BINARY LAMBDA-SET FUNTION AND RELIABILITY OF AIRLINE Y. Paramonov, S. Tretyakov, M. Hauka Ra Tehnal Unversty, Aeronautal Insttute, Ra, Latva e-mal: yur.paramonov@mal.om serejs.tretjakovs@mal.om mars.hauka@mal.om ABSTRAT A efnton of bnary - set funton s ntroue. It s use for the nspeton nterval plannn n orer to lmt a probablty of fatue falure rate (FFR) of an arlne (AL). A soluton of ths problem s base on a proessn of the result of the aeptane full - sale fatue test of a new type of an arraft. Numeral example s ven. INTRODUTION Ths paper s really the aton to the prevous author paper [] evote to the relablty of an arraft (A) an presentaton [2]. Here we onser the relablty of the proess of operaton of an arlne when after the spefe lfe s reahe (retrement tme), fatue falure sovery or fatue falure takes plae a new arraft s aqure an the operaton of arlne s ontnue up to nfnty. We onser aan the ase when for soluton of the problem of lmtaton of fatue falure rate (FFR) of arlne we know the type of strbuton funton of fatue lfe of A but o know the parameter of ths funton, for estmaton of whh we have the result of the aeptane full - sale fatue test of a new type of an arraft. Despte of all the smplty, the equaton a( t) exp( Qt) ves us rather omprehensble esrpton of fatue rak rowth n the nterval ( t, t ), where (we reall) t s the tme when a fatue rak beomes etetable ( a( t ) a ) an t s the tme when the rak reahes ts rtal sze ( a( t ) a ) an fatue falure takes plae. It an be assume that orresponn ranom varables T (lo a lo ) / Q / Q an T (lo a lo ) / Q / Q have the lonormal strbuton beause, as t s assume usually, normal strbuton of lot an take plae only f ether both lo an loq are normally strbute or f one of these omponents s normally strbute whle another one s onstant. We suppose also, that vetor ( X, Y ) (lo( Q),lo( )) has two mensonal normal strbuton wth vetor-parameter (,,,, ). It s worth to X Y X Y r note, that for the ase when a an a are onstants then f of s ompletely efne by the strbuton of beause, where =lo( a / a ). Frst, we onser soluton of the problem for the known strbuton parameter an then for the unknown one. Numeral example wll be ven. 37
2 PROBABILITY MODEL FOR THE KNOWN θ Just as n paper [] for the known, there are two esons: ) the arraft s oo enouh an the operaton of ths arraft type an be allowe, 2) the operaton of the new type of A s not allowe. A reesn of A shoul be mae. In the ase of the frst eson, the vetor t: n t,, tn, where t,,..., n, s the tme moment of -th nspeton, shoul be efne also. If s known the fferent rules an be offere for the hoe of struture of the vetor t :n : ) every nterval between the nspetons s equal to the onstant tsl / ( n ), where tsl s the arraft spefe lfe (SL) (the retrement tme), 2) the probabltes of a falure n every nterval are equal to the same value In ths paper we onser the frst approah, but really, our onseratons an be apple n more eneral ase when the vetor t :n s efne by two parameters, the fxe t SL an the the number of nspetons, n, n suh a way that probablty of falure tens to zero when n tens to nfnty. For the substantaton of the hoe of nspeton number we shoul know FFR an the an of AL as a funtons of vetor t :n. For ths purpose the proess of an operaton of AL an be onsere as an Markov han (Mh) wth n 4 states. The states E, E2,, En orrespon to an A operaton n the tme ntervals t, t, t, t2,, t n, t n, t, tn t. States SL En 2, En 3 an En 4 orrespon to the follown events : the spefe lfe SL s reahe, fatue falure (FF) or fatue rak eteton (D) take plae. In all these three ases the aquston of new A takes plae. E E 2 E 3... E n- E n E n+ E n+2 (SL) E n+3 (FF) E n+4 (D) E u... q v E 2 u 2... q 2 v 2 E 3... q 3 v 3................................. E n-... u n- q n- v n- E n... u n q n v n E n+... u n+ q n+ v n+ E n+2 (SL) E n+3 (FF) E n+4 (D)......... Fure. Matrx of transton probabltes P AL. 38
In the orresponn transton probablty matrx, P AL, let be the probablty of a rak eteton urn the nspeton number, let q be the probablty of the falure n serve tme nterval t, t, an let u q be the probablty of suessful transton to the next state. In our moel we also assume that an arraft s sare from a serve at t SL even f there are no any rak sovere by nspeton at the tme moment t SL. Ths nspeton at the en of n -th nterval (n state En ) oes not hane the relablty but t s mae n orer to know the state of an arraft (whether there s a fatue rak or there s no fatue rak). Here t s suppose that fatue rak s sovere wth probablty equal to unt f nspeton s mae n nterval ( T, T ). It an be shown that u P( T t T t ), q P( t T T t T t ), u q,,, n. () In the three last lnes of the matrx P AL there are three unts n the frst olumn, orresponn to renewal of an operaton of the arlne (the AL operaton returns to the frst nterval). All the other entres of ths matrx are equal to zero, see F.. Usn the theory of sem-markov proess wth rewars an efnton of P AL we an et the vetor of statonary probabltes, (,..., n 4) whh s efne by the equaton system an the arlne an P AL n4, (2) n4, (3) n n where au b q,,, n, n. (4), n 2,, n 4, a s the rewar efne by the suessful transton from one operaton nterval to the follown one an the ost of one nspeton; b, an orrespon to transton to the states En 3 (FF), En 4 (D) an then to the state E (the ost of FF of A, fatue rak eteton, aquston of new A). Let us note that f a t t, b then n4 n (5) n n n ( t t ) t an Lj t ( n, ) / j efnes the mean step number of Mh to return to the same state E j, F ( n, ) / Ln 3 t, where t tsl / ( n ), s the FFR. If s known we alulate the an as a funton of n, n (, ), an hoose the number orresponn to the maxmum of the an : n ( ) armax n,. (6) Then we alulate FFR as funton of n, F ( n, ), an hoose n n suh a way that for any n the funton ( n, ) wll be equal or less than some value : n F n n An fnally n (, ) mn n: ( n, ), for all n n (, ). (7) F n n (, ) max( n, n ). (8) 39
3. SOLUTION FOR AN UNKNOWN θ In [] the problem of a lmtaton of fatue falure probablty n an operaton of one A (FFP) was onsere usn the efnton of bnary p-set funton 3.. Bnary p-set funton Now let us take nto aount that we onser the ase when the for the estmate of unknown parameter, ( x,..., ) x n, the result of aeptane test s use an the operaton of a new type of arraft wll not take plae f the result of the fatue test n a laboratory s too ba (prevously, the reesn of the new type of A shoul be mae). We say that n ths ase the event, takes plae (for example, f the test fatue lfe T s lower than some lmt; or np (, ) s too lare, ). Let us efne some bnary set funton n ( ), ( S n f S,, n), (9), f where S ( n) {( t, t): t t, t t}, t tsl / ( n ),,..., n; s an empty set. It an be shown that for very we rane of the efnton of the set an the requrements to lmt FFP by the value p, where ( p ) s a requre relablty, there s a prelmnary esne allowe FFP, p, suh that orresponn set funton S(,, n( p, )) s bnary p -set funton of the level p for the vetor Z ( T, T ) on the base of the estmate : n P Z S n p p. () sup ( ( (, )) ) Ths means that FFP wll be lmte by the value p for any unknown. 3.2. Bnary -set funton In smlar way, t an be shown that for very we rane of the efnton of the set an the requrements to lmt FFR of AL by the value, where s a requre fatue falure ntensty, there s a prelmnary esne allowe FFR,, suh that orresponn set funton S(,, n (, )) s a bnary -set funton of the level for the vetor Z ( T, T) on the base of the estmate : sup E(( ( n (, )) ) P( )). () Ths means that FFR wll be lmte by the value for any unknown. Let us note, that nstea of the wors a bnary -set funton we shoul use the wors bnary - set funton f nstea of n (, ) we use n (, ). For the requrement of a hh relablty the hoe of an nspeton number wll be efne by the lmtaton of FFR. For very hh ost of FF of A t wll be efne by the maxmum of the an. 4
4 NUMERIAL EXAMPLE The example of the soluton of the relablty problem of arraft fleet s onsere n [, 2]. Here we onser only the problem of relablty of AL. We use the follown efntons of the omponents of an AL nome: for all,..., n a a ( n ) nsptsl, where a ( n) a tsl / ( n ), - s the rewar relate to suessful transton from one operaton nterval to the follown one, a efnes the rewar of operaton n one tme unt (one hour or one flht); nsptsl s the ost of one nspeton (neatve value) whh s suppose to be proportonal to t SL ; b btsl s relate to FF (neatve value), a ( n ) s the rewar relate to transtons from any state E,..., En to the state En 4 (t s suppose to be proportonal to a beause t s a part of a ); tsl s neatve rewar, the absolute value of whh s the ost of new arraft aquston after events SL, FF or D an transton to E takes plae. In numeral example we have use the follown values: tsl 4, b =-.3; nsp = -.5; a =; =.; =-.3. Suppose we have the follown estmate of parameter (, X Y, X, Y, r ) : =(-8.5868844,.942468,.55,.778895,.796437) (see F 2.2 an Table 2. n [3]). It was assume that the set orrespons to the eson to make reesn f the estmate of rtal tme to falure t exp( ) s too small: t.3t. Y X SL Fure 2. The value of w(,, ) for fve value of Y (.245 Y 2.6435) n vnty of maxmum value of w(,, ) whh s equal to.94-6 for (, X Y, r ) =(.5528668,.778895,.796437). alulaton of w(,, ) E{ F (,, )} was mae for (7.229 X 9.979), assumn that the vetor (,, ) s the same for all fferent vetors (.3972 Y 2.4877) X Y r ( X, Y). It was foun that for =.-6 the maxmum value of w(,, ) s equal to.94-6. Suppose that the value.94-6 s requre relablty. Then for the known estmate of the parameter the alulaton of n (, ) for =.-6 ves us the requre number of nspeton. It s equal to 6. For the onsere estmate of t ( realzaton of T s equal to 4
niav Y. Paramonov, S. Tretyakov, M. Hauka - BINARY LAMBDA-SET FUNTION AND RELIABILITY OF AIRLINE 37.4574e+3 so the reesn s not neee. After the neessary alulaton of n (, ) t s foun n 4. So fnally, the requre number of nspetons max(, ) s equal to 6. n n n 8 7 6 5 my my2 my3 my4 my5 4 Fure 3. The value of n (, ) for fve value of Y (.245 Y 2.6435) for =.-6 as funton of equvalent mean value of T whh was alulate as exp( ) ONLUSIONS The problem of nspeton plannn on the bases of the result of aeptane full-sale fatue test of A struture s the hoe of the sequene t, t2,..., tn, t SL provn the lmtaton of FFR of AL f some requrements to the result of aeptane full-sale fatue test are met. If these requrements are not met the reesn of the new type of an arraft shoul be mae. The efnton of bnary -set funton s ntroue for esrpton of orresponn mathematal proeure, base on the observaton of some fatue rak urn the aeptane full-sale fatue test of arraft struture. In eneral ase the the esre to nrease the an of arlne serve an be taken nto aount but uner onton that requre relablty s alreay prove. The lmtaton of FFR s prove for any unknown parameter of the fatue rak moel. The metho of neessary alulaton s prove. REFERENES 3 2 5 5 mean t x 4. Paramonov Yu, Tretjakov S, Hauka M. Fatue-prone arraft fleet relablty base on the use of a p-set funton. RT&A, #(36) (Vol. ) 25, pp.4-49. 2. Hauka M, Tretjakov S, Paramonov Yu (24). Mnmax nspeton proram for relablty of arraft fleet an arlne. In: Proeens for 8 th IMA Internatonal onferene on Moelln n Inustral Mantenane an Relablty (MIMAR), Oxfor, Insttut of Mathemats an ts Applatons, 2 July 24, pp. 2-24 3. Paramonov Yu., Kuznetsov A., Klenhofs M. (2). Relablty of fatue-prone arframes an omposte materals. Ra: RTU. (http://neenko-forum.or/lbrary/paramonov/relablty_ Paramonov.pf) Y X 42