23working_repair - Supplement for Repairable System...

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Supplement for Repairable System Reliability PDF = probability_density_function = f t ( ) Reference: Probability and Reliability for Engineers, Miller, TA340.M648 1985 t primarily chapter 15 CDF = cumulative_distribution_function = ( ) = ( ) NIST e-book Engineering Statistics Handbook, F t f x dx 0 sections labelled 8.1.2.2 Reliability or Survival function Reliability_function = probability_unit_survives_beyond_t R t ( ) = 1 F t ( ) F t ( ) = 1 R t ( ) or ... 8.1.2.3 Failure (or Hazard) rate h t ( ) = failure_rate = f t ( ) = f t ( ) conditional probability 1 F t ( ) R t ( ) therefore ... f t = R t ( ) ( ) ( ) h t now ... R t ( ) = 1 F t ( ) d F t ( ) = d R t ( ) = f t ( ) dt dt d R t ( ) h t ( ) = f t ( ) = dt = d ( ( ) ln R t ) R t ( ) R t ( ) dt t integrate from 0 to t h x dx = ( ( )) ( ) ln R t 0 t h x dx ( ) exponentiate .. R t ( ) = e 0 t h x dx ( ) therefore ... f t = R t ( ) = h t ( ) e 0 ( ) ( ) h t now .. if assume (observe) failure rate h(t) = constant = λ h t ( ) := λ t h x dx ( ) f t ( ) := ( ) e 0 h t ) t t ( − λ ) F t ( ) t := t λ⋅ e ( − λ ) x dx e ( − λ + 1 F t ( ) := 1 e λ f t ( ) λ⋅ e 0
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