MTBF Understanding Its Role in Reliability Engineering

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1 MTBF Undersanding Is Role in Reliabiliy Engineering A Whie Paper February, 7 By David C. Wilson CEO/Founder Wilson Consuling Services, LLC dave@wilsonconsulingservices.ne Copyrigh 7. All righs reserved

2 Table of Conens Descripion Page Inroducion Reliabiliy Mahemaics7 Bahub Curve Failure Rae Compuaions3 MTBF Compuaions3 MTTF Compuaions38 MTTR Compuaions 4 Relaionships Summary4 Reliabiliy Models44 Summary46 References47 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

3 Inroducion The objecive is o enable an individual who is unfamiliar wih he use of he MTBF reliabiliy parameer o undersand is relaionship in produc predicions, failure raes, field performance, ec. Afer compleing his uorial, he paricipan will know and undersand how MTBF relaes o produc performance over ime. In order for he praciioner o speak inelligenly and auhoriaively on he parameer, MTBF, i imporan a a minimum ha a cursory undersanding of he mahemaical conceps involved wih Reliabiliy be masered. Therefore, mahemaical and pracical reamens relaive o MTBF are included in his uorial using he exponenial disribuion model. Addiionally, o achieve a horough undersanding of saisics and reliabiliy, many saisical expers lis four kinds of undersanding as shown below.*. Compuaional/Numerical Visual/Graphical 3. Verbal/Inerpreive 4. Srucural/deducive WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3

4 Ø Wha is MTBF? Inroducion con d Measure of rae of failure wihin he design life. Ø Wha is design life? Inended period of use which is expeced o be failure free. Wha do hese erms mean? Reliabiliy? Failure? Failure Rae? Hazard Rae? MTBF/MTTF? WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4

5 Inroducion con d Assumpion of a consan failure rae ØWhen using Mean Time Beween Failure (MTBF or Mean Time o Failure (MTTF, a consan failure rae is assumed and he exponenial disribuion model prevails. The exponenial disribuion is among one of he mos common and useful life disribuion models. The exponenial P.D.F occurs frequenly in reliabiliy engineering. Describes he siuaion wherein he hazard rae is consan. I is he disribuion of ime o failure for a grea number of elecronic sysem pars. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 5

6 Reliabiliy Definiions: Inroducion con d RELIABILITY [R(] The probabiliy ha an iem will perform is inended funcion wihou failure under saed condiions for a specified period of ime. FAILURE The erminaion of he abiliy of an iem o perform is required funcion as specified. FAILURE RATE (FR The raio of he number of failures wihin a sample o he cumulaive operaing ime. HAZARD RATE [l, h(] The "insananeous" probabiliy of failure of an iem given ha i has survived up unil ha ime. Someimes called he insananeous failure rae. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 6

7 Reliabiliy Mahemaics Wha is he Probabiliy Densiy Funcion (P.D.F.? Descripion of is meaning Frequency disribuion and cumulaive disribuion are calculaed from sample measuremens. Since samples are drawn from a populaion, he quesion is wha can be said abou he populaion? The ypical procedure sugges a mahemaical formula, which provides a heoreical model (p.d.f. for describing he way he populaion values are disribued. A hisogram and cumulaive frequency funcions are hen esimaes of hese populaion models. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 7

8 Reliabiliy Mahemaics con d P.D.F. con d A posiive coninuous random variable follows an exponenial disribuion if he probabiliy densiy funcion (P.D.F.is as shown: Thus he P.D.F. f ( x ì ï ae í ïî ax ( For x³ ( For x< I is imporan in reliabiliy work because i has he same Cenral Limi Theorem relaionship o Life Saisics as he Normal disribuion has o NonLife Saisics. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 8

9 Reliabiliy Mahemaics con d P.D.F. con d I is a probabiliy densiy funcion (P.D.F. f ( le l Lambda l is a consan and is called he failure rae f( Figure WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 9

10 Reliabiliy Mahemaics con d Wha is he Cumulaive Disribuion Funcion (C.D.F.? The cumulaive disribuion corresponds o a populaion model called cumulaive disribuion funcion C.D.F. and is donaed by F(. I is relaed o he P.D.F. via he following relaionship. F(. P( T ò f ( d Use as a dummy variable; Le, hen d d C.D.F., F( Figure A WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

11 Reliabiliy Mahemaics con d C.D.F. Relaionship o P.D.F. Ø Reliabiliy deals wih failure imes,, which are nonnegaive values. C.D.F. for populaion failures ime is relaed o P.D.F. The P.D.F., which f( can be inegraed o obain he cumulaive disribuion funcion F(, and he hazard funcion h( can be inegraed o obain he cumulaive hazard funcion H(. The P.D.F. for he exponenial disribuion f ( e l l The C.D.F for he exponenial disribuion l e F( WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

12 Reliabiliy Mahemaics con d C.D.F. mahemaical derivaion. Probabiliy densiy funcion (p.d.f. f ( le l. C.D.F. is derived by inegraing p.d.f. F( P( T ò F( le F( e F( e l l \ F( R( QED l d ò f ( d is F( \The probabiliy ha a componen fails in he inerval e l WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

13 Reliabiliy Mahemaics con d. Hazard funcion l f ( f ( e h( l l F( R( e. Cumulaive hazard funcion: (Inegraing he hazard funcion o obain he cum hazard funcion. Use dummy variable of inegraion l f ( e H ( l ( ò h d ò d ò d d R( e ò l l l \ H ( ln R( Taking ln R( \ l can he lne be naural l l log of expressed as WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3 l boh ln( e sides : R( l e l ln R( l QED

14 Hazard Rae The exponenial P.D.F. is a valid useful life ime o failure model for many debugged elecronic componens. P.D.F. f ( < < le l Where l represens a consan failure rae ha does no vary wih ime. For reliabiliy purposes, he C.D.F. is designaed F( raher han F(x and F( is defined as he probabiliy of failure in he inerval < T <. h( l( Reliabiliy Mahemaics con d Useful Life h( e e l f ( l l R( l Figure WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4 ime

15 Reliabiliy Mahemaics con d. R( Illusraions: Exponenial Reliabiliy Curve R( e l ime Figure 3 f( Probabiliy Densiy Funcion Curve Densiy f ( le l ime Figure 4 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 5

16 Reliabiliy Areas of Probabiliies Illusraion. R( Reliabiliy Mahemaics con d Unreliabiliy area F( F( ò f ( d P( T Reliabiliy exponenial disribuion plo Reliabiliy area R( R( ò f ( d 3 4 Figure 5 F l ( e \ F( e l R( R( l e F( WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 6

17 Reliabiliy Mahemaics con d Example : For exponenial disribuion, ake he inegral of f(, where f( le l, where F( ò f ( d < Soluion F( ò l l e d < F( ò l e l d e l e WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 7

18 Reliabiliy Mahemaics con d Example : An elecronic device conains discree ransisors. Each ransisor has a consan failure rae of l 5 failure rae/hour. Wha is he probabiliy ha a single ransisor will survive a mission of 4 hours? R( F( Soluion R( \ R( F( R( e l e ç æ è 5 ö ç æ øè e l 4 ö ø.95 R( WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 8

19 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 9 Wha is he probabiliy ha i will survive a mission of 3 hours?..99 ( (.99 ( ( (. 3 5 \ ø ö ç è æ ø ö ç è æ R F R e R e R Example 3: Soluion Reliabiliy Mahemaics con d

20 Bahub Curve Reliabiliy Bahub curve for consan failure rae Failures infan moraliy consan failure rae wearou Figure 6 Time Infan moraliy ofen due o manufacuring defecs In elecronics sysems, here are models o predic MTBF for he consan failure rae period (Bellcore Model, MILHDBK7F, ohers Undersanding wearou requires daa on he paricular device Semiconducor devices should no show wearou excep a long imes Elecrical devices which wearou: relays, EL caps, fans, connecors, solder WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

21 Bahub Curve WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

22 Graph: Cumulaive disribuion funcion (c.d.f. for he exponenial disribuion funcion. % Average Failure Rae Failure Rae Hazard Rae Consan wih respec o ime A funcion of ime An average Insananeous Probabiliy of Failure..95 Average Failure Rae Insananeous Hazard Rae Reliabiliy: Average Failure Rae vs. Hazard Rae.9 Figure Time WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page

$for, AFR oof : Pr [ ] [ ] QED ( ln ( ln ( ln ( ln : ( ln ( ( ( : Pr R R AFR R R AFR Then R H If H H AFR oof \ Average Failure Rae$

23 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3 R R H H d h AFR ( ln ( ln ( ( ( ò Average Failure Rae (AFR for, AFR oof : Pr [ ] [ ] QED ( ln ( ln ( ln ( ln : ( ln ( ( ( : Pr R R AFR R R AFR Then R H If H H AFR oof \ Average Failure Rae con d

24 Average Failure Rae con d Example..98 R( Exponenial Disribuion.95 T e x ime in hours Figure 9 AFR AFR AFR ln R( ln R( ln(.98 ln( ( E 6 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4

25 Average Failure Rae con d Illusraion Exponenial Disribuion AFR can be represened by lambda (l Therefore: l 3.55E6, which is he hazard funcion h( h( l 3.55E6 hours Figure WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 5

26 Average Failure Rae con d Esimaing Failure Rae Ø Lambda (l can be obained as an esimae (l ha of he rue populaion for all operaing hours for all unis esed including failed and hose ha compleed he es wihou failing. I is he bes esimae for complee or censored sample: lˆ number of failures oal uni es ime The denominaor is obained by adding up all operaing hours on es of all unis esed, including hose ha failed and hose ha compleing he es wihou failing. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 6

27 Average Failure Rae con d Example : Five elecronics Subsysems failed from a sample of which were used consanly for 9 days. Wha is he Failure Rae? Esimae of failure rae for λ ^ λ Failure rae 5 failures 5 *4*9,59, failures / hour.93e6 failures/hour Oher expressions Failures per million hours (Fpmh Fpmh l *E + 6 (.93E 6(E Percen per housand hours This rae rae can be expressed by muliplying l (E+5 resuling in he average failure rae AFR.93%/ hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 7

28 Failure Rae con d Sysem failure rae Sysem failure funcion h s ( is he sum of n componen failure rae funcions h (, h (,,h n (. When he componens have exponenial lifeimes wih parameers l, l,, l n, hen he sysem has a consan failure rae equal l s n å i l i WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 8

29 Average Failure Rae con d Ø Facors o conver h( and he AFR PPM or FIT when period is in hours. Failure rae in %/K [E+5][(h(] AFR in %/K [E+5][AFR(T,T ] Failure rae in FITS [E+4][failure rae in %/K] AFR in FITS [E9][AFR(T,T ] WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 9

30 MTBF Compuaions MTBF is no Life Design Life: MTBF: Inended period of use which is expeced o be failure free. Measure of rae of failure wihin he design life. EXAMPLES: Iem Design Life MTBF Conacor 5, cycles 55, cycles Pushbuon 3 million op s million op s CPUPanel 5 years 37 years Do no confuse MTBF wih Design Life of an iem WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3

31 MTBF y l d dy l MTBF ò l l l By subsiuion le : y l ò MTBF l \ MTBF l MTBF Compuaions con d e y l d e y Inegraing by pars l dy [ y ( y+ ] e QED Mahemaical Proof Inegraing by pars soluion: Recall: e ò le u v by udv uv y & du dy e ò ye ye y y y & dv dv e [ ] y ( y + e e ye y facoring ò vdu y y dv òe y dy WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3

32 Example MTBF Compuaions con d Mean Time Beween Failures [MTBF] For a repairable iem, he raio of he cumulaive operaing ime o he number of failures for ha iem. AFR can be represened by lambda (l Therefore: l 3.55E6, which is he hazard funcion h( h( l 3.55E6 Figure hours MTBF MTBF ò le l d 3.55E 6 l 8,757hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 3

MTBF Compuaions con d Mean Time Beween Failures (MTBF Example : If a moor is repaired and reurned o service six imes during is life and provides 45, hours of service.

33 MTBF Compuaions con d Mean Time Beween Failures (MTBF Example : If a moor is repaired and reurned o service six imes during is life and provides 45, hours of service. Also l MTBF failures ime l 6 45, ,5hours Toal operaing ime 45, MTBF 7, 5hours # of failures 6 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 33

34 Example 3 MTBF Compuaions con d Using Chisquare model o find MTBF MTBF Upper and lower bound is calculaed such as 5%, 8% 9%, 95%, ec. Daa from example, previous page. MTBF MTBF lower upper c c T a,( n n T a,( n n n º Number of defecs T º Toal operaing ime n º Degrees f freedom n a º Significance level WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 34

35 MTBF Compuaions con d Chisquare model con d Daa from example 3, use 9% confidence or % level of significance. Soluion lower bound MTBF MTBF MTBF MTBF lower lower Upper bound lupper lupper * 45, T c a,( n n c., ν ( 6 T c a,( n n * 45, c.,ν (6 9, c., 9, c.9, 9, 8.5 9, 6.3 4,86 hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 35 4,865 hours Can be found in any Chisquare disribuion able

36 MTBF Compuaions con d Example 4 Ø An elecronics assembly has a goal of.99 reliabiliy for one year. ü Wha is he MTBF ha he designer should work owards o mee he goal? Reliabiliy equaion: R( e l and MTBF /l Solve for MTBF Soluion: ln R( MTBF* ln R( MTBF Hence : R( e MTBF MTBF ln R( 876 ln.99 87, hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 36

37 MTBF Compuaions con d Example 5: If MTBF for an auomobile is, miles... Reliabiliy as a Funcion of Mission Time Mission Lengh, (miles *Consan hazard rae Reliabiliy*, 99.%, 9.5% 5, 6.7%, 36.8%, 3.5% There is only a 36.8% chance of surviving pas he period of one MTBF (i.e. when MTBF WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 37

38 MTTF Compuaions Mean Time To Failure [MTTF] For nonrepairable iems, he raio of he cumulaive operaing ime o he number of failures for a group of iems. Example : monioring devices are operaed for 9 days. During ha ime, five failures occur. failures 5 5 λ 9. 9E ime * 4* 9, 59, MTTF 58, 43 hours λ 9. 9E 7 7 Also, MTTF Toal Operaing Time * 4* 9 58, Toal Failures 5 43 hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 38

39 Example MTTF Compuaions con d ØAn elecronics assembly has a goal of.99 reliabiliy for one year. Wha is he MTTF ha he designer should work owards o mee he goal? ü Reliabiliy equaion: R( e l and MTTF /l ü Solving for MTTF MTTF ln R( MTTF 876 ln R(.99 MTTF 876 (.5 87,64 hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 39

MTTR Compuaions Mean Time To Repair (MTTR his is correcive mainenance, which includes all acions o reurn a sysem from a failed o an operaing or available sae. I is difficul o plan.

40 MTTR Compuaions Mean Time To Repair (MTTR his is correcive mainenance, which includes all acions o reurn a sysem from a failed o an operaing or available sae. I is difficul o plan. I enails, for example:. Preparaion ime: finding a person for he job, ravel, obaining ools and es equipmen, ec.. Acive mainenance ime, i.e., doing he job 3. Delay ime ( logisic ime: waiing for spare pars., once he job has been sared. MTTR å å ( l r l Summaion of expeced imes of individual failures modes Summaion of individual failure raes Availabiliy MTBF MTBF + MTTR + mean prevenive main enance ime WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4

41 Relaionships Summary Reliabiliy Parameers Descripion Hours Reliabiliy C.D.F. Unreliabiliy Produc MTBF Hazard rae (failure rae/hour P.D.F. Avg. failure rae (AFR Failure rae in PPM Fails in ime Failures per million hours R( F( h( f( % / K hrs. PPM / K hrs. FIT Fpmh AMPS E6.5983E CPU E6.988E CPU E6 3.58E NCMW E6.443E NCA E6.585E WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4

42 Relaionships Summary con d I. Definiions: Failure rae The raio of he number of failures wihin a sample o he cumulaive operaing ime. Example: PPM / K hrs. MTBF Mean ime beween failure, which means ha 63.% of he produc would have failed by his ime. Example: MTBF òo f ( d òo le l d Reliabiliy R( The probabiliy ha an iem will perform is inended funcion wihou failure under saed condiions for a specified period of ime. Example: R( f ( d Unreliabiliy F( Commonly referred o as cumulaive disribuion funcion (CDF The probabiliy ha an iem will no perform is inended funcion wihou failure under saed condiions for a specified period of ime. Also, commonly referred o as cumulaive densiy funcion. Example: Hazard rae h( The "insananeous" probabiliy of failure of an iem given ha i has survived up unil ha ime. Someimes called he insananeous failure rae. I is he failure rae per uni ime. Example:.8E6 / hour, ò ò F ( f ( d o h ( f ( R ( WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 4

43 Relaionships Summary con d Hazard rae h( The "insananeous" probabiliy of failure of an iem given ha i has survived up unil ha ime. Someimes called he insananeous failure rae. I is he failure rae per uni ime. Example:.8E6 / hour, h ( f ( R ( Probabiliy Densiy Funcion (PDF Commonly referred o as f( I is denoed by f( where is a he variable of ineresed where f(d is he fracion of failure imes of he populaion occurring in he inerval d. Basically, i assumes a mahemaical formula ha provides a heoreical model describing he way he populaion values are disribued. The definie inegral of is domain mus equal. Example: f ( le l, < II. Examples. (% / hrs. One percen per housand hours would mean an expeced rae of fail for each unis operaing hours. Example:.8%/ hours means ha 8 failures each million unis operaing for hours.. (PPM / hrs. One per million per housand hours means fail is expeced ou of million componens operaing for hours. Example: 8 pars per million per housand hours means ha 8 failures are expeced ou of million componens operaing 3. Failure in ime (FIT: FIT Failure in One Billion Hours Example: 8 FITS 8 failures in one billion hours WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 43

44 Reliabiliy Models Type of Disribuion Parameers Probabiliy densiy funcion, f( Reliabiliy funcion, R( F( Hazard funcion (insananeous failure rae, h( f( / R( Normal Mean, µ Sandard deviaion, s Numerous applicaions. Useful when i is equally likely ha readings will fall above or below he average. f (.4 R ( h ( ( f e µ ( p ò f ( R ( s p f ( d h ( R ( Exponenial Failure rae, l MTBF, q q l Describes consan Failure rae condiions. Applies for he useful Life cycle of many Producs. Frequenly, ime( Is used for x. l f ( 3 l f ( l e R ( l R ( e 4 h ( l 3 h( l q Weibull Used for many reliabiliy applicaions. Can es for he end infan moraliy period. Can also describe he normal and exponenial disribuions. Shape, b Scale (characerisic life, q Locaion (minimum life, g Curves shown for g f( b3 b3 f ( R (.5.5 b.5 b b b b ( g b b b.. g g exp R( exp q b q q h h ( b3 b b.5 3 b.( h( g b q b WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 44

45 Reliabiliy Models con d Type of Disribuion Gamma Parameers Describes a siuaion when parial failures can exis. Used o describe random variables bounded a one end. The parial failures can be described as sub failures. Is an appropriae model for he ime required for a oal of exacly? independen evens o ake place if evens occur a a consan rae l. Failure rae, l a.5 Evens per failure, or Time o ah f ( a failure.5 a µ l s a l Noe: when a is an ineger I (a (a! Probabiliy densiy funcion, f( a 4 a l ( l l f ( G( a e Reliabiliy funcion, R( F( R (.5 a a a.5 4 a l a l e R( G( a ò Hazard funcion (insananeous failure rae, h( f( / R( h( a ( l d a.5 a a 4 h( f( R( Lognormal Mean, µ.8 The Lognormal disribuion is ofen a good model for imes o failure when failures are caused by faigue cracks. Le T be a random variable wih a Lognormal disribuion. By definiion he new random variable X ln T will have a normal disribuion. Sandard deviaion, s f ( f( R (.5. exp ln( µ R( s..(. p.5. s 3 f( d h ( 3 h( f( R( WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 45

46 Summary This paper did elaborae on he value of using he MTBF parameer. However, here have been remendous improvemens in solidsae devices over he years. In earlier imes, elecronic componens were fragile, used glass ubes, filamens, ec., had inheren wear ou mechanisms. By he same oken, earlier solidsae devices had mechanisms ha would cause failures in ime, such as chemical conaminaion, meallizaion defecs, and packaging defecs, which resuled in corrosion and delaminaing. Many of hese defecs were acceleraed by high emperaure, which resuled in successful use of he burnin process o weed ou infan moraliy. Saisical predicion during ha period was valid and acceped because designs a ha ime consiss mosly of discree componens; herefore, reliabiliy saisical esimaes of he life of a new design had for he mos par a reasonable correlaion o he acual MTBF. Today hundreds of new componens are inroduced o he marke almos every week and hundreds are probably aken off he marke every week; herefore, i is impossible o make an accurae predicion based on a summaion of pars reliabiliy. Example: MilSd7 Today s componens do no have wearou modes ha are wihin mos elecronics echnologically useful life. Therefore, he vas majoriy of failures is due o defecs in design or inroduced in manufacuring. Unplanned evens in manufacuring such as ECN, change in machine operaors, or change in vendor s capabiliies of design, inroducion of cos reduced pars, ec., any of hese or combinaion can inroduce a decrease in design margins. Hence: his affecs reliabiliy and increase field reurns. Many expers feel ha i is bes o spend ime, no on saisical predicions raher on discovering he real capabiliies and idenifying he weak links in he design or manufacuring process, and improving hem. This approach will help realize significan improvemen in reliabiliy. The end user environmen is even more unconrolled. The enduser will always push he limis; herefore, a robus design will have a higher survivor rae for hese exremes. WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 46

47 References References. Parick D.T. O Connor, Pracical reliabiliy Engineering, hird ediion John Wiley & Sons 99. Paul A. Tobias 7 David C. Trindade, Applied Reliabiliy, second ediion Chapman & Hall/CRC 995 WCS, LLC MTBF Undersanding Is Role in Reliabiliy Engineering Page 47

Errata and Updates for ASM Exam MLC (Fourteenth Edition) Sorted by Page

Errata and Updates for ASM Exam MLC (Fourteenth Edition) Sorted by Page Erraa for ASM Exam MLC Sudy Manual (Foureenh Ediion) Sored by Page 1 Erraa and Updaes for ASM Exam MLC (Foureenh Ediion) Sored by Page Pracice Exam 7:25 (page 1386) is defecive, Pracice Exam 5:21 (page