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Concept of Reliability

Prepared By

Dr. M. S. Memon

Department of Industrial Engineering and Management

Mehran University of Engineering and Technology

Jamshoro, Sindh, Pakistan

• Reliability is the probability of a product performing its intendedfunction for a stated period of time under certain specifiedconditions.

• Four aspects of reliability are apparent from this definition.– First, reliability is a probability-related concept; the numerical value of this

probability is between 0 and 1.

– Second, the functional performance of the product has to meet certainstipulations.

– Third, reliability implies successful operation over a certain period of time.

– Fourth, operating or environmental conditions under which product usetakes place are specified.

RELIABILITY

• Most products go through three distinct

phases from product inception to wear-out.

• Figure shows a typical life-cycle curve for

which the failure rate λ is plotted as a

function of time.

• This curve is often referred to as the

bathtub curve; it consists of the

debugging phase, the chance-failure phase,

and the wear-out phase.

LIFE-CYCLE CURVE AND PROBABILITY DISTRIBUTIONS IN

MODELING RELIABILITY

FIGURE: Typical life-cycle curve

• The debugging phase, also known as the infant-mortality phase,exhibits a drop in the failure rate as initial problems identifiedduring prototype testing are ironed out.

• The chance failure phase, between times t1 and t2, is thenencountered; failures occur randomly and independently. Thisphase, in which the failure rate is constant, typically represents theuseful life of the product.

• Following this is the wear-out phase, in which an increase in thefailure rate is observed. Here, at the end of their useful life, partsage and wear out.

RELIABILITY

• Exponential Distribution The life-cycle curve shown in above figureshows the variation of the failure rate as a function of time. For the chance-failure phase, which represents the useful life of the product, the failure rateis constant.

• As a result, the exponential distribution can be used to describe the timeto failure of the product for this phase.

• The probability density function of exponential distribution is given by

• where λ denotes the failure rate

Probability Distributions to Model Failure Rate

• The mean time to failure (MTTF) for the exponential distribution is givenas

• Thus, if the failure rate is constant, the mean time to failure is the reciprocalof the failure rate.

• For repairable equipment, this is also equal to the mean time betweenfailures (MTBF).

• There will be a difference between MTBF and MTTF only if there is asignificant repair or replacement time upon failure of the product.

Probability Distributions to Model Failure Rate

• The reliability at time t, R(t), is theprobability of the product lasting up to atleast time t. It is given by

• where F(t) represents the cumulativedistribution function at time t.

• Adjacent Figure shows the reliabilityfunction, R(t), for the exponential failuredistribution.

• At time 0, the reliability is 1, as it should be.Reliability decreases exponentially with time.

Probability Distributions to Model Failure Rate

FIGURE: Reliability function for the

exponential time-to-failure

distribution

• In general, the failure-rate function r(t) is givenby the ratio of the time-to-failure probabilitydensity function to the reliability function. Wehave

• For the exponential failure distribution

implying a constant failure rate, as mentionedearlier.

Probability Distributions to Model Failure Rate

FIGURE: Reliability function for the

exponential time-to-failure

distribution

• An amplifier has an exponential time-to-failure distribution

with a failure rate of 8% per 1000 hours. What is the reliability

of the amplifier at 5000hours? Find the mean time to failure.

Example 1

• What is the highest failure rate for a product if it is to have a probability

of survival (i.e., successful operation) of 95% at 4000 hours? Assume

that the time to failure follows an exponential distribution.

Example 2

• The availability of a system at time t is the probability that the

system will be up and running at time t. To improve availability,

maintenance procedures are incorporated, which may include

periodic or preventive maintenance or condition-based

maintenance. An availability index is defined as

Availability

• Downtime may consist of active repair time, administrative time (processing ofnecessary paperwork), and logistic time (waiting time due to lack of parts). It isobserved that maintainability is an important factor in influencing availability.Through design it is possible to increase the reliability and hence operationalprobability of a system. Further, downtime can be reduced through adequatemaintenance plans. For a steady-state system, denoting the mean time to repair(MTTR) to include all the various components of downtime, we have

• In the situation when the time-to-failure distribution is exponential (with a failurerate λ) and the time-to-repair distribution is also exponential (with a repair rateμ), the availability is given by μ /(λ + μ).

Availability

• Most products are made up of a number of components. The reliability ofeach component and the configuration of the system consisting of thesecomponents determines the system reliability (i.e., the reliability of theproduct).

• Although product design, manufacture, and maintenance influencereliability, improving reliability is largely the domain of design.

• One common approach for increasing the reliability of the system isthrough redundance in design, which is usually achieved by placingcomponents in parallel: As long as one component operates, the systemoperates.

• Here we demonstrate how to compute system reliability for systems thathave components in series, in parallel, or both.

SYSTEM RELIABILITY

• Figure shows a system with three components(A, B, and C) in series. For the system tooperate, each component must operate. It isassumed that the components operateindependent of each other (i.e., the failure of onecomponent has no influence on the failure of anyother component).

• In general, if there are n components in series,where the reliability of the ith component isdenoted by Ri, the system reliability is

Systems with Components in Series

A B C

FIGURE:

System with components in

series.

• A module of a satellite monitoring system has 500 components in series.

The reliability of each component is 0.999. Find the reliability of the

module. If the number of components in series is reduced to 200, what is

the reliability of the module?

Example 3

• Use of the Exponential Model If the system components can be

assumed to have a time to failure given by the exponential

distribution and each component has a constant failure rate, we

can compute the system reliability, failure rate, and mean time

to failure.

• As noted earlier, when the components are in the chance-failure

phase, the assumption of a constant failure rate should be

justified.

Systems with Components in Series

• Suppose that the system has n components in series, each withexponentially distributed time-to-failure with failure rates λ1, λ2, … λn . Thesystem reliability is found as the product of the component reliabilities:

• implies that the time to failure of the system is exponentially distributedwith an equivalent failure rate of σ𝑖=1

𝑛 𝜆𝑖. Thus, if each component that failsis replaced immediately by another that has the same failure rate, the meantime to failure for the system is given by

Systems with Components in Series

• When all components in series have an identical failure rate,

say λ, the MTTF for the system is given by

Systems with Components in Series

• The automatic focus unit of a television camera has 10 components in series.Each component has an exponential time-to-failure distribution with a constantfailure rate of 0.05 per 4000 hours. What is the reliability of each component after2000 hours of operation? Find the reliability of the automatic focus unit for 2000hours of operation. What is its mean time-to-failure?

Example 4

• Refer to Example 4 concerning the automatic focus unit of a televisioncamera, which has 10 similar components in series. It is desired for thefocus unit to have a reliability of 0.95 after 2000 hours of operation. Whatwould be the mean time to failure of the individual components?

Example 5

• System reliability can be improved by placingcomponents in parallel. The components areredundant; the system operates as long as atleast one of the components operates. The onlytime the system fails is when all the parallelcomponents fail.

• Figure demonstrates an example of a systemwith three components (A, B, and C) inparallel.

• All components are assumed to operatesimultaneously.

Systems with Components in Parallel

A

B

C

• Suppose that we have n components in parallel, with the

reliability of the ith component denoted by Ri, i=1, 2, ..., n.

• Assuming that the components operate randomly and

independently of each other, the probability of failure of each

component is given by Fi = 1- Ri. Now, the system fails only if

all the components fail. Thus, the probability of system failure

is

Systems with Components in Parallel

• Thus, the probability of system failure is

Systems with Components in Parallel

• Use of the Exponential Model If the time to failure of eachcomponent can be modeled by the exponential distribution, eachwith a constant failure rate λi, i= 1, ..., n, the system reliability,assuming independence of component operation, is given by

• In the special case where all components have the same failure rateλ, the system reliability is given by

Systems with Components in Parallel

• the mean time to failure for the system with n identical

components in parallel, assuming that each failed component is

immediately replaced by an identical component, is given by

Systems with Components in Parallel

• Find the reliability of the system shown with three components (A, B, andC) in parallel. The reliabilities of A, B, and C are 0.95, 0.92, and 0.90,respectively.

• Note that the system reliability is much higher than that of the individualcomponents. Designers can increase system reliability by placing morecomponents in parallel, but the cost of the additional componentsnecessitates a trade-off between the two objectives.

Example 6

• For the system shown in Figure, determine thesystem reliability for 2000 hours of operation,and find the mean time to failure.

• Assume that all three components have anidentical time-to-failure distribution that isexponential, with a constant failure rate of0.0005/ hour.

• What is the mean time to failure of eachcomponent? If it is desired for the system tohave a mean time to failure of 4000 hours,what should the mean time to failure be foreach component?

Example 7

A

B

C

• By placing three identical components in parallel, the system MTTF has been increased byabout 83.3%.

Example 7

Example 7

Systems with Components in Series and in Parallel

• Complex systems often consist of components that are both in

series and in parallel.

• Reliability calculations are based on the concepts discussed

previously, assuming that the components operate

independently.

Example 8

• Find the reliability of the eight-

component system shown in

Figure; some components are in

series and some are in parallel. The

reliabilities of the components are

as follows: RA1 = 0.92, RA2 = 0.90,

RA3 = 0.88, RA4 = 0.96, RB1 = 0.95,

RB2 = 0.90, RB3 = 0.92, and RC1 =

0.93.

B1

B2

B3

A2

A3

A1

A4

C1

Example 8

Example 9

• Find the system failure rate and the mean time to failure for the

eight component system shown in previous Figure of example

8. The failure rates (number of units per hour) for the

components are as follows: λA1 = 0.0006, λA2 = 0.0045, λA3 =

0.0035, λA4 = 0.0016, λB1 = 0.0060, λB2 = 0.0060, λB3 = 0.0060,

and λC1 = 0.0050.

Example 9