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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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9 123456789101112131415 161718192021222324252627282930 313233343536373839404142434445 464748495051525354555657585960 616263646566676869707172737475 767778798081828384858687888990 919293949596979899100101102103104105 106107108109110111112113114115116117118119120 121122123124125126127128129130131132133134135 136137138139140141142143144145146147148149150 151152153154155156157158159160161162163164165 166167168169170171172173174175176177Example 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 16 27 314 416 526 630 738 839 951 FailureTrial 1055 1164 1271 1379 1498 15108 16129 17145 18148

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11 FailureTrial Trials Between Failures 166 271 3147 4162 52610 6304 7388 8391 95112 There appears to be reliability growth. Example FailureTime Trials Between Failures 10554 11649 12717 13798 149819 1510810 1612921 1714516 181483 Less trials between failures. More trials between failures.

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12 0.9254Example Applying Reliability Growth Methodology, we get the following curve:.

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13 0.9254Example 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 reading400020 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 reading400020 Calculating α (the growth rate)

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22 First calculate the cumulative MTBF for each reading. Time (hrs) Total Failures MTBF First reading5005100 Last reading400020200 Calculating α (the growth rate)

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23 Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading5005100Ln(500)Ln(100) Last reading400020200Ln(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 reading5005100Ln(500)Ln(100) Last reading400020200Ln(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, 4.605 ) ( 8.294, 5.298 ) Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading5005100Ln(500)Ln(100) Last reading400020200Ln(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.40.27.46.64 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.40.27.46.64 TITI 100 MIMI 50 MFMF 150 T total 4351823285113 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 110 240 390 4160 5250 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 110 24020 39030 416040 525050 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 110 2402030 3903050 41604070 52505090 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 110 31 240203043 390305052 4160407059 5250509066 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 110 31 240203043 390305052 4160407059 5250509066 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 10 100 1000 10,000 10 100 Inst vs. Cum MTBF

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39 This is how the graphs look In standard Cartesian coordinate MTBF Time (or Test), t 300 100 5001000 1500 2000 200 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 0.40. 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 0.40. 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 0.40. 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 100150200250 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 100150200250 α.40 MIMI 50 MFMF 150 T total 4356528691087 Why?

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60 2505007501000 50 100 150 TITI 100250 α.40 MIMI 50 MFMF 150 T total 4351087 Sensitivity of Initial Time Time MTBF TITI

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61 2505007501000 50 100 150 TITI 100250 α.40 MIMI 50 MFMF 150 T total 4351087 Growth is more rapid the smaller T I is! Sensitivity of Initial Time MTBF Time TITI

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62 MIMI 50257085 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 50257085 α.40 TITI 100 MFMF 150 T total 4352459187115 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 = 0.9200 Discrete PM2 Growth Plan Example 041712-Sims-Reliability Growth (TE Class)

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74 Continuous PM2 Growth Plan Example M I = 190 500 M R = 200 M GP = 782 415 041712-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 10.027B 20.015B 30.033B 40.001A 50.013B

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81 Management Strategy Factor Example: What is the MS here? Failure mode Failure mode rate Mode Type 10.027B 20.015B 30.033B 40.001A 50.013B Total B-modes 0.088 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 10.027B 20.015B 30.033B 40.001A 50.013B μ, Fix Effectiveness Factor Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? λ A = 0.001 λ B = 0.088 λ System = 0.089

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88 Failure mode Failure mode rate Mode Type 10.027B 20.015B 30.033B 40.001A 50.013B 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 = 0.001 λ B = 0.088 λ System = 0.089 Updated λ A = 0.001 λ B = 0.088 * (1- 0.75) = 0.022 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 – 0.96. 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 – 0.96. 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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