110.4 Mean Time To Failure(MTTF The mean or expected value of the continuous random variable"time-to-failure"is the MTTE This is a very useful parameter and is often enough to assess the suitability of components. It can be obtained using either the failure density function f(r) or the reliability function R(t) as follows: MTTF=tf( dt or r(tdt (110.6) In the case of repairable components, the repair time can also be considered as a continuous random variable with an expected value of MTTR The mean time between failures, MTBE is the sum of MTTF and MTTR. Since for well-designed components MTTR<<MTTE MTBF and MTTF are often used interchangably. 110.5 Average Failure Rate The average failure rate over the time interval o to t is defined as In rlt AFR(O, T)= AFR(T) 110.6 A Posteriori Failure Probability When components are subjected to a burn-in(or wearin)period of duration T, and if the component survives during(0, T), the probability of failure during(T, T+n) is called the a posteriori failure probability Q(n. It can be found using f(E)dE (110.8) f(sds The probability of survival during(T, T+t)is R(n)=1-Q()= f(5R(T+) R(T) -4((109 f(s)ds 110.7 Units for Failure rates Several units are used to express failure rates. In addition to i(r) which is usually in number per hour, %/K is used to denote failure rate in percent per thousand hours and PPM/K is used to express failure rate in parts per million per thousand hours. The last unit is also known as FIT"fails in time". The relatio these units are given in Table 110.2 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 110.4 Mean Time To Failure (MTTF) The mean or expected value of the continuous random variable “time-to-failure” is the MTTF. This is a very useful parameter and is often enough to assess the suitability of components. It can be obtained using either the failure density function f(t) or the reliability function R(t) as follows: (110.6) In the case of repairable components, the repair time can also be considered as a continuous random variable with an expected value of MTTR. The mean time between failures, MTBF, is the sum of MTTF and MTTR. Since for well-designed components MTTR<<MTTF, MTBF and MTTF are often used interchangably. 110.5 Average Failure Rate The average failure rate over the time interval 0 to T is defined as (110.7) 110.6 A Posteriori Failure Probability When components are subjected to a burn-in (or wearin) period of duration T, and if the component survives during (0, T), the probability of failure during (T, T+t) is called the a posteriori failure probability Qc(t). It can be found using (110.8) The probability of survival during (T, T+t) is (110.9) 110.7 Units for Failure Rates Several units are used to express failure rates. In addition to l(t) which is usually in number per hour, %/K is used to denote failure rate in percent per thousand hours and PPM/K is used to express failure rate in parts per million per thousand hours. The last unit is also known as FIT for “fails in time”. The relationships between these units are given in Table 110.2. MTTF = t f(t)dt R(t)dt • • Ú Ú 0 0 or AFR T AFR T R T T 0, ln ( ) º ( ) = - ( ) Q t f d f d c T T t T ( ) = ( ) ( ) + • Ú Ú x x x x R tT Q t f d f d R T t R T d c T t T T T t ( ) = - ( ) = ( ) ( ) = ( + ) ( ) = - ( ) È Î Í ˘ ˚ ˙ + • • + Ú Ú Ú 1 x x x x exp l x x
TABLE 110.2 Relationships Between Different Failure Rate Units PPM/K(FIT) 105(%/K) %/K=105λ 0-(PPM/K) PPMK(FIT)=109λ 104(%/K PPM/K 110.8 Application of the Binomial Distribution In an experiment consisting of n identical independent trials, with each trial resulting in success or failure with probabilities of p and g, the probability P, of r successes and(n-r) failures is P,=Cp(1-p) (110.10) If X denotes the number of successes in n trials, then it is a discrete random variable with a mean value of (np) and a variance of (npg) In a system consisting of a collection of n identical components with a probability p that a component is defective, the probability of finding r defects out of n is given by the P, in Eq (110.10). If p is the probability of success of one component and if at least r of them must be good for system success, then the system reliability (probability of system success)is given by R=∑(-p)y For systems with redundancy, r< n 110.9 Application of the Poisson Distribution For events that occur"in-time"at an average rate of n occurrences per unit of time, the probability P(t) of exactly x occurences during the time interval(0, t) is given by 2()=c (110.12) The number of occurrences X in(0, r)is a discrete random variable with a mean value u of (ar)and a standard deviation o of vAr. By setting X=0 in Eq (110.12), we obtain the probability of no occurrence in(0, t)as e If the event is failure, then no occurrence means success and e-r is the probability of success or system reliability This is the well-known and often-used exponential distribution, also known as the constant-hazard model. 110.10 The Exponential Distribution A constant hazard rate(constant n)corresponding to the useful lifetime of components leads to the single parameter exponential distribution. The functions of interest associated with a constant n are f( =λe-,t>0 (110.13) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 110.8 Application of the Binomial Distribution In an experiment consisting of n identical independent trials, with each trial resulting in success or failure with probabilities of p and q, the probability Pr of r successes and (n-r) failures is (110.10) If X denotes the number of successes in n trials, then it is a discrete random variable with a mean value of (np) and a variance of (npq). In a system consisting of a collection of n identical components with a probability p that a component is defective, the probability of finding r defects out of n is given by the Pr in Eq. (110.10). If p is the probability of success of one component and if at least r of them must be good for system success, then the system reliability (probability of system success) is given by (110.11) For systems with redundancy, r < n. 110.9 Application of the Poisson Distribution For events that occur “in-time” at an average rate of l occurrences per unit of time, the probability Px(t) of exactly x occurences during the time interval (0 , t) is given by (110.12) The number of occurrences X in (0, t) is a discrete random variable with a mean value m of (lt) and a standard deviation s of . By setting X = 0 in Eq. (110.12), we obtain the probability of no occurrence in (0,t) as e–lt . If the event is failure, then no occurrence means success and e–lt is the probability of success or system reliability. This is the well-known and often-used exponential distribution, also known as the constant-hazard model. 110.10 The Exponential Distribution A constant hazard rate (constant l) corresponding to the useful lifetime of components leads to the singleparameter exponential distribution. The functions of interest associated with a constant l are: (110.13) TABLE 110.2 Relationships Between Different Failure Rate Units l(#/hr) %K PPM/K (FIT) l = l 10–5 (%/K) 10–9(PPM/K) %/K = 105 l %/K 10–4(PPM/K) PPM/K (FIT) = 109 l 104 (%/K) PPM/K P C p p r n r r n r = - ( ) - 1 R C p p n k k n k k r n = - ( ) - = Â 1 P t t e x x x t ( ) = ( ) - l l ! lt f t e t t ( ) = > - l l , 0