Accelerated Life Testing in Interference Models with Monte-Carlo Simulation

Transcription

1 Global Journal of ure and ppled Mathematcs. ISSN Volume 3, Number (07), pp esearch Inda ublcatons ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka 3* ssstant rofessor, College of Fsheres, ssam grcultural Unversty, ssam, Inda. rofessor, Department of Statstcs, Dbrugarh Unversty, ssam, Inda. 3* ssstant rofessor, Department of Statstcs, Dbrugarh Unversty (*Correspondng uthor) bstract Here we have presented two accelerated Lfe Testng (LT) models for nterference theory of relablty. We have assumed that nstead of a sngle stress, as faced by a system n normal operatng condtons, for accelerated condtons a number of stresses are appled to the system smultaneously. In the frst model we assume that the system fals f the sum of the stresses exceeds the strength of the system and assumpton n the second s that the system fals f maxmum of the stresses exceeds the strength. For the frst model assumng that both stress and strength follow ether exponental or gamma or normal dstrbutons and n the second, generalzed exponentals, we have obtaned the relablty, say (), of the system n the accelerated condton. The expressons show that when the number of stresses ncreases, decreases or falure probablty ncreases and one may get more falure data quckly, justfyng the models. Usng Monte-Carlo smulaton we have estmated. nother estmate of s obtaned from proporton of successes. From we obtaned estmates of relablty, say, at use level. Some numercal values of are tabulated for partcular values of the parameters of stress-strength dstrbutons. The numercal values also justfy the use of the models. Keywords: Stress-strength, relablty

2 734.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka. INTODUCTION The more relable a devce s, the more dffcult t s to measure ts relablty. For the systems whose relablty s studed from stress-strength models.e. nterference models the stuaton s stll more desperate. Ths s so because many years may be decades or century, of testng under actual operatng condtons would be requred to obtan numercal measures of ts relablty. Even f such testng were feasble, the rate of techncal advance s so great that those parts would be obsolete by the tme ther relablty had been measured. One approach to solvng ths predcament s to use accelerated lfe tests, n whch parts are operated at hgher stress levels than requred for normal use, Mann et al. (974). Today s manufacturers face strong pressure to develop new, hgher-technology products n record tme, whle mprovng productvty, product feld relablty and overall qualty. Estmatng the falure tme dstrbuton or long term performance of components of hgh relablty product s partcularly dffcult. Most modern products are desgned to operate wthout falure for years, decades or longer. Thus few unts wll fal or degrade apprecably n a test of any practcal length of tme at normal use condtons. For example, the desgn and constructon of a communcatons satellte may allow only eght months testng components that are expected to be n servce for 0 or 5 years. For such applcatons ccelerated Lfe Testng (LT) are used n manufacturng ndustres to asses, to demonstrate components and subsystems relabltes, to certfy components, to detect falure modes so that they can be correlated, to compare dfferent manufacturers, and so on and so forth. LTs have become ncreasngly mportant because of rapdly changng technologes, more complcated products wth more components, hgher customers expectatons for better relablty and the need for rapd product development, Escobar and Meeker (006). n LT s one whch s conducted at a stress level usually hgher than that whch we expect to occur n the use envronment. For nstance t s sometmes sad that hgh temperature s the enemy of relablty. Increasng temperature s one of the most commonly used methods to accelerate a falure mechansm. In a bologcal context t could be the number of rad of radaton delvered to anmals n radaton experments desgned to extrapolate lfe tme test results at hgh stress or dose to lfetmes at low-radaton-dose envronments, Barlow (98). But Evans (977) made an mportant pont that the need to make rapd relablty assessments and the fact that LTs may be the only game n town are not suffcent to justfy the use of the method. Justfcaton must be based on physcal models or emprcal evdence. Gven test results, the man problem s to relate lfetmes at varous stress levels and to predct lfe at usually much lower stress level, Barlow (98). Some accelerated tests use more than one acceleratng varables (or stresses). Such tests mght be suggested when t s known that two or more potental acceleratng

3 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton 735 varables contrbute to degradaton and falure. Usng two or more varables may provde needed tme acceleraton wthout requrng levels of the ndvdual acceleratng varables to be too hgh. For example t s a common knowledge that a certan jont may stand a partcular shock or vbraton at a low temperature but t may fal even by a smaller shock at hgher temperature. Thus n LT more severe stresses than the normal s used n the hope to get falure data quckly. In nterference models stress and strength both are random varables, so more severe stress has no meanng. But f we use a sum of a number of random stresses nstead of a sngle stress we may expect to get more falure data.e. the combnaton of more than one stresses may gve more falure data. The combnaton of stresses may be lnear or non-lnear. Here, we shall assume that the effect of dfferent stresses on the system, under consderaton, s lnear. Here n Sec. we have consdered an LT for Interference models where nstead of a sngle stress a lnear combnaton of m stresses work, smultaneously, on the system under study. ll the stresses and the strength of the system are represented by ndependent random varables. For smplcty we have assumed that all the stresses are dentcally dstrbuted and obtaned system relablty at combned stresses and then obtaned relablty for a sngle stress.e. for use condton. Of course one may also use dfferent dstrbutons for stresses. The dstrbutons consdered here are exponental, gamma and normal. We have obtaned the estmates of the parameters nvolved by usng Monte Carlo Smulaton (MCS) and there by gettng estmates of system relablty. In Sec.3 we have consdered another model where more than one stresses are appled smultaneously and the system works f maxmum of these stresses s less than the system strength. In partcular we have assumed that stressstrength are generalzed exponental varates. Most of the LT studes n the lterature are for tme-to-falure (TTF) models. Here, generally, a falure tme dstrbuton of the unt under study s consdered and then a relatonshp between the parameters of ths dstrbuton and envronmental condtons (stress) s assumed. But we have not come across any LT studes for Interference models accept an old study vz. Kakat and Srwastav (986).. N LT FO INTEFEENCE MODELS Suppose we use a sum of m (fxed) ndependently and dentcally dstrbuted stresses Y, Y,, Ym, nstead of a sngle stress Y (say), on a component (or system) wth strength X (say). Let Y = Y+Y+ + Ym. Then the relablty,, of the system wll be gven by = [X > Y]. (.). Exponental stress-strength Let us assume that X and each Y s follow exponental dstrbutons wth parameters and (wthout loss of generalty), respectvely. Then we know that Y wll follow a

4 736.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka gamma dstrbuton wth degrees of freedom m, so the p.d.f. s of X and Y, vz. f(x) and g(y), are gven by f(x) = x e and g(y) = m y y e (m). (.) Then from Eq.(.), the relablty of the system at accelerated stress Y s gven by = m. (.3) From Eq.(.3) t s obvous that as m (the number of stresses) ncreases the relablty decreases and 0 as m and so the probablty of falure (= ) ncreases and expectedly we may get more falure data quckly. The relablty,, at the use level or actual level of stress s gven by = = ()/m. (.4) We use Monte Carlo Smulaton (MCS) to estmate. To use MCS we proceed as follows: We estmate and n two ways by MCS. () For a partcular values of and m we have taken random samples of sze M from exponental ( ) and gamma (m) dstrbutons for X and Y, respectvely, usng MTLB. Means of samples of X and Y gve the estmates of the parameters and m, respectvely. Then substtutng these estmates n Eq.(.3), we get the estmates (say) of. Then from Eq.(.4) the estmate of, say, s gven by the m th root of. () lternatvely, for the same sample as n () each value of X s compared wth correspondng value of Y and whenever an X s greater than or equal to Y, t wll be called a success. Then another estmate of, say, s gven as Number of Successes (.5) n Obvously, for a gven m, the correspondng estmate of, say s gven by = ( ) /m (.6) If the above two processes are replcated k tmes then we get estmated relablty data sets,, and of sze k. For partcular values of and m takng M = 00, 500, 000 and k = 00, we have obtaned means and s.d. s of,, and tabulated n the Table. Then, we have drawn normal probablty plot (N) graphs for each data set of estmated relablty and for dfferent parameter values of and m as gven n Table. ll N

5 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton 737 graphs suggests that the dstrbuton of and are normal. For llustraton purpose, we have gven only one N graph for data sets and when = and m = n Fg.. We have also appled tests of sgnfcance vz. z-test (snce k > 30) for a gven and m, to test the sgnfcance of dfference between true gven by Eq.(.4) and correspondng. Smlarly, we have used z-test to test the sgnfcance of the dfference between and for the same and m. ll these values of z s vz. z for and, z for and Note: In Fg., we have wrtten Estmated for are also gven n Table. ; smlarly n all the graphs = m = k = 00 M = = m = k = 00 M = robablty 0.50 robablty Estmated Fgure : (Exponental S-S, Sec..) M k m Table : Exponental stress-strength Mean of SD of Mean of SD of z for z for

6 738.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka In Table, we see that for both the estmator and, z and z are nsgnfcant except for M = 00, z s sgnfcant when =, m =, k =00. We have ncreased M to 500 and then z for the same values of, m and k, becomes nsgnfcant.e. accuracy of estmator ncreases wth n whch s expected; but for M = 000 there s not much change. So, we have decded to take M = 500 throughout. Obvously apprecably decreases wth m.e. falure probablty ncreases so we may get more falure data. For example, = for m = but = for m = 3 when =, M = 500 and k = 00.. Gamma stress-strength Suppose X ~ (, ) and Y ~ (, ), =,,, m, then from the reproductve property of gamma dstrbuton Y (=Y+Y+ +Ym) ~ (,m ). Therefore, f s an nteger [andt and Srwastav (975)], or f = ( m ). (.7) m 0 (m )( )! and are not necessarly ntegers, Kapur and Lamberson (977), then = Beta(, m )Beta / (, m ) (.8) = Beta(, )Beta / (, ) (.9) From Eq.(.7) we see that f m s much larger than then (m ) ~ ( + m ) and then ~ m 0( )!, (.0) whch 0 as m. So that as m ncreases relablty decreases or falure probablty ncreases, thereby we may get more falure data quckly. Further, from Eq.(.0) ~ ( )!, for m =. (.) 0 Then, from Eq.(.0) and Eq.(.) ~ 0 m m 0 whch mples that n ths case also, (.)

7 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton 739 ~ m (). (.3) We may expect smlar relaton to hold approxmately for and gven by Eq.(.8) and Eq.(.9) also. Now, as n Sec.. from MCS we can estmate of and m. Snce the estmates of and m may not be ntegers, so we use Eq.(.8) nstead of Eq.(.7). Substtutng estmated values of and m n Eq.(.8) we get the estmate, say, of relablty at accelerated stress. The estmate of for use level of stress, s gven by ˆ = m, (.4) m and the estmate, say, of relablty at use level of stress s gven by substtutng ˆ for and ˆ for n Eq.(.9). Smlarly as n Eq.(.5) and Eq.(.6), by MCS, we can get an alternatve estmate of by. For gven n, k,, we have obtaned means and s.d. s of and, and tabulated n Table. We have drawn N graphs for each data set of estmated relablty and for dfferent values of, and m as gven n Table. ll N graphs suggest that the data sets of and follow normal dstrbuton. For llustraton purpose, we have gven a N graph for data sets and when =, = and m = 3 n Fg.. We have performed z-test also, as n the last secton, and the values of z s are gven n the same Table. Table : Gamma stress-strength M k m Mean of SD of Mean of SD of z for z for

8 740.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka We have taken M = 500, as we saw n the last secton that for M = 500, the estmates are good enough. In Table 6., we see that z s sgnfcant but z s nsgnfcant for k =00 when =, = and m =. So, for achevng better estmator, we have taken k = 00 throughout. Then we observe that all z values are nsgnfcant for all cases. For example, z = and z = for =, =, m = 3 s smaller than the tabulated z =.96 at 5% level of sgnfcance. We also observe that the values of decrease when m ncreases whch s expected. For example, = for m = and = for m = 3 when =, =. The nsgnfcant value of z also ndcate the relaton Eq.(.3) s true robablty M = 500 k = 00 = = m = 3 robablty M = 500 k = 00 = = m = Estmated Fgure : (Gamma S-S, Sec..).3 Normal stress-strength Suppose X ~ N (, ) and Y ~ N (, ), =,,, m, then we know that Y ~N (m, m ) where Y=Y+Y+ +Ym. Then, andt and Srwastav (975), = m m. (.5) Obvously, as n the above cases, 0 as m. So the falure probablty of the system wll ncrease wth m and hopefully gvng more falure data quckly. Further, from Eq.(.5)

9 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton 74 =, for m =. (.6) From Eq.(.5) and Eq.(.6) we see that here we cannot fnd a relaton lke Eq.(.3).e. n ths case m (), (.7) whch s reflected n numercal evaluaton also [see Table 3(a)] s above for gven,,,, m, n and k usng MCS we can estmate, and m. Then the estmates of and, at the use level of stress, vz. are gven by ˆ = (m ) m and, m ˆ and ˆ, ˆ = (m ). (.8) m Substtutng ˆ and ˆ for and n Eq.(6..7) we get system relablty at use level of stress., the estmate of obtaned as n Eq.(.5) gves another estmate of. But estmate of from cannot be obtaned from Eq.(.6) because of Eq.(.7). Here, we have obtaned from usng Eq.(.6) and note that the dfference between and s hghly sgnfcant [see Table 3(a)]. Even ncreasng the values of M does not mprove the estmator. Of course, correspondng z s nsgnfcant. Thus data also show that Eq.(.7) s true. Table 3 (a): Normal stress-strength n k m Mean of SD of Mean of SD z for z for Further, we have obtaned means and s.d. s of and whch are tabulated n the Table 3(b). We have drawn N graphs for each data set of estmated relablty and for dfferent values of,, and m as gven n Table 3(b). The N graphs suggest that the dstrbuton of and are normal. For llustraton, we have gven only one such graph n Fg.3 correspondng to =, =, =, =, m =. s earler, we have used z-test for each case; z values are also tabulated n Table 3(b).

10 74.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka robablty = = = = m = M = 500 k = 00 robablty = = = = m = M = 500 k = Estmated Fgure 3: (Normal S-S, Sec..3) Table 3 (b): Normal stress-strength M k m Mean of SD of Mean of SD z for z for In Table 3(b), we see that all z-values for both the estmator for and for are nsgnfcant n all cases. For example, z = 0.3 and z =.4600 for =, = 3, =, = and m = are smaller than tabulated z =.96, so, z and z are nsgnfcant. We also see that value decreases wth ncreasng m whch s expected. For example, = 0.4 for m = but = 0.07 for m = 3 when =, =, = and =.

11 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton NOTHE LT MODEL FO INTEFEENCE THEOY Kakaty and Srwastav (986) used an LT for nterference models n a dfferent way. They assumed that a number of random stresses are appled, smultaneously, to a system under test. If the maxmum of these stresses s more than the strength of the system then the system fals otherwse t survves. They assumed that all the stresses are..d. exponental varates and strength s another exponental varate. Here we have consdered the same model for generalzed exponental dstrbutons, Gupta and Kundu (997). 3. Mathematcal formulaton Let Y, Y,, Ym be m..d. random varables representng m stresses smultaneously appled on a system whose strength s represented by another r.v. X. Let G(y) be be the c.d.f. of Y s and f(x) be the p.d.f. of X. Then as per assumpton of the model the system survves f Max (Y, Y,, Ym) < X. Then relablty of the system at accelerated stress s gven by = [Max (Y, Y,, Ym) < X] = G(x) m f(x)dx. (3.) We note that Eq.(3.) s a decreasng functon of m.e. as m, the number of stresses appled smultaneously, ncreases the survval probablty of the system decreases and 0 as m. In other words falure probablty (= ) ncreases wth m and hopefully we shall get more falure data quckly. Ths justfes the use of more than one stresses smultaneously. 3. Stress - strength are generalzed exponental varates Let us assume that both X and Y, =,,,m follows generalzed exponental dstrbutons (GED) wth same scale parameters but dfferent shape parameters wth p.d.f. s f(x) and g(y), respectvely, gven by x x f(x,, ) e e ; x 0, 0, >0 (3.) where, s the scale parameter and s the shape parameter. y y g (y,, ) e e ; y 0, 0, > 0 (3.3) where, s the scale parameter and s the shape parameter.

12 744.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka Obvously, the c.d.f. of Y s y G (y,, ) e. (3.4) Let Y = Max( Y, Y,, Ym) (3.5) Then, the c.d.f. of Y, G(y), s gven by Suppose G(y) = m [G (y )] (3.6) G(y) = G(y) = = Gm(ym) = G(y), say Then G(y) = [G(y)] m = y m ( e ), (3.7) whch s the c.d.f. of a generalzed exponental varate wth scale parameter and shape parameter m Then, from Eq.(3.) the relablty of the system at accelerated stress s m 0 x x x e e e dx. (3.8) Substtutng z = x e n Eq.(4.3.8) we get = z m 0 dz = m. (3.9) Obvously, 0 as m, so as number of stresses (m) ncreases relablty decreases.e. falure probablty ncreases hopefully gvng more falure data quckly. Now from Eq.(3.9) the relablty of the system by the applcaton of a sngle stress.e. relablty of the system at use level of stress, (say), s gven by =, when m = (3.0) If ˆ and ˆ are estmates of and then the estmate of n Eq.(3.9) s gven by = ˆ ˆ m (3.) and that of n Eq.(4.3.0) by = ˆ ˆ ˆ. (3.)

13 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton 745 If we have samples of szes M from populatons gven by Eq.(3.) and Eq.(3.3), respectvely, then the MLE s of and are gven by Kundu and Gupta (005) ˆ M M x log( e ) and ˆ M M y log( e ), respectvely. (3.3) Snce s a one-to-one functon of, hence as per the propertes of MLE s s a MLE of. It seems dffcult to obtan the mean and the varance of, analytcally. We have used Monte-Carlo Smulaton to obtan and ts mean and varance. To use MCS we proceed as follows: The c.d.f. of X from Eq.(3.) s where Z s unform n (0, ). x F(x) e Z (say), (3.4) x x Now, z e or e z or x log( z ) x log( z ) (3.5) Thus by takng M unformly dstrbuted random values of Z, from Eq.(3.5) we can obtan a random sample of sze M of the r.v. X followng generalzed exponental dstrbuton. Now we apply m stresses, whch are dentcally dstrbuted as generalzed exponental varates wth parameters (, ), smultaneously, and we assume that f the maxmum of these stresses exceeds the strength of the system the system fals. From Eq.(3.5) Y represents ths maxmum and ts dstrbuton s gven by Eq.(3.7) wth parameters (,m ). So, frst we take a set of m unformly dstrbuted values of Z and obtan m yj s from the formula [as n Eq.(3.5)]. yj = log( z ), j =,,,m. (3.6) Then the maxmum of frst set of m yj s gven by Eq.(3.6) s, say y. We repeat ths process M tmes and get y, y,, ym. s noted above, the dstrbuton of these y s s generalzed exponental wth parameters (, m ). Then the estmate of m s gven by [see Eq.(3.3)]

14 746.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka m M M y log( e ) where y s are the maxmums of yj s gven by Eq.(3.6) Then the estmate of s gven by, (3.7) ˆ m (3.8) m The estmates ˆ of s gven by Eq.(3.3) where x s are gven by Eq.(3.5). Consequently estmates of relablty wth m (known) smultaneous stresses and wth a sngle stress s gven respectvely by Eq.(3.) and Eq.(3.), respectvely. lternatvely, we can also estmate by where = Number of cases out of M where [X > Y] / M. (3.9) s earler the whole process s repeated k tmes. We have obtaned relabltes estmates gven by Eq.(3.), Eq.(3.) and Eq.(3.9) for dfferent values of,,, m, M and k and obtaned ther mean and standard devaton and tabulated them n the Table 4. We have plotted N graphs n each case and found that the dstrbutons of and are normal. For llustraton purpose we have gven one such N graph n Fg.4 correspondng to = 5, m =, =, =, M = 500, k = 00. s earler we have used z-test to test the sgnfcance of dfference between, and,. The z-values are also gven n the same Table 4. Table 4: Generalzed exponental stress-strength M k m Mean of SD of Mean of SD of z for z for In Table 4, we note that z s nsgnfcant but z s sgnfcant when M = 500, k = 00, = 5, m =, = and =. So for achevng better estmator we have taken k =00 and found that all z-values are nsgnfcant. For example, z = 0.46, z =

15 ccelerated Lfe Testng n Interference Models wth Monte-Carlo Smulaton are smaller than z <.96 when k = 00, m =, = 5, = 3 and =. So we have taken M = 500 and k = 00 throughout. In Table 4, we see that system relablty at accelerated stress ncreases when ncreases. For example, = for = but = for =3 when = 5 and =. Smlarly, decreases when decreases. For example, = for = but = for = 3, = 5 and = M = 500 k = 00 = = = m = M = 500 k = 00 = = = m = robablty robablty Estmated Fgure 4: (Generalzed exponental S-S, Sec.3.) 4. CONCLUSION We have studed here two LT models and the numercal values of relablty, n each case, justfy them as such. Here nstead of a sngle stress a number of stresses are appled to get the accelerated effect. In the st model the effect of stresses s addtve. In the second model we have assumed that maxmum of the stresses cause the system to fal. We have consdered exponental, gamma and normal dstrbutons for the frst model and generalzed exponental for the second model as stress-strength dstrbutons. ny other dstrbuton may also be consdered for both the models. For some systems even a mnmum level of stress s requred for the system to work, so mnmum of the stress s appled may also be consdered.

16 748.N. atowary, J. Hazarka, G.L. Srwastav and.j. Hazarka EFEENCES [] Barlow,.E. (98): ccelerated Lfe Tests and Informaton, adaton esearch, Vol. 90, pp [] Escober, L.. and Meeker, W.Q. (006): evew of ccelerated Test Models, Statstcal Scence, Vol., No. 4, pp [3] Evans,.. (977): ccelerated Testng: The only Game n Town, IEEE Transacton on elablty, Vol.6, pp.4. [4] Gupta,.D. and Kundu, D. (999): Generalzed Exponental Dstrbutons, ustralan and New Zealand Journal of Statstcs, Vol.4, pp [5] Kakaty, M. C. and Srwastav, G. L., (986): ccelerated Lfe Testng under Stress-strength Model, Indan ssocaton of roductvty, Qualty and elablty Transacton, Vol., No.-, pp [6] Kapur, K. C. and Lamberson, L.., (977): elablty n Engneerng Desgn, John Wley and Sons, New York. [7] Kundu, D. and Gupta,. D., (005): Estmaton for Generalzed Exponental Dstrbuton, METIK, Vol.6, pp [8] Mann, N.., Schafer,. E. and Sngpurwalla, N. D., (974): Methods for Statstcal nalyss of elablty and Lfe Data, John Wley and Sons, New York. [9] andt, S. N. N. and Srwastav, G. L., (975): Studes n Cascade elablty- I, IEEE Transacton on elablty, Vol.-4, No., pp