Transcript Time to failure - livingreliability.com

Time to failure
Probability, Survival, the Hazard
rate, and the Conditional Failure
Probability
f(t) is the Probability Density
Function (PDF).
PDF
f(t)
1. Failures do not occur at fixed times.
2. They occur randomly according to a probability
distribution.
3. The PDF is the usual way of representing a
failure distribution (also known as an “agereliability relationship”).
Working Age t
f(t) is the Probability Density
Function (PDF).
PDF
1. As density equals mass per unit of volume,
probability density is the probability of failure per
unit of time.
2. When multiplied by the length of a small time
interval T at t, the product is the probability of
failure in that interval.
3. The PDF is the basic description of the time to
failure of an item.
f(t)
T x f(t)
T
Working Age t
f(t) is the Probability Density
Function (PDF).
PDF
1. The PDF is often estimated from real life data. It
resembles a histogram of the failures of an item
in consecutive age intervals.
2. All other functions related to an item’s reliability
can be derived from the PDF. For example,
3. The area Σ(t x f(t)) under the PDF curve
between time 0 and time t1 is the (cumulative)
probability F(t) of failing prior to t1.
f(t)
T T T T T
t1
Working Age t
Finding the PDF equation
• The easiest (standard) way to get the PDF
equation is through Weibull Analysis. Weibull
Analysis assumes that the equation has the
form:

t
f (t )   
  
 1
e
t 
  
 
•  is the shape factor,  is the scale factor
• Given a sample of life cycles we can estimate 
and  using numerical methods.
All other functions related to an item’s reliability
can be derived from the PDF. For example:
• F(t) is the cumulative distribution function (CDF).
It is the area under the f(t) curve from 0 to t.. (Sometimes called the
unreliability, or the cumulative probability of failure.)
F (t ) 
t
 f (t )dt
• R(t) is the survival function. (Also called the reliability0
function.) R(t) = 1-F(t)
• h(t) is the hazard rate. (At various times called the hazard
function or failure rate.) h(t) = f(t)/R(t)
• H(t) is the conditional probability of failure H(t)=
(R(t)-R(t+L))/R(t). It is the probability that the item fails in a time interval [t to
t+L] given that it has not failed up to time t. Its graph resembles the shape of
the hazard rate curve.
• MTTF is the average time to failure. (Also called the
mean time to failure, expected time to failure, or average life.)

MTTF   tf (t )dt
0
Conditional Failure Probability
versus Failure Rate
• Often, the two terms "conditional probability of failure" and "hazard
rate" are used interchangeably in many RCM and practical
maintenance references.
• They sometimes define both terms as:
“The probability that an item will fail during an
age interval given that the item enters (or survives) to
that age interval.“
• This definition is not the one usually meant in reliability theoretical
works when they refer to “hazard rate” or “hazard function”.
• Nowlan and Heap point out that the hazard rate
may be considered as the limit of the ratio (R(t)R(t+L))/(R(t)*L) as the age interval L tends to
zero.
• This is shown on the next slides.
To summarize, "hazard rate" and "conditional probability
of failure" are often used interchangeably (in more
practical maintenance books).
The “hazard rate” is commonly used in most reliability
theory books. The conditional probability of failure is
more popular with reliability practitioners and is used
in RCM books such as those of N&H and Moubray.
The two definitions are called:
1. "hazard rate“, and
2. "conditional probability of failure“.
The two definitions
1. h(t) = f(t)/R(t)
2. H(t) = (R(t)-R(t+L))/R(t).
1. where L is the length of an
age interval.
1. h(t) = f(t)/R(t)
2. H(t) = (R(t)-R(t+L))/R(t).
• When you divide equation 2 by
L and let L tend to 0, you get
equation 1, as follows:
h(t) = f(t)/R(t)
(Hazard, Failure rate func.)
H(t) = (R(t)-R(t+L))/R(t). (Conditional failure prob.)
F(t)=1-R(t)
(Cumulative failure prob.)
• Differentiating F(t)=1-R(t)
f(t)= -dR(t)/d(t)
• Dividing the right side of
H(t) = (R(t)-R(t+L))/R(t) by L and and then
applying the definition of a limit as L tends to 0.
Lim R(t)-R(t+L) = (1/R(t))( -dR(t)/dt) = f(t)/R(t) = h(t)
L0
LR(t)
This applies not only to hazard
• Actually, not only the hazard function, but
f(t), F(t), and R(t) also have two versions
of their defintions as above.
• The first version is defined over a
continous range of age t while the second
one is defined over discrete age intervals,
e.g., (0,100), (100,200), (200,300), ...
• Roughly, we can say the second definition
is a discrete version of the first definition.
How are h(t) and H(t) used?
• h(t) = f(t)/R(t) is useful in reliability theory and
is mainly used for theoretical development.
•
H(t) = (R(t)-R(t+L))/R(t) is useful for
reliability practitioners, since in practice people usually
divide the age horizon into a number of equal age
intervals.
• The PDF, cdf, and Survival function may all be calculated
using age intervals.
• The results would be similar to histograms, rather than
continous functions obtained using the failure rate
defintion.