what is the probability that an item fails before a specified time?

Transcription

1 7 Relably There are varous ssues n relably We have already looked a deermnng he relably of a sysem f he relables of s componens are known Here we are neresed n he me ll falure of pars Informaon abou falure mes s needed n order o guaranee he qualy of producs Typcal quesons nclude: wha s he mean me ll falure? wha s he probably ha an em fals before a specfed me? Tesng Modern producs are usually expeced o las for many years Havng desgned a produc, how do you quckly ge an dea of s me ll falure? You can modfy he mehod of esng, acceleraed lfe esng: compressed-me esng: produc s esed under he usual condons, bu more nensvely eg a washng machne used almos connuously advanced-sress esng; produc s esed under harsher condons han wll suffer n regular use so ha falure wll end o occur earler eg refrgeraor moor run a a hgher speed han f operang whn a frdge Need o assume a relaon beween me ll falure under sressed condons and under normal condons eg 1 week under sress 1 year under normal Ths adds o he uncerany of nferences Whchever mehod of esng s used, s lkely ha you wll no have me o wa unl every sngle em has faled How do you deal wh ems whch are sll workng a he end of he es programme? Maxmum lkelhood mehod for censored Exponenal daa The daa s called censored f s ncomplee, n hs case we can wa long enough for all pars o fal, so we don know all he falure mes If a par has a consan probably of falure per un me, we showed prevously ha he me ll frs falure s gven by an Exponenal dsrbuon Recall ha f 1, 2,, n are a random sample from an Exponenal dsrbuon wh parameer ν, each me has pdf P = νe ν The oal lkelhood s he produc of he ndependen pdf s: Lν P { } ν = ν n exp ν Suppose ha we observe pars for up o me and of he ems fal up o hs me, wh falure mes 1, 2 nf The oher n w = n ems are sll 1

2 workng a me For each em P T > = P Sll workng a me = ν exp ν d = [ exp ν] = e ν as expeced, hs s jus he Posson probably for no evens wh mean ν Ths s he conrbuon o he lkelhood a me from non-faled pars; for he faled pars we know he falure me so he oal lkelhood s [e Lν = ν ν exp ν ] nw = ν exp ν f we defne = for workng pars We can now fnd he maxmum-lkelhood esmaor ˆν for ν: dl n dν = n f ν 1 exp ν ν exp ν = n = ˆν 1 = ˆν = ˆν = Example: 5 componens are esed for wo weeks 2 of hem fal n hs me, wh an average falure me of 12 weeks Wha s he mean me ll falure? Answer: n = 5, = 2, = = 84 weeks So he esmae of he falure frequency s ˆν = = 2 84 = 238/week Hence he mean me ll falure s esmaed o be 1/ˆν = 1/238 = 42 weeks Relably funcon and falure rae or hazard funcon Alhough he pdf, f, descrbes he me ll falure compleely, does no drecly ndcae eher he chance of he par connung o work for a gven perod of me or how he chance of falure depends on he age of he par So we defne 2

3 Relably funcon: R = P T > = fxdx = 1 F = probably of survvng a leas ll age where F s he cumulave dsrbuouncon Falure rae, ϕ = f = rae of falure gven survval ll age R Jusfcaoor he defnon of ϕ Suppose ha he par has survved unl me and we wan o evaluae he condonal probably ha fals before me + δ P T + δ T > = = P T + δ and T > P T > 1 R +δ fxdx δf R In a me δ we expec rae δ = ϕδ falures, so δf R = ϕδ = ϕ = f/r Example: Fnd he falure rae of he Exponenal dsrbuon Soluon The relably s R = νe νx dx = e ν as before Falure rae, ϕ = νe ν e ν = ν, a consan The fac ha he falure rae s consan s a specal lack of ageng propery of he exponenal dsrbuon Some componens behave lke hs, bu for many he falure rae ncreases wh age We need a more flexble model o descrbe falure mes han he Exponenal Deducng he dsrbuorom he falure rae ϕ = f R = df d 1 F = d [ln1 F ] d = ln [1 F ] = ϕd 3

4 Snce F =, ln [1 F ] = ϕxdx Ths can be used o fnd F and hence R and f from ϕ Example Say ϕ = consan = ν Ths gves ln1 F = νdx = ν = F = 1 e ν Then we can ge f usng f = df/d = νe ν, he exponenal dsrbuon, as expeced The Webull dsrbuon The Webull dsrbuon has wo parameers, m and ν, boh posve Falure rae s gven by ϕ = mν m 1 for > So for m = 1 he rae s consan, for m > 1 he rae ncreases wh me ageng, and for m < 1 he rae decreases wh me defecve componens are gradually weeded ou The dsrbuon can herefore be used o model falures where he rae s no consan n me 4

5 Usng he falure rae we have ln1 F = mνx m 1 dx = ν m = F = 1 e νm So he relably s R = e νm and f = mν m 1 e νm The m parameer s a shape parameer and ν s a scale parameer as for he exponenal For m = 1 he Webull dsrbuon gves he exponenal dsrbuon as a specal case For a Webull dsrbuon ln [ lnr] = lnν+m ln so plong ln [ lnr] agans ln should gve a sragh lne, approxmaely Esmang m and ν A graphcal mehod s gven n Chafeld A beer mehod, whch copes wh he case of censored daa, s o use maxmum lkelhood Suppose ha we observe pars for up o me and of he ems fal up o hs me, wh falure mes 1, 2 nf The oher n w = n ems are sll workng a me If he dsrbuon s Webull he lkelhood s gven by: Lν, m = nf = m ν mν m 1 e νm nf [ ] nw e νm m 1 exp ν m The log-lkelhood s ln Lν, m = ln m + ln ν + m 1 ln ν hence he maxmum lkelhood s gven by ln L ν = n ˆν m = ˆν = n m The maxmum wr m generally has o be found numercally m 5