Supplement for Repairable System Reliability Reference Probability and Reliability for Engineers Miller TA340 M648 1985 primarily chapter 15 NIST e book Engineering Statistics Handbook sections labelled PDF probability density function f t t CDF cumulative distribution function F t f x dx 0 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 1 F t f t conditional probability R t f t R t h t therefore d d F t R t f t dt dt R t 1 F t now h t f t R t d R t dt R t d ln R t dt t integrate from 0 to t h x dx ln R t 0 t exponentiate h x dx 0 R t e t f t R t h t h t e therefore h x dx 0 now if assume observe failure rate h t constant t h t f t h t e h x dx 0 t f t e x t F t e dx e 1 F t 1 e t 0 have exponential assumption of probability of failure times 1 11 27 2006 exponential pdf and cdf example PDF of time to next failure f t e t 1 t 0 10000 1000 f t e t probability density of time to next failure lambda 1 1000 probability 0 001 5 10 4 0 0 2000 4000 6000 1 10 8000 4 time to next failure CDF F t 1 e t CDF of waiting time to next failure F t 1 e t or CDF of interarrival time between failures cumulative probability density of time to next failure lambda 1 1000 probability 1 0 5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1 10 4 time to next failure exponential pdf and cdf example reset variables interpret time to failure as a waiting time it can be shown that this can be represented as a Poisson process if a component which fails is immediately replaced with a new one having the same failure rate Some results from this observation mean waiting ttime between successive failures 1 MTBF Some results for exponential model 2 11 27 2006 R t 1 F t Reliability or Survival function R t e Reliability function probability unit survives beyond t t 0 05 e g if component has a failure rate of 0 05 1000 hours probability that it will survive at least 10 000 hrs is given by e 10000 1000 0 607 n components in series if a system consists of n components in series with respective failure rates 1 2 n n n Rs t e i t e i i t 1 so it also is an exponential distribution and the MTBF for the system is i 1 MTBF series system 1 1 n n i i 1 1 MTBF i i 1 for a parallel system with respective failure rates 1 2 n in this case we need to deal with unreliabilities F 1 R i is probability component i will fail i n probability all will fail unreliability Fp Fi i 1 n Rp t 1 Fp t 1 and probability of survival R p t i n F t 1 i 1 1 Ri t i 1 in this case exponential probability of failure Fi t 1 e n Fp t n i 1 h p t fp t Rp t i t this will not show exponential distribution i 1 d Fp t dt difficult to evaluate but notice at least it is f t Rp t Rp t difficult to obtain in general but when all components have same failure rate 3 11 27 2006 n Rp t 1 t t 1 e 1 1 e n 1 R t 1 i i 1 i 1 n choose k n k 1 1 e t n 1 1 n choose k n 1 e Rp t n choose k n 1 e t binomial coefficient k n k n n t n choose k n 1 e n choose k n 2 e t 2 t 1 n 1 n t e binomial theorem from mathworld wolfram com BinomialTheorem html it can be shown see reference page 460 after differentiating to find fp t fp t d Rp t dt and then calculating the mean MBTF parallel MBTF parallel 1 1 1 2 1 n e g if use two identical components in parallel 4 1 3 MBTF parallel 2 increase of 50 not double k 1 1 k 2 083 four to double another example if time permits on board 4 11 27 2006