Random Event, Probability of Failure and Failure Rate What is the random event?
Random event in reliability engineering means random failure which is a predict chance or average event, not a specific event in specific time due to, in fact they are limitation of statistical operating time or some variables are not completely understood.
The predictions are based on statistical data gathered from a large number of sources and varieties condition. So, “random failure event” or “time of failure event” often cannot be preciously predicted.
Random event are classified typically into two (2) type; 1) Discrete variable when certain value is available and 2) Continuous value from calculated numerical value within a range.
Why is the important of random event?
The random event or random variable relates to Probability Density Function (PDF) P(x) or probability of getting a value and further to Cumulative Distribution Function (CDF) in reliability engineering. Anyhow, the basic mathematic property of probability is summation of all probability, sigma of all probability shall be equal to one for discrete value. And for continuous value, the integral of all probability density function shall be equal to one same.
Cumulative Distribution Funtion (CDF) F(x) represents the cumulative probability of getting a value in the interval time which is Probability of Failure (PF) in process safety engineering. So, it is the integral of Probability Density Funtion which means the area under the PDF curve accumulated up to the present value.
F(t)=∫_(-∞)^(+∞)▒f(t)dt
However, in reliability engineering there are several to represent the CDF such as normal distribution, exponential distribution, lognormal distribution, weibull distribution, gramma distribution and etc . Typically, to use which type of distribution it depends on area of concerned.
Exponential distribution is generally used for non-repairable system.
Normal distribution is generally used for high stress components, science area of concerned
Lognormal distribution is generally used for repairable system.
The below is the represent curve of PDF f(t) and CDF F(t) of several kind of distribution such as exponential distribution, normal distribution and lognormal distribution.
How to estimate the Probability of Failure
The most used of distribution in process safety engineering is exponential distribution. So, the PDF of exponential distribution is;
f(t)=ke^(-kt),for t more than 0 and
=0,for t lower than 0
And the CDF is represented by capital F(T) and calculated as;
F(T)=∫▒〖ke^(-kt) dt=〗 1-e^(-kt)
K is constant value, in view point of process safety engineering it is a represent by failure rate or lamda (). So, the equation of CDF or Probability of Failure (PF) will be;
PF=1-e^(-t) for t 0
The below is the picture presents the PDF curve and CDF curve, the maximum CDF will reach at one.
What is the failure rate?
From the above equation, the probability is directly relied on the failure rate or lamda (). Failure is typically defined as fails to perform it’s intend function. In fact, failure is classified into 2 categories 1) Random Failure and 2) Systematic Failure.
The picture below is the bathtub curve that represent the life cycle of electronic part.
Failure rate, often it is called “hazard rate”. Failure rate is separated into three (3) regions 1) Infant Mortality 2) Useful Life and 3) Wearout.
In reliability engineering, the failure rate that is used in calculation is failure rate in “Useful Life” or “Constant failure rate” the other regions such as infant mortality will be removed during commissioning or start-up phase.
How to calculate the failure rate
For discrete values sets, the instantaneous failure rate is a calculated from number of failures per unit of time divided from a total quantity of components exposed to failure.
λ_i=(Failures per unit time )/(Total quantity exposed)
However, the actual recording of failure rate is doing during plant is on operation. It calls time-dependent failure rate. The difference with above basic formula is the total quality exposed or surviving unit is reduced based on time increasing.
So, the time dependent failure rate is ratio of the changing number of failure unit during time period and number of survival multiple with the time difference between each failure
λ=(ΔNumber of failure unit during time period )/(Number of survived unit x ΔTime period between failure )
From above equation, it means some of number unit is still alive during collecting data. For conservative calculation in reliability engineering all sampling shall be taken into calculation, typically it is assumed that for survival units will at least one more hours can be operated, then all sampling will be taken into account for calculate failure rate.
The below is sample of calculation the time-dependent failure rate. The items number 7 to 10 are assumed to be failure during 1801 hours.
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