# 失效率

## 离散定义下的失效率

${\displaystyle \lambda (t)={\frac {f(t)}{R(t)))}$，其中${\displaystyle f(t)}$为（第一次）失效发生时间的分布（失效密度函数），而${\displaystyle R(t)=1-F(t)}$.
${\displaystyle \lambda (t)={\frac {R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\cdot R(t_{1})))={\frac {R(t)-R(t+\triangle t)}{\triangle t\cdot R(t)))\!}$

## 连续定义下的失效率

${\displaystyle h(t)=\lim _{\Delta t\to 0}{\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t))).}$

${\displaystyle \operatorname {Pr} (T\leq t)=F(t)=1-R(t),\quad t\geq 0.\!}$

${\displaystyle F(t)=\int _{0}^{t}f(\tau )\,d\tau .\!}$

${\displaystyle h(t)={\frac {f(t)}{1-F(t)))={\frac {f(t)}{R(t))).}$

${\displaystyle F(t)=\int _{0}^{t}\lambda e^{-\lambda \tau }\,d\tau =1-e^{-\lambda t},\!}$

${\displaystyle h(t)={\frac {f(t)}{R(t)))={\frac {\lambda e^{-\lambda t)){e^{-\lambda t))}=\lambda .}$

## 失效率递减

DFR的随机变数混合后仍为DFR[4]，而指数分布的随机变数混合后也是为DFR[5]

## 失效率资料

### 举例

${\displaystyle {\frac {6{\text{ failures))}{7502{\text{ hours))))=0.0007998{\frac {\text{failures)){\text{hour))}=799.8\times 10^{-6}{\frac {\text{failures)){\text{hour))},}$

## 参考资料

1. ^ 中国规范术语 - 检索结果
2. ^ MacDiarmid, Preston; Morris, Seymour; 等. Reliability Toolkit Commercial Practices. Rome, New York: Reliability Analysis Center and Rome Laboratory. n.d.: 35–39.
3. ^ Introduction. Springer, London. 2008: 1–7 [2018-04-02]. ISBN 9781848009851. doi:10.1007/978-1-84800-986-8_1 （英语）.
4. ^ Mark Brown. Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes. The Annals of Probability. April 1980, 8 (2): 227–240 [2018-04-02]. ISSN 0091-1798. doi:10.1214/aop/1176994773 （英语）.
5. Frank Proschan. Theoretical Explanation of Observed Decreasing Failure Rate. Technometrics: 375–383. doi:10.1080/00401706.1963.10490105.
6. ^ J. C. BAKER, G. A.SR. BAKER. Impact of the space environment on spacecraft lifetimes. Journal of Spacecraft and Rockets: 479–480. doi:10.2514/3.28040.
7. ^ On Time, Reliability, and Spacecraft. Wiley-Blackwell. : 1–8 [2018-04-02]. doi:10.1002/9781119994077.ch1 （英语）.
8. ^ Adam Wierman, Nikhil Bansal, Mor Harchol-Balter. A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues. Operations Research Letters: 73–76. [2018-04-02]. doi:10.1016/s0167-6377(03)00061-0.
9. ^ Gautam, Natarajan. Analysis of Queues: Methods and Applications. CRC Press. 2012: 703. ISBN 1439806586.
10. ^ 电子产品制造技术 - 第 25 页 - Google 图书结果