deoted as r ( t). Let the variates X 1, X, X3,... represet the ith time to failure ( TTF i ) be idepedetly ad idetially distributed (iid) with the same uderlyig failure desity f ( x) havig mea MTTF μx ad variae σ x ; fuher, Y 1, Y, Y3,... represet Reeived: September 6, 14; Aepted: November 4, 14 1 Mathematis Subjet Classifiatio: 6-XX. Keywords ad phrases: types 1 ad reewal futios, Laplae trasforms, ovolutios, umulative hazard, reliability, reewal ad availability futios.

2 66 Dilu B. Helvai ad Saeed Maghsoodloo the ith time to restore ( TTR i ), i 1,, 3, 4,... with the same pdf r ( y) havig mea MTTR μ y ad variae σ y. The T i Xi Yi represets the time betwee yles (TBCs) whih are also iid whose desity is give by the ovolutio g() t f () t r( t), ad whose Laplae trasform (LT) is give by L { g ( t) } g( s) f ( s) r( s). Clearly, the mea ad variae of the yle-times T i s are μ x μ y ad σ x σ y. As desribed by [] there will be two types of reewals: (1) A trasitio from a Y-state (i.e., whe system is uder repair) to a X-state (at whih the system is operatig reliably). () A trasitio from a X-state (or operatig-reliably state) to a Y-state (where system will go uder repair or restoratio). Let M 1 () t represet the expeted umber of yles (or umber of reewals of type 1), ad M ( t) represet the mea umber of failures (or reewals of type ). The, as prove by [3] ad later by [], the LTs of the two reewal futios (RNFs), respetively, are give by M1 M g( s) f ( s) r( s) ( s) s[ 1 g( s) ] s[ 1 f ( s) r( s)], (1a) f ( s) f ( s) () s s[ 1 g() s ] s[ 1 f () s r()] s. (1b) by The orrespodig LTs of RNIFs (reewal-itesity futios) are give f ( s) r( s) f ( s) ρ1 ( s) ad ρ () s 1 f ( s) r( s) 1 f () s r() s. () It is essetial to ote that authors i Stohasti Proesses refer to iversetrasforms of equatios () as the reewal desities. As a example, suppose TTFi ~ Exp(, λ), i.e., expoetial with zero miimum-life ad ostat hazard rate h ( t) λ, ad TTR ~ Exp(, r); i

3 Reewal ad Availability Futios 67 the as has bee doumeted both i Stohasti Proesses ad Reliability Egieerig literature, f ( s) λe e dt λ ( λ s) re e st dt r ( r s). λt st ad r () s Suh a proess is refereed as a alteratig Poisso Proess [], whih we aroym as APP. O substitutig the above LTs ito equatio (1a), we obtai the well-ow M ( s) 1 λr λr λr λr s[ ( λ s)( r s) λr] sξ s ξ ξ ( s ξ), where ξ λ r, ad M 1 1 1() t L { M1( s)} L λr sξ λr s ξ λr ξ ( s ξ) λr ξ λr t ξ λr e ξ ξt, whih gives the expeted umber of trasitios from a repair-state to a operatioal-state (or the mea umber of yles). Similarly, M ( s) f ( s) λ λr λ s[ 1 f ( s) r( s)] sξ s ξ ξ ( s ξ), λ λr λ ξt whih upo iversio yields M () t t e, represetig the ξ ξ ξ expeted umber of failures durig the iterval (, t ). For example, if λ.5 hour ad the ostat repair-rate r. 5 per hour, the ξ λ r.55, M 1 ( t 5 hours) , while M ( 5) Note that the limit of both above RNFs M 1 ( t) ad M ( t) as repair-rate r ( i.e., MTTR ) is exatly equal to the expoetial RNF M () t λt, as expeted. Fuher, a ompariso of M ( t) with M 1 ( t) reveals that

4 68 Dilu B. Helvai ad Saeed Maghsoodloo M () t > M1( t) for all t >, whih is ituitively meaigful beause the expeted umber of failures must exeed the expeted umber of yles for all t >, as we are assumig that time zero is whe a system stas i the last reewed state X.. (a) Types 1 ad Reewal Futios Usig Laplae Trasforms Elsayed [4] obtais the (poit) availability futio (AVF) for a system omprisig of similar ompoets A ad B, usig their RNFs M A () t ad M B (), t where his P A () t represets the Pr that ompoet A is i use at time t. Usig a similar argumet, we first obtai M ( t) ad M 1 ( t) by iveig equatios (1), ad we later use these futios i order to obtai the availability (AVL) futio A ( t). Equatio (1a) shows that ( s)[ g( s)] g( s) M ( s) s 1 g( s) M ( s) g( s) s M1 1 1 t M1 () t G() t M1( t x) g( x) dx, where G () t is the df of TBCs. Equatio (1b) ow shows that ( s)[ g( s)] f ( s) M s ( s) f ( s) M s M 1 t M () t F() t M ( t x) g( x) dx; ( s) g( s) thus, i geeral the well-ow expeted umber of yles is give by t M1 () t G() t M1( t x) g( x) dx. (3a) While the orrespodig well-ow expeted umber of failures durig (, t) is give by t M () t F() t M ( t x) g( x) dx. (3b)

5 Reewal ad Availability Futios 69 (b) The Normal TTF ad TTR Suppose time betwee failures TBFs ~ N( μ MTBF, σ ) ad TTR is y also N ( μ y, σ ); the TBCs ~ N( μ x μ y, σ σ ). We are maig the tait assumptio that both oeffiiet of variatios are suffiietly small (say CV < 15%) suh that the ormal distributio a qualify as a failure ad repair desities; fuher, the system iitially stas i a X-state. As a result, equatio (3a) shows that () t μ M 1 t Φ 1 σ, where μ μx μ y, σ σ x σ y, ad M 1 () t gives the expeted umber of yles. However, beause a system is uder repair a small (but ot egligible) fratio of the iterval (, t ), the () t μ M t Φ x. I order to obtai a 1 σx approximatio for M () t ad the resultig availability futio (AVF), A (), t defied later, we may argue that the expeted duratio of time a system is uder repair durig the iterval (, t) is give by M 1 ( t) MTTR; lettig t t M1() t MTTR, the equatio (3b) shows that the expeted umber of failures, assumig that the system stas i the X-state, is approximately give by () t μ μ Φ x t Φ M t x. σx σ x x 3. Poit Availability Beause we are assumig that a system a be either i a operatioalstate ( X ), or uder repair, the it has bee well-ow that the reliability futio, R (), t must be replaed by the istataeous (or poit) AVF at time t, deoted A (), t whih represets the probability (Pr) that a repairable system (or uit) is futioig reliably at time t. Thus, if restoratio-time is egligible, the AVF is simply A ( t) R( t). However, if a system (or a ompoet of a system) is repairable, the there are two mutually exlusive x y x

6 7 Dilu B. Helvai ad Saeed Maghsoodloo possibilities [5]: (1) The system is reliable at t, i whih ase A ( t) R( ). 1 t () The system fails at time x, < x < t, gets reewed (or restored to almost as-good-as-ew) i the iterval ( x, x Δx) with uoditioal Pr elemet ρ ( x) dx, ρ () t beig the RNIF of TBCs, ad the is reliable from time x to time t. This seod Pr is give by A () t t ρ( x) dxr( t x) ; beause the above two ases are mutually exlusive, the A() t A1 () t A () t R() t R( t x) ρ( x) dx. (4) Taig Laplae trasform of the above equatio (4) (ad observig that the itegral is the ovolutio of R ( t) with ρ ( t) [6]) yields the very wellow LT of AVF A( s) R( s) R( s) ρ( s) R( s)[ 1 ρ( s)] t f ( s) r( s) R( s) R() s 1 1 f () s r() s 1 f () s r() s, (5) where r () s is the LT of r (), t the desity (or pdf ) of repair-time. For the ase whe the TTF (of a ompoet or system) has a ostat failure-rate λ ad time to repair (TTR) is also expoetial at the rate r (i.e., a APP), R ( s) λt st e e dt λ ( λ s), ad hee the Laplae trasform of AVL from equatio (5) is give by 1 ( λ s) r s A() s 1 [ λ ( λ s) ][ r( r s) ] s[ s ( λ r) ] r λ ξ 1 r λ ξt A() t L { A()} s e, ξs s ξ ξ ξ where ξ λ r, whih is provided by may authors i Reliability

7 Reewal ad Availability Futios 71 Egieerig suh as [4, 7, 8] ad may other otables. For example, give that the failure-rate λ. 5 ad r repair-rate. 5 per hour, the ξ λ r. 55 ad the Pr that a etwor is available (i.e., ot.5 uder restoratio) at t 5 hours is give by A ( 5) ( 5 e ).99991, while R (at 5 hours with miimal-.55 repair) e < A( 5).991. Thus, restoratio has improved AVL by 7.31%. As stated by umerous authors i Stohasti Proesses ad Reliability Egieerig, as t, R ( t) for all failure desities, while for a APP A () t r ξ r ( λ r) A MTTF ( MTTF MTTR) if.991, where restoratio iludes admiistrative, logisti ad ative repair-times. Note that i the APP ase (i.e., both rate-parameters are ostats), we a also obtai the AVF, A ( t), diretly from equatio (4) as follows: where t t 1 1, λt λ t x ρ A() t R() t R( t x) ρ ( x) dx e e ( ) ( x) dx d λr r r ξx r r ξx ( x) dm ( x) dx λ λ λ λ ρ1 1 x e e dx ξ ξ ξ ξ ξ is the RNIF of the umber of yles. Upo substitutio of this RNIF ito the expressio for A (), t we obtai as before. λt λ () ( t x) λr ξx A t e e ( 1 e ) dx ξ t r ξ λ ξt e, ξ

8 7 Dilu B. Helvai ad Saeed Maghsoodloo As poited out by [4], we also observe that st st R( s) e R() t dt e [( 1 F() t )] dt 1 st e s 1 F() t dt F () s. s 1 f () s Hildebrad [6] proves that F ( s) f ( s) s so that R() s ; o s substitutio ito equatio (5), we obtai A( s) 1 f ( s) 1 s[ 1 f ( s) r( s)] s[ 1 f ( s) r( s)] f ( s) s[ 1 f ( s) r( s)] 1 f () s r( s) f ( s) s s[ 1 f () s r()] s s[ 1 f () s r()] s. Iveig these last 3 LTs from equatios (1), we obtai ( t) 1 M ( t) M ( t) (6) A 1 for all uderlyig failure desities f ( t) ad TTR-desity r ( t). Equatio (6) is idetial to that of Elsayed [4] atop his page 467, whih he derived usig a system of alteratig ompoets. Fuher, equatios (3) imply that M () t M 1 () t yields the uoditioal Pr that a system is uder repair at time t, ad hee equatio (6) is ituitively appealig beause ( t) 1 [ M () t A M 1 ()]. t For the above expoetial example with λ. 5 ad repairrate r.5, equatio (6) shows that A ( t 5) , as before. Example 1. Our experiee shows that i the ase of ormal TTF ad TTR the approximate value of t t M1( t) MTTR is a bit too small. If.5 MTTR MTTF.5, the t t [ M1( t).475] MTTR; however, if.5 < MTTR MTTF.1, the t t [ M1( t). 45] MTTR is a better approximatio. These values were obtaied suh that the

9 Reewal ad Availability Futios 73 limitig AVL, give by A if MTTF ( MTTF MTTR), is approximately equal to A() t 1 M1() t M ( t) at t 1 MTTF to 3 deimals. For example, if TTF ~ N ( 5 hours, 16 hours ) ad also TTR ~ N ( hours, 576 hours ), the ( ) 6 5 M 1 6 Φ σ 1 expeted yles, where σ , M () t 6 μ μ Φ x t Φ x expeted failures, σx σx where t t [ M1() t.475] MTTR , A ( 6, hours) 1 M1( t) M ( t) , whih is lose to A if Similar alulatios will show if TTR ~ N ( 4 hours, 34) so that MTTR MTTF.8, the A ( t) M ( t) M ( t).9586, t 1 1 t [ M1 () t.45] MTTR , ad A if It should be oted that if < MTTR MTTF <.5, the from a pratial stadpoit the reewal proess approximately redues to the miimal-repair ase (i.e., a ohomogeeous Poisso proess) for whih M () t M ( ). Fuher, the ormal A ( t) geerally dereases o the iterval 1 t [, MTTF) as t ireases, seems to attai its worst value aroud the MTTF, teds to irease with ireasig time beyod MTTF, ad the overges toward A if. 4. Marov Aalysis Whe Oly Repair-rate r is a Costat The Marov aalysis of AVF, A ( t), for ase of ostat failure- ad

10 74 Dilu B. Helvai ad Saeed Maghsoodloo repair-rates (i.e., the APP) has bee repoed by early all authors i Stohasti Proesses ad Reliability Egieerig. Our objetive is to mae a slight geeralizatio to whe the hazard futio (HZF) is time-depedet, i.e., h () t λ, where λ is the CFR (ostat failure-rate). We a obtai the AVL of a simple o ad off (or up-time ad dow-time) system from Figure 1, where state represets a system i the reliable-state ad 1 represets the same system uder repair. The trasitio-rate i Figure 1 shows that its Kolmogorov equatio is give by dp ( t) dt h( t) P ( t) rp ( ), where 1 t P () t A() t represets the uoditioal Pr of fidig the system i the operatioal state at time t, ad similarly for P 1( t). Beause P 1 () t 1 () t for all t, we obtai dp ( t) dt h( t) P ( t) r( ), ad hee P 1 P dp () t dt [ h() t r] P () t r. This last is a simple differetial equatio [ h() t r] dt H with the itegratig fator () t e e, where H ( t) H is the atiderivative of h (); t it should be oted that the atiderivative h () t dt does ot seem to math the defiitio of the umulative HZF h( x) dx. However, if the umulative hazard at miimum-life is zero, whih is expeted, the H () t is also the umulative HZF. It is widely ow that the geeral solutio of the above differetial equatio is give by ( H ) H ( t) ( H ) t P () t e re dt Ce, (7a) where the ostat of itegratio will be omputed as usual from the H boudary oditio P ( t δ) 1, δ beig the miimum-life. Beause () t e 1 R(), t the (7a) is modified to () () ( H ) P t A t e C [ ()] re R t dt e R() t C [ ()]. re R t dt (7b)

11 Reewal ad Availability Futios 75 Ufouately, there is o exat solutio to (7b) for the geeral lasses of failure distributios, F() t 1 R( t), beause the idefiite-itegral I () t [ re R()] t dt re [ 1 F()] t dt has o losed-form atiderivative for all uoutably ifiite umber of failure distributios. However, we may obtai a exat solutio for a few failure distributios F ( t), ad the have to approximate equatio (7b) for others. We sta with the simplest ase of λ -parameter expoetial () ( t δ F t 1 e ) ad the solve (7b) ase by ase as listed below, ireasig solutio diffiulty. Fuher, merely for writig simpliity we let F F( t), R R( t), ad as stated above the repair-rate stays ostat at r. Figure 1. The trasitio-rate diagram for a o ad off system. Case (a). Whe the HZF is a CFR but miimum-life δ is ot eessarily 1, t δ zero, the the reliability futio R() t λ( t δ) ad applyig e, δ t < the boudary oditio P ( t δ) 1, equatio (7b) after extesive algebra r λ ξ yields () () ( t δ P ) t A t e, ξ λ r, t δ, whih at δ is ξ ξ the same futio give i Setio 3 for a APP. Clearly, A ( t) 1 for t δ. For example, suppose a etwor s TTF ~ Exp( δ 4, λ. 5 hours ) ad ostat repair-rate r. 5 per hour. The, the harateristilife ow improves to 1 λ δ 4 hours, whih is also equal to MTTF of the Exp ( 4,.5), ad λ is the rate-parameter. As before, ξ λ r

12 76 Dilu B. Helvai ad Saeed Maghsoodloo.55 ad the Pr that the etwor is available (i.e., ot uder repair) at t 5 hours is give by.5.5 ( ) ( 1 A 5 e.55 ) , while the value of reliability futio is R(5, miimal-repair) Case (b). Seodly, suppose TTF is uiformly distributed over the realiterval [ a, b], i.e., U ( a, b), where a is the miimum-life, b maximum-life > a ad b a > is the uiform-desity base. The the df F () t ( t a), R() t ( b t), a t < b, R( t) 1 for all t a, ad R () t for all t b, at whih poit the system will be trasitio to the repair-state. For t a, the substitutio of R ( t) 1 ito (7b) ad applyig the boudary oditio P ( t ) 1 will ot yield the value C beause the idefiite-itegral o the far RHS of (7b) has to be evaluated first. Whe t b, R( t) results i a idetermiate form for the RHS of (7b), e R() t [ re R()] t dt. Thus, we will have to ompute the value of the ostat C after obtaiig the geeral solutio for A ( t). Next, for a t < b, R ( b t), < F ( t a) < 1, b a > ad substitutio ito (7b) yields P () t A() t e R() t C [ re ( 1 F )] R() t C re e F dt dt where e R() t { C I() t }, I () t re F dt [ re F ] dt. (8)

13 Reewal ad Availability Futios 77 Note that the alterative proedure I() t [ re R()] t dt [ re ( b t) ] dt re [ b( 1 t b) ] dt, ad usig the geometri series for 1 ( 1 t b), will lead to the same exat result for A (). t We ow obtai the atiderivative I ( t) of equatio (8) as follows: () [ ] 1 I t re F dt e F e F ( df dt) dt 1 e F e F ( 1 ) dt, (9) where F ( t a) ; the itegral uder the summatio o the far RHS of equatio (9) is valid oly for the uiform-ttf. Repeated itegratio by pas, as show above, will show that at a speifi, ( re F ) dt e [( 1 ) ( P ) F ( r) ] e [( ) ( ) 1 r P F ], (1) where F F() t ( t a), P! ( )!,, is the permutatio of objets tae at a time, ad! Γ( 1) 1. Substitutig equatio (1) ito (9) yields I () t [ re F ] dt e [( 1 r) ( P ) F ]. (11) Combiig equatios (11) ad (8) results i P () t A() t e R() t C e [( 1 r) ( P ) F ]. (1)

14 78 Dilu B. Helvai ad Saeed Maghsoodloo So far we have argued that A ( t) 1 for t a, ad A ( t) is give by equatio (1) oly for a t < b; so, what is the AVF for t b? Reall that TBC TTF TTR so that the suppo of TBC is t < ; this is due to the fat that we are assumig a CFR with expoetial repair distributio futio 1, t <. Before providig the overall AVF, we first obtai the value of the ostat C i equatio (1) by applyig the boudary oditio P ( at t a) P ( at F ) 1. I order to examie ad evaluate P () t A() t at t δ, we rewrite the double-sum o the far RHS of equatio (1) separatig out the ostat terms from those whose expoet of F exeeds zero, e I () t e [( 1 ( r)) ( P ) F ] 1 1 [( 1 ( r )) ( P ) F ] [( 1 ( r)) ( P )], (13) where P!. Equatio (13) learly shows that lim F I () t e lim I () t e 1 1 ( r) ( r) 6 ( r) 3 t a 4 4 ( r) [! ( r) ], whih we deote by A. Ufouately, this last alteratig ifiite-sum A [! ( r) ] does ot overge o matter how large r is; the larger r is, the more aurate value of C a be obtaied. However, equatio (1) will provide fairly aurate AVF if the summatio over a be termiated at < 171. It should be highlighted that at > 17 MATLAB will ot ompute P! ( )!,, ad hee the ifiite double sum i equatio (1) has to termite at some reasoable value of, say 6 1; this i

15 Reewal ad Availability Futios 79 tur will resolve the divergee problem with A whe tae as a sum with the double-sum i equatio (1). It should also be oted that the exat b 1 dt average (or expeted) hazard rate for the uiform-desity does a b t ot exist, ad the use of approximate value 1 MTTF ( a b) for h () t i equatio (7a) redues the proess to the ase of ostat failure-rate durig the iterval [ a, b], whih is ot realisti. Fially, substitutig t a i equatio (1) yields 1 Ce ra A ; orrespodig AVF is give by ra 1 hee, C e ( A ), ad the 1, t a r () ( t a ) r ( ( t a R t e A 1 e ) ) A() t 1 1 ( P ) F 1 r r( t b) 1 e, b t <., a t < b, (14) I equatio (14), R() t ( b t) ad F ( t a). As disussed i Setio 3, the widely-ow log-term AVL for a APP is A if MTTF ( MTTF MTTR ), where the suppo for TBC is [, ). Beause the suppo for TTF i equatio (14) is the fiite iterval [ a, b], taig the limit as t is ot warrated. Equatio (14) shows that at t b, the system fails with eaity ad goes uder repair ad will be AVL with a Pr of 1 r( t b e ), b t <, at whih poit oe yle is ompleted. However, we a asse with eaity that the glb for average availability is A a ( b 1 r), while the lub is b ( b 1 r), i.e., a ( b 1 r) Aave b ( b 1 r), where A ave gives the propoio of time that the system is operatioal. If we examie the AVL for the time itervals [, a ), [ a, b], ad ( b, b 1 r], the it follows that the system has a AVL 1 with approximate Pr of a ( b 1 r); it has a AVL [( a b) ] [( a b) 1 r] with approximate Pr of ( b a) ( b 1 r), ad ave

16 8 Dilu B. Helvai ad Saeed Maghsoodloo a AVL of zero with approximate Pr of ( 1 r ) ( b 1 r). Hee, the weightedaverage (or expeted) AVL is give by A a ( a b) b a 1 r 1 b 1 r ( a b) 1 r b 1 r b 1 r ave abr a rb ( a b r)( br 1). (15) For example, if miimum-life is a hours, maximum-life is b 1 hours, i.e., TTF ~ U (, 1), ad repair-rate is. per hour, the equatio (14) gives A ( 7 hours) Fuher, equatio (15) shows that A.9667, the lub o AVL is.96, while A ave MTTF ( MTTF MTTR) Case (). Suppose TTF is distributed lie gamma with miimum-life δ, shape α ad sale β 1 λ; as before the repair-rate is a ostat at r. It a easily be verified that 1, for t δ, R() t λ ( λ ) ( t δ 1 ) where x t δ x e, δ t <, ad the HZF is h() t λ( λx)( 1 λx). Clearly, the AVF for the iterval [, δ] is equal to 1. I order to obtai the exat expressio for A ( t) durig [ δ, ) give i equatio (7b), agai we have to obtai the atiderivative λ( t δ ) ] I() t [ re R()] t dt re [( 1 λx) e dt if δ [ ξ ( ) re r e x 1 1 λ x ] dx, (15a) where we have trasformed t δ to x so that dt dx ad ξ λ r. 1 Expadig ( 1 λx ), x t δ, geometrially i (15a) we obtai rδ ξx rδ ξx I () t re e ( λx) dx re [ e ( λx) ] dx. (15b)

17 Reewal ad Availability Futios 81 Bearig i mid that the overgee-radius of ( λx ) is λ( t δ) < 1, repeated itegratio by pas, ad lettig ω r ξ, will redue equatio (15b) to rδ ξx I () t ωe e ( P ) ω ( λx), (15) where we remid the reader that x t δ. Separatig out the ostat term from the double-sum o the RHS of (15) yields 1 () ( ) ( ) (! ). 1 rδ ξx I t ωe e P ω λx ω (15d) As a result the AVF for t δ from equatio (7b) is give by 1 rδ ξx A () t e R() t C ωe e ( P ) ω ( λx) C, 1. (16) where C (! ω ) I order to solve for ostat C, we require the rδ 1 iitial-oditio that A ( x ) 1; this yields C e ( ωc ), where < ω λ ξ 1. Substitutig for C ito equatio (16) ad bearig i mid that x t δ, we obtai 1, t δ, A() t ( 1 λx) ( 1 ωc ) e ω 1 1 ξx ωc ( P ) ω ( λx), δ t <. (17)

18 8 Dilu B. Helvai ad Saeed Maghsoodloo Beause the gamma mea at shape α is give by MTTF δ λ μ, it a be argued as i the previous ase, where we ow divide AVL itervals ito [, δ ), [ δ, μ) ad [ μ, μ 1 r], that the expeted propoio of time the above system is available is give by A ave μ δ r. (18) ( μ 1 r) For example, if the TTF ~ gamma ( δ 5 hours, α, sale β 1 λ 15 hours) ad repair-rate is a ostat at r., the μ MTTF 5.8 5,5 hours, ad equatio (18) gives Aave.99618, whih is lose to A MTTF ( MTTF 1 r) if For t 85, r., ad λ.8, our MATLAB program usig equatio (17) gives the poit AVL of A(85) , while at t 1, A ( 1 ) Note that equatio (17) will ot give meaigful aswers for A ( t) if λx λ( t δ) 1 beause ( λx ) diverges for λx 1. Case (d). Suppose TTF is distributed lie Weibull (W) with miimumlife δ, harateristi-life θ, ad shape (or slope) β, 3, 4, 5,..., i.e., a exat positive iteger; as before the repair-rate is a ostat at r. The Rayleigh failure desity is a speial ase of the W ( δ, θ, β ). It has bee widely ow sie the early 195 s that the uderlyig failure β distributio is give by () 1 ( λx ) 1, t δ, F t e, R() t β ( λx) e, δ t <, β 1 where x t δ, the HZF is h () t βλ( λx), the umulative hazard is β H () t ( λx), ad λ 1 ( θ δ). Clearly, the AVF for the iterval [, δ] is equal to 1. I order to obtai the exat expressio for A ( t) durig [ δ, )

19 Reewal ad Availability Futios 83 give i equatio (7b), agai we have to obtai the atiderivative I() t [ re R()] t dt [ re e ( ) ] dt β λx β β β ( ) λ x ( λx) re dt re dt. (19a)!! Beause we are restritig the shape oly to positive itegers 1,, 3, 4,..., repeated itegratio by pas will show that at a speifi the value of the idefiite-itegral uder the summatio i equatio (19a) is give by ( λx)! β β β re dt! λ e β β [( 1 r) ( P ) x ] β ( λx) e! β [( P ) ( 1 ( rx) ) ]. β Substitutig this last atiderivative ito equatio (19a) yields β β ( λx) e I () t [( β ) ( 1 ( )) ].! P rx (19b) As a result the AVF for t δ from equatio (7b) is give by β () ( ) β λx λ β A t e e C e [( β ) ( 1 ) ]! P r x β ( ) β λ β x λ β C e [( β ) ( 1 ) ]! P r x e e 1 β 1 ( λ ) ( β )! β r e.! (a) I order to solve for the ostat C, we require that A ( t δ) 1; this

20 84 Dilu B. Helvai ad Saeed Maghsoodloo rδ ( ) ( ) yields C e ( 1 B ), where ω β β! B, < ω λ r 1.! Substitutig for C ito equatio (a), we obtai 1, t δ, β () ( ) λx rx rx A t e e B( 1 e ) β β 1 ( λx) [( ) ( ( )) ] δ < β P 1 rx, t.! 1 (b) For example, if the TTF ~ W (,, ) ad repair-rate is a ostat at r. 4 per hour, the μ δ ( θ δ) Γ( 1 1 β) MTTF Γ ( 1 1 ) hours, ad equatio (18) gives A ave , whih is lose to A MTTF ( MTTF 1 r) For if t 15, r.4, ad λ.5, our MATLAB program usig equatio (b) gives the poit AVL of A ( 15 ) , while at t 3, A ( 3 ) It must be highlighted that there are omputatioal problems with equatio (b) for larger values of t ad slope β, as MATLAB will ot do omputatios for fatorials beyod 17; thus we were uable to verify that for very large values of t that equatio (b) gives results that are lose to MTTF ( MTTF 1 r). However, we have prove that at shape β 1, r λ ξ equatio (b) idetially redues to () () ( t δ) P t A t e, ξ λ ξ ξ r; MATLAB also verifies this laim omputatioally. Case (e). Suppose TTF is distributed lie Weibull (W) with δ miimum-life, harateristi-life θ, ad shape (or slope) β that is ot a exat iteger; as before the repair-rate is a ostat at r. For example,

21 Reewal ad Availability Futios 85 suppose TTF ~ W ( δ, θ, β 1.5); the from equatio (15a), I () t 1.5 ( λx) re dt. It should be lear that at 1, the! [ re ( λx) 1. 5 ] dt has o losed-form atiderivative ad hee o exat solutio for A () t a be obtaied. Fuher, approximatig this last atiderivative by expadig re i a Malauri series ad usig oly the first 1 terms of the ifiite series will ot lead to a adequate approximatio. Wor will be i progress to develop a method to approximate A ( t) for the geeral lass of failure distributios. 5. The Reewal ad Availability Futios whe TTF is Gamma ad TTR is Expoetial It is well ow that the LT of a uderlyig gamma failure desity with α α shape α ad sale β 1 λ is give by f () s λ ( λ s) ; ote that oly whe α is a positive iteger this last losed-form is valid. Whe α is ot a exat positive iteger, there is o losed-form solutio for the LT of a gamma desity beause the itegratio-by-pas ever termiates. Thus, i the ase of shape beig a exat positive iteger, i.e., Erlag uderlyig failure desity, we have the well-ow LT of AVL: A( s) 1 f ( s) s[ 1 f ( s) r( s)] α λ 1 α ( λ s) α λ r s 1 α ( λ s) r s α α ( λ s) ( s r) λ ( s r). (1) α α s[( λ s) ( r s) λ r] At α, equatio (1) redues to s ( λ r) s λr A( s) s[ s ( λ r) s λ λr] 1 s s r1 3, s r

22 86 Dilu B. Helvai ad Saeed Maghsoodloo where r 1 ad r are the roots of the polyomial s ( λ r) s λ λr. Thus, r1 ( λ r ) ( r ) λr, r ( λ r ) ( r ) λr, 1 r ( r λ), λ( λ r r1 ), ad ( λ r) r 4λr 3 λ( λ r r ). ( λ r) r 4λr Iveig ba to the t-spae, we obtai A () t r ( r λ) 1 e 3e. This last AVF learly shows that as t, A () t r ( r λ) r ( r λ ) MTTF ( MTTF MTTR), ad fuher, A ( ) 1, as expeted. Example. Suppose a system has a uderlyig gamma failure distributio with shape α, sale β 1 λ 1 hours ad TTR has a ostat repair-rate r.5, the the availability at 5 hours is give by A( 5) ; while the same system with miimal-repair has a ( ) ( ) λx λt A t 5 R 5 λ( λx) e dx e ( 1 λt) That t is, repair will improve availability by 9.4%. We also used our equatio (17) with termiated at 13 ad obtaied A ( 5) for the same system of δ, α, sale β 1 λ 1 hours ad r.5. The steady-state (or log-term) AVL of suh a system as disussed by may other authors is A.5 (.5. 5) At α, the LT of expeted umber of yles redues to M ( ) s s s s r1 s r rλ s [ s ( λ r) s λ λr] where r 1 ad r are the same roots, 4 r t r t r( λ r), 5 λr ( λ r), ( λ r) 5 4r 6, ad 4r Upo iversio, we obtai M 1 () t r 4λr r 4λr r1 t t e e. For the same parameters as the above example, ,

23 Reewal ad Availability Futios 87 we obtai M 1( t 1, hours) expeted yles. Similarly, it a be show that the LT of the expeted umber of failures is give by M ( ) s s s s s r1 s r λ ( r s) [ s ( λ r) s λ λr] ( λ r ) ( λ r r ) where 8, ( ( λ r) 9 λr λ r), , ad r 4λr ( λ r r ) Upo iversio to the t-spae, we obtai r 4λr r t r t M () 1 t 8 9t 1e 11e. The value of expeted umber of failures durig a missio of legth 1, hours is M ( t 1) , whih exeeds M 1( 1) , as expeted. Fuher, M ( 1) M ( 1) , whih is idetial to the value 1 of AVF obtaied from A() t r ( r λ) e 1 e at t 1. r t r t 3 Ufouately, whe TTF is Erlag at α 3, 4, 5, 6 ad 7 ad a ostat repair-rate r, the orrespodig LT deomiators D ( s) s[ 1 f ( s) r( s)] have at least omplex roots, whih have bee well ow to be omplex ojugate pairs. Yet, after paial-fratioig, the LT s a be iveed to yield real-valued M 1 () t ad M ( ), as demostrated below. At α 3, M ( s) t f ( s) r( s) λ r λ r s s[ 1 f ( s) r( s)] 3 ( λ ) r s 3 s ( λ s) r s λ r 3 s[( λ s) ( r s) rλ 3 ] 3 3 λ r 3 3 s [ s ( 3λ s) s ( 3λr 3λ ) s λ 3rλ ] 1 s s s 3 r1 s 4 r 5 s r 3, 3,

24 88 Dilu B. Helvai ad Saeed Maghsoodloo where the root r 1 will be real, while r ad r 3 will be omplex ojugates, i.e., both r r3 ad r r3 will for eai be real umbers. I order to maitai equality i the above paial fratio, it a be show that 3 λ r r r r 1 3, 1 r r 1 r1 r3 r r r r r ; fuher, lettig the ostats a ( r r r ) ( r r r r r ), r3 a 1 ( r1 r r3 ), the 3, 4, ad 5 are the uique solutios 1 give by C [ ] A, where C is the 3 1 solutio vetor, b is a 3 1 vetor b a1 b a ad the 3 3 matrix 1 r r3 r1 r3 r1 r A r r3 r1 r3 r1 r A MATLAB program was devised to obtai the expeted umber of yles M 1 () t as outlied above. The program also uses similar proedure to ompute M () t ad the resultig A ( t). The MATLAB program has the apability to ompute the 3 reewal measures M 1( t), M ( t), ad A () t for shapes α, 3, 4, 5, 6 ad Colusios This aile provided the exat RNFs ad AVF for the ase of ormal TTF ad TTR. The Marov aalysis was used to obtai the AVL futios for the ases of -parameter expoetial TTF, the uiform, the gamma at shape, ad Weibull with positive iteger shape TTF, while the repair-rate was held ostat at r. Fuher, LTs were used to obtai both RNFs ad the AVF of the gamma at shapes, 3, 4,..., 7 at ostat repair-rate. The gamma AVL obtaied i Setios 4 ad 5 were the same up to 4 deimal auraies..

25 Reewal ad Availability Futios 89 Referees [1] S. Maghsoodloo ad D. Helvai, Reewal ad reewal-itesity futios with miimal repair, J. Quality ad Reliability Egieerig 14 (14), Aile ID , 1 pp. [] U. N. Bhat, Elemets of Applied Stohasti Proesses, d ed., Joh Wiley & Sos I., [3] D. R. Cox ad H. D. Miller, The Theory of Stohasti Proesses, Wiley, New Yor, [4] E. A. Elsayed, Reliability Egieerig, d ed., Wiley, Hoboe, N.J., 1. [5] K. S. Trivedi, Probability ad Statistis with Reliability, Queuig ad Computer Siee Appliatios, Pretie Hall, New Yor, 198. [6] F. Hildebrad, Advaed Calulus for Appliatios (196 Editio), Pretie-Hall, 196. [7] L. M. Leemis, Reliability: Probabilisti Models ad Statistial Methods, d ed., Aseded Ideas, 9. [8] C. E. Ebelig, A Itrodutio to Reliability ad Maitaiability Egieerig, d ed., Wavelad Press, Log Grove, Ill, 1.