Maintenance Engineering(KXGP 6307)

“Constant Failure Rate” model

Lecturer: Professor Dr. Ezutah Udoncy Olugu

Done by: Mohammad Amin Ezazi (KGC110016)

Due date: 6th March 2013

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Abstract

This research, considers the terms “Failure” and “Failure Rate” as the basis of all analysis and studies.

The objective of the “Failure Rate” analysis is to investigate and elaborate the failures occurred to a system

relative to time factor. Considering the aging tendency of the equipments, we will take a look at the practical

chart called “Bath-tub” and will specifically analyze it’s constant fragment. In this way, we will utilize the

time-based distribution functions like: Exponential and Weibull along with common terms like: Mean Time

To Failure(MTTF) and Mean Time Between Failure(MTBF). It should be mentioned that, not all the time-

based distribution functions, can always model the real world phenomena in the best manner, but they all

have drawbacks and advantages.

Keywords: Bath-tub curve, Constant failure rate, Mean time to failure, Mean time between failure.

1. Introduction

a)Failure

To start with this research, it can be crucial to clarify the term of “Failure” and “Failure Rate”. Based on

the failure types, we may classify the failure into two categories: Functional and Potential. Functional failure

is the inability of an item (or the component containing it) to meet a specified performance standard,

whereas potential failure is an identifiable physical condition indicating an imminent functional failure, in

another word, failure is the termination of the ability of an item (i.e., any part, component, device,

subsystem, functional unit, component, or system that can be individually considered) to perform a required

function.

The natural process of a mechanical component/system failure relates to its age, specifically the wear and

tear effects. In many cases, increasing in aging is followed by reduced component performance

(degradation). According to the researches done so far, the consequences of failure are many and varied, but

most failures create economic impact. In manufacturing industries, production machine failures can create

many inconveniences. It may be responsible for machine downtime, low availability, customer

dissatisfaction, increased maintenance time and production costs, lower product quality, and delivery time

delay. In chemical or nuclear plants, failures might be very expensive (catastrophic effects) and in some

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cases, are not allowed at all due to safety issues. Although, failure cannot be totally avoided, the related risks

and effects can be minimized and controlled by implementing effective and efficient maintenance practice.

b)Failure Rate(𝝀)

It is a term that usually used to point at the number of failures or defects face with the manufactured

products or machineries per unit of time. Depending on the manufacturing procedures and conditions, this

rate can change or remains constant. In general, the components failure rate, will follow almost similar

patterns which is like a “Bath-tub” when sketched. Briefly, this curve illustrates the stepwise components

life time. This curve comprises three main sections.

The first section which usually shows a decreasing tendency, is the result of poor design, the use of

substandard components, or lack of adequate controls in the manufacturing process.

The second section with the constant incline, illustrates the useful life period which is characterized by an

essentially constant failure rate. This is the period dominated by chance failures. It is during this period of

time that the lowest failure rate occurs. The useful life period is the most common time frame for making

reliability predictions.

The third section with increasing rate, is a result of equipment deterioration due to age or use.

We can see a “Bath-tub” curve including these three distinct sections below:

0<𝛽<1 𝛽 = 1 𝛽 > 1

In this research, we will mainly focus on the Constant Failure Rate area.

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2. Discussion on the ”Constant Failure Rate” model The failure rate, can be easily modeled by time-based Weibull distribution function. Weibull Analysis

can be used as a method of determining where a population of modules is on the bathtub curve. Weibull

distribution is used to represent increasing, constant and decreasing failure rates .The Weibull distribution is

a 3-parameter distribution(whereas Exponential distribution has only 1 parameter). This distribution is given

by:

𝒇 𝒕 =  𝜷𝜼

𝑻𝜼

𝜷!𝟏

𝒆𝑻𝜼𝜷

The Weibull parameter (𝛽) is the slope. It signifies the rate of failure. When 𝛽 < 1, the Weibull distribution

models early failures of parts. When  𝛽 = 1, the Weibull distribution models the exponential distribution with

parameter =1.The exponential distribution is the model for the useful life period, signifying that random

failures are occurring. When 𝛽 = 3, the Weibull distribution models the normal distribution. The sketch

below, illustrates how the 𝛽 factor will determine the total shape of Bath-tub graph:

𝒇 𝒕 =   𝜷𝜼

𝑻𝜼

𝜷!𝟏𝒆

𝑻𝜼

𝜷

-----> if 𝜷 = 1 -----> 𝒇 𝒕 =   𝟏𝜼𝒆

𝑻𝜼 -----> as we can see, having the new

condition, the Weibull is changed to Exponential with parameter =1.

As we can see in the graph above, if 𝛽=1, the rate at which the failures occure will become constant and in

fact, the Weibull distribution behaves like an Exponential distribution with a constant parameter.

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the useful life period, is characterized by an essentially constant failure rate. This is the period dominated by

chance failures. Chance failures are those failures that result from strictly random or chance causes. They

cannot be eliminated by either lengthy burn-in periods or good preventive maintenance practices. Equipment

is designed to operate under certain conditions and up to certain stress levels. When these stress levels are

exceeded due to random unforeseen or unknown events, a chance failure will occur. The figure above is

somewhat deceiving, since Zone 2(CFR) is usually of much greater length than Zones 1(DFR) or 3(IFR).

The time when a chance failure will occur cannot be predicted, however, the likelihood or probability that

one will occur during a given period of time within the useful life can be determined by analyzing the

equipment design. If the probability of chance failure is too great, either design changes must be introduced

or the operating environment made less severe.

This CFR period is the basis for application of most reliability engineering design methods. Since it is

constant, the exponential distribution of time to failure is applicable. It describes the time between events in

a Poisson process, i.e. a process in which events occur continuously and independently at a constant average

rate. An important assumption for effective maintenance is that components will eventually have an

Increasing Failure Rate. Maintenance can return the component to the Constant Failure Region. Thus, for an

exponential failure distribution, the failure rate is a constant with respect to time (that is, the distribution is

"memory-less"). For other distributions, such as a Weibull distribution or a log-normal distribution, the

failure function may not be constant with respect to time.

The simplicity of the approach utilizing the exponential distribution makes it extremely attractive.

Fortunately, it is widely applicable for complex equipments and systems. If complex equipment consists of

many components, each having a different mean life and variance which are randomly distributed, then the

system malfunction rate becomes essentially constant as failed parts are replaced. Thus, even though the

failures might be wearout failures, the mixed population causes them to occur at random time intervals with

a constant failure rate and exponential behavior.

The figure below indicates this concept for a population of incandescent lamps in a factory. This has been

verified for many equipments from electronic systems to bus-motor overhaul rates.

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𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙  "Probability  Distribution  Function"  -­‐-­‐-­‐-­‐-­‐>  f(t) =  𝝀𝒆!𝝀𝒕   𝝀 > 𝟎      ,      𝒕 > 𝟎

Reliability -----> R(t) = 1 – F(t) -----> R(t) = 𝒆!𝝀𝒕

If we apply the Exponential function “Memoryless” property -----> R(x I t) = P(T>t+x I T>t) = 𝒆!𝝀(𝒕!𝒙)

𝒆!𝝀𝒕 =

𝒆!𝝀𝒙 = R(x)

Also -----> 𝝀 𝒕 =   𝒇(𝒕)𝑹(𝒕)

=  𝝀 -----> As we can clearly see, the      𝜆  , is not function of time anymore and is a

constant independent value.

In case of Weibull function we can see also -----> 𝝀 𝒕 =   𝒇(𝒕)𝑹(𝒕)

-----> 𝝀 𝒕 = =   𝜷𝜼

𝒕𝜼

𝜷!𝟏 -----> 𝜷 = 𝟏

----->  𝝀 𝒕 =   𝟏𝜼  -----> which is a constant rate as well

constant failure rate of incandescent lamps in a factory

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3. Disadvantages of “Constant Failure Rate” model The exponential distribution models the behavior of units that fail at a constant rate, regardless of the

accumulated age. Although this property greatly simplifies analysis, it makes the distribution inappropriate

for most “good” reliability analyses because it does not apply to most real world applications. Like the

theory that the world is flat, the hypothesis of a constant failure rate provides mathematical models that can

be easily implemented and explained, yet leads us away from the benefits that can be gained by adopting

models that more accurately represent real world conditions. we can also look around the physical world we

live in and find irrefutable evidence that the failure rates of most, if not all, real world products are not

constant. To clarify this Exponential function’s deficiency, I chose an appropriate example as below:

Eg. We find that if the human mortality rate (failure rate) were constant, a significant percentage of the

population (10% based on the data sample used) would be dead by age 10, while another 10% would be

alive and well beyond 175 years of age, and a lucky 1% of us would continue to live well past 350 years of

age! These calculations clearly disagree with our observation of human mortality in the real world and the

following figure, demonstrates the discrepancy. This graph displays the human mortality data analyzed with

both the Exponential and Weibull distributions. It shows that the Weibull distribution models the behavior

better, while the exponential distribution overestimates the initial failure rate and significantly

underestimates the rate in later stages of life only because of “Constant Failure Rate” property.

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In order to complement the previous example and clarify it better, one can assume the effect of car mileage

on the fuel consumption. If cars exhibited a constant failure rate, then the vehicle’s mileage would not be a

factor in the price of a used car because it would not affect the subsequent reliability of the vehicle. In other

words, if a product can wear out over time, then that product does not have a constant failure rate.

Unfortunately, most items in this world do wear out, even electronic components and non-physical products

such as computer software. Electronic components have been shown to exhibit degradation over time and

computer software also exhibits wear-out mechanisms. For example, a freshly rebooted PC is less likely to

crash than a PC that has been running for a while, indicating an increasing software failure rate during each

run.

Despite the inadequacy of the exponential distribution to accurately model the behavior of most products in

the real world, it is still widely used in today’s reliability practices, standards and methods. As an example,

the first term learned by most people when they are introduced to reliability is MTBF (mean time between

failures). This term is so ingrained in current reliability science that it forms the basis for many comparisons

and most reliability standards, is widely used in reliability specifications and is the desired result of many

reliability, maintainability and availability analyses.

The exponential distribution is also widely used, although inappropriately, in the development of preventive

maintenance strategies. In many cases, the MTBF is used to determine a preventive maintenance interval for

a component. However, the use of the MTBF metric implies that the data were analyzed with an exponential

distribution since the mean will only fully describe the distribution when the exponential distribution is used

for analysis. The use of the exponential distribution, in turn, implies that the component has a constant

failure rate. This now begs the question of why anyone would preventively replace a component that has a

constant failure rate and does not experience wear-out over time! With a constant failure rate assumption,

preventive maintenance actions do not improve the reliability of the component, but rather waste time and

parts.

However, if the data are modeled by another distribution, then the mean is not sufficient to describe the data

and is, in many cases, a poor reliability metric. In addition, the term itself has led to many erroneous

assumptions and confusions about its relationship to other terms, such as MTTF (mean time to failure or the

mean of the data based on the assumed distribution). The reason for the confusion is that both of these terms

are equal if you assume a constant failure rate. In other words, if a product experiences one failure per hour,

the MTTF is one hour and so is the MTBF. However, if the failure rate is not constant, then each metric has

a different meaning and a different result. Thus, in the majority of cases, most practitioners are really

looking for and solving for the MTTF, regardless of what they choose to call it. In the analysis of repairable

systems, one might argue that the MTBF is a valuable metric because we record the times between failures

for the system (the random variable) and compute the mean of these times.

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This simple example demonstrates that preventive maintenance actions do not improve the overall reliability

of components that fail at a constant rate (i.e. follow an exponential distribution).

Eg. Two components follow an exponential distribution with MTTF = 100 hrs (or Lambda = 0.01).

Component 1 is preventively replaced every 50 hrs, while component 2 is never maintained.

Now we can compare the reliabilities of the components from 0 to 60 hrs:

• With PM: The reliability from 0 to 60 hrs is based on the reliability of the original component for 50 hrs,

R(t=50)=60.65%, multiplied by the reliability of the new component for 10 hrs, R(t=10)=90.48%. The

overall result is 54.88%.

• Without PM: Without PM, the reliability from 0 to 60 hrs is based on the reliability of the original

component operating to 60 hrs, R(60)=54.88%.

Comparing the reliabilities of the components from 50 to 60 hrs:

• With PM: The reliability from 50 to 60 hrs is based on the reliability of the new component,

R(t=10)=90.48%.

• Without PM: Without PM, the reliability from 50 to 60 hrs is based on the conditional reliability of the

original component operating to 60 hrs, having already survived to 50 hrs, or

RC(T=60|50)=R(60)/R(50)=90.48%.

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4. Conclusion

The Exponential function is used widely to model the “Constant Failure Rate” of the equipments due to

the simplicity of this approach and its high applicablity for complex equipments and systems. However, as

this method ignores the accumulated age of the equipments, , it is inappropriate for most “good” reliability

analyses, because it can not be applied to most real world applications. Thus, in many cases that precise

estimations and investigations are required, it can be better to use the alternative approaches like: Weibull-

based analysis instead of utilizing the Exponential function.

5. References

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108.

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distribution: a bathtub-shaped hazard for discontinuous failure data. Reliability Engineering and

System Safety 106, 37–44.

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flood data application. Communications in Statistics—Theory and Methods 25, 3059–3083.

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flood data application. Communications in Statistics—Theory and Methods 25, 3059–3083.

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• Pal, M., Ali, M.M., Woo, J., 2006. Exponentiated Weibull distribution. Statistica 66, 139–147.

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• Silva, R.B., Barreto-Souza, W., Cordeiro, G.M., 2010. A new distribution with decreasing,

increasing and upside-down bathtub failure rate. Computational Statistics and Data Analysis 54,

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