Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION Suinde Kuma, Mukesh Kuma 2,2 Depamen of Applied Saisics, Babasaheb Bhimao Ambedka Univesiy, Lucknow-226025, (India) ABSTRACT Genealized invese Weibull disibuion is widely used in he field of eliabiliy and engineeing fo esimaion of failue ae. The poblem of sess-sengh eliabiliy model is sudied hough esablishing he elaionship among he paamees of he disibuions elaed o sess and sengh of he manufacuing iems.life esing models ae invese exponenial disibuion, invese Rayliegh disibuion and invese Weibull disibuion ae specific cases of genealized invese Weibull disibuion. In his pape, iis consideed ha he sess follows genealized invese Weibull disibuion and sengh follows a powe funcion disibuion. Keywods- Genealized Invese Weibull Disibuion, Hazad Rae, Powe Funcion Disibuion, Sess-Sengh Reliabiliy. I. INTRODUCTION The poblem of inceasing eliabiliy of any sysem is now a well-ecognized and apidly developing banch of engineeing. Reliabiliy sudy is consideed essenial fo pope uilizaion and mainenance of engineeing sysems and equipmens. Thus, i has become moe significan in many fields of indusy, anspo, communicaions echnology, ec., wih he complex mechanizaion and auomaion of indusial pocesses. Unde esimaion and ove esimaion of facos associaed wih eliabiliy may engende gea losses. In he eliabiliy engineeing he pacice of sess-sengh esing is an impoan and ineesing opic of connoaion fo he eseaches. One of he saisical modelsof he sess-sengh esing is he pobabiliy P X Y, which epesens he pefomance of an iem of sengh Y subjec o a sess X. The em sess- sengh was fis inoduced by Chuch and Haies (970). Since hen a lo of wok has been done in his diecion by vaious auhos. Fo a bief eview, one may efe o Downon (973), Tong (974),Kelly (976),Sahe and Shah (98), Chao (982), Awad(986),Chauvedi and Rani(997), Chauvedi and Suinde (999). In he pesen pape, we conside he following genealized invese Weibull disibuion poposed by de. Gosmaoe. al (2009).Thepobabiliy densiy funcion (p. d. f.) is given ( ) f ( ) e x p, 0, whee,, ae hepaamees. (.) 75 P a g e
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 I is assumed ha he andom vaiable X epesens he sess ha an iem faces, follows he (p. d. f.) given a (.) and sengh Y follows Powe funcion disibuion wih (p. d. f.) y g y ; 0 y, 0 (.2) Whee and ae he scale and shape paameesespecively. II. STRENGTH RELIABILITY FOR FINITE STRENGTH A finie ime disibuion should be capable of descibing he andom vaiaions in failue ime of equipmens. The designedlifeime (sengh) of equipmen should only be limied o a finie ange. This is because he sengh of manufacued poduc is always a funcion of he combinaion of a se of he subcomponens ae no likely o have an infinie lifeime. Since, he maximum possible value of Sengh disibuion is. The oal uneliabiliy of he iems is heefoe obained by P X. Alam and Roohi (2003)have emed i as pobabiliy of disase. 2. Theoem If he andom vaiables X and Y follow genealized invese Weibull disibuion given a (.) and Powe funcion disibuion given a (.2), especively hen P X is given by exp, (2.) P X m whee m. 2.2 Poof ( ) P X e x p d x.... (2.2) On subsiuing, we ge x exp( ), whee m. P X m Hence he heoem follows. REMARKS 2.: Table (2.) depics he pobabiliy of disase fo genealized invese Weibull disibuion. I is ineesed o noe ha he pobabiliy of disase inceases wih he incease in value of m. 752 P a g e
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 TABLE(2.) Pobabiliy of disase P ( X ) fo Genealized Invese Weibull disibuion. P ( X ) m γ=0.5 γ= γ=2 γ=5 γ=7 γ=0 γ=20 0. 0.04877 0.0956 0.826 0.39346 0.5034 0.6322 0.86466 0.2 0.0956 0.826 0.32968 0.6322 0.75340 0.86466 0.9868 0.4 0.826 0.32968 0.55067 0.86466 0.9399 0.9868 0.99966 0.8 0.32968 0.55067 0.7980 0.9868 0.99630 0.99966 2.0 0.6322 0.86466 0.9868 0.99995 0.99999 5.0 0.979 0.99326 0.99995 7.0 0.96980 0.99908 0.99999 0.0 0.99326 0.99995 20.0 0.99995 2.2: On consideing, (.) educes o invese Weibull disibuion,, i becomes invese exponenial disibuion and on aking, 2 i gives he (p.d.f.) of invese Rayleigh disibuion. Table (2.2) shows he pobabiliy of uneliabiliy fo he afoesaid disibuions fo diffeen values of m. TABLE (2.2) Specific cases of Genealized InveseWeibull Disibuion. P ( X ) m INVERSE WEIBUL L INVERSE INVERSE RAYLEIGH γ= EXPONENTIAL γ=, β=2 γ=, β= 0. 0.0956 0.0956 0.00995 0.2 0.826 0.826 0.0392 0.4 0.32968 0.32968 0.4785 0.8 0.55067 0.55067 0.47270 2.0 0.86466 0.86466 0.9868 5.0 0.99326 0.99326 7.0 0.99908 0.99908 0.0 0.99995 0.99995 20.0 753 P a g e
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 III. STRESS AND STRENGTH RELIABILITY Fo he sess-sengh model, P Y X when he andom vaiable X and Y follows he (p.d.f s) (.) and (.2), especively is given by he following heoem. 3. Theoem P Y X is given by P ( Y X ) e x p m m e x p ( ) d, (3.) m whee m. PROOF: P Y X f ( x ) g ( y ) d xd y x 0 (3.2) x 0 ( ) x y e x p x d x d y ( ) ( ) x e x p d x x x e x p d x x x 0 0 (3.3) On subsiuing x, in (3.3),we ge e x p ( ) d e x p ( ) d,... (3.4) Le m, hen equaion (3.4) can be wien as ex p ( ) d ex p ( ) d. m m Thus, P ( Y X ) e x p ( m ) m e x p ( ) d. m (3.5) Hence he heoem follows. 754 P a g e
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 3.2 Remak Taking diffeen values of m and in eq. (3.5),obain sess-sengh eliabiliy fo genealized invese Weibull disibuion which is shown in able (3.) whee a n d m 0..TABLE 3.2 Sengh-eliabiliy of an iem fo genealized invese Weibull disibuion. m = =2 =3 =4 =5 =6 =7 =8 0.00 0.992669 0.998008 0.99850 0.99865 0.998757 0.99880 0.998834 0.998858 0.002 0.986723 0.996029 0.997006 0.997337 0.997503 0.997603 0.997669 0.99777 0.004 0.97625 0.99203 0.994024 0.994683 0.99503 0.99522 0.995345 0.995439 0.006 0.966749 0.98827 0.99053 0.992036 0.992530 0.992827 0.993025 0.99367 0.008 0.957959 0.984368 0.988094 0.989397 0.990053 0.990448 0.9907 0.990900 0.0 0.94967 0.980553 0.98547 0.986766 0.987583 0.988075 0.988403 0.988638 0.02 0.9305 0.96935 0.970579 0.973728 0.975330 0.976297 0.976944 0.977407 0.03 0.88672 0.943995 0.956286 0.960883 0.963239 0.964666 0.965622 0.966307 0.04 0.853539 0.926648 0.942560 0.948226 0.95307 0.953779 0.954435 0.955336 0.05 0.827835 0.909838 0.928483 0.935755 0.939532 0.94834 0.94338 0.94449 IV. THE BEHAVIOUR OF HAZARD RATE Conside genealized densiy of invese Weibull disibuion given a (.). The eliabiliy funcion R() fo and specified mission ime (>0) is R ( ) P ( X ) The hazad ae f ( ;,, ) h( ) R( ) ; hen he hazad ae of genealized invese Weibull disibuion i.e. h( ) ( ) e x p e x p (4.) Fom he eq. (4.) we can geneae specific cases of genealized invese Weibull disibuion i.e. REMARKS. On subsiuing,, in eq. (4.), we obain hazad ae of Invese exponenial Disibuion and depiced in figue. 755 P a g e
h() h() Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 h( ) ex p ex p ; 2 (4.2) Fig.. shows ha when inceases hazad ae of invese exponenial disibuion deceases. 0.6 0.5 0.4 0.3 0.2 0. h() θ=0.5 h() θ=0.7 h() θ=0.9 0.0 0 5 0 5 Fig:.Hazad ae of Invese exponenial disibuion. 2. On subsiuing in eq.(4.), we ge hazad ae of Invese Weibull disibuion and depiced in figue 2. ( ) h ( ) ex p ex p... (4.3) 2 The vaious values of,,, we ge Fig.2. Shows he when inceases hazad ae of Invese Weibull disibuion deceases. 0.5 0.4 0.3 0.2 0. 0 0 0 20 h() {β=0.5} h() {β=} h() {β=.5} Fig: 2.Hazad ae of Invese Weibull disibuion fo θ=0.5. 3. On subsiuing, 2 in eq. (4.3), hazad ae of Invese Rayleigh disibuion and depiced in figue3. 756 P a g e
h() Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 2 e x p 3 2 h( ), e x p 2 2... (4.4) Fig.3. Shows he when inceases hazad ae of Invese Rayleigh disibuion deceases. 2.5 0.5 0 0 5 0 5 h() θ=0.5 h() θ= h() θ=.5 REFERENCES Fig: 3.Hazad ae of invese Raleigh disibuion fo θ=0.5,,.5. [] Alam and Roohi (2003): On facing an exponenial sess wih sengh having powe funcion disibuion. Aligah J. of Sa. 23, 57-63. [2] Awad, A.M. and Chaaf, M.K. (986): Esimaion of P(Y<X) in he Bu case: A compaaive sudy. Commun. Sais.-Simul., 5, 389-403. [3] Chao, A. (982): On compaiing esimaos of P(X>Y) in he exponenial case. IEEE Tans. Reliabiliy, R-26, 389-392. [4] Chuch, J.D. and Hais, B. (970): The esimaion of eliabiliy fom Sess-Sengh elaionships. Technomeics, 2, 49-54. [5] Chauvedi and Rani (997): Esimaion pocedues fo a family of densiy funcions epesening vaious life-esing models. Meika ; Vol. 46, (997). [6] Chauvedi and Suinde (999): Chauvedi, A. and Suinde, K. (999): Fuhe emaks on esimaing he eliabiliy funcion of exponenial disibuion unde Type-I and Type-II censoings. Bazilian Jounal of Pobabiliy and Saisics, 3, 29-39. [7] De.Gusmao F.R.S., Oega M.M.E and Cadeio M.G.(2009): The genealized invese Weibull disibuion: 6, 625.Spinge - [8] Downon, F. (973): The esimaion of P{Y<X} in he Nomal case. Technomeics, 5, 55-558. [9] Kelly, G.D., Kelley, J.A. and Schucany, W.R. (976): Efficien esimaion of P(Y<X) in he exponenial case. Technomeics, 8, 359-360. [0] Sahe, Y.S. and Shah, S.P. (98): On esimaing P(X<Y) fo he exponenial disibuion. Commun.Sais., A0, 39-47. [] Tong, H. (974): A noe on he esimaion of P(Y<X) in he exponenial case. Technomeics, 6, 625. 757 P a g e