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Appendix
1. Table 1: Standard normal distribution table. 2. Table 2: t distribution table 3. Table 3: Gamma function table 4. Table 4: Sigma level, Yield and DPMO table
Appendix
Table 1: Area under standard normal distribution N(0, 1)
Appendix
367
Table 2 : Critical t values with n degrees of freedom
Value of aN 0.100 0.050 0.025 0.010 0.0051 3 .078 6.314 12 .706 31 .821 63 .6572 1 .886 2.920 4.303 6.695 9.9253 1 .639 2.353 3.182 4.541 5 .84 14 1 .533 2.132 2 .776 3 .747 4.6045 1 .476 2.015 2 .571 3 .365 4.0326 1 .440 1 .943 2 .447 3.143 3 .7077 1 .415 1 .895 2 .365 2 .998 3 .4998 1 .397 1 .860 2 .306 2 .896 3 .3559 1 .383 1 .833 2 .262 2 .821 3 .25010 1 .372 1 .812 2 .228 2 .764 3 .16911 1 .366 1 .796 2 .201 2 .718 3 .10 612 1 .356 1 .782 2.179 2.681 3 .05513 1 .350 1 .771 2.160 2.650 3 .01 214 1 .345 1 .761 2.145 2.624 2.99715 1 .341 1 .753 2.131 2.602 2 .94716 1 .337 1 .746 2.120 2.583 2.92 117 1 .333 1 .740 2.110 2.567 2.89818 1 .330 1 .734 2.101 2.552 2.87819 1 .328 1 .729 2.093 2.539 2.86 120 1 .325 1 .725 2.086 2.528 2.84521 1 .323 1 .721 2.080 2.518 2 .83 122 1 .321 1 .717 2.074 2.508 2.81923 1 .319 1 .714 2.069 2.500 2.80724 1 .318 1 .711 2.064 2.492 2.79725 1 .316 1 .708 2.060 2.485 2.78726 1 .315 1 .706 2.056 2.479 2.79927 1 .314 1 .703 2.052 2.473 2 .77 128 1 .313 1 .701 2.048 2.467 2.76329 1 .311 1 .699 2.045 2.462 2.756
1 .282 1 .645 1 .960 2.326 2.576
Appendix
Table 3: Gamma Function Table
Appendix
Table 4: Sigma Level, DPMO and Yield table
PROCESS SIGMA(Z) (including the 1.5
shift)
DPMO YIELD (Y in %)
370 Appendix
Table 4 continued
(including the 1.5
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Index
Accelerated life testing (ALT), 215,254 Accelerated testing, 165
Shake and bake tests, 165 Achieved availability, 314, 323 Achieved availability, 323 Aggregated Cumulative Hazard, 236 AHP. See Analytic hierarchy process Analytic hierarchy process, 351 Analyze Phase, 358 Arrhenius relationship, 172 Availability, 313 - 320
Availability and Sigma level, 3 18 Average availability, 3 17 Axioms of probability, 14
Basic Reliability Measures, 63 Bathtub curve, 85 Benard's approximation, 263 Bernoulli trials, 30 Boeing 737,209 Bombay dabbawallas, 1 ,2 Burn-in, 165,167
Case study, 194 Aircraft engine components, 244 Armoured vehicle, 101 Aircraft emergency stopping system,
194 ARIANE 5,309 Engineer tank, 28 1 Freight cars, 344 DMAIC methodology, 360
Censored data, 253 - 255,263,264 Left, 255 Right, 255
Central limit theorem, 42 Chapman-Kolmogrov, 48 Cold standby system, 134, 136 Conditional probabilities, 17, 121
Confidence Interval, 256 Continuous probability distributions, 35 Control phase, 359 Correlation co-efficient, 267
Cost breakdown structure, 338 Cost estimation relationships, 338 Criticality analysis, 174, 182, 183, 205 Cumulative distribution function, 24
Index
Cumulative hazard function, 84 Cyclic exchange rate. 107
Defects per million opportunities,S-l0,59 Defects per Unit, 59 Define phase, 356,357 Demonstration of constant failure rate,
333 Design for Six Sigma, 7, 143,363 Design for six-sigma. 143 Distribution
Binomial, 31 Exponential, 36 Lognormal, 43 Normal, 38 Poisson, 33
Weibull, 44,4546 DMADV, 363 DMAIC, 7,351-355 DPMO. See Defects per million
opportunities Drenick, 212 Duane Model, 335 Duty cycle, 107
Elementary event, 13 Elementary Renewal Theorem. 55 Engine monitoring systems, 228 Environmental stress screening
Tests, 168 Environmental Stress Screening, 166 ESS. See Environmental stress screening Estimation of logistic support cost, 343 Estimation of Maintenance Cost, 342 Estimation of operating cost, 340 Estimation of parameters, 255 Event, 13
Failure Mechanism, 70
Failure function, 65 Applications, 67,68
Failure mode effects analysis. 182 Failure Mode Effects and Criticality
Analysis, 181, 188 Failure rate, 5 Failure reporting, 254 Failure reporting analysis and corrective
action system, 192 Fault tolerant software, 307 Fault tree analysis, 174, 175 Fault tree of a breath analyzer, 181 Field data, 254 FMEA. See Failure modes effects
analysis FMECA. See Failure mode, effects and
criticality analysis FTA. See Fault tree analysis Functional FMEA, 187
General Electric, 6 Goel-Okumoto Model, 301 Graceful degradation, 137 Green belt, 364
Hardware FMEA, 187,196,202 Hazard analysis, 188 hazard function, 117,128 Hazard function, 80,117,128
Applications, 8 1 Highly accelerated life testing, 173 Highly accelerated life testing (HALT),
254 Hot standby system, 136 House of Quality. See Quality function
deployment
Index
Improve phase, 359 Inherent availability, 314,317 Inservice mtbf, 217 inservice reliability, 209,211 Inverse Power Law Relationship, 172
Jelinski Moranda model, 302 Joint event, 16
KLOC, 294
Kolmogrov, 14 k-out-of-n systems, 131
LCC. See Life cycle cost Life Cycle Cost, 338 life exchange rate matrix, 11 8
Life testing, 254 Line replaceable unit, 214 Linear regression, 253 Location parameter, 36 Lower and upper control limits, 4 LRU. See line replaceable unit
Maintenance Free Operating Period, 98
Maintenance recovery period (MRP), 98 Market effectiveness, 10 Markov chain, 3 14
Continuous time, 47 Markov process, 47,48,49 Maximum likelihood estimator, 253,272
exponential distribution, 274 Weibull distribution, 276
McDonnell Douglas., 302 mean. 27
Mean operating time between failures. See MTBF
Mean residual life, 93 Mean Residual Life, 93 Mean Time Between Critical Failure, 64 . See MTBF, Mean time between maintenance. See
MTBM Mean Time Between Unscheduled
Removal, 64 mean time to failure, 113
. See MTTF, Measure Phase, 357 Measures of central tendency, 25
Measures of dispersion, 25 median, 27,72 Memory less property, 37 MFOP, 64,98
for Weibull distribution, 99 Maximum length, 100
MIL-HDBK-217,9,97 Minimal repair, 84 Mission reliability, 78 MLE. See maximum likelihood estimator mode, 27 Motorola, 2,3,5,6,9 MTBF, 96,97 MTBM, 323 MTTF, 89 MTTF and Sigma level, 91 Musa's execution time model, 295
NFF. See No fault found No fault found, 245, 249
Non-homogeneous poisson process, 51,
333 N-version programming, 309
Operational availability, 314,325
Index
Palm, 218 Parallel system, 122 Parts life tracking, 223
Point availability, 314
Poisson process
non-homogeneous (NHPP), 5 1 Probability
density function, 22
mass function, 22
Probability distribution, 18,20
Probability distributions
Fitting, 253
Probability papers, 253
Probability terminology, 12 Process capability, 57,58 Process capability index, 57
Process capability index, 57 Process shift, 5
Product Effectiveness, 10
Production effectiveness, 10
Prognostics, 237
Putnam's model. 297
QFD. See Quality function deployment, Quality function deployment, 10, 144
Random experiments, 12
Random variable, 18
Expectation of a, 26
Continuous, 20
Discrete, 20 Recovery blocks, 307
Regenerative process, 316 Regression, 253,- 272-282
Relevant Condition Indicators, 243
Relevant Condition Parameters, 238
Reliability allocation, 152
Equal apportionment technique, 153
Factors, 159 Minimum effort algorithm, 154
Reliability Apportionment. See Reliability allocation
Reliability Block Diagram, 108 Reliability centered maintenance, 85
Reliability demonstration, 330
Reliability Demonstration Testing, 331 Reliability estimation, 254
Reliability function, 72 Applications, 73
Properties, 73
Reliability function and Six Sigma, 77
Reliability Growth Program, 334
Reliability management, 330
Reliability measures, 63,64
Basic, 57 Mission, 63
Operational, 64 Reliability prediction, 106
Reliability Qualification Testing, 33 1
Renewal Function, 53
Renewal process, 52
Rules of probability, 15
Safety and hazard analysis, 174 Safety critical parts, 226
Sample space, 13
Scale parameter, 36
Series system, 109 Series-parallel combination, 129
Shape parameter, 36 Sigma level and MTTF, 1 14
Sigma quality level, 8 ,60
SINCGARS, 96 Six Sigma, 2-10
Six Sigma black belt, 364
Six Sigma measure, 57
Six Sigma process, 4
Six sigma project management, 350
384
Index
software reliability, 290-299.301, 303 Software reliability
Exponential model, 299 Metrics, 291
Software testing, 303 Sonar system, 138 Source parameter, 36 STAN-00-40,97 Standard deviation, 5,29 Standby redundancy, 133 Stochastic process, 46
TCO. See Total cost of ownership Temperature-Humidity Relationship, 173 Three sigma process, 4 Time to failure analysis, 233 Total cost of ownership, 3,9
Impact of sigma level, 349 Total cost of ownership, 343 Total Cost of Ownership, 338
Total cost of ownership model, 339
Total Quality Management, 7 Transition Probability, 48
Unscheduled maintenance, 247,249 Upper control limits, 3 Usage monitoring, 228
Value of z and sigma level, 41
William Smith, 3
Yield, 58