Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions

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1 Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur college, P.G. Cetre, Waragal Departmet of Mathematcs, Kaatya Uversty, Waragal. ABSTRACT: I ths paper cosdered a stuato where stress ad stregth follow fte mxture of expoetal dstrbutos to fd the relablty of a system. It has bee studed whe stress follow expoetal dstrbuto ad stregth follow fte mxture of expoetal dstrbutos ad both stress-stregth follow the fte mxture of expoetal dstrbutos. The geeral expresso for the relablty of a system s obtaed. The relablty s computed umercally for dfferet values of the stress ad stregth parameters. We estmate the parameters of the relablty stress-stregth models by the method of maxmum lelhood estmato. The role of fte mxture of expoetal dstrbutos s llustrated usg a real lfe data o tme to death of two groups of leuaema patets. KEYWORDS: Expoetal dstrbuto, Fte mxture of expoetal dstrbutos, Maxmum Lelhood Estmato, Relablty, Stress-Stregth model.. INTRODUCTION Relablty of a system s the probablty that a system wll adequately perform ts teded purpose for a gve perod of tme uder stated evrometal codtos []. I some cases system falures occur due to certa type of stresses actg o them. Thus system composed of radom stregths wll have ts stregth as radom varable ad the stress appled o t wll also be a radom varable. A system fals wheever a appled stress exceeds stregth of the system. I a fte mxture model, the dstrbuto of radom quatty of terest s modelled as a mxture of a fte umber of compoet dstrbutos varyg proportos []. The flexblty ad hgh degree of accuracy of fte mxture models have bee the ma reaso for ther successful applcatos a wde rage of felds the bologcal physcal ad socal sceces. The estmato of relablty based o fte mxture of pareto ad beta dstrbutos was studed by Maya, T. Nar (007)[]. I relablty theory, the mxture dstrbutos are used for the aalyss of the falure tmes of a sample of tems of coheret laser used telecommucato etwor. I a expermet, oe hudred ad three laser devces were operated at a temperature of 70 degree Celsus utl all had faled. The expermet was ru loger tha oe year before all the devces had faled, because most of the devces were extremely relable. The sample thus cossts of two dstct populatos, oe wth a very short mea lfe ad oe wth a much loger mea lfe. Ths ca be cosdered as a example of a mxture of two expoetal dstrbutos wth probablty desty fucto of the form f ( x) p exp( x) ( p) exp( x), 0 p, 0,, The above model wll be useful to predct how log all maufactured lasers should be lfe tested to assure that the fal product cotaed o devce from the fat mortalty populato. I the preset paper we dscuss the statstcal aalyss of fte mxture of expoetal dstrbutos the cotext of relablty theory. We gve the defto ad propertes of the fte mxture of expoetal dstrbutos. We derve the relablty, whe the stregth X follows fte mxture of expoetal ad the stress Y taes expoetal ad fte mxture of expoetal. We dscuss estmato procedure for fte mxture of expoetal dstrbutos by the method of maxmum lelhood estmato ad also estmato of stress-stregth relablty. We llustrate the method for a real data o survval tmes of leuaema patets ad fally gve the cocluso. Iss November 0 Page 9

2 Estmato Of Stress- Stregth Relablty Model... II. STATISTICAL MODEL: The assumptos tae ths model are () The radom varables X ad Y are depedet. () The values of stress ad stregth are o-egatve. If X deotes the stregth of the compoet ad Y s the stress mposed o t, the the relablty of the compoet s gve by [], x R P( X Y) g( y) dy f ( x) dx 00 () where f(x) ad g(y) are probablty desty fuctos of stregth ad stress respectvely. A fte mxture of expoetal dstrbuto wth -compoets ca be represeted the form f ( x) p f ( x) p f ( x)... p f ( x) () The r th where p 0,,,..., p momet of the mxture of two expoetal dstrbutos ( r r E x ) x p exp( x) p exp( x) dx Whe r =, Whe r =, 0 ( r) ( r) p ( p ) r r Ex ( ) p p p ( p ) Ex ( ) 0 p, 0,, Thus the varace s gve by p ( p ) ( p )( p ) p ( p ) V( x) I ths paper we are cosderg two cases. They are () Stress follows expoetal dstrbuto ad stregth follows fte mxture of expoetal dstrbutos. () Stress ad stregth follows fte mxture of expoetal dstrbutos. III. RELIABILITY COMPUTATIONS: Let X be the stregth of the -compoets wth probablty desty fuctos f (x); =,,..,. Stregth X follows fte mxture of expoetal dstrbuto wth pdf f ( x) p exp( x), 0, x 0, p 0,,,..., ; p..case() The stress Y follows expoetal dstrbuto: Whe the stress Y follows expoetal wth pdf g( y) exp( y), y 0, 0 As a specal case of () wth =, we have f ( x) p exp( x) ( p ) exp( x), 0, x 0,(,) Ad f X ad Y are depedet, the the relablty R from () x exp( ) exp( ) ( ) exp( ) R y p x p x dydx 00 () Iss November 0 Page 40

3 Estmato Of Stress- Stregth Relablty Model... R p ( p ) (4) As a specal case of () wth =, we have f ( x) p exp( x) p exp( x) p exp( x), 0, x 0,(,,); p x exp( ) exp( ) exp( ) exp( ) R y p x p x p x dydx 00 R p p p (5) f ( x) p f ( x) p f ( x)... p f ( x) I geeral from (), we get R where p 0,,,..., p p (6) From table ad table ad fgs. ad, t s observed that f stress parameter creases the the value of relablty creases, f stregth parameter creases the the value of relablty decreases...case () The stress Y follows fte mxture of expoetal dstrbutos: For =, we have f ( x) p exp( x) ( p ) exp( x),, 0 g( y) p exp( x) ( p ) exp( x),, Ad f X ad Y are depedet, the the relablty R from () x exp( ) ( ) exp( ) exp( ) ( ) exp( ) R p y p y p x p x dydx p p p ( p ) ( p ) p ( p )( p ) For =, we have 4 4 f ( x) p exp( x) p exp( x) p exp( x), 0, x 0,(,,); p The x R p exp( y) p exp( y) p exp( y) p exp( x) p exp( x) p exp( x) dydx R p p p p p p (7) Iss November 0 Page 4

4 R Estmato Of Stress- Stregth Relablty Model... p p p p p p p p p p p p pj p j j I geeral from (), The R pj p j j f ( x) p f ( x) p f ( x)... p f ( x) where p 0,,,..., p From table ad table 4 ad fgs. ad 4, t s observed that f stress parameter creases the the value of relablty decreases, f stregth parameter creases the the value of relablty creases... Hazard Rate: Let t deotes lfe tme of a compoet wth survval fucto S(t). The the survval fucto of the model s obtaed as S( t) p exp( t) ( p ) exp( t), 0, t 0, 0 p For the model the hazard rate h(t) s gve by f() t ht () st () I geeral, p exp( t ) ( p ) exp( t) p exp( t) ( p )exp( t) ht () p exp( t) p exp( t) () Fte mxture of expoetal possesses decreasg hazard rate ad costat hazard rate depedg upo the values of the parameters. Fg 5, show the behavour of hazard rate at varous tme pots. IV. ESTIMATION OF PARAMETERS: We estmate the parameters of the models by the method of maxmum lelhood estmato.cosder the stuato whe there are oly two sub populatos wth mxg proportos p &(-p ) ad f (x) ad f (x) are expoetal destes wth parameters & respectvely. The lelhood fucto s gve by L(,, p / y) p exp( y j) ( p) exp( y j) j () Where y j deoted the falure tme of the j th ut belogg to the th sub populato j=,,... ; =, ad y y, y,... y ; y, y,... y Maxmzato of log lelhood fucto of () w.r.t the parameters yelds the followg equato L p ( p ) exp( y ) exp( y ) j j j j (8) (9) (0) Iss November 0 Page 4

5 Estmato Of Stress- Stregth Relablty Model... L c p ( p ) exp y j y j j j log( L) log c log( p ) log( p ) y y j j j j 0, 0, 0 yj yj j j p p j y j j y j p where The above results ca be geeralzed for ay, gvg the followg estmators p & j y j () (4) (5) (6) (7) 4.. Estmato of Stress Stregth relablty: () Whe the stregth X follows fte mxture of expoetal dstrbutos wth parameters ad p ad the stress Y follows expoetal dstrbuto wth parameter, the the M.L.E of R s gve as R p (8) () Whe the stregth X follows fte mxture of expoetal dstrbutos wth parameters ad p ad the stress Y follows fte mxture of expoetal dstrbuto wth parameter j ad p j the the M.L.E of R s gve as R pj p j (9) V. DATA ANALYSIS: We cosder a data o tme to death of two groups of leuaema patets whch s gve Table 5 (see Fegl ad Zele, 965) to llustrate the procedure of estmato. We the estmate the parameters usg M.L.E techque. Table 6 provdes the values of the estmates by M.L.E method. Table 7 provdes the maxmum lelhood estmate of survval fucto at varous tme pots. Table 8 provdes the hazard rate fucto at varous tme pots. Iss November 0 Page 4

6 Estmato Of Stress- Stregth Relablty Model... Case () Stress has expoetal dstrbuto ad Stregth has mxture two of expoetal dstrbutos: Table Table P = R P R P =P Case() Stress-Stregth has mxture two of expoetal dstrbutos: Table Table 4 4 R P =P = 4 R Table 5 Survval tmes of leuaema patets AG +ve AG ve Table 6 Estmates of parameters of survval tmes of leuaema patets p 5 Iss November 0 Page 44

7 Relablty Relablty Relablty Estmato Of Stress- Stregth Relablty Model... Table 7 Maxmum lelhood estmate of survval probablty at varous tme pots T St () Table 8 Hazard rate at varous tme pots T h(t) Case () Stress has expoetal dstrbuto ad Stregth has mxture two of expoetal dstrbutos: 0.95 Fgure Stress Parameter 0.9 Fgure Stregth Parameter Case() Stress-Stregth has mxture two of expoetal dstrbutos: 0.9 Fgure Stress parameter Iss November 0 Page 45

8 Hazard rate Relablty Estmato Of Stress- Stregth Relablty Model Fgure Stregth parameter Hazard rate fucto: 5 Fgure Lfe tme VI. CONCLUSION: The role of fte mxture of expoetal dstrbutos relablty aalyss s studed. We derve the relablty, whe the stregth X follows fte mxture of expoetal ad the stress Y taes expoetal ad fte mxture of expoetal. It has bee observed by the computatos ad graphs, case() f stress parameter creases the the value of relablty creases, f stregth parameter creases the the value of relablty decreases. Where as case() f stress parameter creases the the value of relablty decreases, f stregth parameter creases the the value of relablty creases. We developed estmates of parameters usg Maxmum lelhood estmato. The role of fte mxture of expoetal dstrbutos s llustrated usg a real lfe data o tme to death of two groups of leuaema patets. REFERENCES: [] Kapur, K.C ad Lamberso, L.R.: Relablty Egeerg Desg, Joh Wley ad Sos, Ic., U.K. (997). [] Sha, S.K. (986). Relablty ad Lfe Testg, Wley Easter Lmted, New Delh. [] Maya, T.Nar (007). O a fte mxture of Pareto ad beta dstrbutos. Ph.D Thess submtted to Coch Uversty of Scece ad Techology, February 007. [4] Fegal, P. ad Zele, M. (965). Estmato of survval probabltes wth cocomtat formato. Bometrcs, Iss November 0 Page 46

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