# SIST EN 61703:2002

(Main)## Mathematical expressions for reliability, availability, maintainability and maintenance support terms

## Mathematical expressions for reliability, availability, maintainability and maintenance support terms

Provides mathematical expressions for reliability, availability, maintainability and maintenance support measures. - Non-repaired items and - repaired items with zero and non-zero time to restoration are considered separately in this standard.

## Mathematische Ausdrücke für Begriffe der Funktionsfähigkeit, Verfügbarkeit, Instandhaltbarkeit und Instandhaltungsbereitschaft

## Expressions mathématiques pour les termes de fiabilité, de disponibilité, de maintenabilité et de logistique de maintenance

Fournit des expressions mathématiques pour les caractéristiques liées à la fiabilité, à la disponibilité, à la maintenabilité et à la logistique de maintenance. - Les entités non réparées et - les entités réparées à temps de panne nul et non nul sont traitées séparément dans la présente norme.

## Mathematical expressions for reliability, availability, maintainability and maintenance support terms (IEC 61703:2001)

### General Information

### Relations

### Standards Content (Sample)

SLOVENSKI STANDARD

SIST EN 61703:2002

01-september-2002

Mathematical expressions for reliability, availability, maintainability and

maintenance support terms (IEC 61703:2001)

Mathematical expressions for reliability, availability, maintainability and maintenance

support termsMathematische Ausdrücke für Begriffe der Funktionsfähigkeit, Verfügbarkeit,

Instandhaltbarkeit und Instandhaltungsbereitschaft

Expressions mathématiques pour les termes de fiabilité, de disponibilité, de

maintenabilité et de logistique de maintenance

Ta slovenski standard je istoveten z: EN 61703:2002

ICS:

01.040.07 Matematika. Naravoslovne Mathematics. Natural

vede (Slovarji) sciences (Vocabularies)

07.020 Matematika Mathematics

21.020 =QDþLOQRVWLLQQDþUWRYDQMH Characteristics and design of

VWURMHYDSDUDWRYRSUHPH machines, apparatus,

equipment

SIST EN 61703:2002 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------SIST EN 61703:2002

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SIST EN 61703:2002

EUROPEAN STANDARD EN 61703

NORME EUROPÉENNE

EUROPÄISCHE NORM January 2002

ICS 03.120.30;21.020

English version

Mathematical expressions for reliability, availability,

maintainability and maintenance support terms

(IEC 61703:2001)

Expressions mathématiques pour Mathematische Ausdrücke für Begriffe

les termes de fiabilité, de disponibilité, der Funktionsfähigkeit, Instandhaltbarkeit

de maintenabilité et de logistique und Verfügbarkeitde maintenance (IEC 61703:2001)

(CEI 61703:2001)

This European Standard was approved by CENELEC on 2001-12-04. CENELEC members are bound to

comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European

Standard the status of a national standard without any alteration.Up-to-date lists and bibliographical references concerning such national standards may be obtained on

application to the Central Secretariat or to any CENELEC member.This European Standard exists in three official versions (English, French, German). A version in any other

language made by translation under the responsibility of a CENELEC member into its own language and

notified to the Central Secretariat has the same status as the official versions.

CENELEC members are the national electrotechnical committees of Austria, Belgium, Czech Republic,

Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands,

Norway, Portugal, Spain, Sweden, Switzerland and United Kingdom.CENELEC

European Committee for Electrotechnical Standardization

Comité Européen de Normalisation Electrotechnique

Europäisches Komitee für Elektrotechnische Normung

Central Secretariat: rue de Stassart 35, B - 1050 Brussels

© 2002 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.

Ref. No. EN 61703:2002 E---------------------- Page: 3 ----------------------

SIST EN 61703:2002

EN 61703:2002 - 2 -

Foreword

The text of document 56/747/FDIS, future edition 1 of IEC 61703, prepared by IEC TC 56,

Dependability, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as

EN 61703 on 2001-12-04.The following dates were fixed:

– latest date by which the EN has to be implemented

at national level by publication of an identical

national standard or by endorsement (dop) 2002-09-01

– latest date by which the national standards conflicting

with the EN have to be withdrawn (dow) 2004-12-01

Annexes designated "normative" are part of the body of the standard.

Annexes designated "informative" are given for information only.

In this standard, annex ZA is normative and annexes A, B, C and D are informative.

Annex ZA has been added by CENELEC.__________

Endorsement notice

The text of the International Standard IEC 61703:2001 was approved by CENELEC as a European

Standard without any modification.__________

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SIST EN 61703:2002

- 3 - EN 61703:2002

Annex ZA

(normative)

Normative references to international publications

with their corresponding European publications

This European Standard incorporates by dated or undated reference, provisions from other

publications. These normative references are cited at the appropriate places in the text and the

publications are listed hereafter. For dated references, subsequent amendments to or revisions of any

of these publications apply to this European Standard only when incorporated in it by amendment or

revision. For undated references the latest edition of the publication referred to applies (including

amendments).NOTE When an international publication has been modified by common modifications, indicated by (mod), the relevant

EN/HD applies.Publication Year Title EN/HD Year

IEC 60050-191 1990 International Electrotechnical --

Vocabulary (IEV)

Chapter 191: Dependability and quality

of service

ISO 3534-1 1993 Statistics - Vocabulary and symbols--

Part 1: Probability and general statistical

terms

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SIST EN 61703:2002

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SIST EN 61703:2002

NORME CEI

INTERNATIONALE IEC

61703

INTERNATIONAL

Première édition

STANDARD

First edition

2001-09

Expressions mathématiques pour les termes

de fiabilité, de disponibilité, de maintenabilité

et de logistique de maintenance

Mathematical expressions for reliability,

availability, maintainability and

maintenance support terms

IEC 2001 Droits de reproduction réservés Copyright - all rights reserved

Aucune partie de cette publication ne peut être reproduite ni No part of this publication may be reproduced or utilized in

utilisée sous quelque forme que ce soit et par aucun procédé, any form or by any means, electronic or mechanical,

électronique ou mécanique, y compris la photocopie et les including photocopying and microfilm, without permission in

microfilms, sans l'accord écrit de l'éditeur. writing from the publisher.International Electrotechnical Commission 3, rue de Varembé Geneva, Switzerland

Telefax: +41 22 919 0300 e-mail: inmail@iec.ch IEC web site http://www.iec.ch

CODE PRIX

Commission Electrotechnique Internationale

PRICE CODE

International Electrotechnical Commission

Pour prix, voir catalogue en vigueur

For price, see current catalogue

---------------------- Page: 7 ----------------------

SIST EN 61703:2002

61703 © IEC:2001 – 3 –

CONTENTS

FOREWORD .......................................................................................................................... 7

INTRODUCTION ....................................................................................................................9

1 Scope ............................................................................................................................ 11

2 Normative references..................................................................................................... 11

3 Definitions...................................................................................................................... 13

4 Glossary of symbols and abbreviations .......................................................................... 15

4.1 Non-repaired items ............................................................................................... 15

4.2 Repaired items with zero time to restoration ......................................................... 17

4.3 Repaired items with non-zero time to restoration .................................................. 19

5 Assumptions .................................................................................................................. 25

5.1 General remarks ................................................................................................... 25

5.2 Assumptions for non-repaired items...................................................................... 25

5.3 Assumptions for repaired items............................................................................. 27

6 Mathematical expressions.............................................................................................. 33

6.1 Non-repaired items ............................................................................................... 33

6.1.1 Presentation ............................................................................................. 33

6.1.2 Reliability [191-12-01] ............................................................................... 33

6.1.3 Instantaneous failure rate [191-12-02]....................................................... 35

6.1.4 Mean failure rate [191-12-03] .................................................................... 37

6.1.5 Mean time to failure [191-12-07] ............................................................... 39

6.2 Repaired items with zero time to restoration ......................................................... 41

6.2.1 Presentation ............................................................................................. 41

6.2.2 Reliability [191-12-01] ............................................................................... 41

6.2.3 Instantaneous failure intensity [191-12-04]................................................ 43

6.2.4 Mean failure intensity [191-12-05] ............................................................. 45

6.2.5 Mean time to failure [191-12-07] ............................................................... 47

6.2.6 Mean time between failures [191-12-08].................................................... 47

6.2.7 Mean operating time between failures [191-12-09] .................................... 49

6.2.8 Mean up time [191-11-11] ......................................................................... 49

6.3 Repaired items with non-zero time to restoration .................................................. 49

6.3.1 Presentation ............................................................................................. 49

6.3.2 Reliability [191-12-01] ............................................................................... 51

6.3.3 Instantaneous failure intensity [191-12-04]................................................ 53

6.3.4 Mean failure intensity [191-12-05] ............................................................. 59

6.3.5 Mean time to failure [191-12-07] ............................................................... 61

6.3.6 Mean time between failures [191-12-08].................................................... 63

6.3.7 Mean operating time between failures [191-12-09] .................................... 63

6.3.8 Instantaneous availability [191-11-01] ....................................................... 65

6.3.9 Instantaneous unavailability [191-11-02] ................................................... 67

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61703 © IEC:2001 – 5 –

6.3.10 Mean availability [191-11-03] .................................................................... 69

6.3.11 Mean unavailability [191-11-04] ................................................................ 71

6.3.12 Asymptotic availability [191-11-05]............................................................ 73

6.3.13 Asymptotic unavailability [191-11-07] ........................................................ 75

6.3.14 Mean up time [191-11-11] ......................................................................... 75

6.3.15 Mean down time [191-11-12]..................................................................... 77

6.3.16 Maintainability [191-13-01] ........................................................................ 79

6.3.17 Mean repair rate [191-13-03]..................................................................... 81

6.3.18 Mean repair time [191-13-05] .................................................................... 83

6.3.19 Mean active corrective maintenance time [191-13-07] ............................... 85

6.3.20 Mean time to restoration [191-13-08]......................................................... 87

6.3.21 Mean administrative delay [191-13-11]...................................................... 89

6.3.22 Mean logistic delay [191-13-13]................................................................. 89

Annex A (informative) Performance aspects and descriptors ............................................... 93

Annex B (informative) Summary of measures related to time to failure ................................ 95

Annex C (informative) Comparison of some dependability measures forcontinuously operating items ................................................................................................ 99

Annex D (informative) Software dependability aspects ...................................................... 101

Bibliography ....................................................................................................................... 103

Figure 1 – Sample realization of a non-repaired item............................................................ 27

Figure 2 – Sample realization of a repaired item with zero time to restoration....................... 29

Figure 3 – Sample realization of a repaired item with non-zero time to restoration................ 31

Figure 4 – Comparison of an up time for a COI and an IOI ................................................... 33

Figure A.1 – Performance aspects and descriptors............................................................... 93

Table B.1 – Relations among functional measures of time to failure of continuously

operating items .................................................................................................................... 95

Table B.2 – Summary of measures for some probability distributions of time to failure

of continuously operating items ............................................................................................ 97

Table C.1 – Comparison of some dependability measures of continuously operatingitems with constant failure rate λ and restoration rate μ ...................................................... 99

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61703 © IEC:2001 – 7 –

INTERNATIONAL ELECTROTECHNICAL COMMISSION

____________

MATHEMATICAL EXPRESSIONS FOR RELIABILITY,

AVAILABILITY, MAINTAINABILITY AND

MAINTENANCE SUPPORT TERMS

FOREWORD

1) The IEC (International Electrotechnical Commission) is a worldwide organization for standardization comprising

all national electrotechnical committees (IEC National Committees). The object of the IEC is to promote

international co-operation on all questions concerning standardization in the electrical and electronic fields. To

this end and in addition to other activities, the IEC publishes International Standards. Their preparation is

entrusted to technical committees; any IEC National Committee interested in the subject dealt with may

participate in this preparatory work. International, governmental and non-governmental organizations liaising

with the IEC also participate in this preparation. The IEC collaborates closely with the International

Organization for Standardization (ISO) in accordance with conditions determined by agreement between the

two organizations.2) The formal decisions or agreements of the IEC on technical matters express, as nearly as possible, an

international consensus of opinion on the relevant subjects since each technical committee has representation

from all interested National Committees.3) The documents produced have the form of recommendations for international use and are published in the form

of standards, technical specifications, technical reports or guides and they are accepted by the National

Committees in that sense.4) In order to promote international unification, IEC National Committees undertake to apply IEC International

Standards transparently to the maximum extent possible in their national and regional standards. Any

divergence between the IEC Standard and the corresponding national or regional standard shall be clearly

indicated in the latter.5) The IEC provides no marking procedure to indicate its approval and cannot be rendered responsible for any

equipment declared to be in conformity with one of its standards.6) Attention is drawn to the possibility that some of the elements of this International Standard may be the subject

of patent rights. The IEC shall not be held responsible for identifying any or all such patent rights.

International Standard IEC 61703 has been prepared by IEC technical committee 56:

Dependability.The text of this standard is based on the following documents:

FDIS Report on voting

56/747/FDIS 56/771/RVD

Full information on the voting for the approval of this standard can be found in the report on

voting indicated in the above table.This publication has been drafted in accordance with the ISO/IEC Directives, Part 3.

Annexes A, B, C and D are for information only.The committee has decided that the contents of this publication will remain unchanged

until 2006. At this date, the publication will bereconfirmed;

withdrawn;

replaced by a revised edition, or

amended.

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SIST EN 61703:2002

61703 © IEC:2001 – 9 –

INTRODUCTION

Part 1 of IEC 60050-191 provides definitions for dependability and its influencing factors,

reliability, availability, maintainability and maintenance support, together with definitions of many

other terms commonly used in this field. Some of these terms relate to specific measures of the

individual performance characteristics, which can be expressed mathematically.This standard, used in conjunction with IEC 60050-191, provides practical guidance essential

for the quantification of those performance measures. For those requiring further information,

for example on detailed statistical methods, reference should be made to the IEC 60605

series of standards.Annex A provides a diagrammatic explanation of the relationships between some basic

mathematical terms, related random variables, probabilistic descriptors and measures.

Annex B provides a summary of measures related to time to failure.Annex C compares some dependability measures for continuously operating items.

Annex D explains some of the software dependability aspects.

The bibliography gives references for the mathematical basis of this standard.

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SIST EN 61703:2002

61703 © IEC:2001 – 11 –

MATHEMATICAL EXPRESSIONS FOR RELIABILITY,

AVAILABILITY, MAINTAINABILITY AND

MAINTENANCE SUPPORT TERMS

1 Scope

This International Standard provides mathematical expressions for reliability, availability,

maintainability and maintenance support measures defined in IEC 60050-191. The following

classes of items are considered separately in this standard:– non-repaired items;

– repaired items with zero time to restoration;

– repaired items with non-zero time to restoration.

In order to keep the mathematical formulae as simple as possible, the following basic

mathematical models are used to quantify dependability measures:– random variable (time to failure) for non-repaired items;

– simple (ordinary) renewal process for repaired items with zero time to restoration;

– simple (ordinary) alternating renewal process for repaired items with non-zero time to

restoration.To facilitate location of the full definition, the IEC 60050-191 reference for each term is shown

(in parenthesis) immediately following each term, for example:mean time to restoration [191-13-08]

The application of each dependability measure is illustrated by means of a simple example.

NOTE This standard is mainly applicable to hardware dependability, but many terms and their definitions may be

applied to items containing software. Some of the software dependability aspects are explained in annex D.

2 Normative referencesThe following normative documents contain provisions which, through reference in this text,

constitute provisions of this International Standard. For dated references, subsequent

amendments to, or revisions of, any of these publications do not apply. However, parties to

agreements based on this International Standard are encouraged to investigate the possibility

of applying the most recent editions of the normative documents indicated below. For undated

references, the latest edition of the normative document referred to applies. Members of IEC

and ISO maintain registers of currently valid International Standards.IEC 60050-191:1990, International Electrotechnical Vocabulary (IEV) – Chapter 191:

Dependability and quality of serviceISO 3534-1:1993, Statistics – Vocabulary and symbols – Part 1: Probability and general

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SIST EN 61703:2002

61703 © IEC:2001 – 13 –

3 Definitions

For the purpose of this International Standard, the terms and definitions of IEC 60050-191

and ISO 3534-1 apply. In addition, the following terms and definitions are used.3.1

instantaneous restoration intensity v(t)

limit, if it exists, of the quotient of the mean number of restorations of a repaired item in the

time interval (t, t + Δt) and the duration of this interval, Δt, when the duration of the time

interval tends to zeroNOTE The instantaneous restoration intensity is expressed by the formula as

[ ]

E N (t + Δt) − N (t)

v(t) = lim

Δt →0+ Δt

where

N (t) is the number of restorations in the time interval (0, t);

E denotes the expectation.

3.2

asymptotic failure intensity z(∞)

for modelling purposes, limit, if it exists, of the instantaneous failure intensity z(t), when time t

tends to infinity3.3

up-time distribution function F (t)

function giving, for every value of t, the probability that an up time will be less than, or equal

to, t3.4

complement to the up-time distribution function R (t)

function giving, for every value of t, the probability that an up time will be greater than t

For convenience, this function is referred to as the up-time survival function in this

standard.NOTE 1 R (t) = 1 − F (t).

U U

NOTE 2 If the up time is exponentially distributed, then

R (t) = exp(−t/MUT)

where MUT is the mean up time.

In this case, the reciprocal of MUT is denoted by λ :

λ = 1/MUT

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SIST EN 61703:2002

61703 © IEC:2001 – 15 –

4 Glossary of symbols and abbreviations

NOTE The symbols given below are widely used and recommended but are not mandatory.

4.1 Non-repaired itemsCOI Continuously operating item

IOI Intermittently operating item

MTTF Mean time to failure

MTTF Point estimate of the mean time to failure

R(t) Reliability function, i.e. the probability of survival until time t:

R(t) = R(t , t ) for t = 0 and t = t

1 2 1 2

Rt() Point estimate of the reliability function at time t

R(t , t ) Reliability for the time interval (t , t )

1 2 1 2

R(t, t + x | t) Conditional reliability for the time interval (t, t + x), assuming that the

item survived to time tTTF Observed time to failure of item i

f(t) Probability density function of the (operating) time to failure

f (t)

Point estimate of the probability density function of the (operating) time

to failure at time t

n Number of (non-repaired) items in the population that are operational at

the instant of time t = 0

n (t) Number of (non-repaired) items that are still operational at the instant of

time t (n (0) = n)n (t) − n (t + Δt) Number of items that fail in the time interval (t, t + Δt)

S S

Constant failure rate, i.e. the reciprocal of the mean time to failure

(MTTF) when the times to failure are exponentially distributed

Point estimate of the constant failure rate

(t) Instantaneous failure rate

λ(t)

Point estimate of the instantaneous failure rate at time t

λ (t , t )

Mean failure rate for the time interval (t , t )

1 2 1 2

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SIST EN 61703:2002

61703 © IEC:2001 – 17 –

4.2 Repaired items with zero time to restoration

COI Continuously operating item

IOI Intermittently operating item

MTBF Mean operating time between failures

MTTF Mean time to failure

MTTF Point estimate of the mean time to failure

MUT Mean up time

N(t) Number of failures in the time interval (0, t)

R(t) Reliability function, i.e. the probability of survival until time t:

R(t) = R(t , t ) for t = 0 and t = t

1 2 1 2

R (t)

Up-time survival function, i.e. the distribution function of the up times is

1 − R (t)

R(t , t ) Reliability for the time interval (t , t )

1 2 1 2

ˆ Point estimate of the reliability for the time interval (t , t )

1 2

R(t , t )

1 2

Z(t) Expected number of failures in the time interval (0, t)

Z(t) = E[N(t)], where E denotes the expectation

f(t) Probability density function of the (operating) times to failure

f (t) Probability density function of the up times

NOTE For COIs, f (t) = f(t).

(n)

Probability density function of calendar time to the nth failure, n ≥ 1

h (t)

CTTF

k Number of failures during a given period of observation

Number of items in the population

n (t, t + Δt) Number of failures observed in the time interval (t, t + Δt)

n (t , t ) Number of failures observed in the time interval (t , t )

F 1 2 1 2

n (t , t ) Number of items that were operational at the instant of time t and

S 1 2 1

operated without failure during the time interval (t , t )

1 2

z(t) Instantaneous failure intensity

z(∞) Asymptotic failure intensity

Point estimate of the instantaneous failure intensity at time tzt()

Mean failure intensity for the time interval (t , t )

z(t , t ) 1 2

1 2

ˆ Point estimate of the mean failure intensity for the time interval (t , t )

1 2

z(t , t )

1 2

Constant failure rate, i.e. the reciprocal of the mean time to failure

(MTTF) when the times to failure are exponentially distributed

Constant rate of transition from an up state to a down state, i.e. the

reciprocal of the mean up time (MUT) when the up time is exponentially

distributed

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SIST EN 61703:2002

61703 © IEC:2001 – 19 –

4.3 Repaired items with non-zero time to restoration

A Asymptotic availability

A(t) Instantaneous availability (availability function), i.e. the probability of the

item being in an up state at the instant of time tA(t ,t ) Mean availability for the time interval (t , t )

1 2

1 2

A(t ,t ) Point estimate of the mean availability for the time interval (t , t )

1 2

1 2

COI Continuously operating item

G (t) Distribution function of the times to restoration

IOI Intermittently operating item

M(t) Maintainability function, i.e. the probability of completing active

maintenance action within time t: M(t) = M(t , t ) for t = 0 and t = t

1 2 1 2

M(t , t ) Maintainability for the time interval (t , t )

1 2 1 2

N(t) Number of failures occurring in the time interval (0, t)

N (t) Number of restorations occurring in the time interval (0, t)

R(t) Reliability function, i.e. the probability of survival until time t

R(t) = R(t , t ) for t = 0 and t = t

1 2 1 2

R (t) Up-time survival function, i.e. the distribution function of the up times is

1 − R (t)R(t , t ) Reliability for the time interval (t , t )

1 2 1 2

MACMT Mean active corrective maintenance time, i.e. the expectation of the

active corrective maintenance time

MACMT Point estimate of the mean active corrective maintenance time

MAD Mean administrative delay

MAD Point estimate of the mean administrative delay

MADT Mean accumulated down time

MADT Point estimate of the mean accumulated down time

MAMT Mean active maintenance time, i.e. the expectation of the active

maintenance time

MCMT Mean corrective maintenance time, i.e. the expectation of the corrective

maintenance time

MDT Mean down time

MDT Point estimate of the mean down time

MLD Mean logistic delay

MLD Point estimate of the mean logistic delay

MRT Mean repair time

MRT Point estimate of the mean repair time

MTBF Mean operating time between failures

MTD Mean technical delay, i.e. the expectation of the technical delay

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SIST EN 61703:2002

61703 © IEC:2001 – 21 –

MTTF Mean time to failure

MTTF Point estimate of the mean time to failure

MTTR Mean time to restoration

MTTR Point estimate of the mean time to restoration

MUFT Mean undetected fault time, i.e. the expectation of the undetected fault

time

MUT Mean up time

MUT Point estimate of the mean up time

U Asymptotic unavailability

U(t) Instantaneous unavailability (unavailability function)

U (t ,t ) Mean unavailability for the time interval (t , t )

1 2

1 2

U (t ,t ) Point estimate of the mean unavailability for the time interval (t , t )

1 21 2

Z(t) Expected number of failures in the time interval (0, t) Z(t) = E[N(t)], where E

denotes the expectationf(t) Probability density function of the (operating) times to failure

f (t) Probability density function of the up times

f (t) Probability density function of the sum of the up times and the corres-

U+R

ponding times to restoration

g (t) Probability density function of the active corrective maintenance times

ACM

g (t) Probability density function of the administrative delays

g (t) Probability density function of the active maintenance times

g (t) Probability density function of the down times

g (t) Probability density function of the logistic delays

g (t) Probability density function of the times to restoration

g (t) Probability density function of the repair times

Rep

(n)

h (t) Probability density function of calendar time to the nth failure, n ≥ 1

CTTF

k Number of active corrective maintenance actions during a given period of

ACM

observation

k Number of administrative delays during a given period of observation

k Number of down times during a give

**...**

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