Dependability & Maintainability Theory and Methods Part 1: Introduction and definitions

A. Bobbio Reggio Emilia, June 17-18, 2003 1

Dependability & Maintainability Theory and Methods

Part 1: Introduction and definitions

Andrea BobbioDipartimento di Informatica

Università del Piemonte Orientale, “A. Avogadro”15100 Alessandria (Italy)

[email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA

IFOA, Reggio Emilia, June 17-18, 2003

mailto:[email protected]





Dependability: DefinitionDependability: Definition

Dependability is the property of a system to be dependable in time, i.e. such that reliance can justifiably be placed on the service it delivers.

Dependability extends the interest on the system from the design and construction phase to the operational phase (life cycle).


What dependability theory and practicewants to avoid


dependability

measures

reliabilityavailabilitymaintainabilitysafetysecurity

means fault forecastingfault tolerancefault removalfault prevention

threats faults errorsfailures

Dependability: TaxonomyDependability: Taxonomy


Quantitative analysisQuantitative analysis

The quantitative analysis aims at numerically evaluating measures to characterize the dependability of an item:

Risk assessment and safety

Design specifications

Technical assistance and maintenance

Life cycle cost

Market competition


Risk assessment and safetyThe risk associated to an activity is given proportional to the probability of occurrence of the activity and to the magnitute of the consequences.

A safety critical system is a system whose incorrect behavior may cause a risk to occur, causing undesirable consequences to the item, to the operators, to the population, to the environment.

R = P M


Design specifications

Technological items must be dependable.

Some times, dependability requirements (both qualitative and quantitative) are part of the design specifications:

Mean time between failures

Total down time


Technical assistance and maintenance

The planning of all the activity related to the technical assistance and maintenance is linked to the system dependability (expected number of failure in time).

planning spare parts and maintenance crews;

cost of the technical assistance (warranty period);

preventive vs reactive maintenance.


Market competition

The choice of the consumers is strongly influenced by the perceived dependability.

advertisement messages stress the dependability;

the image of a product or of a brand may depend on the dependability.


Purpose of evaluation

Understanding a system– Observation– Operational environment– Reasoning

Predicting the behavior of a system– Need a model– A model is a convenient abstraction– Accuracy based on degree of extrapolation


Methods of evaluation

Measurement-Based Most believable, most expensive Not always possible or cost effective during system design

Model-Based Less believable, Less expensive Analytic vs Discrete-Event Simulation Combinatorial vs State-Space Methods


Measurement-BasedMost believable, most expensive;

Data are obtained observing the behavior of physical objects.

field observations; measurements on prototypes; measurements on components (accelerated tests).


Closed-formAnswers

NumericalSolution

Analytic

Simulation

All models are wrong; some models are useful

Models


Methods of evaluation

Measurements + Models data bank


The probabilistic approachThe probabilistic approachThe mechanisms that lead to failure a technological object are very complex and depend on many physical, chemical, technical, human, environmental … factors.

The time to failure cannot be expressed by a determin-istic law.

We are forced to assume the time to failure as a random variable.

The quantitative dependability analysis is based on a probabilistic approach.


ReliabilityReliability

The reliability is a measurable attribute of the dependability and it is defined as:

The reliability R(t) of an item at time t is the probability that the item performs the required function in the interval (0 – t) given the stress and environmental conditions in which it operates.


Basic Definitions: cdfLet X be the random variable representing the time to failure of an item.

The cumulative distribution function (cdf) F(t) of the r.v. X is given by:

F(t) = Pr { X t }

F(t) represents the probability that the item is already failed at time t (unreliability) .


Basic Definitions: cdf

Equivalent terminoloy for F(t) :

CDF (cumulative distribution function)

Probability distribution function

Distribution function


Basic Definitions: cdf

1

0

F(t)

ta

F(b)

F(a)

b

F(0) = 0lim F(t) = 1t

F(t) = non-decreasing


Basic Definitions: ReliabilityLet X be the random variable representing the time to failure of an item.

The survivor function (sf) R(t) of the r.v. X is given by:

R (t) = Pr { X > t } = 1 - F(t)

R(t) represents the probability that the item is correctly working at time t and gives the reliability function .


Basic Definitions

Equivalent terminology for R(t) = 1 -F(t) :

Reliability

Complementary distribution function

Survivor function


Basic Definitions: Reliability

1

0

R(t)

ta b

R(0) = 1lim R(t) = 0t

R(t) = non-increasing

R(a)


Basic Definitions: density

Let X be the random variable representing the time to failure of an item and let F(t) be a derivable cdf:

The density function f(t) is defined as:

d F(t)f (t) = ——— dt

f (t) dt = Pr { t X < t + dt }


Basic Definitions: Density

0

f (t)

ta b

f(x) dx = Pr { a < X b } = F(b) – F(a) a

b


Basic Definitions: Density

1

0

f (t)

t

00

dttRdtttfXEMTTF


Basic Definitions

Equivalent terminology: pdf

probability density function

density function

density

f(t) = dtdF ,)(

)()(

0

t

t

dxxf

dxxftF

For a non-negativerandom variable


Quiz 1:The higher the MTTF is, the higher the

item reliability is.1. Correct2. Wrong

The correct answer is wrong !!!


Hazard (failure) rate

h(t) t = Conditional Prob. system will fail in (t, t + t) given that it is survived until time t

f(t) t = Unconditional Prob. System will fail in (t, t + t)

)(1)(

)()()(

tFtf

tRtfth


is the conditional probability that the unit will fail in the interval given that it is functioning at time t.

is the unconditional probability that the unit will fail in the interval

Difference between the two sentences:– probability that someone will die between 90 and 91, given that he

lives to 90– probability that someone will die between 90 and 91

The Failure Rate of a Distribution

tΔth),( ttt

ttf ),( ttt

30Reggio Emilia, June 17-18, 2003A. Bobbio

DFR IFR

Decreasing failure rate Increasing fail. rate

h(t)

t

CFRConstant fail. rate

(useful life)

(infant mortality – burn in) (wear-out-phase)

Bathtub curve


Infant mortality (dfr)Also called infant mortality phase or reliability growth phase. The failure rate decreases with time.

Caused by undetected hardware/software defects; Can cause significant prediction errors if steady-state failure rates are used;Weibull Model can be used;


Useful life (cfr)The failure rate remains constant in time (age independent) .

Failure rate much lower than in early-life period.

Failure caused by random effects (as environmental shocks).


Wear-out phase (ifr)The failure rate increases with age.

It is characteristic of irreversible aging phenomena (deterioration, wear-out, fatigue, corrosion etc…)

Applicable for mechanical and other systems.

(Properly qualified electronic parts do not exhibit wear-out failure during its intended service life)

Weibull Failure Model can be used


Cumul. distribution function:

Reliability :

Density Function :

Failure Rate (CFR):

Mean Time to Failure:

0 1 tetF t

0 t tetf

0 ttR e t

tRtfth

1MTTF

Exponential DistributionFailure rate is age-independent (constant).


2.50

The Cumulative Distribution Function of an Exponentially Distributed Random

Variable With Parameter = 1

F(t)1.0

0.5

0 1.25 3.75 5.00 t

F(t) = 1 - e - t


2.50

The Reliability Function of an Exponentially Distributed Random

Variable With Parameter = 1

R(t)1.0

0.5

0 1.25 3.75 5.00 t

R(t) = e - t


Exponential Density Function (pdf)

f(t)

MTTF = 1/


Memoryless Property of the Exponential Distribution

Assume X > t. We have observed that the

component has not failed until time t

Let Y = X - t , the remaining (residual) lifetime

y

t

etXPtyXtPtXtyXP

tXyYPyG

1)(

)()|(

)|()(


Memoryless Property of the Exponential Distribution (cont.)

Thus Gt(y) is independent of t and is identical to the original exponential distribution of X

The distribution of the remaining life does not depend on how long the component has been operating

An observed failure is the result of some suddenly appearing failure, not due to gradual deterioration


Quiz 3: If two components (say, A and B) have independent

identical exponentially distributed times to failure, by the “memoryless” property, which of the following is

true? 1. They will always fail at the same time2. They have the same probability of failing at time

‘t’ during operation3. When these two components are operating

simultaneously, the component which has been operational for a shorter duration of time will survive longer


0

0

0 1

1

tetR

tettf

tetF

t

t

t

Weibull Distribution

Distribution Function:

Density Function:

Reliability:


1

1

0 1

)(

)( ttth

tR

tf

Weibull Distribution : shape parameter;

: scale parameter.

Failure Rate:

1 DfrCfr

Ifr


Failure Rate of the Weibull Distribution with Various Values of


Weibull Distribution for Various Values of

Cdf density


We use a truncated Weibull Model

Infant mortality phase modeled by DFR Weibull and the steady-state phase by the exponential

0 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520Operating Times (hrs)

Failu

re-R

ate

Mul

tiplie

r

76543210

Figure 2.34 Weibull Failure-Rate Model

Failure Rate Models


Failure Rate Models (cont.)

This model has the form:

where:steady-state failure rate

is Weibull shape parameter

Failure rate multiplier =

SS

W tCt

1)(760,8760,81

tt

SSWC ,11

SSW t)(



There are several ways to incorporate time dependent failure rates in availability modelsThe easiest way is to approximate a continuous function by a piecewise constant step function

2,190 4,380 6,570 10,950 13,140 15,330 17,520Operating Times (hrs)

Failu

re-R

ate

Mul

tiplie

r

76543210

Discrete Failure-Rate Model

8,7600

1

2 SS



Here the discrete failure-rate model is defined by:

ss

W t

2

1)(

760,8760,8380,4380,40

ttt


A lifetime experimentA lifetime experiment

N i.i.d components are put in a life test experiment.

1

2

3

4

N

t = 0

X 1

X 2X 3

X 4

X N


A lifetime experimentA lifetime experiment1234

N

X 1X 2

X 3X 4

X N

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Dependability & Maintainability Theory and Methods Part 1: Introduction and definitions