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Presented by: Mark E. Sims Reliability S&T Engineer Aviation and Missile Research, Development and Engineering Center UNCLASSIFIED Intro Reliability Growth "Approved for public release; distribution unlimited. Review completed by the AMRDEC Public Affairs Office 11 Oct 2013; PR0073."

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2 Mil-HDBK-189 Definition Reliability Growth The positive improvement in a reliability parameter over a period of time due to changes in product design or the manufacturing process. MIL-HDBK-189 is a Department of Army Handbook for Reliability Growth Management

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3 J.T. Duane was an engineer at the Aerospace Electronics Department of the General Electric Company. He published a paper in 1964 that applied a “learning curve approach” to reliability monitoring. He observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper. The learning (i.e., growing) is accomplished through a “test, analyze, and fix” (TAAF) process. Beginnings

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4 log-log paper graphing Normal graphing Graphs Duane Postulate: The cumulative MTBF versus cumulative operating time is a straight line on log-log paper.

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5 Continuous Growth Continuous means time. You can plot failure rate or MTBF against the total test hours.

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6 Discrete Growth Discrete means trials.

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7 Discrete Growth Reliability Growth follows a Learning Curve approach. Note: More rapid growth occurs earlier in the process then flattens out!

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8 Why Reliability Growth?

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Example A System has 18 Failures in 177 Trials

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10Example A system has 18 failures in 177 trials. The failures are listed the tables below. FailureTrial FailureTrial

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11 FailureTrial Trials Between Failures There appears to be reliability growth. Example FailureTime Trials Between Failures Less trials between failures. More trials between failures.

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Example Applying Reliability Growth Methodology, we get the following curve:.

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Example Applying Reliability Growth Methodology, we get the following curve:.

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14 Why Reliability Growth? Saves Assets Reduces Test Time Saves $$$$$$$

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15 Duane Model Power Law Formulation for reliability growth

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16 Duane Postulate During Reliability Growth, Graphing the log of time (or tests) against its corresponding log of MTBF Will be a straight line with slope α. MTBF Cum = Cumulative Mean-Time-Between-Failure t = Time K = Constant for Power Law Equation α = Growth parameter

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17 Duane Postulate Slope, α Time (or Trial), t MTBF (Ln(t 1 ), Ln(M 1 )) (Ln(t 3 ), Ln(M 3 )) (Ln(t 2 ), Ln(M 2 )) TimesMTBF Cum t1t1 M1M1 t2t2 M2M2 t3t3 M3M3

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18 Duane Postulate Linear relationship: y = αx + b Has a linear log-log relationship!

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19 Calculating α the growth rate

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20 Calculating α (the growth rate) Time (hrs) Total Failures First reading5005 Last reading We will determine α from these two readings.

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21 We will determine α from these two readings. Time (hrs) Total Failures First reading5005 Last reading Calculating α (the growth rate)

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22 First calculate the cumulative MTBF for each reading. Time (hrs) Total Failures MTBF First reading Last reading Calculating α (the growth rate)

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23 Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Take logs of the readings. Calculating α (the growth rate)

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24 Slope, α x-axis y-axis ( Ln(500), Ln(100) ) ( Ln(4000), Ln(200) ) Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Plot the logs of the readings. Calculating α (the growth rate)

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25 α = 0.33 x-axis y-axis ( 6.215, ) ( 8.294, ) Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Calculating α (the growth rate)

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26 α = 0.33 x-axis y-axis Growth is indicated when 0 < α < 1 Calculating α (the growth rate)

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27 Duane Parameters α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time These parameters go into the Duane equation. If you know 4 of the parameters, you can calculate the other.

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28 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What is the Total Test time if we are given these 4 parameters? α.40 TITI 100 MIMI 50 MFMF 150 T Total ?

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29 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time How does changing the growth parameter α affect the total test time? α.40 TITI 100 MIMI 50 MFMF 150 T total 435

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30 α TITI 100 MIMI 50 MFMF 150 T total 435 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time How does changing the growth parameter α affect the total test time?

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31 α TITI 100 MIMI 50 MFMF 150 T total Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time The α is very sensitive to the Total Time! How does changing the growth parameter α affect the total test time?

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32 Instantaneous vs Cumulative Duane MTBF Equation Finding the true estimate of a system’s MTBF using reliability growth.

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33 Failure Number Failure Time Inst vs. Cum MTBF What is the true estimate of the MTBF at 250 hours?

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34 Failure Number Failure Time MTBF Cum Inst vs. Cum MTBF Is the MTBF 50 at time 250?

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35 Failure Number Failure Time MTBF Cum Time Between Failures Inst vs. Cum MTBF Or would you say the MTBF is 90 at 250 hours?

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36 Failure Number Failure Time MTBF Cum Time Between Failures MTBF Inst Inst vs. Cum MTBF Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool will give these numbers.

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37 Failure Number Failure Time MTBF Cum Time Between Failures MTBF Inst Inst vs. Cum MTBF Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool to get these numbers. So, 66 is the true MTBF at 250 operating hours, if reliability growth is occurring.

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38 MTBF Time (or Test), t On Log-Log Graph Paper , Inst vs. Cum MTBF

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39 This is how the graphs look In standard Cartesian coordinate MTBF Time (or Test), t Inst vs. Cum MTBF

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40ExerciseExercise 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?

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41 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?ExerciseExercise

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42 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?ExerciseExercise

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43 Reliability Growth Formulas Failure Rate MTBF Reliability

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44 M(t) = 1 / r(t) MTBF is the reciprocal of the failure rate.

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45 r I = Initial failure rate t I = Initial time corresponding to r I α = Growth rate parameter Failure Rate Formula Initial Conditions

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46 M I = Initial MTBF t I = Initial time corresponding to M I α = Growth rate parameter MTBF Formula Initial Conditions

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47 R I = Initial Reliability N I = Initial number of trials corresponding to R I α = Growth rate parameter Reliability (Discrete) Initial Conditions

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48 Deriving r(t) Formula r(t) is sometimes called the Hazard Rate.

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49 Deriving r(t) Formula K = Constant for Power Law Equation First, start with the Duane Postulate.

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50 Insert initial conditions M I at T I, and solve for K. Deriving r(t) Formula t I is the Initial Test Time. M I is the Initial MTBF at time t I. t I is the Initial Test Time. M I is the Initial MTBF at time t I.

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51 Now substitute for K. Deriving r(t) Formula

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52 The failure rate, r, is the inverse of the MTBF, so r(t) = 1 / M(t). Deriving r(t) Formula

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53 Deriving r(t) Formula Now we will simplify and take the derivative.

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54 Deriving r(t) Formula Now we will simplify and take the derivative.

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55 M I = Initial MTBF t I = Initial time corresponding to M I α = Growth rate parameter Deriving M(t) Formula

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56 Deriving M(t) Formula Recall MTBF = 1/r, so take the inverse of r(t).

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57 The Sensitivity of Duane’s Initial Conditions T I and M I on the Total Test Time.

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58 TITI α.40 MIMI 50 MFMF 150 T total 435??? α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What if we increase the initial time for a planning curve? Sensitivity of Initial Time

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59 Sensitivity of Initial Time What if we increase the initial time for a planning curve? A higher initial time significantly increases T total ! TITI α.40 MIMI 50 MFMF 150 T total Why?

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TITI α.40 MIMI 50 MFMF 150 T total Sensitivity of Initial Time Time MTBF TITI

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TITI α.40 MIMI 50 MFMF 150 T total Growth is more rapid the smaller T I is! Sensitivity of Initial Time MTBF Time TITI

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62 MIMI α.40 TITI 100 MFMF 150 T total 435??? α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What if we change the initial MTBF for a planning curve? Sensitivity of Initial MTBF

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63 What if we change the initial MTBF for a planning curve? A higher initial MTBF significantly decreases T total ! MIMI α.40 TITI 100 MFMF 150 T total Sensitivity of Initial MTBF

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64 R I = Initial Reliability N I = Initial number of trials corresponding to R I α = Growth rate parameter Deriving Reliability Formula

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65 Deriving Reliability Formula R cum = Cumulative Reliability F = Number of Failures N = Number of Trials r is the failure rate.

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66 Deriving Reliability Formula Recall failure rate formula.

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67 Deriving Reliability Formula Subtract from 1.

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68 Deriving Reliability Formula Make substitutions.

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69 Initially, System A has 3 failures after 100 firings. If you expect a growth rate of 0.25, what would be the expected reliability after 1000 flight tests? Exercise

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70 Initially, System A has 3 failures after 100 firings. If you expect a growth rate of 0.25, what would be the expected reliability after 1000 flight tests? Exercise

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71 Inst / Cum Conversions

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72 AMSAA-Crow Model Projection Method for reliability growth planning

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73 R R = Discrete PM2 Growth Plan Example Sims-Reliability Growth (TE Class)

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74 Continuous PM2 Growth Plan Example M I = M R = 200 M GP = Sims-Reliability Growth (TE Class)

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75 Continuous Curve Equation Continuous curve is plotted using this equation. MTBF(T) = System Mean-Time-Between-Failures at time T MTBF I = Initial MTBF MS = Management Strategy µ = Average Fix Effectiveness Factor (FEF) β = Shape parameter

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76 R(N) = System Reliability at trial N. R A = The portion of the system reliability not impacted by the correction action effort R B = The portion of the system reliability addressed by the correction action effort MS = Management Strategy µ = Average Fix Effectiveness Factor (FEF) n = Shape parameter of the beta distribution representing pseudo trials Discrete Curve Equation Discrete curve is plotted using this equation.

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77 Management Strategy Factor Management Strategy (MS) is the fraction of the overall system failure rate to be address by the corrective action plan. λ = Failure rate. For various reasons (prohibitive cost, improbability of reoccurrence), some failure modes will not have a corrective action.

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78 Management Strategy Factor A-Mode: Failures that are not fixed. B-Mode: Failures that will have a fix. A “fix” means a reliability improvement corrective action, not just a remove and replace of the same component.

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79 Management Strategy Factor λ A = Failure rate of A-modes λ B = Failure rate of B-modes λ A + λ B = Overall system failure rate

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80 Management Strategy Factor Example: What is the MS here? Failure mode Failure mode rate Mode Type B B B A B

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81 Management Strategy Factor Example: What is the MS here? Failure mode Failure mode rate Mode Type B B B A B Total B-modes Total System 0.089

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82 μ, Fix Effectiveness Factor Mil-HDBK-189 Definition: Fix Effectiveness Factor, μ = A fraction representing the reduction in an individual initial mode failure rate due to implementation of a corrective action. Essentially Fix Effectiveness Factors discount failures. A couple examples will follow.

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83 Number of tests = 20 Successful tests = 18 Hardware Failure Software Failure What is the reliability? X X μ, Fix Effectiveness Factor

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84 Number of tests = 20 Successful tests = 18 X X Software Failure Hardware Failure μ, Fix Effectiveness Factor

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85 Number of tests = 20 Successful tests = 18 X X What is the updated reliability? μ 1 = 100% μ 2 = 75% Software Failure Hardware Failure μ, Fix Effectiveness Factor

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86 Number of tests = 20 Successful tests = 18 Hardware Software X X 100% Fix 75% Fix μ, Fix Effectiveness Factor

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87 Failure mode Failure mode rate Mode Type B B B A B μ, Fix Effectiveness Factor Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? λ A = λ B = λ System = 0.089

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88 Failure mode Failure mode rate Mode Type B B B A B μ, Fix Effectiveness Factor Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? Original λ A = λ B = λ System = Updated λ A = λ B = * ( ) = λ System = 0.023

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89 Shape Parameter, β β = Shape parameter T T = Total Test Time M G = MTBF Goal M GP = MTBF Growth Potential M I = Initial MTBF

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90 η = Shape parameter of the beta distribution representing pseudo trials N T = Total Number of Trials R G = Reliability Goal R GP = Reliability Growth Potential R I = Initial Reliability Shape Parameter, β

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91 Growth Potential M GP = MTBF Growth Potential The theoretical upper limit on MTBF

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92 Growth Potential M GP = MTBF Growth Potential The theoretical upper limit on MTBF For example: MS = 0.95 μ = 0.80 M I = 190 For example: MS = 0.95 μ = 0.80 M I = 190

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93 R A = The portion of the system reliability not impacted by the correction action effort PM2 Curve Equation MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – R I = Initial Reliability

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94 R B = The portion of the system reliability addressed by the correction action effort PM2 Curve Equation MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – R I = Initial Reliability

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95 Management Strategy Factor A-Mode: Failures that are not fixed. B-Mode: Failures that will have a fix. λ = Failure rate. Fraction of failures to be addressed by the corrective action plan.

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96 n = Shape parameter of the beta distribution representing pseudo trials PM2 Growth Plan R GP = Reliability Growth Potential R G = Reliability Goal (to meet requirement) N T = Total trials before going into IOT phase

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97 Reliability Growth Potential R GP = Reliability Growth Potential The theoretical upper limit on system reliability

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98 Reliability Growth Potential R GP = Reliability Growth Potential The theoretical upper limit on system reliability For example: MS = 0.95 μ = 0.80 M I = 190 For example: MS = 0.95 μ = 0.80 M I = 190

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99 Summary Reliability Growth applies a “Learning Curve” Approach System must undergo Test-Analyze-And-Fix for reliability to grow. Initial Conditions are sensitive to a growth plan.

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100 ASMSA-Crow/Duane Equations 1. Single Shot Systems Expected Failures: 2. Continuously Operating Systems Expected Failures:

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