IEC 61703:2016

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

IEC 61703:2016 provides mathematical expressions for selected reliability, availability, maintainability and maintenance support measures defined in IEC 60050­192:2015. In addition, it introduces some terms not covered in IEC 60050-192:2015. They are related to aspects of the system of item classes (see hereafter). According to IEC 60050-192:2015, dependability [192-01-22] is the ability of an item to perform as and when required and an item [192-01-01] can be an individual part, component, device, functional unit, equipment, subsystem, or system. To account for mathematical constraints, this standard splits the items between the individual items considered as a whole (e.g. individual components) and the systems made of several individual items. It provides general considerations for the mathematical expressions for systems as well as individual items but the individual items which are easier to model are analysed in more detail with regards to their repair aspects. This standard is mainly applicable to hardware dependability, but many terms and their definitions may be applied to items containing software. This second edition cancels and replaces the first edition published in 2001. This edition constitutes a technical revision. This edition includes the following significant technical changes with respect to the previous edition:
- standard made as self containing as possible;
- item split between individual items and systems;
- generalization of the dependability concepts for systems made of several components [introduction of the conditional failure intensity (Vesely failure rate);
- introduction of the state-transition and the Markovian models;
- generalization of the availability to production availability];
- introduction of curves to illustrate the various concepts. Keywords: mathematical expressions for dependability

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

L'IEC 61703:2016 fournit des expressions mathématiques pour les mesures sélectionnées liées à la fiabilité, à la disponibilité, à la maintenabilité et à la logistique de maintenance définies dans l'IEC 60050-192:2015. En outre, elle présente certains termes non couverts par l'IEC 60050-192:2015. Ces termes sont liés aux aspects relatifs aux items de la classe système (voir ci-après). Selon l'IEC 60050-192:2015, la sûreté de fonctionnement [192-01-22] est l'aptitude d'une entité à fonctionner quand et tel que requis; une entité [192-01-01] peut être une pièce isolée, un composant, un dispositif, une unité fonctionnelle, un équipement, un sous-système ou un système. Pour tenir compte des contraintes mathématiques, la présente norme effectue une distinction entre les entités individuelles considérées dans leur ensemble (par exemple, des composants individuels) et les systèmes composés de plusieurs entités individuelles. Elle fournit des considérations générales sur les expressions mathématiques liées aux systèmes et aux entités individuelles. Les entités individuelles plus facilement modélisables sont, quant à elles, analysées plus en détail pour ce qui concerne leurs aspects de réparation. La présente norme s'applique principalement à la sûreté de fonctionnement du matériel, mais de nombreux termes et leurs définitions peuvent être appliqués à des entités contenant du logiciel. Cette deuxième édition annule et remplace la première édition parue en 2001. Cette édition constitue une révision technique. Cette édition inclut les modifications techniques majeures suivantes par rapport à l'édition précédente:
- norme aussi autonome que possible;
- entité scindée entre entités individuelles et systèmes;
- généralisation des concepts de sûreté de fonctionnement pour les systèmes constitués de plusieurs composants;
- [introduction de l'intensité conditionnelle de défaillance (taux de défaillance de Vesely);
- introduction des modèles états-transitions et des modèles de Markov;
- généralisation de la disponibilité à la disponibilité de production];
- introduction de courbes pour représenter les différents concepts. Mots clés: des expressions mathématiques pour la fiabilité

General Information

• Status: Published
• Publication Date: 11-Aug-2016
• ICS: 03.120.30 - Application of statistical methods
• ICS: 21.020 - Characteristics and design of machines, apparatus, equipment
• Technical Committee: TC 56 - Dependability
• Drafting Committee: WG 1 - TC 56/WG 1
• Current Stage: PPUB - Publication issued
• Start Date: 12-Aug-2016
• Completion Date: 31-Aug-2016

Relations

• Revises: IEC 61703:2001 - Mathematical expressions for reliability, availability, maintainability and maintenance support terms
• Effective Date: 05-Sep-2023

Overview

IEC 61703:2016 is an international standard developed by the International Electrotechnical Commission (IEC) that specifies mathematical expressions for reliability, availability, maintainability, and maintenance support terms. It builds upon the terminologies defined in IEC 60050-192:2015 and extends to new terms related to system item classes. The standard serves as an essential reference for defining and modeling dependability parameters mathematically, focusing mainly on hardware but also applicable to software-containing items.

This second edition revises the original 2001 version comprehensively to provide a self-contained, generalized framework for analyzing individual items and systems composed of multiple components. It introduces advanced concepts such as conditional failure intensity and Markovian state-transition models, with illustrative curves to enhance conceptual understanding.

Key keywords for search include: mathematical expressions for dependability, reliability modeling, availability measures, maintainability terms, IEC reliability standard.

Key Topics

• Scope and Definitions
  IEC 61703:2016 covers the mathematical modeling of dependability attributes including reliability, availability, and maintainability of hardware and software items. It distinguishes between individual items (components or parts) and systems comprising multiple items, aligning with IEC 60050-192 vocabulary.

• Dependability Concepts
  Dependability is defined as the ability of an item to perform as required over time. The standard formulates mathematical expressions for failure intensity, instantaneous availability, mean time between failures (MTBF), mean repair time, and more.

• System and Item Modeling
  The document addresses the modeling of non-repairable and repairable items, considering instantaneous and non-instantaneous repair scenarios. It introduces state-transition diagrams and Markov models to represent system state evolution and failure-repair processes.

• Key Mathematical Expressions
  Expressions are provided for:

  • Availability (instantaneous, mean, asymptotic, production availability)
  • Reliability over time intervals
  • Failure intensity, including Vesely’s conditional failure intensity
  • Maintainability metrics like mean repair time, administrative and logistic delays
  • Renewal processes and failure distributions for both single items and complex systems

• Visualization and Curves
  Graphical illustrations such as state-transition graphs, failure rate curves, and probability evolutions aid in interpreting key dependability measures and validating mathematical models.

Applications

IEC 61703:2016 is widely applicable to industries where reliability, availability, and maintainability (RAM) analysis is critical for system design, operation, and maintenance planning. Typical application areas include:

• Electrical and Electronic Equipment
  Facilitates reliability engineering of components, devices, subsystems, and integrated systems in power, telecommunications, aerospace, and defense sectors.

• Maintenance Support Planning
  Provides quantifiable metrics for maintenance scheduling, logistics support, and resources allocation to optimize system uptime.

• System Safety and Risk Assessments
  Enables precise calculation of failure probabilities and repair impacts, essential for risk-informed decision-making and safety certification.

• Software-Integrated Hardware Systems
  Although primarily hardware-focused, concepts can extend to support dependability assessment of embedded software systems.

• Performance Prediction and Lifecycle Management
  Assists engineers in predicting system behavior over operational life, enabling preventative measures and cost-effective lifecycle management.

Related Standards

• IEC 60050-192 (Dependability and Quality of Service Vocabulary)
  Provides the terminology foundation for dependability measures that IEC 61703 mathematically expresses.

• IEC 60300 Series (Dependability Management)
  Covers the framework and requirements for dependability management systems aligned with IEC 61703 metrics.

• ISO 14224 (Reliability and Maintenance Data)
  Related to collection and analysis of reliability data complementing IEC 61703 modeling approaches.

• IEC 61508 (Functional Safety of Electrical/Electronic Systems)
  Uses dependability measures like reliability and availability, with IEC 61703 providing mathematical tools supporting such safety standards.

---

By adopting IEC 61703:2016, organizations can ensure standardized, accurate mathematical modeling of reliability, availability, and maintainability parameters. This standard supports robust engineering practices, driving improvements in product dependability and enhanced maintenance strategies across industries.

Standards Content (Sample)

IEC 61703 ®
Edition 2.0 2016-08
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Mathematical expressions for reliability, availability, maintainability and
maintenance support terms
Expressions mathématiques pour les termes de fiabilité, de disponibilité, de
maintenabilité et de logistique de maintenance

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IEC 61703 ®
Edition 2.0 2016-08
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Mathematical expressions for reliability, availability, maintainability and

maintenance support terms
Expressions mathématiques pour les termes de fiabilité, de disponibilité, de

maintenabilité et de logistique de maintenance

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
INTERNATIONALE
ICS 03.120.30; 21.020 ISBN 978-2-8322-3558-4

– 2 – IEC 61703:2016 © IEC 2016
CONTENTS
FOREWORD . 6
INTRODUCTION . 8
1 Scope . 9
2 Normative references. 10
3 Terms and definitions . 10
4 Glossary of symbols and abbreviations . 13
4.1 General . 13
4.2 Acronyms used in this standard . 13
4.3 Symbols used in this standard . 15
5 General models and assumptions . 18
5.1 Constituents of up and down times . 18
5.2 Introduction to models and assumptions . 19
5.3 State-transition approach . 20
5.4 Model and assumptions for non-repairable individual items . 22
5.5 Assumptions and model for repairable individual items . 23
5.5.1 Assumption for repairable individual items . 23
5.5.2 Instantaneous repair . 24
5.5.3 Non-instantaneous repair . 25
5.6 Continuously operating items (COI) versus intermittently operating individual
items (IOI) . 26
6 Mathematical models and expressions . 27
6.1 Systems . 27
6.1.1 General . 27
6.1.2 Availability related expressions . 29
6.1.3 Reliability related expressions . 36
6.1.4 Mean operating time between failures [192-05-13] and mean time
between failures . 40
6.1.5 Instantaneous failure rate [192-05-06] and conditional failure intensity
(Vesely failure rate) . 41
6.1.6 Failure density and unconditional failure intensity [192-05-08] . 44
6.1.7 Comparison of λ(t), λ (t), z(t) and f(t) for high and small MTTRs . 47
V
6.1.8 Restoration related expressions . 47
6.2 Non-repairable individual items . 49
6.2.1 General . 49
6.2.2 Instantaneous availability [192-08-01] . 50
6.2.3 Reliability [192-05-05] . 50
6.2.4 Instantaneous failure rate [192-05-06] . 51
6.2.5 Mean failure rate [192-05-07] . 52
6.2.6 Mean operating time to failure [192-05-11] . 53
6.3 Repairable individual items with zero time to restoration . 54
6.3.1 General . 54
6.3.2 Reliability [192-05-05] . 54
6.3.3 Instantaneous failure intensity [192-05-08]. 56
6.3.4 Asymptotic failure intensity [192-05-10] . 58
6.3.5 Mean failure intensity [192-05-09] . 59
6.3.6 Mean time between failures (see 3.3) . 60

6.3.7 Mean operating time to failure [192-05-11] . 60
6.3.8 Mean operating time between failures [192-05-13] . 61
6.3.9 Instantaneous availability [192-08-01], mean availability [192-08-05]
and asymptotic availability [192-08-07] . 61
6.3.10 Mean up time [192-08-09] . 61
6.4 Repairable individual items with non-zero time to restoration . 62
6.4.1 General . 62
6.4.2 Reliability [192-05-05] . 62
6.4.3 Instantaneous failure intensity [192-05-08]. 64
6.4.4 Asymptotic failure intensity [192-05-10] . 67
6.4.5 Mean failure intensity [192-05-09] . 68
6.4.6 Mean operating time to failure [192-05-11] . 69
6.4.7 Mean time between failures (see 3.3) . 70
6.4.8 Mean operating time between failures [192-05-13] . 71
6.4.9 Instantaneous availability [192-08-01] . 71
6.4.10 Instantaneous unavailability [192-08-04] . 73
6.4.11 Mean availability [192-08-05] . 74
6.4.12 Mean unavailability [192-08-06] . 76
6.4.13 Asymptotic availability [192-08-07] . 78
6.4.14 Asymptotic unavailability [192-08-08] . 78
6.4.15 Mean up time [192-08-09] . 79
6.4.16 Mean down time [192-08-10] . 81
6.4.17 Maintainability [192-07-01] . 82
6.4.18 Instantaneous repair rate [192-07-20] . 84
6.4.19 Mean repair time [192-07-21] . 86
6.4.20 Mean active corrective maintenance time [192-07-22] . 87
6.4.21 Mean time to restoration [192-07-23] . 88
6.4.22 Mean administrative delay [192-07-26] . 89
6.4.23 Mean logistic delay [192-07-27] . 90
Annex A (informative) Performance aspects and descriptors . 91
Annex B (informative) Summary of measures related to time to failure . 92
Annex C (informative) Comparison of some dependability measures for continuously
operating items . 95
Bibliography . 97

Figure 1 – Constituents of up time . 18
Figure 2 – Constituents of down time. 19
Figure 3 – Acronyms related to failure times . 19
Figure 4 – Simple state-transition diagram . 21
Figure 5 – Sample realization (chronogram) related to the system in Figure 4 . 22
Figure 6 – State-transition diagram of a non-repairable individual item . 22
Figure 7 – Sample realization of a non-repairable individual item . 23
Figure 8 – State-transition diagram of an instantaneously repairable individual item . 24
Figure 9 – Sample realization of a repairable individual item with zero time to
restoration . 25
Figure 10 – State-transition diagram of a repairable individual item . 25

– 4 – IEC 61703:2016 © IEC 2016
Figure 11 – Sample realization of a repairable individual item with non-zero time to
restoration . 26
Figure 12 – Comparison of an enabled time for a COI and an IOI . 26
Figure 13 – Equivalent operating time for IOI items . 27
Figure 14 – State-transition graph for a simple redundant system . 27
Figure 15 – Markov graph for a simple redundant system . 28
Figure 16 – Evolution of the state probabilities related to the Markov model in Figure 15 . 28
Figure 17 – Evolution of A(t) and U(t) related to the Markov model in Figure 15 . 29
Figure 18 – Evolution of the Ast (0, t) related to the Markov model in Figure 15 . 31
i
Figure 19 – Instantaneous availability and mean availability of a periodically tested item . 33
Figure 20 – Example of a simple production system . 34
Figure 21 – Evolution of A(t) and K(t) . 35
Figure 22 – Illustration of a system reliable behaviour over [0, t] . 36
Figure 23 – Illustration of a system reliable behaviour over time interval [t , t ] . 37
1 2
Figure 24 – State-transition and Markov graphs for reliability calculations . 37
Figure 25 – Evolution of the state probabilities related to the Markov model in Figure 24 . 38
Figure 26 – Evolution of R(t) and F(t) related to the Markov model in Figure 24 . 39
Figure 27 – Evolution of Ast (0, t) related to the Markov model in Figure 24 . 40
i
Figure 28 – Time between failures versus operating time between failures . 40
Figure 29 – Comparison between λ(t) and λ (t) related to the model in Figure 24 . 43
V
Figure 30 – Comparison between z(t) and f(t) . 46
Figure 31 – Comparison of λ(t), λ (t), z(t) and f(t) for high and small values of MTTRs . 47
V
Figure 32 – Illustration of reliable behaviour over [t , t ] for a zero time to restoration
1 2
individual item . 55
Figure 33 – Sample of possible number of failures at the renewal time t . 56
Figure 34 – Illustration of reliable behaviour over [t t ] for a non-zero time to
1 , 2
restoration individual item . 62
Figure 35 – Evolution of R(t, t + 1/4) . 64
Figure 36 – Sample of possible number of failures at the renewal time t . 64
Figure 37 – Evolution of the failure intensity z(t) . 66
Figure 38 – Evolution of the mean failure intensity z(t, t + 1/4) . 69
Figure 39 – Illustration of available behaviour at time t for a non-zero time to

restoration individual item . 71
Figure 40 – Evolution of the instantaneous availability A(t) . 73
Figure 41 – Illustration of unavailable behaviour at time t for a non-zero time to

restoration individual item . 73
Figure 42 – Evolution of the instantaneous unavailability U(t) . 74
Figure 43 – Evolution of the mean availability A (t, t + 1/ 4) . 76
Figure 44 – Evolution of the mean unavailability U (t, t +1/ 4) . 77
Figure 45 – Sample realization of the individual item state . 80
Figure 46 – Plot of the up-time hazard rate function λ (t) . 80
U
Figure 47 – Evolution of the maintainability M(t, t+16h) . 84
Figure 48 – Evolution of the lognormal repair rate µ(t) . 86

Figure A.1 – Performance aspects and descriptors . 91

Table B.1 – Relations among measures related to time to failure of continuously
operating items . 92
Table B.2 – Summary of characteristics for some continuous probability distributions of
time to failure of continuously operating items . 93
Table B.3 – Summary of characteristics for some probability distributions of repair time . 94
Table C.1 – Comparison of some dependability measures of continuously operating
items with constant failure rate λ and restoration rate µ . 95
R
– 6 – IEC 61703:2016 © IEC 2016
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
MATHEMATICAL EXPRESSIONS FOR RELIABILITY,
AVAILABILITY, MAINTAINABILITY AND
MAINTENANCE SUPPORT TERMS
FOREWORD
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patent rights. 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.
This second edition cancels and replaces the first edition published in 2001. This edition
constitutes a technical revision.
This edition includes the following significant technical changes with respect to the previous
edition:
a) standard made as self containing as possible;
b) item split between individual items and systems;
c) generalization of the dependability concepts for systems made of several components;
– introduction of the conditional failure intensity (Vesely failure rate);
– introduction of the state-transition and the Markovian models;

– generalization of the availability to production availability;
d) introduction of curves to illustrate the various concepts.
The text of this standard is based on the following documents:
FDIS Report on voting
56/1682/FDIS 56/1693/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 2.
This International Standard is to be used in conjunction with IEC 60050-192:2015.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC website under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
– 8 – IEC 61703:2016 © IEC 2016
INTRODUCTION
IEC 60050-192 provides definitions for dependability and its influencing factors, reliability,
availability, maintainability and maintenance support, together with definitions of other related
terms commonly used in this field. Some of these terms are measures of specific
dependability characteristics, which can be expressed mathematically.
It is important for the users to understand the mathematical meaning of those expressions and
how they are established. This is the purpose of the present International Standard which,
used in conjunction with IEC 60050-192, provides practical guidance essential for the
quantification of those measures. For those requiring further information, for example on
detailed statistical methods, reference should be made to the IEC 60605 series [23] .
Annex A provides a diagrammatic explanation of the relationships between some basic
dependability terms, related random variables, probabilistic descriptors and modifiers.
Annex B provides a summary of measures related to time to failure.
Annex C compares some dependability measures for continuously operating items.
The bibliography gives references for the mathematical basis of this standard, in particular,
the mathematical material is based on references [2], [6], [8], [9], [13], [14] and [18]; the
renewal theory (renewal and alternating renewal processes) can be found in [6], [8], [9], [10],
[11], [13], [15], and [17]; and more advanced treatment of renewal theory can be found in
references [1], [3], [12], [16], [19] and [20]. More information on the theory and applications of
Markov processes can be found in references [3], [9], [10], [15], [16], [17] and [19].

____________
Numbers in brackets refer to the Bibliography.

MATHEMATICAL EXPRESSIONS FOR RELIABILITY,
AVAILABILITY, MAINTAINABILITY AND
MAINTENANCE SUPPORT TERMS
1 Scope
This International Standard provides mathematical expressions for selected reliability,
availability, maintainability and maintenance support measures defined in
IEC 60050-192:2015. In addition, it introduces some terms not covered in IEC 60050-
192:2015. They are related to aspects of the system of item classes (see hereafter).
According to IEC 60050-192:2015, dependability [192-01-22] is the ability of an item to
perform as and when required and an item [192-01-01] can be an individual part, component,
device, functional unit, equipment, subsystem, or system.
To account for mathematical constraints, this standard splits the items between the individual
items considered as a whole (e.g. individual components) and the systems made of several
individual items. It provides general considerations for the mathematical expressions for
systems as well as individual items but the individual items which are easier to model are
analysed in more detail with regards to their repair aspects.
The following item classes are considered separately:
• Systems;
• Individual items:
– non-repairable [192-01-12];
– repairable [192-01-11]:
i) with zero (or negligible) time to restoration;
ii) with non-zero time to restoration.
In order to explain the dependability concepts which can be difficult to understand, keep the
standard self-contained and the mathematical formulae as simple as possible, the following
basic mathematical models are used in this standard to quantify dependability measures:
• Systems:
– state-transition models;
– Markovian models.
• Individual items:
– random variable (time to failure) for non-repairable items;
– simple (ordinary) renewal process for repairable items with zero time to restoration;
– simple (ordinary) alternating renewal process for repairable items with non-zero time to
restoration.
The application of each dependability measure is illustrated by means of simple examples.
This standard is mainly applicable to hardware dependability, but many terms and their
definitions may be applied to items containing software.

– 10 – IEC 61703:2016 © IEC 2016
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and
are indispensable for its application. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC 60050-192:2015, International electrotechnical vocabulary – Part 192: Dependability
(available at http://www.electropedia.org)
ISO 3534-1:2006, Statistics – Vocabulary and symbols – Part 1: General statistical terms and
terms used in probability
3 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-192:2015,
ISO 3534-1 and the following apply.
NOTE To facilitate the location of the full definition, the IEC 60050-192 reference for each term is shown [in
square brackets] immediately following each term, for example:
mean time to restoration [192-07-23]
The terms and definitions given in Clause 3, which do not appear in IEC 60050-192, are used in order to facilitate
the presentation of mathematical expressions of some IEC 60050-192 terms. Some terms have been taken from
IEC 60050-192 and modified for the needs of this standard.
3.1
instantaneous restoration intensity
restoration intensity
restoration frequency
ν(t)
limit, if it exists, of the quotient of the mean number of restorations [192-06-23] of a repairable
item [192-01-11] within time interval [t, t + ∆t], and ∆t, when ∆t tends to zero, given that the
item is as good as new at t = 0
E[N (t +Δt) − N (t) | as good as new at t = 0]
R R
v(t) = lim
Δt→0+ Δt
where
N (t) is the number of restorations in the time interval [0, t];
R
E denotes the expectation.
Note 1 to entry: The difference between the restoration intensity and the repair rate comes from the conditions:
the item is as good as new at time t = 0 for the restoration intensity and for the repair rate the repair starts at time
t = 0. From a mathematical point of view, the restoration intensity is similar to the unconditional failure intensity
(see 3.8).
Note 2 to entry: The unit of measurement of instantaneous restoration intensity is the unit of time to the power-1.
3.2
instantaneous repair rate
repair rate
µ(t)
limit, if it exists, of the quotient of the conditional probability that the repair is completed within
= 0 and
time interval [t, t + ∆t] and ∆t, when ∆t tends to zero, given that the repair started at t
had not been completed before time t

Note 1 to entry: The difference between the restoration intensity and the repair rate comes from the conditions:
the item is as good as new at time t = 0 for the restoration intensity and for the repair rate the repair starts at time
t = 0. From a mathematical point of view, the repair rate is similar to the failure rate (see 3.6).
[SOURCE: IEC 60050-192:2015, 192-07-20, modified — Note 1 to entry has been replaced
and Note 2 to entry deleted]
3.3
mean time between failures
METBF
expectation of the time which elapses between successive failures
Note 1 to entry: The concept of mean time between failures has been omitted from IEC 60050-192. It was defined
in IEC 60050-191 as “the expectation of time between failures”. The definition has been modified to explain the
acronym METBF (mean elapsed time between failures) which is used in this standard to avoid any confusion
between the mean time between failure (METBF) and the mean operating time between failures (MTBF or MOTBF).
[SOURCE: IEC 60050-191:1990, 191-12-08, modified — acronym and Note 1 to entry added]
3.4
up-time distribution function
function giving, for every value of t, the probability that an up-time will be less than, or equal
to, t
Note 1 to entry: If the up-time is (strictly) positive and a continuous random variable, then F (0) = 0 and
U
t
 
F (t) = 1− exp− λ (x)dx
U U
 ∫ 
 
where
λ (t) is the instantaneous up-time hazard rate function.
U
Note 2 to entry: The up-time distribution function is the general up-time distribution valid for both COI
(continuously operating item) and IOI (intermittently operating item). For COIs, F (t) = F(t)
U
Note 3 to entry: If the up time is exponentially distributed, then
F (t) = 1 − exp(−t/MUT)
U
where MUT is the mean up-time.
In this case, the reciprocal of MUT is denoted by λ and λ = 1/MUT
U U
3.5
instantaneous up-time hazard rate function
up-time hazard rate function
λ (t)
U
limit, if it exists, of the quotient of the conditional probability that the up-time will end within
time interval [t, t + ∆t] and ∆t, when ∆t tends to zero, given that the up-time started at t = 0
and had not been finished before time t
Note 1 to entry: The instantaneous up-time hazard rate function is expressed by the formula:
1 F (t +Δt) − F (t) f (t)
U U U
λ (t) = lim =
U
Δt →0+Δt 1 − F (t) 1 − F (t)
U U
where F (t) is the up-time distribution function and f (t) is the probability density function of the up-time.
U U
Note 2 to entry: λ (t) is the general hazard rate for the up times valid for both COI and IOI. For COIs, λ (t) = λ (t)
U U
(see 3.6).
Note 3 to entry: If the up time is exponentially distributed, then the instantaneous up-time hazard rate function is
constant in time and is denoted by λ .
U
– 12 – IEC 61703:2016 © IEC 2016
Note 4 to entry: The unit of measurement of instantaneous up-time hazard rate function is the unit of time to the
power −1.
3.6
instantaneous failure rate
failure rate
λ(t)
limit, if it exists, of the quotient of the conditional probability that the failure of an item occurs
within time interval [t, t + ∆t], by ∆t, when ∆t tends to zero, given that failure has not occurred
within time interval [0, t]
Note 1 to entry: The instantaneous failure rate is expressed by the formula:
1 F(t +Δt) − F(t) f (t)
λ (t) = lim =
Δt→0+Δt R(t) R(t)
where F(t) and f(t) are, respectively, the distribution function and the probability density at the failure instant, and
where R(t) is the reliability function, related to the reliability R(t , t ) by R(t) = R(0, t).
1 2
Note 2 to entry: The restriction to non-repairable items in the definition provided by IEC 60050-192 can be
removed to generalize this definition for any kind of items i.e. systems or individual items, repairable or not.
Note 3 to entry: The instantaneous failure rate is the up-time hazard rate for COIs. In this case, λ(t) = λ (t) (see
U
3.5).
Note 4 to entry: When ∆t →0+, the failure rate is the conditional probability per unit of time that the item fails
between t and t + ∆t, given it is in up state all over the time interval [0, t]. It is usually assumed that the item is as
good as new at time 0.
Note 5 to entry: The instantaneous failure rate may be also expressed by the formula:
E[N(t +Δt) − N(t) | up state over [0, t]]

λ t
( ) = lim
Δt→0+ Δt
where N(t) is the number of failures in the time interval [0, t], where E denotes the expectation.
This form of the definitions allows comparisons of the failure rate to the conditional failure intensity and to the
unconditional failure intensity.
[SOURCE: IEC 60050-192:2015, 192-05-06, modified — generalized to systems with
repairable components and Notes to entry added]
3.7
conditional failure intensity
Vesely failure rate
λ (t)
v
limit, if it exists, of the quotient of the mean number of failures of a repairable item within time
interval [t, t + ∆t], by ∆t, when ∆t tends to zero, given that the item is in up state at time t and
as good as new at time 0
Note 1 to entry: The instantaneous failure intensity is expressed by the formula:
E[N(t +Δt) − N(t) | up state at t and as good as new at time 0]

λ (t) = lim
V
Δt →0+ Δt
where N(t) is the number of failures in the time interval [0, t], where E denotes the expectation.
Note 2 to entry: When ∆t →0+, the conditional failure intensity is the probability per unit of time that the item fails
between t and t + dt, given it is in up state at time t and as good as new at time 0. In particular cases (quick
restoration of failures), it provides good approximations of the failure rate. This parameter introduced in 1970 by W.
E. Vesely [26] is also called Vesely failure rate.
Note 3 to entry: According to the definitions, λ (t) and z(t) are linked by the formula: λ (t) = z(t)/Α(t). where Α(t)
V V
is the item instantaneous availability at time t.

3.8
unconditional failure intensity
instantaneous failure intensity
failure intensity
failure frequency
z(t)
limit, if it exists, of the quotient of the mean number of failures of a repairable item within time
interval [t, t + ∆t], by ∆t, when ∆t tends to zero, given that the item is as good as new at time
t= 0
Note 1 to entry: The instantaneous failure intensity is expressed by the formula:
E[N(t +Δt) − N(t) | as good as new at t = 0]

z(t) = lim
Δt →0+ Δt
where N(t) is the number of failures in the time interval [0, t], where E denotes the expectation and where the
implicit condition that the item is as good as new at time t = 0 has been added.
Note 2 to entry: The unconditional failure intensity is the failure intensity as defined in IEC 60050-192:2015 [192-
05-08]. It is also sometimes named ROCOF (rate of occurrence of failure).
Note 3 to entry: When ∆t →0+, the unconditional failure intensity is the probability per unit of time that the item
fails between t and t + dt given that the item is in up state at time t = 0. Here the item may be in any state at time t
and this is why the adjective unconditional is used.
Note 4 to entry: According to the definitions, λ (t) and z(t) are linked by the formula: z(t) = Α(t). λ (t) where Α(t) is
V V
the item instantaneous availability at time t.
[SOURCE: IEC 60050-192:2015, 192-05-08, modified — synonyms have been added and the
entry has been revised]
3.9
continuously operating item
COI
item for which operating time [192-02-05] is equal to its enabled time [192-02-17]
3.10
intermittently operating item
IOI
item for which operating time [192-02-05] is less than its enabled time [192-02-17]
Note 1 to entry: In this case, the enabled time of the item is made of the sum of the times spent in the operating
[192-02-04], idle [192-02-14] and standby [192-02-10] states.
4 Glossary of symbols and abbreviations
4.1 General
The symbols and abbreviations given in Clause 4 are widely used and recommended,
however, they are not mandatory. For consistency of presentation, the notation in this
document may differ from that used in a referenced document.
4.2 Acronyms used in this standard
Acronym/abbreviation Meaning
COI Continuously operating item
IOI Intermittently operating item
MACMT Mean active corrective maintenance time, i.e. the expectation of
the active corrective maintenance time

– 14 – IEC 61703:2016 © IEC 2016
Acronym/abbreviation Meaning
∧
MACMT
Point estimate of the mean active corrective maintenance time
MAD Mean administrative delay
∧
Point estimate of the mean administrative delay
MAD
MADT(t , t ) Mean accumulated down time over the time interval [t , t ]
1 2 1 2
∧
Point estimate of the mean accumulated down time over the time
MADT(t ,t )
1 2 interval [t , t ]
1 2
MAUT(t , t ) Mean accumulated up time over the time interval [t , t ]
1 2 1 2
∧
Point estimate of the mean accumulated up time over the time
MAUT(t , t )
1 2 interval [t , t ]
1 2
MDT Mean down time
∧
MDT Point estimate of the mean down time
METBF Mean (elapsed) time between failures
MFDT Mean fault detection time, i.e. the expectation of the fault
detection time
MLD Mean logistic delay
∧
MLD Point estimate of the mean logistic delay
MMAT Mean maintenance action time, i.e. the expectation of a given
maintenance action time
MRT Mean repair time
∧
Point estimate of the mean repair time
MRT
MOTBF Mean operating time between failures
MTD Mean technical delay, i.e. the expectation of the technical delay
MTTF Mean time to failure
∧
MTTF Point estimate of the mean time to failure
MTTR Mean time to restoration
∧
Point estimate of the mean time to restoration
MTTR
MUT Mean up time
∧
MUT
Point estimate of the mean up time
RT Observed repair time of item i
i
TTF Time to failure of item i
i
2 2
VRT The variance of repair time, VRT = Var[ζ] = E[ζ ] − MRT , where
ζ is a random variable representing the repair
...