Random failure and the MTTF

Posted on October 3, 2014 by Murray Wiseman

Nowlan and Heap said that the (conditional) probability of occurrence of most failure modes is constant throughout the life of a component. They described this behavior as being that of “random” failure. What is the Mean Time to Failure (MTTF) of a randomly failing part?

In cases of random failure the formula for a component’s reliability (survival) over time is one of exponential decay [1] expressed as:

R\left ( t \right )=e^{-Lt}

The relationship between the reliability R(t), and the failure rate function h(t) [2] can be expressed by:

h\left ( t \right )=f\left ( t \right )/R\left ( t \right )=\frac{dF(t)/dt}{R(t)}=\frac{d(1-R(t))/dt}{R(t)}=\frac{-dR(t)/dt}{R(t)}

Substituting for R(t) in the case of exponential decay:

h(t)=-\frac{de^{-Lt}/dt}{e^{-Lt}}=--\frac{Le^{-Lt}}{e^{-Lt}}=L

That is to say, for components whose Reliability decreases exponentially with age, the failure rate is constant and equal to L.

But what is L?

By definition the mean time to failure (mttf) is the expected time to failure:

MTTF=\int_{0}^{\infty}tf(t)dt=\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}\frac{f(t)}{H(t)}dt [3]

But for exponential decay we have shown that h(t)=L, therefore

MTTF=\frac{1}{L}\int_{0}^{\infty}f(t)dt=1/L=1/h(t)

Hence for random failure: h(t) =L =1/MTTF [4]

and R\left ( t \right )=e^{-Lt}=e^{-\frac{t}{MTTF}}

© 2014 – 2016, Murray Wiseman. All rights reserved.

  1. [1]this was demonstrated graphically in the article here.↩
  2. [2]Or roughly speaking a discretized version of the failure rate, i.e. the Conditional Probability of Failure. For more on the difference between Hazard and Conditional Failure Probability see here and here↩
  3. [3]See here why MTTF is the area under the survival curve R(t)↩
  4. [4]Not exactly true if h(t) is taken as synonymous with  the conditional failure probability. See here.↩
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