RELIABILITY ASSESSMENT OF SYSTEMS WITH PERIODIC MAINTENANCE UNDER RARE FAILURES OF ITS ELEMENTS

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1 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS Yakov Geis Boough of Mahatta Commuity College City Uivesity of New Yok, USA yashag5@yahoocom Abstact Thee is ivestigated а model of a system with the highly eliable elemets, whee the peiods of fuctioig ae chaged by the peiods of maiteace The system must be opeatioal oly i the peiods of fuctioig although the estoatio i these peiods is ot povided The system is completely estoed i the eaest peiod of maiteace Sice the elemets of system ae highly eliable, the eseve of system is aely exhausted duig each peiod of fuctioig Theefoe it is possible to use the esults, obtaied fo the systems with fast estoatio, fo the eliability assessmet of system, which is ot estoable i the peiods of the fuctioig The estimatios of idices of failue-fee pefomace ad maitaiability of such systems ae obtaied Itoductio ad motivatio Thee is examied a system with peiodic maiteace The peiods of fuctioig of the system, ae chaged by the peiods, whe the system is tued off ad thee is poduced its maiteace ad the estoatio of all failed elemets The elemets of the system fail oly i the peiod of fuctioig, ad thei estoatio is poduced i the peiod of maiteace Use is iteested i the high eliability of the system i the peiods of fuctioig Examples of such systems ae aicafts, eusable spacecafts, etc So aicafts i a flight ae cayig out thei flight missio Afte ladig thei complete maiteace is caied out Afte this, the aicafts ae eady to the fulfillmet of a ew flight missio Aothe example is a uclea powe plat whee uses as a ule have o access o a limited access to its eacto compatmet betwee peiods of plaed maiteace At the time of the maiteace uclea eacto ca be adusted ad eewed to cotiue its missio Fo these systems the peiods of fuctioig, whee the system woks ad fulfills the assiged fuctios, ae chaged by the peiods of maiteace, whe the system is tued off ad the maiteace is poduced The estoatio of all failed elemets as well as the elimiatio of all faultiesses, accumulated duig the fuctioig peiod, is poduced oly i the secod peiod ad it maages to ed up to the ext fuctioig peiod I the secod peiod the eliability of opeatioal elemets is ot chaged Afte gluig togethe the fuctioig peiods of the system, we will obtai the model of the system with the peiodic istataeous maiteace I the wok thee is give assessmet of the idices of failue-fee pefomace ad maitaiability ude the coditio that elemets of system fail aely i the fuctioig peiod 2 Model desciptio 47

2 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach The duatio of the peiods of the time fom iitial momet of the fuctioig of the system till the eaest maiteace peiod o betwee the ed of the peiod of maiteace ad the begiig of the followig peiod of maiteace is a adom vaiable ξ with distibutio fuctio (DF) P{ ξ < x} =Φ ( x) ad mea value Mξ = T The elemets of the system ca fail oly i the fuctioig peiods of the system, ad thei estoatio is poduced oly i the eaest iteval of maiteace The failue of the system occus whe betwee the fist failue of some elemet of the system ad the eaest iteval of maiteace thee is completely exhausted the etie eseve of the system, icludig a time eseve, if it exists The failues of elemets ad the failue of the system ae istatly detected If the eseve of the system is spet ot completely i the peiod of fuctioig, the the failue of the system is ot obseved, the eseve of the system is estoed i the eaest iteval of maiteace ad the system is deived fo the ew peiod of fuctioig If the failues of elemets do ot occu i a cetai peiod of the fuctioig, the the eliability of the system i the eaest peiod of maiteace is ot chaged The itevals of fuctioig ca have diffeet duatio i sese of thei mea value Sice i the itevals of fuctioig the system is ot estoed, the failue of the system is moe pobable i the moe pologed iteval of fuctioig Afte takig the wost (most pologed i aveage) iteval of fuctioig we ca obtai the estimatio of the wost eliability of the system Theefoe without limitig the geeality we will coside that all itevals of fuctioig ae distibuted equally ad coespod to the most pologed by its mea value iteval of fuctioig We will examie the highly eliable systems, i which the failues of elemets i each iteval of fuctioig occu aely This i paticula meas that the poduct of the maximum of summay failue ate of the system elemets by the maximum duatio of the fuctioig iteval is cosideably less tha oe Actually this case coespods to pactice We ae iteested to study the behavio of the system oly i the summay iteval of fuctioig, obtaied as a esult of igoig all itevals of maiteace This summay iteval ca be cosideed as a ew time axis with the poit flow T Each poit of the flow coespods to the iteval of the maiteace, whee the system is estoed istatly ad completely O this time axis it is possible to idicate the momets of the failues of the elemets of the system Afte the fist failue of a elemet i ay fuctioig iteval a iteval of malfuctio (IM) is begu IM eds at the momet of the begiig of the eaest iteval of maiteace Let us ame a IM as a failue IM, if the failue of system occus i it Let us ame a poit of flow T as a failue poit if it is cotiguous with a failue IM Afte such a failue poit the system is fully estoed (!) We have a poit flow T with aefactio whee some poits may be failue poits with pobability coveged to zeo But this meas that thee is possible to use a asymptotic appoach fo eliability assessmet of the system Sice o the fuctioig itevals the estoatios of elemets ae ot poduced the failue of the system i IM is developed oly alog the mootoic path Mootoic path is such a taectoy of the failue of the system o which fom the momet of the failue of the fist elemet i IM (the begiig of IM) ad to the momet of the failue of the system i this IM oe estoatios wee fiished 48

3 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach Let us estimate the behavio of the system to the fist failue of the system ad betwee two adacet failues of the system, whe the elemets of the system ae highly eliable This meas that the pobability of the failue of system i ay peiod of fuctioig is small ad moe tha zeo, which coespods to the coditios fo fast estoatio, itoduced ad studied i [] ad [2] Thus the estimatio of the eliability of system with the peiodic maiteace, whee thee ae o estoatios o the itevals of fuctioig, ca be ad will be actually poduced with usig a appaatus, developed fo the systems with the fast estoatio It souds paadoxically, but this is actually so because the elemets of the system ae highly eliable, the peiods of fuctioig have elatively small duatio ad the poit iteval of maiteace follows them 3 Mathematical fomulatio of poblem Peset aticle est upo the esults of the woks оf Geis [] ad [2] I the wok [] thee is give the geeal appoach to the estimatio of the eliability of systems with fast estoatio I the wok [2] thee is give the moe detailed detemiatio of the citeio of fast estoatio Fo coveiece of the eade let us give some details of the systems, aleady descibed i [] ad [2], which ae used i this wok System cosists of elemets Each elemet of the system ca be oly i the opeatioal o iopeative state Each opeatioal elemet ca be located i the loaded o uloaded egime (lighteed egime fo the bevity is omitted) Let F i (x), f i (x), и m i ae accodigly the distibutio fuctio, the desity of distibutio (DD), ad the mea value of the time of failue-fee opeatio of the i-th elemet i the system, m i <, i, Withi the famewok of this wok we will examie the systems of the -st ad the 3-d types, descibed i [] Systems of the -st type ae systems with the expoetially distibuted times of failue-fee opeatio of elemets Systems of the 3-d type ae the systems, whose elemets have a limited DD of the failue-fee opeatio time; thee is equied also, that these DDs i zeo ae ot equal to zeo, f i () = c i, i, Thee ae o limitatios to the stuctue of the system Thee is assiged the citeio of the system s failue, which ca iclude ad the coditio of time edudacy The state of the elemets of system at the momet z is descibed by the vecto ν ( z) = { v( z),, v( z)}, whee each compoet ca take the values of {,,, } Numbe coespods to failed elemets; umbes fom to coespod to opeatioal elemets These umbes make it possible to uambiguously assig the ode of the eplacemet Vecto ν ( z) helps to estimate the eliability of a cocete system Let E is the set of the states of the system, { v( z)} = E = E+ E, whee E + is the aea of the fully opeatioal, ad E is the aea of the befoe failue states of the system The system is cosideed as befoe failed at the momet z if v( z) E ad failed if its malfuctio lasts time ot smalle the η, P{ η < x} = H( x) I the absece of the time eseve ( η ) the aea of the befoe failed states of system is coveted ito the aea of the failues of the system All elemets of the system wee ew ad fuctioed popely at the iitial momet of time 49

4 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach Let b N is a cetai state vecto of elemets of the system diectly befoe IM, ad b is the state vecto of the elemets of the system o the same IM immediately afte the momet of passig the state vecto of system fom the aea E + ito the aea E Let us ame the way π leadig fom N b E + ito the state b E o the IM the sequece of the state vectos of elemets, begiig fom the vecto b N, which diectly pecede the begiig of the IM, ad edig with the vecto b, which coespods to the fist befoe failed state of the system o this IM The path legth is equal to the umbe of elemets, which failed o this way Let us ame way mootoic, if o it thee ae o estoatios of elemets Let us ame mootoic way miimal fo b if its legth lb ( ) equals to the miimum of the path legths, which lead fom b to E The the miimum umbe of elemets, whose failue ca cause the system s failue, equals s = mi l( b) o b E + The value s 2 coespods to a fault-toleat system with a stuctual eseve If s = a failue of some elemet may lead to a system failue uless thee is ot itoduced a time eseve that will ot be exhausted till the eaest momet of a maiteace peiod It is cosideed that the system woks i the coditios of the fast estoatio Pactically this meas that the mea time of the iteval of fuctioig is substatially less tha the mea time betwee ay two failues of elemets i the system (see sectio 4) Let i the steady-state opeatio sectio of wok with the estoatio disciplie d with the staight ode of maiteace ad oe epai uit, β ( d,) is the estimatio of the system s failue ate takig ito accout oly mootoic paths of the failue (but i the system with peiodic " maiteace thee ae o othe paths), τ ( d,) is the adom system s estoatio time afte failue, T (,) d is the mea value of this time, K (,) A d is the availability fuctio of the system I all of these idices the fist paamete is the type of estoatio disciplie ad the secod oe is the umbe of estoatio uits i the system The poblem cosists i estimatig of the idices of failue-fee pefomace ad maitaiability of the descibed system ude the coditios of the fast estoatio A system with the peiodic maiteace fails, if the eseve of the system is completely exhausted duig the peiod betwee two maiteaces I this case the failue of the system occus oly alog the mootoic paths Let DF Φ ( x) is absolutely cotiuous I the steady-state opeatig coditios of the system DF of time fom the begiig of IM to the eaest momet of the system s maiteace is equal to Ax ( ) = Φ( udu ) / T (3) x Fo the failue of the system duig a IM thee must be spet the eseve of the system Theefoe the behavio of the descibed system is aalogous to the behavio of the system of sectio 2 [] with estoatio disciplie d = d with the staight ode of maiteace FIFO, oe epai uit k =, ad with idetical fo all elemets distibutio fuctio of the estoatio time G ( x) = G( x) = A( x), 5 i

5 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach i =,, with the additioal assumptio that oly mootoic paths of the system s failue ae pemitted Ideed, fo such epaiable system the fist failed elemet will ot be estoed duig IM time, i the same time the eseve of system should be spet, ad ude the coditios of fast estoatio the failue of system will be developed alog the mootoic path I [] ad [2] fo establishig the citeia of the system s behavio ad fomulatig the esults of the study thee wee used the idices of failue-fee pefomace of the elemets m i ad c i, DF of estoatio time G(x) (if it is idetical fo all elemets) ad shifted fo the time η momets DF G(x) ( k) k η m ( ) = kx G( x + u) dxdh ( u), m ( k) ( k ) m () =, m = m (32) (), whee fo ay DF Γ ( x) Γ ( x) = Γ ( x) I the peset wok let us eplace Gx ( ) by Ax ( ) fom (3) ad let us itoduce the desigatios ( k) k η m% ( ) = kx Φ ( x+ u) dxdh( u), m% % (33) ( k) ( k ) = m () The followig lemma makes it possible to pass fom (32) to (33) Lemma 3 Fo ay umbes k m ( η) = m% ( η), (34) ( k + ) T ( k) ( k+ ) if ( k+ m% ) ( η) exists The poof of Lemma 3 is give i the Appedix 4 Asymptotic appoximatio Citeio of the fast estoatio Let s is the miimum umbe of elemets, whose failue ca cause the malfuctio of the system; λ ad λ ae the maximum ad the miimum failue ates of elemets i the opeatioal system [] Let us say that the coditio of the fast estoatio is satisfied fo the system if λ >, α = [ λ m / ( s ) ( ) s m ] = % %, (4) ( s+ ) (2) s [ λ( m /( s+ ) T)/( m /2 T) ] ad fo all DF F i (x), i =,, the limited distibutio desities must exist [, 2] I the pactically impotat cases m ( ) s C (m )s, whee C is a cetai costat I these cases the coditio fo fast estoatio (4) is educed to 5

6 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach (2) m m /2T α = λ = λ % (42) Fo the fast estoatio it is actually equied that the pobability of the system s failue o the peiod of fuctioig would be moe tha zeo (it is eough to demad λ > ), ad the pobability of the system s failue o IM coveges to zeo ( α ) 5 Estimatio of the idices of failue-fee pefomace Let τ i( t) is the iteval fom the momet t to the eaest momet of the system s failue afte momet t Details of the detemiatio of time to the fist failue of the system τ () t ad time betwee (-)-th ad th system s failues τ ( t), 2, ae give i sectio 5 of [] Takig ito accout the theoem 53 of [] we will obtai that fo the dicussed system it is fullfilled Coollay 5 I the steady-state opeatig coditios of the system with peiodic maiteace ad with stuctual eseve fo i ude the coditios of the fast estoatio λ > ad α P{ τ ( t) x} exp{ β ( d,) x} (5) i Let us ame the itevals betwee the ed of oe IM ad the begiig of the ext IM i the ew time axis as itevals of opeability (IO) Let fom v( z) = b E +, z IO, thee ae possible miimal paths l = l( b ), leadig ito the cetai state b E Evey of state b is chaacteized by the set l of umbes of failed elemets, belogig to the set J = J+ J, whee J + ad J ae accodigly the sets of the umbes of those elemets, which i the state b wee located i the loaded ad uloaded egime Let the set of miimum paths leadig fom b ito b is Π Let Λ =Πc Π / m (52) k J k i J + Let ( b) is the umbe of miimal paths of the legth l = l( b ), leadig fom b ito b E ; s is the miimum umbe of elemets, whose failue ca cause the befoe failue system state; λ( b ) ( b ) ad q ae espectively the summay failue ate of elemets ad the pobability of the occuece of a failue IM i the steady-state opeatig coditios of the system i coditio that the system diectly befoe IM was i the state b The thee is valid Theoem 5 Ude the coditios of the fast estoatio with λ > ad α i the steady-state opeatig coditios of the system with the peiodic maiteace ad stuctual eseve ( ) b ( l) λ( bq ) ( b) Λ m% ( η)/ l! (53) T b E i 52

7 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach The pove of Theoem 5 is give i the Appedix Obsevatio 5 I theoem 5 thee is possible to emove the equiemet of absolute cotiuity of DF Φ ( x) Usig a appaatus fo a study of ae evet i the egeeatig pocess [3], it is possible to obtai estimatio (53) afte equiig existece of fiite fist ad secod momets fo the peiod ξ This makes it possible to exted the theoem 5 to the case, whe ξ T, that has a geat pactical value I the sectio 8 of [] it is show that ude the coditios of the fast estoatio the estimatio of the idices of the eliability of a complex system ca be bought to the estimatios of the idices of the eliability of its seies-coected i the sese of the eliability schemes p out of m, calculated ude the assumptio of thei autoomous opeatio The scheme p out of m has m elemets Its malfuctio occus with the failue of ot less the p elemets fom m, p m, ad its failue begis whe the malfuctio of scheme lasts ot less tha time η, P{ η < x} = H( x) Theefoe withi the famewok of this aticle we will coside that the system is a scheme p out of m, which failue ate is β p Usig (53), thee is possible to obtai the estimatios β p = β p( d,) fo the system p out of with the peiodic maiteace fo vaious types of stuctual eseve I paticula let us examie schemes out of, coespodig to the paallel i the sese of eliability coectio of elemets The scheme becomes defective, if all elemets fail Coollay 52 Fo the system out of with the peiodic maiteace ad the loaded eseve β ( ) m η T mi i= % ( )/ (54) System out of with the uloaded eseve cosists of the basic elemet used i the loaded egime, ad ( - ) eseve elemets, which ae located i the uloaded egime With the failue of the basic elemet its place takes the elemet fom the uloaded eseve, which stayed i the eseve the geatest time The malfuctio of system begis whe the basic elemet fails ad thee is abset a eseve elemet that is capable to eplace the failed oe Coollay 53 Fo the system out of with the peiodic maiteace ad the uloaded eseve β ( ) m η ci T m k= i k = % ( ) /! (55) 53

8 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach 6 Idices of maitaiability ad availability fuctio Thee is examied the system, descibed above, with η h = cost I this case ( k) k m% ( y) = kz Φ ( z+ y) dz Fo the system out of with the peiodic maiteace ad the loaded eseve (the paallel i the sese of eliability coectio of the elemets) P x m% h x m% h, (6) " ( ) ( ) { τ } ( + )/ ( ) T m% ( h)/( + ) m% ( h), (62) ( + ) ( ) ( + ) A ( )/( + ) i i= K m% h T m (63) Fo the system out of with the peiodic maiteace ad the uloaded eseve the estimatios of " P{ τ x} ad T ae the same, as fo the system with the loaded eseve, but the estimatio of availability fuctio takes the fom ( ) A + ( ) /( + )! k i= i k= K m% h c T m (64) Summay The eceived esults allow the developig of a pactical methodology of eliability assessmet of systems descibed i the fist sectio of this aticle Togethe with techologist who kows the system vey good it should be established the citeia of the system failue Afte that the system s eliability assessmet is povided If esults ae ot satisfactoy thee is itoduced a additioal eseve i the system ad eliability assessmet is epeated The pocess is cotiued util satisfactoy esults ae obtaied 54

9 Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach Appedix Poof of the Lemma 3 Takig ito accout (3) fo ay value u G( x + u) = Φ ( y) dy = Φ ( y + u) dy T x+ u T x I this case fo ay itege k k kx G( x + u) dx = T k kx Φ ( y + u) dydx x Afte eplacig the ode of itegatio fo < x< ad x<y< to <y< ad <x<y, ad the eplacig the vaiable of itegatio y to x we will obtai k kx G( x + u) dx = T ( ) y k Φ y + u kx dxdy = ( k ) k ( k + ) x Φ ( x+ u) dx + T (A) Afte addig to (A2) itegatio by value u with measue H(u) we will obtai the assetio (34) of Lemma 3 Poof of the Theoem 5 We will use the Theoem 6 of [] takig ito accout that the disciplie of the estoatio i the system is d, the umbe of epai uits is k =, ad fo ay elemet i failed the fist o IM the DF G ( ) ( ) ( ) i x = G x = A x The we will obtai that fom (64) of [] follows λ l 2 ( b) ( bq ) Λ Gx ( + udxdhu ) ( ) b E π Π ( l 2)! x (A2) ( l Sice fo ay path π Π the itegal i (A2) is the same ad it is equal to m ) ( η)/( l )!, ad the umbe of miimal paths i Π equals ( b ), the λ ( b) ( l ) ( bq ) ( b) Λ m ( η) ( l )! b E (A3) Afte applyig assetio of lemma 3 to (A3) we will obtai assetio (53) of theoem 5 EFEENCES Geis, Y O eliability of eewal Systems with Fast epai Joual of Iteatioal Goup o eliability, eliability: Theoy & Applicatios, No 3 (Vol ), pp 3 47, Septembe 26 ISSN Geis, Y eliability ad isk Assessmet of Systems of Potectio ad Blockig with Fast estoatio Joual of Iteatioal Goup o eliability, eliability & isk Aalysis: Theoy & Applicatio, No (Vol ) 28, pp 4 57, ISSN Geis, Y Stochastic-system behavio descibed by semi-eewal pocess Automatio ad emote Cotol, 4(5), (979) 55