Reliability Estimate using Degradation Data

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1 Reliabiliy Esimae using Degradaion Daa G. EGHBALI and E. A. ELSAYED Deparmen of Indusrial Engineering Rugers Universiy 96 Frelinghuysen Road Piscaaway, NJ USA Absrac:-The use of degradaion daa o esimae reliabiliy is an alernaive o he use of failure daa when no failures or few failures are epeced o occur in a life esing under normal or acceleraed condiions. Models for reliabiliy esimae using degradaion daa can be classified ino wo groups: physics-based (or eperimenallybased and saisics-based models. The applicaion of a physics-based model is limied o hose unis ha ehibi he same underlying degradaion phenomenon. The saisics-based models are more general han he physicsbased models; however, hey assume ha he degradaion pahs are linear and/or he sandard deviaion of he degradaion daa is consan. In his paper, we propose a saisics-based approach ha relaes hese assumpions and illusrae is applicabiliy using numerical eample. Key-ords:- Degradaion, reliabiliy models, saisics-based models INTRODUCTION Noaion S( S resisance drif a ime X random variable represening he degradaion measure, X ime f( ; probabiliy densiy funcion ( pdf of X a ime F( ; cumulaive disribuion funcion (CDF complemen of X r ( ; hazard funcion of f( ;, referred o as he degradaion hazard funcion g (, q ( posiive funcions T random variable represening he ime o degrade o a,b,, λ, θ consans R ( reliabiliy a ime given a hreshold degradaion level R ( T > H ( = ln R ( There are many producs ha ehibi performance degradaion over ime. They include appliances, auomobiles, and many elecronic componens. For eample, consider a resisor wih resisance S a = ha ehibis a change S( of is value wih ime resuling in a drif of S( S. This, in urn, will cause a gradual degradaion in he performance of he sysem in which he resisor is a componen unil he drif reaches a hreshold level ha causes he sysem s failure. Therefore, i is of ineres o observe he degradaion of he sysem s performance over ime and uilize such observaions in esimaing he reliabiliy of he sysem. Elsayed (996 classifies he acceleraed failure daa models as saisics-based models (parameric and nonparameric, physics-saisics based models, and physics-eperimenally based models. Similarly, we classify he degradaion models as physics-based and saisics-based models. The physics-based degradaion models are hose in which he degradaion phenomenon is described by a physicsbased relaionship like Arrhenius law or eperimenally-based resuls such as ho carrier degradaion model in Leblebici and Kang (993. The saisics-based degradaion models are hose in which he degradaion phenomenon is described by a saisical model such as regression.

2 . Model Developmen I is assumed ha he effec of he degradaion phenomenon on he produc performance can be epressed by a random variable called degradaion measure. Typical measures include he amoun of wear of mechanical pars such as shafs and bearings, he drif of a resisor, oupu power drop of ligh emiing diodes, and he propagaion delay of an elecronic chip. I is clear ha unis wih he same age would have differen degradaion measure levels; he degradaion measure is a sochasic process. For a specific ime, he degradaion measure, X, is a random variable ha is disribuion is ime dependen [Papoulis (965]. The degradaion measure disribuion, in general, migh change wih ime in he ype of he disribuion family and is parameers as shown in Figure. The solid curve represens he mean of he degradaion measure versus ime and he areas under he densiy funcions bu above he hreshold level line represen he failure probabiliy a he corresponding imes. In his paper, we assume ha he degradaion measure follows he same disribuion family bu is parameers may change wih ime. I is also assumed ha he degradaion pahs are monoonic funcions of ime: Monoonically Increasing Degradaion Pahs (MIDP or Monoonically Decreasing Degradaion Pahs (MDDP. P P( X < ;, for MIDP ( T > = P( X > ;, ( for MDDP P ( X < ; corresponds o producs ha fail when he degradaion measure reaches above a hreshold level (MIDP such as he case of he IFL devices described in Carey and Koenig (99. On he oher hand, P ( X > ; corresponds o producs ha fail when he degradaion measure reaches below a hreshold level (MDDP such as he case of he srengh of insulaion maerial [Nelson (98]. e prove Eq.( for MDDP as follows: PT ( > = PT ( > X= s; f( s; ds = P( T > X = s; f( s; ds + P( T > X = s; f ( s; ds = + f( s; ds X> ; or P( T > X > ;. I should be noed ha P( T > X = s; equals zero for s < since he uni fails as soon as he degradaion measure reaches a level less han. Likewise, P( T > X = s; equals for s >. Eq.( for MIDP can be easily proven using he same approach.. Degradaion hazard funcion. Relaionship beween he ime o degrade and degradaion measure disribuions The relaionships beween he ime o degrade and degradaion measure disribuion is epressed as e define he degradaion hazard funcion as f( ; r ( ; =. ( F( ; Unlike he radiional failure rae funcion, f ( R(, he degradaion hazard funcion considers boh ime and degradaion measure level. I is useful in developing a reliabiliy funcion based on degradaion daa wihou making assumpions abou he degradaion pahs and/or he sandard deviaion of he degradaion daa. I can also be used o invesigae he effec of a change in he hreshold

3 degradaion level on he reliabiliy of he sysem during is design sage. Le us assume ha he degradaion hazard funcion r ( ; can be epressed as he produc of funcions: r ( ; = g( q(, q( >, g( >, (3 where g( and q( are posiive funcions of he degradaion measure and ime, respecively. This assumpion ensures ha he disribuion family of he degradaion measure does no change wih ime as shown below. From Eq.(3 he raio of he degradaion hazard funcions a wo differen imes, and, can be wrien as: r ( ; q ( = (4 r ( ; q ( which means he change in degradaion hazard funcion wih ime is independen of g ( and is proporional o he raio of q q(... Consrains on g ( ( e uilize he properies of he CDF complemen in order o deermine he consrain( on g (. e rewrie F( ; in erms of he degradaion hazard funcion as follows: F( ; X > ; = ep( r( s; d or (5 F ( ; = ep( q( g( d. (6 The following condiions mus be saisfied for Eq.(6. (i F( ; = ep( q( g( d = (ii F( ; = ep( q( g( d = which implies ha gsds ( =. Therefore, he only consrain on g ( is gsds ( =... Consrains on q ( Consrains on q ( are differen for MDDP and MIDP. e firs assume ha he degradaion pahs are monoonically decreasing, MDDP, and uilize Eqs.( and (6 o deermine he consrains on q ( as follows: R ( T > X > ; = ep( q( g( d (7 The necessary and sufficien condiions for R ( o be a valid reliabiliy funcion are: lim R ( = lim ep( q( g( d = ep( lim q ( gsds ( = which implies ha lim ( = q lim R ( = lim ep( q( g( d = ep( lim q ( gsds ( = which implies ha (8 (9 ( lim q ( = ( Therefore, Eqs.(9 and ( are consrains on q (. The consrain on q ( for he monoonically increasing degradaion pahs, MIDP, is differen when as follows: 3 lim R ( = lim { ep( q( g( d} = which implies ha

4 lim ( = q. In summary, q ( approaches infiniy as ime approaches infiniy for he monoonically decreasing degradaion pahs and q ( approaches zero as ime approaches infiniy for he monoonically increasing degradaion pahs. e now show ha he relaionship beween he degradaion pahs and degradaion hazard funcion is as q ( > implies monoonically decreasing degradaion pahs. q ( < implies monoonically increasing degradaion pahs. Proof- Eq.(6 can be wrien as F ( ; F ( ; = ep[ q( ep[ q( = ep[( q( where < <. g( ds] g( ds] q( g( ds] (..3 Inerpreaions of he degradaion hazard funcion The degradaion hazard funcion can have monoonically decreasing or increasing pah. e show he analysis for he increasing pah below...3. Monoonically increasing degradaion pahs Assuming ha he degradaion pahs are monoonically increasing, he condiional probabiliy of a small change in he degradaion measure given ha is value is less han a specified level a ime can be wrien as: PX [ (, ; ] P[ X (, X < ; ] = PX [ < ; ] f(; s ds f( ; = F( ; F( ; (3 Dividing boh sides of Eq.(5 by and aking limis as approaches zero yields: d f ( ; u ( ; = P[ X (, X < ; ] =. (4 d F( ; Assuming ha failure occurs when he degradaion measure reaches, hen u ( ; is he failure rae a ime in erms of degradaion measure. Assuming F( ;, u( ; can be wrien in erms of degradaion hazard funcion as f ( ; F ( ; F ( ; u( ; = = r( ;. (5 F( ; F ( ; F( ; Eq.(5 reveals ha in he monoonically increasing case he degradaion hazard funcion is no he failure rae in erms of he degradaion measure. This can be also undersood from he relaionship beween he cumulaive failure rae and cumulaive degradaion hazard funcion for he monoonically increasing degradaion pahs as shown in Eq.(6. The reliabiliy funcion for he monoonically increasing degradaion pahs is R ( = ep( r( s; d = ep( rs ( ; d(ep( rs ( ; d Taking he logarihm of boh sides yields: H ( = r( s; ds lnψ ( ; (6 o where ψ ( ; = ep( r( s; d. 4

5 In summary, he degradaion hazard funcion represens he failure rae in erms of he degradaion measure when he degradaion pahs are monoonically decreasing bu i only has a conribuion o he failure rae when he degradaion pahs are monoonically increasing. 3. APPLICATION e now uilize he proposed approach o predic he reliabiliy of an insulaion maerial. e use he degradaion daa presened in Table aken from Nelson (98. I is imporan o noe ha each observaion in Table comes from a differen specimen; herefore, he daa are assumed o be independen. The dielecric srengh of he insulaion maerial is considered as he degradaion measure. e consider he following degradaion hazard funcion: r ( ; = qg ( ( ee k where Diele cric sren gh (kv b >>. Table - Degradaion daa ee k Diele cric sren gh (kv eek Diel ecri c sren gh (kv ee k q ( =, g ( = bep( a Diel ecri c sren gh (kv , and The parameric form of q ( is deermined based on graphing he degradaion daa versus ime. Moreover, he disribuion of he degradaion daa in each week is sudied in order o deermine g (. I is realized ha g ( = is a good fi for all 5 weeks. The corresponding degradaion measure disribuion for his degradaion hazard funcion is a eibull disribuion wih a ime-dependen scale parameer as f ( ; = θ ( ep(, > θ ( where θ( = bep( a is he scale parameer. (7 e fi he eibull model given in Eq.(7 o he degradaion daa shown in Table. The Maimum Likelihood mehod was uilized o esimae he parameers of he model as follows: m L(, a, b; = ( bep( a ni i= i m n i ij ij i= j= b ai ep( ep( (8 where m is he number of weeks, n i is he oal number of degradaion observaions in week i and ij is he degradaion measure of uni j in week i. Taking he logarihm of Eq.(8 we obain: m m m ln L= n ln n ln b+ na i i i i i= i= i= m ni m n i ij ( ln ij i= j= i= j= b ai + ep( (9 Taking he parial derivaives of Eq.( wih respec o, a and b yields a =., b = 4993, = Subsiuing he values of q( and g( ino Eq.(7, he reliabiliy funcion can hen be deermined as R R ( X > ; = ep[ ] b ep( a 5.65 ( = ep[ ]. ( 4994ep(-. For =, R ( equals o and R ( equals o zero. Therefore, he necessary and sufficien condiions are saisfied for his model.

6 I is of ineres o deermine effec of changes in he hreshold degradaion level on he componen lifeime a he design sage where he componen specificaions such as he hreshold degradaion level are deermined based on he design requiremens and cos consideraions. 6. Conclusion e inroduce a degradaion hazard funcion which is used o esimae reliabiliy using degradaion daa wihou making assumpions abou he degradaion pahs and he sandard deviaion of he degradaion daa. The developed reliabiliy funcion includes he hreshold degradaion level a which failure occurs as a parameer. Therefore, sensiiviy of he reliabiliy o he hreshold degradaion level can be easily deermined. The proposed saisics-based approach is uilized in esimaing reliabiliy of an insulaion maerial. The reliabiliy esimae is hen compared wih a physics-based reliabiliy esimae of he insulaion maerial. REFERENCES [] I. F. Blake and. C. Lindsey, Level Crossing Problems for Random Processes, IEEE Transacions on Informaion Technology Vol. IT-9 No. 3 (973, [] M. B. Carey and R. H. Koenig, Reliabiliy Assessmen Based on Acceleraed Degradaion: A Case Sudy, IEEE Transacions on Reliabiliy Vol. 4 No. 5 (99, [3] S. E. Chick and M. B. Mendel, An Engineering Basis for Saisical Lifeime Models wih an Applicaion o Tribology, IEEE Transacions on Reliabiliy Vol. 45 No. (996, 8-4. [4] S. L. Chuang, A. Ishibashi, S. Kijima, N. Nakayama, M. Ukia, and S. Taniguchi, Kineic Model for Degradaion of Ligh Emiing Diodes, IEEE Journal of Quanum Elecronics Vol. 33 No. 6 (997, [5] K. A. Doksum and A. Hoyland, Models for Variable-Sress Acceleraed Life Tesing Eperimens Based on iener Processes and he Inverse Gaussian Disribuion, Technomerics Vol. 34 No. (99, [6] M. Domine, Momens of he firs passage ime of a iener process wih drif beween wo elasic barriers, J. Appl. Prob. Vol. 3 (995, 7-3. [7] M. Domine, Firs passage ime disribuion of a iener process wih drif concerning wo elasic barriers, J. Appl. Prob. Vol. 33 (996, [8] E. A. Elsayed, Reliabiliy Engineering, Addison- esley (996. [9] A. A. Feinberg and A. idom, Connecing Parameric Aging o Caasrophic Failure Through Thermodynamics, IEEE Transacions on Reliabiliy Vol. 45 No. (996, [] E. Ioannides, E. Beghini, G. Bergling, J. Goodall and B. Jacobson, Cleanliness and is imporance for bearing performance, Ball Bearing Journal 4 (993, 8-5. [] Y. Leblebici and S-M Kang, Ho carrier Reliabiliy of MOS VLSI Circuis, Boson, Kluwer (993. [] J. C. Lu and. Q. Meeker, Using Degradaion Measures o Esimae a Time-o-Failure Disribuion, Technomerics Vol. 35 No. (993, [3] J. Lu, J. Park and Q. Yang, Saisical Inference of a Time-o-Failure Disribuion Derived From Linear Degradaion Daa, Technomerics Vol. 39 No. 4 (997, 39-4 [4]. Q. Meeker and M. Hamada, Saisical Tools for he Rapid Developmen & Evaluaion of High-Reliabiliy Producs, IEEE Transacions on reliabiliy Vol. 44 No. (995, [5]. Q. Meeker, L. A. Escobar and C. J. Lu, Acceleraed Degradaion Tess: Modeling and Analysis, Technomerics Vol.4 No. (998, [6]. Nelson, Analysis of performancedegradaion Daa from Acceleraed Tess, IEEE Transacions on reliabiliy Vol. R-3 No. (98, [7] A. Papoulis, Probabiliy, Random Variables, and Sochasic Process, McGraw-Hill, (965. [8] V. Pieper, M. Domine, P. Kurh, Level Crossing Problem and Drif Reliabiliy, Mahemaical Mehods of Operaion Research Vol. 45 (997, [9] F. S. Qureshi and A. K. Sheikh, A Probabilisic Characerizaion of Adhesive wear in Meals, IEEE Transacions on Reliabiliy Vol. 46 No. (997,

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based