Mathematical expression for reliability, availability, maintainability and maintenance support terms.pdf
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Literaturhinweise
[1]
Alsmeyer, G., Erneuerungstheorie. Analyse stochastischer Regenerationsschemata , Stuttgart, B.G. Teubner, 1991
[2]
Ascher, H.R., Feingold, H., Repairable System Reliability: Modeling, Inference, Misconceptions and Their Causes, Causes, New York, Marcel Dekker, 1984
[3]
Asmussen, S., Applied S., Applied Probability Probability and Queues Queues,, Second Edition, New York, Springer-Verlag, 2003.
[4]
Aven, T., Reliability and Risk Analysis, Analysis, London, Elsevier Applied Science, 1992
[5]
Aven, T., Jensen, U., Stochastic models in reliability , New York, Springer-Verlag, 1999
[6]
Barlow, R.E., Proschan, F., Mathematical Theory of Reliability , New York, Wiley, 1965. Reprinted: Philadelphia, SIAM, 1996
[7]
Barlow, R.E. Proschan, F., Statistical Theory of Reliability and Life Testing: Probabilistic Models , New York, Holt, Rinehart and Winston, 1975. Reprinted with corrections: Silver Spring, MD, To Begin With, 1981
[8]
Beichelt, F., Franken, P., Zuverlässigkeit und Instandhaltung: mathematische Methoden , Berlin, VEB Verlag Technik, 1983
[9]
Birolini, A., Reliability Engineering: Engineering: Theory and Practice, Practice , Sixth Edition, Berlin, Springer-Verlag, 2010
[10] Cocozza-Thivent, Cocozza-Thivent, C., Processus stochastiques et fiabilité des systèmes, systèmes , Berlin, Springer-Verlag, 1997 [11] Cox, D.R., Renewal Theory , London, Methuen & Co. Ltd, 1962 [12] Feller, W., An Introduction to Probability Theory and Its Applications, Applications, Volume II, Second Edition, New York, Wiley, 1971 [13] Gnedenko, Gnedenko, B.V., Belyayev, Belyayev, Y.K., Solovyew, Solovyew, A.D., Mathematical Methods of Reliability Theory , New York, Academic Press, 1969 [14] Henley, E.J., Kumamoto, Kumamoto, H., Reliability Engineering and Risk Assessment , Englewood Cliffs, Prentice Hall, 1981 [15] Heyman, D.P., Sobel, M.J., Stochastic Models in Operations Research, Research, Volume I, Stochastic Processes and Operating Characteristics, New York, McGraw-Hill, 1982 [16] Resnick, S., Adventures S., Adventures in Stochastic Processes, Processes, Boston, Birkhäuser, 1992 (4th printing, 2005). [17] Ross, S.M., Stochastic Processes, Processes, Second Edition, New York, Wiley, 1996 [18] Villemeur, A., Reliability, Availability, Maintainability and Safety Assessment , Volume 1, Chichester, Wiley, 1982
Für eine vertiefende Behandlung der Zuverlässigkeit von Software wird auf folgende Literaturstellen verwiesen: BS 5760 Part 8:1998, 8:1998 , Reliability of systems, equipment and components – Part 8: Guide to assessment of reliability of systems containing software. IEC 61713:2000-06, Software dependability through the software life-cycle processes – Application guide IEC 62628:2012, Guidance 62628:2012, Guidance on software aspects of dependability. ____________ ____________ 56 Copyright Deutsches Institut für Normung e. V. Provided by IHS under license with DIN No reproduction or networking permitted without license from IHS
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CONTENTS
1 2 3
1
Scope ............................... ............................................... ................................ ............................... ............................... ................................ ............................... ................. .. 7
4
2
Normative references .............................. ............................................. ............................... ................................ ............................... .......................... ........... 7
5
3
Definitions .............................. .............................................. ............................... ............................... ................................ ............................... ........................... ............ 8
6
4
Glossary of symbols and abbreviations ............................... ............................................... ............................... .............................. ............... 9
7
4.1 4. 1
Non-repairable items .............................. ............................................. ............................... ................................ ............................... .................. ... 9
8
4.2 4. 2
Repairable items w ith zero time to restoration ............................ ........................................... ........................... ............ 10
9
4.3 4. 3
Repairable items with non-zero time t o rest oration ........................ ....................................... ........................ ......... 11
10
5
Assu As sump mpti tion on s ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... .. 14
11
5.1 5. 1
General .............................. .............................................. ................................ ............................... ............................... ................................ .................... .... 14
12
5.2 5. 2
As sum ptio pt ions ns for nonno n-re re pa ira ble bl e item it ems s ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... 14
13
5.3 5. 3
As sum ptio pt ions ns for repa re pa irab ir able le item it em s ... ... ... ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... 15
14 15
6
Mathematical expressions ............................... .............................................. ............................... ................................ ............................... ................. 18 6.1 6. 1
Non-repairable items .............................. ............................................. ............................... ................................ ............................... ................. 18
16
6.1.1
General ............................. ............................................. ................................ ............................... ............................... ......................... ......... 18
17
6.1.2
Reliability [191-45-05] ...................................... ...................................................... ................................ .......................... .......... 18
18
6.1.3
Instantaneous failure rate [191-45-06] ...................................... ...................................................... .................. 19
19
6.1.4
Mean failure rate [191-45-07] ............................. ............................................. ................................ ....................... ....... 20
20
6.1.5
Mean (operating) time to failure [191-45-11] ........................... ........................................... ................... ... 21
21
6.2 6. 2
Repairable items w ith zero time to r estoration ...................... ...................................... ................................ .................. 22
22
6.2.1
General ............................. ............................................. ................................ ............................... ............................... ......................... ......... 22
23
6.2.2
Reliability [191-45-05] ...................................... ...................................................... ................................ .......................... .......... 22
24
6.2.3
Instantaneous failure intensity [191-45-08] ............................................ ................................................ .... 23
25
6.2.4
As ymp toti to tic c fail fa ilur ure e inte in te nsi ty [1 91-4 91 -455- 10] 10 ] ...... ... ... ... ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... .. 24
26
6.2.5
Mean failure intensity [191-45-09] ................................ ............................................... .............................. ............... 25
27
6.2.6
Mean (operating) time to failure [191-45-11] ........................... ........................................... ................... ... 25
28
6.2.7
Mean operating time between failures [191-45-13] ............................... ..................................... ...... 26
29
6.2.8
Mean up time [191-48-09] ................... ................................... ................................ ................................ ....................... ....... 26
30
6.3 6. 3
Repairable items with non-zero time t o rest oration ........................ ....................................... ........................ ......... 26
31
6.3.1
General ............................. ............................................. ................................ ............................... ............................... ......................... ......... 26
32
6.3.2
reliability [191-45-05] ............................... ............................................... ................................ ............................... .................. ... 27
33
6.3.3
Instantaneous failure intensity [191-45-08] ............................................ ................................................ .... 28
34
6.3.4
As ymp toti to tic c fail fa ilur ure e inte in te nsi ty [1 91-4 91 -455- 10] 10 ] ...... ... ... ... ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... .. 30
35
6.3.5
Mean failure intensity [191-45-09] ................................ ............................................... .............................. ............... 31
36
6.3.6
Mean (operating) time to failure [191-45-11] ........................... ........................................... ................... ... 32
37
6.3.7
Mean operating time between failures [191-45-13] ............................... ..................................... ...... 33
38
6.3.8
Instantaneous availability [191-48-01] ............................... ............................................... ......................... ......... 33
39
6.3.9
Instantaneous unavailability [191-48-04] ...................... ...................................... .............................. .............. 34
40
6.3.10 Mean availability [191-48-05] .............................. .............................................. ................................ ....................... ....... 35
41
6.3.11 Mean unavailability [191-48-06] ..................................... ..................................................... ............................ ............ 36
42
6.3.12 As ymp toti to tic c avai av aila labil bil it y [ 19119 1- 48-0 48 -0 7] ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... .. 38
43
6.3.13 As ymp toti to tic c unav un avai aila labi bilit lit y [191 [1 91-4 -488- 08] 08 ] ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... .. 38
44
6.3.14 mean up time [191-48-09] ................... ................................... ................................ ................................ ....................... ....... 39
45
6.3.15 Mean down time [191-48-10] ............................... .............................................. ............................... ....................... ....... 41
46
6.3.16 Maintainability [191-47-01] ............................... .............................................. ............................... .......................... .......... 42
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6.3.17 Instantaneous repair rate [191-47-20] ..................... .................................... ............................... .................... .... 43
48
6.3.18 Mean repair time [191-47-21] .............................. .............................................. ................................ ....................... ....... 45
49
6.3.19 Mean active corrective maintenance time [191- 47-22] ............................... ............................... 46
50
6.3.20 Mean time to restoration [191-47-23] ........................... .......................................... .............................. ............... 46
51
6.3.21 Mean administrative delay [191-47-26] .......................................... ...................................................... ............ 47
52
6.3.22 Mean logistic delay [191-47-27] ................................ ............................................... ............................... .................. .. 48
53
Anne An ne x A (inf (i nf orm at ive ) Perf Pe rf or manc ma nce e aspe as pect ct s a nd descr de scr ipt or s ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... . 50
54
Anne An ne x B (inf (i nf orm at ive ) Su mmar mm ar y o f meas me asur ur es re late la ted d to ti me to fa ilu re ... ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ... .. . 51
55 56
Anne An ne x C (inf (i nfor or mati ma tive ve)) Comp Co mp ariso ar iso n of some so me depe de pe ndab nd abili ili ty meas me as ures ur es for fo r cont co ntin inuo uo usly us ly operating items............................................ ............................................................ ................................ ............................... ............................... .................... .... 53
57
Anne An ne x D (inf (i nfor or mati ma tive ve)) Soft So ft war e depe de pe ndab nd abililit it y a spe cts ct s ..... .. ... ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ..... .. ... ... ..... .. ... ... ..... .. ... .. 55
58 59
Figure 1 – Sample realization of a non-repairable item ..................................... ..................................................... .................... .... 15
60
Figure 2 – Sample realization of a repairable item with zero time to restora tion .................... .................... 16
61
Figure 3 – Sample realization of a repairable item with non- zero time to restoration ............. ............. 17
62
Figure 4 – Comparison of an enabled time for a COI and an IOI .......................................... ............................................ 18
63
Figure 5 – S ample re alization of the item state .............................. .............................................. ................................ ....................... ....... 40
64
Figure 6 – Plot of the up-time hazard rate function λ U (t ) .............................. .............................................. ......................... ......... 40
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65
Figure A.1 – Performance aspects and descriptors .............................. ............................................. ............................... .................. .. 50
66 67 68
Table B.1 – Relations among functional measures of time to failure of continuously operating items ................................ ............................................... ............................... ................................ ................................ ............................... ...................... ....... 51
69 70
Table B.2 – Summary of measures for some continuous probability distributions of time to failure of continuously op erating items ............................ ........................................... ............................... ................................ .................. .. 52
71 72
Table C.1 – Comparison of some dependability measures of continuously operatin g items with constant failure rate λ and and restoration rate µ R ............................... ............................................... ...................... ...... 53
73 74 75
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INTERNATIONAL ELECTROTECHNICAL COMMISSION
79 80 81 82
MATHEMATICAL EXPRESSIONS FOR RELIABILITY, AVAILABILITY, MAINTAINABILITY AND MAINTENANCE SUPPORT TERMS
83
FOREWORD
84 85 86 87 88 89 90 91 92 93
1) The International Electrotechnical Electrotechnical Commission (IEC) is a worldwide organization organization for standardization comprising comprising all national electrotechnical committees (IEC National Committees). The object of IEC is to promote international co-operation co-operation on a ll questions concerning standardization standardization in t he electrical and electronic fields. To this end and in addition to other activities, IEC publishes International Standards, Technical Specifications, Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “I EC Publication(s)”). Publication(s)”). 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 nongovernmental organizations liaising with the IEC also participate in this preparation. IEC collaborates closely with the International Organization for Standardization (ISO) in accordance with conditions determined by agreement between the t wo organizations.
94 95 96
2) The formal decisions decisions or agreements of IEC on technical matters matters express, as nearly as possible, possible, an international consensus of opinion on the relevant subjects since each technical committee has representation from all interested IEC National Committees.
97 98 99 100
3) IEC Publications Publications have the form of recommendations for international use and are accepted accepted by IEC National Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any misinterpretation misinterpretation by any end user.
101 102 103 104
4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications transparently to the maximum extent possible in their national and regional publications. Any divergence between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in the latter.
105 106 107
5) IEC itself does not provide any attestation of conformity. conformity. Independent certification certification bodies provide conformity assessment services and, in some areas, access to IEC marks of conformity. IEC is not responsible for any services carried out by independent certification certification bodies.
108
6) All users should should ensure that they they have the latest edition edition of this publication. publication.
109 110 111 112 113
7) No liability shall attach attach to IEC or its directors, employees, employees, servants or agents including individual individual experts and members of its technical committees and IEC National Committees for any personal injury, property damage or other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC Publications.
114 115
8) Attention is drawn to the Normative references references cited in this publication. publication. Use of the referenced publications is indispensable for the correct application of this publication.
116 117
9) Attention is drawn to the possibility possibility that some of the elements of this IEC Publication Publication may be the subject of patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
118 119
International Standard IEC 61703 has been prepared by IEC technical committee 56: Dependability.
120
The text of this standard is based on the following documents:
____________
FDIS
Report on voting
56/XXX/FDIS
56/XXX/RVD
121 122 123
Full information on the voting for the approval of this standard can be found in the report on voting indicated in the above table.
124
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
125
Anne An ne xes xe s A, B, C and an d D are ar e for fo r info in form rmat at ion io n only. on ly.
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The committee has decided that the contents of this publication will remain unchanged until the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data related to the specific publication. At this date, the publication will be
129
•
reconfirmed,
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•
withdrawn,
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•
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•
replaced by a revised edition, or
amended.
133 134 135
The National Committees are requested to note that for this publication the stability date is ....
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THIS TEXT IS INCLUDED FOR THE INFORMATION OF THE NATIONAL COMMITTEES AND WILL BE DELETED AT THE PUBLICATION STAGE .
138 139
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INTRODUCTION
141 142 143 144
IEC 60050-191, Dependability terminology , 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.
145 146 147 148
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.
149 150
Annex A provide s a diagr ammat ic explanation of the relatio nsh ips between som e basic dependability terms, related random variables, probabilistic descriptors and modifiers.
151
Annex B provides a summar y of measures related to time to failure .
152
Annex C compares som e dependabili ty measu res for con tinuously o perating items.
153
Annex D explains some of the sof twa re dependability a spects.
154 155 156 157 158
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) may be found in [6], [8], [9], [10], [11], [13], [15] and [17]; and more advanced treatment of renewal theory may be found in references [1], [3], [12] and [16].
159 160
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MATHEMATICAL EXPRESSIONS FOR RELIABILITY, AVAILABILITY, MAINTAINABILITY AND MAINTENANCE SUPPORT TERMS
161 162 163 164 165 166 167
1
Scope
168 169 170
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:
171
–
non-repairable items;
172
–
repairable items with zer o ( or negligible) time to re sto ration; and
173
–
repairable items with non- zer o tim e to re storation.
174 175
In order to keep the mathematical formulae as simple as possible, the following basic mathematical models are used to quantify dependability measures:
176
–
random var iable ( tim e to failure ) for non-repairable items;
177
–
simple (ord ina ry) renewal p rocess for repairable items w ith zero time to re sto ration;
178 179
–
simple (ordinar y) alternating renew al process for repairable items with non-zer o time to restoration.
180 181
To facilitate location of the full definition, the IEC 60050-191 reference for each term is shown [in brackets] immediately following each term, for example:
182
mean time to restoration [191-47-23]
183
The application of each dependability measure is illustrated by means of a simple example.
184 185 186
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.
187
2
188 189 190
The following referenced documents are applicable to this standard. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced documents (including any amendments) applies.
191 192
IEC 60050-191:XXXX Dependability
193 194
ISO 3534-1:2006, Statistics – Vocabulary and symbols – Part 1: General statistical terms and terms used in probability
Normative references
International
Electrotechnical
Vocabulary
195
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(IEV)
–
Part
191:
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3
Definitions
197 198
For the purpose of this International Standard, the terms and definitions given IEC 60050-191:XXXX and ISO 3534-1 apply.
199 200
In addition, the following terms and definitions, which do not appear in IEV-191, are used in order to facilitate the presentation of mathematical expressions for other IEV-191 terms.
201 202 203 204 205
3.1 instantaneous restoration intensity restoration intensity limit, if it exists, of the quotient of the mean number of restorations (191-46-23) of a repairable item (191-41-11) within time interval ( t , t + ∆ t ) , and ∆ t , when ∆ t tends to zero v(t ) =
206 207 208 209 210 211 212
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
in
E [ N R (t + ∆t ) − N R (t )]
lim
∆t
∆t → 0 +
where N R( t )
is the number of restorations in the time interval (0, t );
E
denotes the expectation.
NOTE 1 to entry: The unit of measurement of instantaneous failure rate is the unit of time to the power − 1. NOTE 2 to entry: Other term used for restoration intensity is “restoration frequency”.
[[Document to be checked for usage of this term, and harmonized with IEV-191 Ed 2]]
213 214 215 216
3.2 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
217
NOTE 1 to entry: If the up time is (strictly) positive and continuous random variable, then F U (0) = 0 and
218
F U (t ) = 1 − exp −
t
∫0 λ U ( x) d x
219
where λ U ( t ) is the instantaneous up-time hazard rate function.
220
NOTE 2 to entry: If the up time is exponentially distributed, then F U ( t ) = 1 − exp( − t / MUT)
221 222
where MUT is the mean up time.
223
In this case, the reciprocal of MUT is denoted by λ U :
224
λ U = 1/MUT
225 226 227 228 229 230 231
3.3 instantaneous up-time hazard rate function up-time hazard rate function λ U ( t ) 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
232
NOTE 1 to entry: The instantaneous up-time hazard rate function is expressed by the formula: λ U (t ) = lim
233
∆t →0
1 F U (t ) − F U (t + ∆t )
∆t
1 − F U (t )
=
f U (t )
1 − F U (t )
234
where F U (t ) is up-time distribution function and f U (t ) is the probability density at the up-time.
235 236
NOTE 2 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 .
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NOTE 3 to entry: The unit of measurement of instantaneous up-time hazard rate function is the unit of time to the power − 1.
239 240 241 242
3.4 continuously operating item COI item for which operating time (191-42-05) is equal to its enabled time (191-42-17)
243 244 245 246
3.5 intermittently operating item IOI item for which operating time (191-42-05) is less than its enabled time (191-42-17)
247
4
248 249 250
The symbols given below are widely used and recommended but are not mandatory. For consistency of presentation, the notation in this document may differ from that used in a referenced document.
251
4.1
Glossary of symbols and abbreviations
Non-repairable items
COI
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 probabilit y of survival until time t : R(t ) = R( t 1, t 2) for t 1 = 0 and t 2 = t
Rˆ (t )
Point estimate of the reliability function at time t
R ( t 1, t 2 )
Reliability for the time interval ( t 1 , t 2 )
R(t , t + x | t )
Conditional reliability for the time interval ( t , t + x ), assuming that the item survived to time t
TTF i
Observed time to failure of item i
f ( t )
Probability density function of the (operating) time to failure
ˆ (t ) f
Point estimate of the probability density function of the (operating) time to failure at time t
n
Number of (non-repairable) items in the population that are operational at the instant of time t = 0
n S ( t )
Number of (non-repairable) items that are still operational at the instant of time t ( n S (0) = n )
n S ( t ) − n S ( t + ∆ t )
Number of items that fail in the time interval ( t , t + ∆ t )
λ
Constant failure rate, i.e. the reciprocal of the mean time to failure (MTTF) when the times to failure are exponentially distributed
λ ˆ
Point estimate of the constant failure rate
λ ( t )
Instantaneous failure rate
λ ˆ(t )
Point estimate of the instantaneous failure rate at time t
λ (t 1, t 2 )
Mean failure rate for the time interval ( t 1 , t 2)
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 10 254
4.2
Repairable items with zero time to restoration
COI
Continuously operating item
IOI
Intermittently operating item
MOTBF
Mean operating time between failures
MTTF
Mean time to failure
∧
MTTF
Point estimate of the mean time to failure
MUT
Mean up time
F U( t )
Up time distribution function
N ( t )
Number of failures in the time interval (0, t )
R ( t )
Reliability function, i.e. the probabilit y of survival until time t : R(t ) = R(t 1, t 2) for t 1 = 0 and t 2 = t
R ( t 1 , t 2 )
Reliability for the time interval ( t 1 , t 2 )
Rˆ (t 1, t 2 )
Point estimate of the reliability for the time interval ( t 1 , t 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 U ( t )
Probability density function of the up times NOTE
For COIs, f U ( t ) = f (t ).
( n) hCTTF (t )
Probability density function of calendar time to the n th failure, n ≥ 1
k F
Number of failures during a given period of observation
n
Number of items in the population
n F( t , t + ∆ t )
Number of failures observed in the time interval ( t , t + ∆ t )
n F( t 1 , t 2 )
Number of failures observed in the time interval ( t 1 , t 2 )
n S( t 1 , t 2 )
Number of items that were operational at the instant of time t 1 and operated without failure during the time interval ( t 1 , t 2 )
z ( t )
Instantaneous failure intensity
z ( ∞ )
Asymptotic failure intensity
ˆ( t ) z
Point estimate of the instantaneous failure intensity at time t
z (t 1, t 2 )
Mean failure intensity for the time interval ( t 1 , t 2 )
ˆ z (t 1, t 2 )
Point estimate of the mean failure intensity for the time interval ( t 1 , t 2 )
λ
Constant failure rate, i.e. the reciprocal of the mean time to failure (MTTF) when the times to failure are exponentially distributed
λ U
Constant up-time hazard rate function, i.e., the reciprocal of the mean up time (MUT) when the up time is exponentially distributed
255 256
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 11 257
4.3
Repairable items with non-zero time to restoration As ymp totic a vaila bil ity
A A ( t )
Instantaneous availability (availability function), i.e. the probability of the item being in an up state at the instant of time t
A(t 1, t 2 )
Mean availability for the time interval ( t 1 , t 2 )
ˆ A(t 1, t 2 )
Point estimate of the mean availability for the time interval ( t 1 , t 2 )
COI
Continuously operating item
F U( t )
Up time distribution function
G ( t )
Distribution function of the repair times
G AC M( t )
Distribution function of the active corrective maintenance time
G R( t )
Distribution function of the times to restoration
IOI
Intermittently operating item
M ( t )
Maintainability function, i.e. the probability of completing a given maintenance action within time t : M ( t ) = M ( t 1 , t 2 ) for t 1 = 0 and t 2 = t
ˆ (t ) M
Point estimate of the maintainability function at time t
M ( t 1 , t 2 )
Maintainability for the time interval ( t 1 , t 2 )
N ( t )
Number of failures occurring in the time interval (0, t )
N R( 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 1 , t 2 ) for t 1 = 0 and t 2 = t
R ( t 1 , t 2 )
Reliability for the time interval ( t 1 , t 2 )
RT i
Observed repair time of item i
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
MAUT
Mean accumulated up time
∧
MAUD
Point estimate of the mean accumulated up time
MDT
Mean down time
∧
MDT
Point estimate of the mean down time
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, maintenance action time
MRT
Mean repair time
i.e.
the
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expectation of
a
given
E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 12 ∧
MRT
Point estimate of the mean repair time
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
∧
MTTR
Point estimate of the mean time to restoration
MUT
Mean up time
∧
MUT U
Point estimate of the mean up time As ymp totic u navaila bilit y
U ( t )
Instantaneous unavailability (unavailability function)
U (t 1, t 2 )
Mean unavailability for the time interval ( t 1 , t 2 )
ˆ U (t 1, t 2 )
Point estimate of the mean unavailability for the time interval ( t 1 , t 2 ) 2
2
VRT
The variance of repair time, VRT = Va r [ζ ] = E [ζ ] − MRT , where ζ is a random variable representing the repair time, Va r denotes the variance and E denotes the expectation
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 U ( t )
Probability density function of the up times
f U+R ( t )
Probability density function of the sum of the up time and the corresponding time to restoration
g ( t )
Probability density function of the repair times
ˆ (t ) g
Point estimate of the probability density function of the repair time at time t
g ACM ( t )
Probability density function of the active corrective maintenance times
g AD ( t )
Probability density function of the administrative delays
g D( t )
Probability density function of the down times
g LD ( t )
Probability density function of the logistic delays
g MA ( t )
Probability density function of the given maintenance action time
g R( t )
Probability density function of the times to restoration
( n) hCTTF (t )
Probability density function of calendar time to the n th failure, n ≥ 1
k
Number of repair times during a given period of observation
k AC M
Number of active corrective maintenance times during a given period of observation
k AD
Number of administrative delays during a given period of observation
k D
Number of down times during a given period of observation
k F
Number of failures during a given period of observation
k LD
Number of logistic delays during a given period of observation
k O
Number of failures while operating during a given period of observation --``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 13
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
k R
Number of times to restoration during a given period of observation
k U
Number of up times during a given period of observation
m
Number of observed maintenance action times
mMAT (t )
Number of maintenance action times with duration greater that ( mMAT (0) = m )
n
Number of items in the population
n D{ t }
Number of items that are in a down state at the instant of time t
n F( t , t + ∆ t )
Number of failures observed during the time interval ( t , t + ∆ t ) , where the time scale includes both up and down times
n F( t 1 , t 2 )
Number of failures observed during the time interval ( t 1 , t 2 ), where the time scale includes both up and down times
n R ( t )
Number of repairable items that are still under repair at the instant of time t ( n R (0) = n )
n R ( t ) − n R ( t + ∆ t )
Number of items with repair completed in the time interval ( t , t + ∆ t )
n S( t 1 , t 2 )
Number of items that were operational at the instant of time t 1 and operated without failure in the time interval ( t 1 , t 2 )
n U{ t }
Number of items that are in an up state at the instant of time t
v ( t )
Instantaneous restoration intensity
z ( t )
Instantaneous failure intensity
z ( ∞ )
Asymptotic failure intensity
ˆ( t ) z
Point estimate of the instantaneous failure intensity at time t
z (t 1, t 2 )
Mean failure intensity for the time interval ( t 1 , t 2 )
ˆ z (t 1, t 2 )
Point estimate of the mean failure intensity for the time interval ( t 1 , t 2 )
t
λ
Constant failure rate, i.e. the reciprocal of mean time to failure (MTTF) when time to failure is exponentially distributed
λ U
Constant up-time hazard rate function, i.e., the reciprocal of the mean up time (MUT) when up times are exponentially distributed
λ U (t )
The up-time hazard rate function
µ
Constant repair rate, i.e. the reciprocal of the mean repair time (MRT) when the repair times are exponentially distributed
µ ( t )
Instantaneous repair rate
ˆ (t ) µ
Point estimate of the instantaneous repair rate at time t
µ AC M
Reciprocal of the mean active corrective maintenance time (MACMT) when the active corrective maintenance times are exponentially distributed
µ AD
Reciprocal of the mean administrative delay (MAD) when the administrative delays are exponentially distributed
µ D
Reciprocal of the mean down time (MDT) when the down times are exponentially distributed
µ LD
Reciprocal of the mean logistic delay (MLD) when the logistic delays are exponentially distributed
µ MA
Constant rate of the completion of a given maintenance action, when the maintenance action time is exponentially distributed
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 14 Constant restoration rate, i.e. the reciprocal of the mean time to restoration (MTTR) when the times to restoration are exponentially distributed
µ R
258 259
5
260
5.1
261 262 263 264
In order to derive correct mathematical expressions for the measures defined in IEC 60050-191, the distinction needs to be made between repairable items [191-41-11] and non-repairable items [191-41-12]. The following classes of items are considered separately in this standard:
265
–
non-repairable items;
266
–
repairable items with zer o time to restoration;
267
–
repairable ite ms with non- zer o tim e to re storation.
268 269
In order to keep the mathematical formulae as simple as possible, the following basic mathematical models are used to quantify dependability measures:
270
–
random var iable ( tim e to failure) for non-repairable items;
271
–
simple (ord ina ry) renewal p ro cess for repairable items w ith zero time to re sto ration;
272 273
–
simple (ordinar y) alternating renew al process for repairable items with non-zer o time to restoration.
274 275 276 277 278 279 280
The simplest mathematical model for the reliability of a non-repairable item is the random variable, time to failure of the item [191-45-01]. One of the widely used reliability measures of non-repairable items is the instantaneous failure rate, λ ( t ) [191-45-06], also referred to as the hazard rate function. It is derived fro m the distribution function of the time to failure. The expression λ ( t ) ∙ ∆ t is, for small values of ∆ t , approximately equal to the conditional probability of failure of an item during the time interval ( t , t + ∆ t ) given that the item has not failed during the interval (0, t ].
281 282 283 284 285
For repairable items, the basic model is a simple renewal process, when the time to restoration of the item may be neglected, or a simple alternating renewal process in which the time to restoration of the item is non-zero. In the latter case, the item alternates between an up state and a down state, and a widely used measure of reliability of the item is the failure intensity, which is equal to the renewal density.
286 287 288 289
The failure intensity [191-45-08] is a measure derived from the expected value of the cumulative number of failures E [ N ( t ) ] of a repairable item occurring during the time interval (0, t ] . The expression z ( t ) ∙ ∆ t is, for small values of ∆ t , approximately equal to the probability of failure of the item during the time interval ( t , t + ∆ t ).
290 291
To avoid improper use of these mathematical expressions, which could yield erroneous results, the specific assumptions detailed in 5.2 and 5.3 should be observed.
292
5.2
293 294
a) At any instant of time, the non-repairable item will be either in an up state or in a down state (see figure 1).
295 296
b) Unless otherwise stated, when the item is in an up state, it is considered to be operating continuously.
297
Assumptions General
Assumptions for non-repairable items
NOTE
The mathematical expressions given in this sub-clause may not always be valid for IOIs.
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 15 298 299
c) At time t = 0, the item is in an operating state, and is as good as new. Latent faults are not considered, which, if present, may invalidate some mathematical expressions.
300 301
d) Preventive maintenance, or other planned actions that render the item incapable of performing a required function are not considered.
302 303
e) The time to failure is a positive and continuous random variable with a probability density function and finite expectation. ` , , ` , ` , , ` , , ` ` ` ` ` , , ` , , , ` , , , ` , , , , , , , ` ` , , ` ` -
State
Up state
Down state
τ
0
304 305
Key
306
τ Time to failure
307
Time
Figure 1 – Sample realization of a non-repairable item
308
5.3
Assumptions for repairable items
309 310
a) At any instant of time, the repairable item will be either in an up state or in a down state (see figures 2 and 3).
311 312
b) At time t = 0, the item is in an up state, and therefore, R (0) = 1, and is as good as new. Latent faults are not considered.
313 314
c) Unless otherwise stated, when the item is in the up state, it is considered to be operating continuously.
315 316 317
d) Consecutive up times of the item are statistically independent, identically distributed, positive, continuous random variables with a common probability density function and finite expectation.
318 319 320
e) In the case of non-zero down-time duration, the consecutive down times of the item are statistically independent, identically distributed, positive, continuous random variables with a common probability density function and finite expectation
321
f)
322 323
g) Preventive maintenance or other planned actions that render the item incapable of performing a required function are not considered.
324 325 326
h) Unless otherwise stated, each other random variable (e.g. time to failure, repair time, logistic delay, and so on) considered in the standard, is a positive and continuous random variable with a probability density function and finite expectation.
327
In summary:
328
–
any tra nsition from an up state to a down sta te is a failure ;
329
–
any tra nsition from a down sta te to an up state is a restoration;
330 331
–
any down sta te is a fault and, in con seq uence, the down time is equal to the time to restoration;
332
–
after each restoration the item is as good as new.
The up times are statistically independent of the down times.
333
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 16 334 335
For continuously operating items (COI) the up state coexists with the operating state, and the up time is concurrent with the operating time.
336 337
The expressions for reliability measures of continuously operating, repairable items may not be true for IOIs (see figure 4).
338 339
NOTE 1 Models assuming zero time to restoration are used when either the up time of the item is the only time of interest in assessing the performance of the item, or the time to restoration is so short that it is negligible.
340 341
NOTE 2 All mathematical expressions for the reliability measures relating to the time to failure of a non-repairable item may also be applied to each time to failure of a continuously operating repairable item.
342 N ( t ) ` , , ` , ` , , ` , , ` ` ` ` ` , , ` , , , ` , , , ` , , , , , , , ` ` , , ` ` -
m+1 m m − 1
2 1 0 S F ,0 = 0
S F ,1
S F ,2
S F, m
S F, m − 1
Time
State
τ U ,2
τ U ,1
τ U ,3
τ U, m − 1
τ U, m
τ U, m + 1
Up state
Down state
S F ,0 = 0
343 344 345 346 347 348
S F ,1
S F ,2
S F, m − 1
S F, m
Key N ( t )
Number of failures during the time interval (0, t ) S F, 1 , S F, 2 , S F ,3 ... Consecutive instants of failure τ U,1 , τ U, 2 , τ U ,3 ... Consecutive up times
Figure 2 – Sample realization of a repairable item with zero time to restoration
349
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Time
E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 17 N R ( t )
m m − 1 m − 2
2 1 0 S F,0 = S F,0 =0
S F ,1
S R ,1
S F ,2
S R ,2
S R, m − 1
S F, m
S R, m
Time
S F ,1
S R ,1
S F ,2
S R ,2
S R, m − 1
S F, m
S R, m
Time
N ( t )
m m − 1
2 1 0 S F,0 = S F,0 =0
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
State
τ U ,1
τ U ,2
τ U ,3
τ U, m
Up state
ξ R ,1
ξ R ,2
ξ R, m − 1
τ U, m +
1
ξ R, m
Down state
S F,0 = S F,0 =0
S F ,1
S R ,1
S F ,2
S R ,2
S R, m − 1
S F, m
S R, m
350 351
Key
352 353 354 355 356 357
N ( t ) N R ( t )
358
Number of failures during the time interval (0, t ] Number of restorations during the time interval (0, t ] S F, 1 , S F, 2, S F ,3 ... Consecutive instants of failure S R,1 , S R,2 , S R ,3 ... Consecutive instants of restoration τ U,1 τ U, 2 , τ U, 3 ... Consecutive up times ξ R,1 , ξ R,2 , ξ R ,3 ... Consecutive times to restoration
Figure 3 – Sample realization of a repairable item with non-zero time to restoration
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Time
E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 18 Continuously operating item (COI)
Enabled time
Intermittently operating item (IOI)
Restoration
Failure
Key
Operating time [191-42-05] Standby time [191-42-13] Idle time [191-42-15]
359 360
361 362 363
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
Figure 4 – Comparison of an enabled time for a COI and an IOI
6
Mathematical expressions
6.1 6.1.1
Non-repairable items General
364
All exp ressions in 6.1 are applica ble to COI s only.
365
For each measure, the following are presented:
366
a) the generic expression;
367
b) the most common expression (for exponentially distributed time to failure of the item);
368
c) a simple example of application where necessary.
369
6.1.2
Reliability [191-45-05] (Symbol R ( t 1 , t 2 ), 0 ≤ t 1 < t 2 )
370 371 372
For non-repairable items, R ( t 1 , t 2 ) for a given time interval ( t 1 , t 2 ), 0 ≤ t 1 < t 2 , is equivalent to the reliability R (0, t 2 ) for the time interval (0, t 2 ).
373 374
More commonly used expressions are the reliability function R ( t ) = R (0, t ) , with R (0) = 1, and the conditional reliability R ( t , t + x | t ) , when no failure has occurred in time [ 0 , t ].
375
a)
376
where
377
λ ( x ) is the instantaneous failure rate of the item;
R(t ) = exp −
t
∫0
λ ( x ) d x =
∞
∫t f ( x) d x
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61703/Ed2/CD © IEC (E) 19 378 379 380
f ( x ) is the probability density function of th e time to failure of the item, i.e. , for small values ∆ x , f ( x ) ⋅∆ x is approximately equal to the probability that the failure of the item will occur during ( x , x + ∆ x ).
381 382
NOTE If observed failure data are available for n non-repairable items, from a homogenous population, the estimated value of R ( t ) is given by n (t ) ˆ R(t ) = S n
383 384
where
385
n S ( t ) is the number of items that are still operational at the instant of time t ( n S (0) = n ).
The probability that the item will fail during the time interval ( t 1 , t 2 ), 0 ≤ t 1 < t 2 , is given by
386
R(t 1) − R(t 2 ) =
387 388 389 390
t 2
∫t f (t ) dt 1
The conditional reliability, R ( t , t + x | t ) , is defined as the conditional probability that an item can perform a required function for a given time interval ( t , t + x ) provided that the item is in an operating state at the beginning of the time interval. (See [9] 1) , page 40.)
R(t , t + x | t ) = exp −
391
t + x
∫t
λ (t ) dt =
R(t + x) R(t )
392 393
b) When λ ( t ) = λ = constant, i.e. when the (operating) time to failure is exponentially distributed
394
R ( t ) = exp( −λ t )
395
R ( t , t + x | t ) = exp( − λ x )
396 397
c) For an item with a constant failure rate of λ = 1 year − and a required time of operation of six months, the reliability is given by 1
R (6 months) = exp( − 1 ×
398
399
6.1.3
400 401
6 ) = 0,606 5 12
Instantaneous failure rate [191-45-06] (Symbol λ ( t ))
a) According to definition [191-45-06]:
1 R(t ) − R(t + ∆t ) f (t ) = R(t ) R(t ) ∆t → 0 ∆t
λ (t ) = lim
402 403 404
For small values of ∆ t , λ ( t ) ⋅∆ t is approximately equal to the conditional probability that failure of the item will occur during ( t , t + ∆ t ) , given that the item has survived to time t .
405 406
NOTE If observed failure data are available for n non-repairable items, from a homogenous population, the estimated value of λ (t ) at time t is given by n (t ) − nS (t + ∆t ) ˆ λ (t ) = S nS (t )∆t
407 408
where
409
n S (t ) is the number of items that are still operational at the instant of time t ( n S (0) = n );
410
n S (t ) − n S ( t + ∆ t ) is the number of items that fail in the time interval (t , t + ∆ t ) .
411
It should be noted that the estimated value of the failure density function f ( t ) , at time t , is given by n (t ) − nS (t + ∆t ) ˆ f (t ) = S n∆t
412 ________ 1)
Figures in square brackets refer to the bibliography. --``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 20 413
The probability that the item will fail during the time interval ( t 1 , t 2 ) is given by
R(t 1) − R(t 2 ) = exp −
414
415
421
422 423 424
∫
t 2
0
λ (t ) dt
f ( t ) = λ exp( − λ t )
and R ( t ) = exp( − λ t ) ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
NOTE If observed failure data are available for n non-repairable items, from a homogenous population, with constant failure rate, then the estimated value of λ is given by ˆ λ =
n n
∑ TTFi i =1
where TTF i is the time to failure of item i .
c) For 10 non-repairable items, from a homogenous population, with a constant failure rate,
425
the observed total operating time to failures of all the items is
∑ i =1TTFi = 2 years. Hence 10
ˆ = 10 = 5 year -1 λ 2
426 427 428
If the time to failure of a non-repairable item has a two-parameter Weibull distribution with scale parameter α > 0 and shape parameter β > 0, then R ( t ) = exp( − ( α t ) ) β
429 430
and f (t ) =
431 432
−d R(t ) dt
β − 1
= αβ ( α t )
exp( − ( α t ) ) β
hence
433
λ ( t ) =
434
(See [9], page 434.)
435
For β = 2 and α = 0,5 year −
f (t ) R(t )
= αβ ( α t ) β −
1
1
λ ( 6 months) = 0,5 × 2 × (0,5 ×
436
6 1 ) = 0,25 year − 12
λ ( 1 year) = 0,5 × 2 × (0,5 × 1) = 0,5 year −
437 438
0
λ (t ) dt − exp −
b) When the time to failure is exponentially distributed, i.e. λ ( t ) = λ for all values of t,
416 417 418 419 420
∫
t 1
6.1.4
439
Mean failure rate [191-45-07] (Symbol
λ (t 1,
t 2 ), 0 ≤ t 1 < t 2 ) λ (t 1, t 2 ) =
1
∫
t 2
λ (t ) dt =
1
440
a)
441
b) When the time to failure is exponentially distributed
442
t 2 − t 1
t 1
t 2 − t 1
ln
R(t 1) R(t 2 )
λ (t 1, t 2 ) = λ
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 21 443
for all values of t 1 and t 2 .
444
c) Let t 1 = 6 months, R ( t 1 ) = 0,8 and t 2 = 12 months, R ( t 2 ) = 0,5, then λ (6, 12) =
445 446
while ( R (0) = 1) λ (0, 6) =
447
448 449 450 451 452
0,47 1 0,8 1 ln = ln(1,6)/6 = = 0,078 3 month − 12 − 6 0,5 6
6.1.5 ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
1 6−0
ln
0,2231 1 1 = ln(1,25)/6 = = 0,037 2 month − 0,8 6
Mean (operating) time to failure [191-45-11] MTTF (abbreviation) MTTF =
a)
n
MTTF = i =1
where
455
TTF i is the time to failure of item i .
MTTF =
1 λ
c) For a non-repairable item with a constant failure rate of λ = 0,5 year − , 1
459
462 463
n
b) When the time to failure is exponentially distributed, i.e. λ ( t ) = λ for all values of t,
457
460 461
∑ TTFi
∧
454
458
∞
NOTE If observed failure data are available for n non-repairable items, from a homogenous population, then an estimate of MTTF is given by
453
456
∞
∫0 tf (t ) dt = ∫0 R(t ) dt
MTTF = 2 years = 17 520 h If the time to failure of a non-repairable item has a two-parameter Weibull distribution with a scale parameter α > 0 and shape parameter β > 0, then R ( t ) = exp( − ( α t ) ) β
and Γ (1 +
464 465
MTTF =
1 ) β
α
where
466
Γ ( x ) =
∞ x −1 − t
∫0 t
e
dt
467
is the complete gamma function. (See [9], page 435.)
468
For β = 2 and α = 0,5 year − : 1
1 ) 2 = 2 × Γ (1 + 1 ) 0,5 2
Γ (1 +
469 470
MTTF = but
471
Γ (1 +
1 1 ) = Γ ( ) / 2 = π / 2 2 2
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61703/Ed2/CD © IEC (E) 22 472
hence
473
π ≈ 1,772 5 years = 21,27 months
MTTF =
474
6.2
475
6.2.1
476
All exp ressions in 6.2 are applicabl e to COI s. Where they are appli cable to IOIs, this is stated.
477
For each measure, the following are presented:
478
a) the generic expression;
479 480
b) the most common expression (for the cases when the times to failure of the item are exponentially distributed);
481
c) a simple example of application where necessary.
482
6.2.2
483 484 485
Repairable items with zero time to restoration General
Reliability [191-45-05] (Symbol R ( t 1 , t 2 ), 0 ≤ t 1 < t 2 )
a) The reliability of an item for the time interval ( t 1 , t 2 ) may be written as (see [9], page 461 and [13], page 105): t 1
∫0 R(t 2 − t ) z (t ) dt
R(t 1, t 2 ) = R(t 2 ) +
486 487
where
488 489 490
the first term, R ( t 2 ), represents the probability of survival to time t 2 , and the second term represents the probability of failing at time t ( t < t 1 ) and, after immediate restoration, surviving to time t 2 ;
491 492 493
z (t ) is the instantaneous failure intensity (renewal density) of the item, i.e., for small values of ∆ t , z (t )⋅∆ t is approximately equal to the (unconditional) probability that a failure of the item occurs during ( t , t + ∆ t ) , and
494
R ( t ) = R (0, t ) is the reliability function of the item
R(t ) =
495
∫
∞
t
f ( s) d s
496 497 498 499 500
where f ( t ) is the probability density function (also referred to as the failure density function) of the times to failure of the item, i.e., for small values of ∆ t , f ( t ) ⋅∆ t is approximately equal to the probability that the item fails during the time interval ( t , t + ∆ t ). More precisely, it is approximately the probability that a given time to failure terminates in the time interval ( t , t + ∆ t ) , assuming that the time to failure started at time t = 0.
501
NOTE 1 R ( t 1 , t 2 ) is also known as the interval reliability.
502 503
NOTE 2 If observed failure data are available for n repairable items, from a homogenous population, then an estimate of R ( t 1, t 2 ) is given by n (t , t ) ˆ R(t 1, t 2 ) = S 1 2 n
504 505 506 507
where
508 509
By setting t 1 = t and t 2 = t + x , one can obtain the asymptotic interval reliability (see [13], page 106):
n S ( t 1 , t 2 ) is the number of items that were operational at the instant of time t 1 and did not fail during the time interval ( t 1, t 2 ).
510
lim R(t , t + x ) =
t → ∞
1 MTTF
∫
∞ x
R( s) d s
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` , , ` , ` , , ` , , ` ` ` ` ` , , ` , , , ` , , , ` , , , , , , , ` ` , , ` ` -
E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 23 511
which, for large values of t , can be used as an approximation of the R ( t , t + x ), where
512
MTTF is the mean time to failure.
513
This asymptotic expression follows from the key renewal theorem (see [9], page 462).
514 515
b) When λ ( t ) = λ and is constant, i.e. when the times to failure are exponentially distributed, R ( t 1 , t 2 ) = exp ( − λ × ( t 2 − t 1 ))
516 517
In this case, the asymptotic interval reliability is given by
lim R(t , t + x ) = exp ( −λ x )
518 519 520
t → ∞
c) For a repairable item with a constant failure rate λ = 1 year − , its reliability over six months is given by 1
R ( t , t + 6) = exp ( − 1 ×
521
522 523 524
6 12
) = 0,606 5
where t is the starting point of the six month interval. 6.2.3
Instantaneous failure intensity [191-45-08] (Symbol z ( t ))
525
The expressions included in this sub-clause also apply to IOIs.
526 527 528
a) By definition [191-45-08], z ( t ) is the derivative of the expected number of failures, Z ( t ) = E [ N ( t ) ], in the time interval (0, t ) where N ( t ) is the number of failures during the time interval (0, t ) and E denotes the expectation. z (t ) =
529 530
lim
Z (t + ∆t ) − Z (t )
∆t
∆t → 0 +
=
d Z (t ) dt
From the renewal theory (see [9], page 457), it follows that z ( t ) may be written as ∞
z (t ) =
531
∑ h (n)
CTTF
(t )
n =1
532
( n ) (t ) is the probability density function of calendar time until the n th failure of the where hCTTF
533
item. This may be calculated by the following recursive relationship: (1)
hCTTF (t ) = f U (t )
534 (n)
hCTTF (t ) =
535
t
∫0 f ( x) h U
( n − 1) CTTF
(t − x ) d x , for n > 1
536 537 538
where f U( t ) is the probability density function of the up times of the item, i.e., for small values of ∆ t , f U ( t ) ⋅∆ t is approximately equal to the probability that a given up time of the item terminates during ( t , t + ∆ t ) , assuming that the up time started at time t = 0.
539 540
NOTE 1
( n ) Let τ U,1 , τ U, 2 ,..., τ U, n, be consecutive up times ( τ U ) of the item. Then hCTTF (t ) is the probability density
function of the sum
541
τ U,1 + τ U,2 +... + τ U, n − 1 + τ U, n
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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NOTE 2 The instantaneous failure intensity, z ( t ) , satisfies the following integral equation (see [9], page 458 and [11], page 54): z (t ) = f U (t ) +
544
t
∫0 f (t − s) z ( s) d s U
545
which may be solved by numerical methods.
546 547
NOTE 3 If observed failure data are available for n repairable items, from a homogenous population, then an estimate of z ( t ) is given by ˆ(t ) = nF (t , t + ∆t ) z n∆t
548 549
where
550
n F ( t , t + ∆ t ) is the number of failures observed during the time interval ( t , t + ∆ t ).
551
When the item operates continuously, f U ( t ) is equal to f ( t ).
552 553
For small values of ∆ t , z ( t ) ⋅∆ t is approximately equal to the (unconditional) probability that a failure of the item occurs during ( t , t + ∆ t ).
554
b) When the up times are exponentially distributed,
555 556 557
z ( t ) = λ U
For a continuously operating item, λ U equals λ . 6.2.4
Asymptotic failure intensity [191-45-10] (Symbol z ( ∞ ))
558 559
The expressions included in this sub-clause also apply to IOIs.
560 561
a) By definition [191-45-10], z ( ∞ ) is the limit, if it exists, of the instantaneous failure intensity z ( t ) , when time t tends to infinity:
562
z (∞) = lim z (t )
563
t → ∞
The asymptotic failure intensity z ( ∞ ) is given by the equation z (∞) = lim z (t ) =
564
t → ∞
1 MUT
565 566 567
Under appropriate assumptions on f U( t ) , the above equation follows from the renewal density theorem (see [5], page 248, [9], page 462, [10], page 196 and [12], page 367). See Note 2.
568
NOTE 1 Using the elementary renewal theorem (see [9], page 461):
569 570
lim
t →∞
t
=
1 MUT
but Z (t ) =
571 572
Z (t )
t
∫0 z ( s) d s
hence z (∞) = lim
573
t →∞
Z (t ) t
574
provided that z ( ∞ ) exists (see [7], page 191).
575 576
NOTE 2 The easiest to check conditions for the existence of z ( ∞ ) are the following: MUT < ∞ ; f U ( t ) is a bounded function on [0, +∞ ) and tends to 0 as t → + ∞ .
577 578
NOTE 3 If observed failure data are available for n repairable items, from a homogenous population, and the time t is large enough, then an estimate of z ( ∞ ) is given by
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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61703/Ed2/CD © IEC (E) 25 ˆ(∞) = z ˆ(t ) = z
579
nF (t , t + ∆t ) n∆t
580
where
581
n F ( t , t + ∆ t ) is the number of failures observed during the time interval ( t , t + ∆ t ).
582 583
For small values of ∆ t and large values of t , z ( ∞ ) ∙ ∆ t is approximately equal to the (unconditional) probability that a failure of the item occurs during ( t , t + ∆ t ).
584
b) When the up times are exponentially distributed,
585
z ( ∞ ) = λ U
586
For a continuously operating item, λ U equals λ .
587
6.2.5
Mean failure intensity [191-45-09] (Symbol z (t 1, t 2 ), 0 ≤ t 1 < t 2 )
588 589
The expressions included in this sub-clause also apply to IOIs.
590
a)
591
z (t 1, t 2 ) =
The integral
592
∫
t 2
1 t 2 − t 1
∫
t 2
z (t ) dt
t 1
z (t ) dt is equal to the expected number of failures of the item in the time
t 1
interval ( t 1 , t 2 ), hence z (t 1, t 2 ) may be interpreted as the expected number of failures per time-unit in ( t 1 , t 2 ).
593 594 595
NOTE
If observed failure data are available for n repairable items, from a homogenous population, then an
estimate of z (t 1, t 2 ) is given by ˆ n (t , t ) z (t 1, t 2 ) = F 1 2 (t 2 − t 1) n
596 597
where
598
n F ( t 1 , t 2 ) is the number of failures observed in the time interval ( t 1 , t 2 ).
599
By setting t 1 = t and t 2 = t + x , the asymptotic mean failure intensity can be obtained:
600
601 602 603
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
lim z (t , t + x ) = t → ∞
which, for large values of t , can be used as an approximation of z (t , t + x ) . This equality is a direct consequence of Blackwell’s theorem (see [9], page 462). b) When the up times are exponentially distributed, then
604
z (t 1, t 2 ) = λ U
605 606
For a continuously operating item, λ U equals λ . 6.2.6
607 608 609 610
1 MUT
Mean (operating) time to failure [191-45-11] MTTF (abbreviation) MTTF =
a)
∞
∫0
tf (t ) dt =
∞
∫0 R(t ) dt
NOTE When all observed operating times to failure of n items, from a homogenous population, are available, then an estimate of MTTF is given by
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 26 n
∧
MTTF =
611
∑ (operating time) i
total operating time
= i =1
k F
k F
612 613
where
614
k F is the total number of failures observed during the given time period;
615
(operating time)i is the aggregate operating time of the i th item during the given time period.
616
total operating time is the aggregate operating time of all n items during a given time period;
b) When the times to failure are exponentially distributed
617 618
λ
c) For a repairable item with a constant failure rate of 0,5 year −
619 620
1
MTTF =
MTTF = 2 years = 17 520 h 6.2.7
621
Mean operating time between failures [191-45-13] MOTBF MOTBF = MTTF =
∞
∫0
622
a)
623
b) If the times to failure are exponentially distributed,
624
625
1
tf (t ) dt =
MOTBF =
6.2.8
626
∞
∫0 R(t ) dt
1 λ
Mean up time [191-48-09] MUT (abbreviation)
627
The expressions in this sub-clause also apply to IOIs.
628
a)
MUT =
∞
∫0
tf U (t ) dt =
∞
∫ 0 (1 − F (t )) dt U
629 630
where f U ( t ) is the probability density function of the up times of the item (including its operating, idle, standby and external disabled times).
631
For a continuously operating item,
632 633
MUT = MTTF = MOTBF b) If the up times are exponentially distributed,
634
MUT =
635
1 λ υ
If the item operates continuously, λ U equals λ .
636
6.3
Repairable items with non-zero time to restoration
637
6.3.1
638
All exp ressions in 6.3 are applicabl e to COI s. Where they are appli cable to IOI s, this is sta ted.
639
For each measure, the following are presented:
General
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a) the generic expression;
641 642
b) the most common expression (for the cases when times to failure, up times, down times, times to restoration and repair times of the item are exponentially distributed);
643
c) a simple example of application where necessary.
644
6.3.2
645 646 647 ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
reliability [191-45-05] (Symbol R ( t 1 , t 2 ), 0 ≤ t 1 < t 2 )
a) The reliability of a repairable item with non-zero time to restoration for the time interval (t 1 , t 2 ) may be written as (see [13], page 113 and [9], page 177): t 1
∫0 R(t 2 − t ) v(t ) dt
R(t 1, t 2 ) = R(t 2 ) +
648 649
where
650 651 652
the first term, R ( t 2 ), represents the probability of survival to time t 2 , and the second term represents the probability of restoration (after a failure) at time t ( t < t 1 ), and surviving to time t 2 ;
653 654 655
v ( t ) is the instantaneous restoration intensity of the item, i.e., for small values of ∆ t , v ( t ) ⋅∆ t is approximately equal to the probability that a restoration of the item occurs during ( t , t + ∆ t ) (see definition 3.1);
656
R ( t ) = R (0, t ) is the reliability function of the item R(t ) =
657
∞
∫t
f ( s) d s
658 659 660 661 662
where f ( t ) is the probability density function of the times to failure of the item, i.e., for small values of ∆ t , f ( t ) ⋅∆ t is approximately equal to the probability that the item fails during the time interval ( t , t + ∆ t ) . More precisely, it is approximately the probability that a given time to failure terminates in the time interval ( t , t + ∆ t ) , assuming that the time to failure started at time t = 0.
663 664
NOTE 1 R ( t 1 , t 2 ) is the (unconditional) probability of failure-free continuous operation of the item in the time interval ( t 1, t 2 ). The expression may not be true for IOIs.
665
NOTE 2
666 667
NOTE 3 If observed failure data are available for n repairable items, from a homogenous population, then a n estimate of R ( t 1, t 2 ) is given by
An alternative name of the reliability given by R (t 1 , t 2 ) is the interval reliability.
n (t , t ) ˆ R(t 1, t 2 ) = S 1 2 n
668 669
where
670 671
n S (t 1 , t 2 ) is the number of items which were operating at the instant of time t 1 and operated without failure in the time interval ( t 1 , t 2).
672 673
By setting t 1 = t and t 2 = t + x , one can obtain the asymptotic interval reliability (See [13], page 113 and [9], page 183): lim R(t , t + x ) =
674
675
t → ∞
1 MTTF + MTTR
∫ x R( s) d s
which, for large values of t , can be used as an approximation of the R ( t , t + x ), where
676
MTTF is the mean time to failure, and
677
MTTR is the mean time to restoration.
678
∞
This expression follows from the key renewal theorem (see [9], page 462).
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When the times to failure are exponentially distributed, then R(t 1, t 2 ) = A(t 1) exp (−λ × (t 2 − t 1))
680 681
where A ( t 1 ) is the instantaneous availability at time t 1 ;
682
and
lim R(t , t + x ) =
683
t → ∞
MTTF exp( −λ x ) MTTF + MTTR
684
(See [9], pages 181 and 186.)
685 686
NOTE 4 The formula for R ( t 1 , t 2 ) above can be related to note 2 in IEC 60050-191, definition 191-45-10, by assuming that t 1 = 0, t 2 = t , R ( t 1 , t 2 ) = R (0, t ) = R ( t ) and A ( t 1 ) = A (0) = 1.
687 688
b) When the times to failure and times to restoration are exponentially distributed, then, using either Markov techniques or the Laplace transform, the following is obtained:
µ R
R(t 1, t 2 ) =
689
λ + µ R
690
+
λ λ + µ R
exp [ −(λ + µ R )t 1 ] exp [ −λ × (t 2 − t 1)]
and
lim R(t , t + x ) =
691
t → ∞
692
µ R λ + µ R
exp( −λ x)
(See [9], pages 181 and 186.) c) For a COI with λ = 2 year − and a restoration rate of µ R = 10 year − : 1
693
R 0,
694
1
1 10 2 1 exp ( −12 × 0) exp − 2 × − 0 = 0,606 531 + = 4 12 12 4
1 1 1 3 3 , = 0,510 475; R , = 0,505 693; R , 1 = 0,505 455 4 2 2 4 4
R
695
lim R t , t +
696
t → ∞
697
6.3.3
698
1 10 1 exp − 2 × = 0,505 442 = 4 12 4
Instantaneous failure intensity [191-45-08] (Symbol z ( t ))
699
The expressions in this sub-clause also apply to IOIs.
700 701 702
a) By definition [191-45-08], z (t ) is the derivative of the expected number of failures, Z (t ) = E [ N (t )], in the time interval (0, t ) , including up and down times, where N ( t ) is the number of failures in the time interval (0, t ) , and E denotes the expectation, thus
703
704 705
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
z (t ) =
lim
Z (t + ∆t ) − Z (t )
∆t
∆t → 0 +
=
d Z (t ) dt
Using the alternating renewal process theory (see [9], pages 180 and 466), it follows that z ( t ) may be written as ∞
706
z (t ) =
∑ h(n)
CTTF (t )
n =1
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( n) where hCTTF (t ) is the probability density function of calendar time to the n th failure and
708
may be calculated by the following recursive relations: (1)
hCTTF (t ) = f U (t )
709 ( )
n hCTTF (t ) =
710
t ( n − 1)
∫0 h
CTTF
( x ) f U+R (t − x ) d x , for n > 1
711
where
712 713 714 715
f U( t ) is the probability density function of the up times of the item (including its operating, idle, standby and external disabled times). For small values of ∆ t , f U ( t ) ⋅∆ t is approximately equal to the probability that a given up time of the item terminates during ( t , t + ∆ t ), assuming that the up time started at time t = 0;
716 717
f U+R ( t ) is the probability density function of the sum of the up time ( τ U) and the corresponding time to restoration ( ξ R), and is given by f U+R (t ) =
718
t
∫0 g (t − s) f ( s) d s R
U
719
where g R (t ) is the probability density function of the times to restoration of the item, i.e.,
720
for small values of ∆ t , g R (t ) ⋅ ∆t is approximately equal to the probability that the item is
721 722
restored from a fault to an up state in the time interval ( t , t + ∆ t ), assuming that a failure resulting in a fault occurred at time t = 0.
723 724
For small values of ∆ t , z ( t ) ⋅∆ t is approximately equal to the (unconditional) probability that a failure of the item occurs during the time interval ( t , t + ∆ t ).
725 726
NOTE 1
Let τ U,1 , ξ R,1 , τ U,2 , ξ R,2 ,..., τ U, n , ξ R, n ... be consecutive up times ( τ U ) and times to restoration (ξ R ) of the
( n ) item. Then hCTTF (t ) is the probability density function of the sum
τ U,1 + (ξ R,1 + τ U,2 ) + (ξ R,2 + τ U,3 ) + ... + (ξ R,n −1 + τ U,n )
727 728
while f U+R ( t ) is the probability density function of the sum ξ R, m − 1 + τ U, m for any m > 1.
729 730
NOTE 2 The instantaneous failure intensity, z ( t ) , and the instantaneous restoration intensity, v (t ) , fulfil the following simultaneous system of linear Volterra integral equations (see [14], page 193):
731
z (t ) = f U (t ) +
732
v(t ) =
t
∫0 f (t − s) v( s) d s U
t
∫0 g (t − s) z ( s) d s R
733
which may be solved by numerical methods.
734 735
NOTE 3 If observed failure data are available for n repairable items, from a homogenous population, then an estimate of z ( t ) is given by ˆ(t ) = z
736
nF (t , t + ∆t ) n∆t
737 738
where n F (t , t + ∆ t ) is the number of failures observed during the time interval ( t , t + ∆ t ) , where the time scale includes both up and down times.
739
When the up times are exponentially distributed, then (see [4], page 317)
740
z ( t ) = A ( t ) λ U
741
where A ( t ) is the instantaneous availability.
742
When the item operates continuously, f U ( t ) = f ( t ) and λ U = λ .
743 744 745
b) When the up times and times to restoration are exponentially distributed, Markov techniques or the Laplace transform can be used to yield (see [14], page 200 and [18], page 66): --``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 30 z (t ) =
746 747 748 749
λ U µ R λ U + µ R
+
2 λ U
exp [ −(λ U + µ R )t ] = A(t )λ U
λ U + µ R
For a COI, λ U equals λ . c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − , and for arbitrarily chosen values of t = 0, ¼,½,¾ and 1 1
1
4 1 1 20 exp − 12 × = 1,683 262 year −1 + = 4 4 12 12
z (0) = 2 year −1; z
750
1 3 −1 −1 −1 = 1,667 493 year ; z = 1,666 708 year ; z (1) = 1,666 669 year 2 4
z
751 752 753
6.3.4
Asymptotic failure intensity [191-45-10] (Symbol z ( ∞ ))
754 755
The expressions included in this sub-clause also apply to IOIs.
756 757
a) By definition [191-45-10], z ( ∞ ) is the limit, if it exists, of the instantaneous failure intensity z ( t ) , when time t tends to infinity:
758
z (∞) = lim z (t )
759
t → ∞
The asymptotic failure intensity z ( ∞ ) is given by the equation 1 MUT + MTTR
z (∞) = lim z (t ) =
760
t → ∞
761 762 763
which, under appropriate assumptions on f U( t ) and g R ( t ) , follows from the renewal density theorem (see [5], page 248, [9], pages 184 and 461, [10], page 196 and [12], page 367). See Note 2.
764
NOTE 1 Using the elementary renewal theorem (see [9], page 461):
765 766
lim
t →∞
t
=
1 MUT + MTTR
but Z (t ) =
767 768
Z (t )
t
∫0 z ( s) d s
hence, if z (∞ ) exists, then z (∞) = lim
769
t →∞
Z (t ) t
770
(see [7], page 191).
771 772
NOTE 2 The easiest to check conditions for the existence of z ( ∞ ) are the following: MUT < ∞ , MTTR < ∞ , at least one of f U ( t ) or g R ( t ) is a bounded function on [0, + ∞ ) tending to 0 as t → + ∞ .
773 774
NOTE 3 If observed failure data are available for n repairable items, from a homogenous population, and the time t is large enough, then an estimate of z ( ∞ ) is given by ˆ(∞) = z ˆ(t ) = z
775
nF (t , t + ∆t ) n∆t
776
where
777
n F ( t , t + ∆ t ) is the number of failures observed during the time interval ( t , t + ∆ t ).
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 31 778 779
For small values of ∆ t and large values of t , z ( ∞ ) ⋅∆ t is approximately equal to the (unconditional) probability that a failure of the item occurs during ( t , t + ∆ t ).
780
When the up times are exponentially distributed, then (see [4], page 317) z ( ∞ ) = Aλ U
781 782
where A is the asymptotic availability.
783
When the item operates continuously, f U ( t ) = f ( t ) and λ U = λ .
784 785
b) When the up times and times to restoration are exponentially distributed, then (see 6.3.3b), z (∞) = lim z (t ) =
786
t → ∞
λ U µ R
=
λ U + µ R
1 1 λ U
787 788
791
µ R
c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − , 1
z (∞) =
6.3.5
1
20 = 1,666 667 year −1 12
Mean failure intensity [191-45-09] (Symbol z (t 1, t 2 ), 0 ≤ t 1 < t 2 )
792
The expressions in this sub-clause also apply to IOIs.
793
a) By definition z (t 1, t 2 ) =
794
795
1
For a COI, λ U equals λ .
789
790
+
The integral
1
∫
t 2
t 2 − t 1 t 1
z (t ) dt
t 2
∫t z (t ) dt is equal to the expected number of failures of the item in the time 1
796 797 798 799
interval ( t 1 , t 2 ). Hence z (t 1, t 2 ) may be interpreted as the expected number of failures per time-unit in ( t 1 , t 2 ). NOTE
If observed failure data are available for n repairable items, from a homogenous population, then an
estimate of z (t 1, t 2 ) is given by ˆ n (t , t ) z (t 1, t 2 ) = F 1 2 (t 2 − t 1) n
800 801 802
where n F ( t 1, t 2 ) is the number o f failures observed during the time interval ( t 1 , t 2), where the time scale includes both up and down times.
803
By setting t 1 = t and t 2 = t + x , the asymptotic mean failure intensity may be obtained:
lim z (t , t + x ) =
804
t → ∞
1 MUT + MTTR
805
which, for large values of t , can be used as an approximation of z (t , t + x ) . This equality is
806
a direct result of Blackwell’s theorem (see [9], page 462).
807
When the up times are exponentially distributed (see [4], page 317),
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 32 z (t 1, t 2 ) = A(t 1, t 2 ) λ U
808 809 810 811
When the item operates continuously, λ U = λ . b) When the up times and times to restoration are exponentially distributed, then (see [14], page 200 and [18], page 66) λ U µ R
812
z (t 1, t 2 ) =
813
For a COI, λ U = λ .
814
λ U + µ R
z 0,
= A(t 1, t 2 )λ U
1 20 4 + = 4 12 144
exp( −12 × 0) − exp − 12 ×
1 4
1 −0 4
1
= 1,772 246 year −1
1 1 1 3 3 , = 1,671923 year −1; z , = 1,666 928 year −1; z , 1 = 1,666 680 year −1 4 2 2 4 4
lim z t , t +
t → ∞
6.3.6
819
1 1 = 1,666 667 year −1 = 4 0,5 + 0,1
Mean (operating) time to failure [191-45-11] MTTF (abbreviation)
a)
MTTF =
∞
∫0
tf (t ) dt =
∞
∫0 R(t ) dt
where R ( t ) is the reliability function of the item R(t ) =
822 823 824
t 2 − t 1
z
817
821
(λ U + µ R )
1
816
820
exp [ −(λ U + µ R )t 1] − exp [ −(λ U + µ R )t 2 ] 2
c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year −
815
818
+
2 λ U
∞
∫t
f ( s ) d s
NOTE When observed (operating) times to failure of n items, from a homogenous population, are available, then an estimate of MTTF is given by n
∑ (operating time) i
∧
total operating time = i =1 MTTF = k O
825
k O
826
where
827
total operating time is the aggregate operating time of all n items during a given time period;
828
k O is the total number of failures of the items while operating during the given time period;
829
(operating time) i is the total operating time of the i th item during the given time period.
830
b) When the times to failure are exponentially distributed,
831 832
MTTF =
1 λ
c) For a COI with a failure rate of λ = 2 year − : 1
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 33 833
834
6.3.7
835 836
Mean operating time between failures [191-45-13] MOTBF (abbreviation)
a) On the basis of the assumptions in 5.3,
837 838 839
1 = 0,5 years = 4 380 h 2
MTTF =
MOTBF = MTTF (See 6.3.6.) b) If the times to failure are exponentially distributed, then
840 841
MOTBF =
844
λ
c) For a continuously operating item with a failure rate of λ = 2 year − , 1
842
843
1
MOTBF =
6.3.8
1 = 0,5 years = 4 380 h 2
Instantaneous availability [191-48-01] (Symbol A ( t ))
845
The expressions in this sub-clause also apply to IOIs.
846 847
a) The instantaneous availability of a repairable item with non-zero time to restoration at an instant of time t may be written as (see [13], page 111, [9], page 175 and [5], page 101): A(t ) = 1 − F U (t ) +
848
t
∫0 (1 − F (t − x))ν ( x) d x U
849
where
850
F U( t ) is the up time distribution function of the item: F U (t ) =
851
t
∫0 f ( s) d s U
852
which is equal to the probability that a given up time will less than or equal to t ;
853
v ( t ) is the instantaneous restoration intensity of the item.
854 855
NOTE 1 The instantaneous availability, A ( t ) , is equal to the probability that the item is in an up state at the instant of time t and is given by the following integral equation (see [9], page 176): A(t ) = 1 − F U (t ) +
856 857
which may be solved by numerical methods,
858
where f U+R (t ) =
859
t
∫0 f
U+R ( s ) A(t − s) d s
t
∫0 g (t − s) f ( s) d s R
U
860
is the probability density function of the sum of the up time and the corresponding time to restoration.
861 862
NOTE 2 If observed up-state data are available for n repairable items, from a homogenous population, then an estimate of A (t ) is given by n {t } ˆ A(t ) = U n
863 864
where n U {t } is the number of items which are in an up state at the instant of time t .
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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61703/Ed2/CD © IEC (E) 34 865 866 867 868
If the item operates continuously, then R U( t ) = R ( t ) and f U( t ) = f ( t ). b) When the up times and times to restoration are exponentially distributed, then using either Markov techniques or the Laplace transform, the following is obtained (see [9], page 181, [14], page 200 and [18], page 66): A(t ) =
869 870 871 872
λ U λ U + µ R
exp[− (λ U + µ R )t ]
c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − , and for arbitrarily chosen values of t = 0, ¼,½,¾ and 1: 1
1
2 1 1 10 exp − 12 × = 0,841 631 + = 4 4 12 12
A(0) = 1; A
1 3 = 0,833 746; A = 0,833 54; A (1) = 0,833 334 2 4
A
874
A = A ( ∞ ) =
875
877
λU + µ R
+
If the item operates continuously, λ U = λ .
873
876
μR
6.3.9
10 = 0,833 333 12
Instantaneous unavailability [191-48-04] (Symbol U ( t ))
878
The expressions in this sub-clause also apply to IOIs.
879 880
a) The instantaneous unavailability of a repairable item with non-zero time to restoration at an instant of time t may be written as (see [5], page 107]]): t
∫0 (1− G
U ( t ) = 1 − A ( t ) =
881
R (t − x )) z ( x ) d x
882
where
883
z ( t ) is the instantaneous failure intensity of the item;
884
G R( t ) is the distribution function of the times to restoration of the item GR (t ) =
885
t
∫0 g ( s) d s R
886 887
which is equal to the probability that a restoration of the item is completed by time t . In this formula, g R( t ) is the probability density function of the times to restoration of the item.
888 889
NOTE 1 The instantaneous unavailability, U ( t ) , is equal to the probability that the item is in a down state at the instant of time t .
890 891
NOTE 2 If observed down-state data are available for n repairable items, from a homogenous population, then an estimate of U ( t ) is given by ˆ (t ) = nD {t } U n
892 893 894 895
where n D {t } is the number of items which are in a down state at the instant of time t .
b) When the up times and times to restoration are exponentially distributed, then (see [14], page 200 and [18], page 66),
896
U (t ) =
λ U λ U + µ R
(1 − exp[ − (λ U + µ R )t ])
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 35 897 898 899
If the item operates continuously, λ U = λ . c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − , and for arbitrarily chosen values of t = 0, ¼,½,¾ and 1: 1
1 1 2 1 − exp − 12 × = 0,158 369 = 4 4 12
U (0) = 0; U
900
1 3 = 0,166 254; U = 0,166 646; U (1) = 0,166 666 2 4
U
901
2 = 0,166 667 12
U = U ( ∞ ) =
902
903
1
6.3.10 Mean availability [191-48-05] (Symbol A (t 1, t 2 ), 0 ≤ t 1 < t 2 )
904 905
The expressions in this sub-clause also apply to IOIs.
906
a)
907
A(t 1, t 2 ) =
The integral
∫
t 2
1
∫
t 2
t 2 − t 1 t 1
A(t ) dt
A(t )dt is equal to the expected up time accumulated in the time interval
t 1
908
( t 1 , t 2 ), hence A (t 1, t 2 ) gives the expected fraction of the time interva l ( t 1 , t 2 ) that the item is
909
in an up state.
910 911
It then follows that the mean availability, A(t 1, t 2 ), and the mean accumulated up time, MAUT, in the time interval ( t 1 , t 2 ) are related as
MAUT =
912
t 2
∫t A(t ) dt = A(t 1, t 2 ) × (t 2 − t 1) 1
913 914 915
NOTE
If observed up times in the interval (t 1 , t 2 ) are available for n repairable items, from a homogenous
population, then an estimate of A(t 1, t 2 ) is given by n
∑
(up time) i ˆ total up time = i =1 A(t 1, t 2 ) = (t 2 − t 1 ) n (t 2 − t 1 ) n
916 917
where
918
total up time is the aggregate up time of all n items during the time interval (t 1 , t 2);
919 920
(up time ) i is the total up time of the i t h item during the time interval (t 1 , t 2 ). An est ima te of the me an acc umu lat ed up tim e, MA UT, in the ti me int erv al (t 1, t 2 ) is given by n
∧
MAUT =
921
total up time n
∑ (up time) i
= i =1
n
922 923
where
924
(up time)i is the total up time of the i th item during the time interval (t 1, t 2 ).
925
Following from the assumptions given in 5.3, the asymptotic mean availability
926
is equal to the asymptotic availability A (see [7], page 191):
total up time is the aggregate up time of all n items during the time interval (t 1 , t 2);
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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lim A (t 1, t 2 )
t 2 → ∞
E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 36
lim A(t 1, t 2 ) = A =
927 928 929
t 2 → ∞
MUT MUT + MTTR
b) When the up times and times to restoration are exponentially distributed, then integrating A ( t ) over the time interval ( t 1 , t 2 ) gives: MAUT =
930
λ {exp [ −(λ U + µ R )t 1 ] − exp [ − (λ U + µ R )t 2 ]} (t 2 − t 1)µ R + U λ U + µ R (λ U + µ R ) 2
Dividing this expression by t 2 − t 1 yields:
931 932
A (t 1, t 2 ) =
=
933
= 934 935 936
µ R λ U + µ R
+
exp [ − (λ U + µ R )t 1 ] − exp [ − (λ U + µ R )t 2 ] t 2 − t 1
λ U
(λ U + µ R ) 2
z (t 1, t 2 )
λ U
If the item operates continuously, λ U = λ . c) For a continuously operating item with a failure rate of λ = 2 year − and a restoration rate 1 of µ R = 10 year − , then 1
A 0,
937
1 10 2 + = 4 12 144
exp(−12 × 0) − exp − 12 × 1 −0 4
= 0,886123
1 1 1 3 3 , = 0,835 962; A , = 0,833 464; A , 1 = 0,833 340 4 2 2 4 4
939
The mean accumulated up time, MAUT, during the first year may be calculated as follows:
MAUT = A(0, 1) × 1 = A 0,
940
941
1 1 1 1 3 3 + A , + A , + A , 1 / 4 4 4 2 2 4 4
= 3,388 889/4 = 0,847 222 years = 7 421,67 h 6.3.11 Mean unavailability [191-48-06] (Symbol U (t 1, t 2 ), 0 ≤ t 1 < t 2 )
943 944
The expressions in this sub-clause also apply to IOIs.
945
a)
946
1 4
A
938
942
MAUT t 2 − t 1
U (t 1, t 2 ) =
The integral
∫
t 2
1
∫
t 2
t 2 − t 1 t 1
U (t ) dt = 1 − A (t 1, t 2 )
U (t ) dt is equal to the expected down time accumulated in the time interval
t 1
948
( t 1 , t 2 ). Hence, U (t 1, t 2 ) gives the expected fraction of the time interval ( t 1 , t 2 ) spent in the down state.
949 950
It then follows that the mean unavailability, U (t 1, t 2 ), and the mean accumulated down time, MADT, in the time interval ( t 1 , t 2 ) are related as
947
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E DIN EN 61703:2013-11
61703/Ed2/CD © IEC (E) 37
MADT =
951
t 2
∫t U (t ) dt = U (t 1, t 2 ) × (t 2 − t 1) 1
952 953
NOTE
If observed down times in the interval (t 1 , t 2 ) are available for n repairable items, from a homogenous
population, then an estimate of U (t 1, t 2 ) is given by n
∑(down time)i
ˆ total down time = i =1 U (t 1, t 2 ) = (t 2 − t 1) n (t 2 − t 1) n
954 955
An est ima te of the me an acc umu lat ed do wn tim e, MA DT, in th e t ime int erv al (t 1 , t 2 ) is given by n
∧
MADT =
956
total down time n
∑ (down time) i
= i =1
n
957
where
958
total down time is the aggregate down time of all n items during the time interval (t 1 , t 2 );
959
(down time) i is the total down time of the i th item during the time interval ( t 1, t 2 ).
960 961 962 963 964
Following from the assumptions given in 5.3, the asymptotic mean unavailability ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
is equal to the asymptotic unavailability U (see [7], page 191):
lim U (t 1, t 2 ) = U =
t 2 → ∞
MADT =
=
967
MADT t 2 − t 1 λ U λ U + µ R
= 1−
969
−
λ U
(λ U + µ R ) 2
exp [ − (λ U + µ R )t 1 ] − exp [ − (λ U + µ R )t 2 ] t 2 − t 1
z (t 1, t 2 )
λ U
If the item operates continuously, λ U = λ . c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − 1
2 2 1 U 0, = − 4 12 144
970
exp (−12 × 0) − exp − 12 × 1 −0 4
1
1
4
= 0,113 877
1 1 1 3 3 , = 0,164 038; U , = 0,166 536; U , 1 = 0,166 660 4 2 2 4 4
U
971 972 973
λ {exp [ −(λ U + µ R )t 1] − exp [ − (λ U + µ R )t 2 ]} (t 2 − t 1)λ U − U λ U + µ R (λ U + µ R ) 2
Dividing this expression by t 2 − t 1 yields: U (t 1, t 2 ) =
968
MTTR MUT + MTTR
b) When the up times and times to restoration are exponentially distributed, then integrating U ( t ) over the time interval ( t 1 , t 2 ) gives:
965
966
lim U (t 1, t 2 )
t 2 → ∞
The mean accumulated down time, MADT, during the first year may be calculated as follows:
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1 1 1 1 3 3 + U , + U , + U , 1 / 4 4 4 2 2 4 4
MADT = U (0, 1) × 1 = U 0,
974
975
= 0,611 111/4 = 0,152 778 years = 1 338,34 h
976
6.3.12 Asymptotic availability [191-48-07]
977
(Symbol A )
978
The expressions in this sub-clause also apply to IOIs.
979
a)
A = lim A(t ) = t → ∞
980
If the item operates continuously, then
981
A =
982
MTTF MTTF + MTTR
b) When the up times and times to restoration are exponentially distributed, then
983
A =
984
If the item operates continuously, λ U = λ .
991
6.3.13 Asymptotic unavailability [191-48-08] ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
(Symbol U ) The expressions in this sub-clause also apply to IOIs. U = lim U (t ) =
a)
t → ∞
MTTR = 1 − A MUT + MTTR
If the item operates continuously, then U =
992 993
MTTR MTTF + MTTR
b) When the up times and times to restoration are exponentially distributed, then
994
U =
995
If the item operates continuously, λ U = λ .
996
1
10 = 0,833 333 12
A =
987
990
λ U + µ R
1
986
989
µ R
c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year −
985
988
MUT MUT + MTTR
λ U λ U + µ R
c) For a COI with a failure rate of λ = 2 year − and a restoration rate of µ R = 10 year − , then:
997
1
U =
1
2 = 0,166 667 12
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6.3.14 mean up time [191-48-09]
999
MUT (abbreviation)
1000
The expressions in this sub-clause also apply to IOIs.
1001
a)
MUT =
∞
∫0
tf U (t ) d t =
∫
∞
0
(1 − RU (t )) d t
1002
where
1003
f U( t ) is the probability density function of the up times of the item;
1004
F U( t ) is the up time distribution function of the item.
1005 1006
NOTE 1 When the item operates continuously, then, according to the assumption 5.3g) (i.e. no preventive maintenance):
1007
MUT = MTTF
1008 1009
However, when function-preventing preventive maintenance is permitted, the relation bet ween MUT and MTTF is more complex, and usually, MUT < MTTF.
1010 1011
NOTE 2 If observed up times are available for n repairable items, from a homogenous population, then an estimate of MUT is given by n
∧
MUT =
1012
total up time k U
∑ (up time) i
= i =1
k U
1013 1014
where
1015
k U is the total number of up times of the items during given period of observation;
1016
(up time)i is the total up time of the i th item during the given period of observation.
1017
EXAMPLE
1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029
Consider an item belonging to IOIs, which operates as described below. The item starts operating at time t = 0 and is required to operate (be in operating state) for fixed time interval [0, X ), X > 0 during which the item can fail with a constant failure rate λ > 0. If the item did not fail in this interval, it becomes not required for subsequent fixed time interval [ X , X + Y ), Y > 0, during which the item cannot fail (i.e. the item is in a failure free idle state during this interval). At time t = X + Y the next required time starts and the functioning process repeats as from the initial time t = 0, independently of previous history of the item functioning process. When the item fail during required time [0, X ) , it is repaired to the state as goods as new and then the functioning process repeats again as at the initial time t = 0, independently of the previous history of the item functioning process. Repair times are mutually independent and also independent of the previous history of the item functioning process. See figure 5 below.
total up time is the aggregate up time of all n items during given period of observation;
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61703/Ed2/CD © IEC (E) 40 State X
X
X
X
Operating state
e t a t s p U
Y
Y
Y
Idle state
Down state t = 0
X + Y
X
2 X + Y
2 X +2 Y
Failure
1030 1031
It is clear from the above description that consecutive up times are statistically independent and identically distributed positive, continuous random variables. Therefore, to calculate MUT, the up time to first failure of the item can be considered. The up-time hazard rate function λ U (t ) is depicted in figure 6. λ U ( t )
1037 1038
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
X
X
λ
1043 1044
Y
0 t = 0
X + Y
X
2 X + Y
2 X +2 Y
3 X +2 Y
3 X +3 Y
4 X +3 Y
Time
Figure 6 – Plot of the up-time hazard rate function λ U (t )
The analytical form of λ U (t ) is:
λ for n ⋅ ( X + Y ) ≤ t < n ⋅ ( X + Y ) + X , n = 0, 1, 2, ... λ U (t ) = 0 for n ⋅ ( X + Y ) + X ≤ t < (n + 1) ⋅ ( X + Y ) Since
1 − F U (t ) = exp −
t
∫0 λ U ( x) d x ,
then by integrating λ U (t ) , the above formula can be written as:
exp( −λ ⋅ (t − n ⋅ Y )) for n ⋅ ( X + Y ) ≤ t < n ⋅ ( X + Y ) + X 1 − F U (t ) = , n = 0, 1, 2, ... exp( −λ ⋅ (n + 1) ⋅ X ) for n ⋅ ( X + Y ) + X ≤ t < (n + 1) ⋅ ( X + Y ) By integrating 1 − F U (t ) over [0, ∞ ), MUT of the item is obtained:
MUT =
1045 1046 1047 1048
X
Y
1041 1042
X
Y
1039 1040
Restoration, the item continues functioning as at t = 0
Figure 5 – Sample realization of the item state
1032 1033 1034 1035
1036
Time
3 X +2 Y
1 λ
+ Y ⋅
exp( −λ ⋅ X ) exp( −λ ⋅ X ) = MOTBF + Y ⋅ 1 − exp( −λ ⋅ X ) 1 − exp( −λ ⋅ X )
The second term of the above formula is equal to the expected accumulated idle time to 1 failure of the item. For λ = 0,01 h − and X = 10 h, we obtain the following values of MUT for some values of Y :
1049
MUT = 195 h for Y = 10 h,
MUT = 290 h for Y = 20 h,
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MUT = 575 h for Y = 50 h,
1051
whereas:
1052 1053
MOTBF = 100 h. b) When the up times are exponentially distributed
MUT =
1054 1055
MUT = 1 051 h for Y = 100 h,
c) For a repairable item with λ U = 2 year −
1056
MUT =
1057
6.3.15 Mean down time [191-48-10]
1058
1 λ U
1
1 = 0,5 years = 4 380 h 2
MDT (abbreviation)
1059
The expressions in this sub-clause also apply to IOIs.
1060
a)
MDT =
∞
∫0 t g (t ) dt D
1061
where g D (t ) is the probability density function of the down times of the item (which are
1062 1063
defined to include the item’s restoration times, after failure and/or function-preventing preventive maintenance times), i.e., for small values of ∆ t , g D (t ) ⋅ ∆t is approximately equal
1064 1065
to the probability that the item returns to its up state from its down state in the time interval ( t , t + ∆ t ) , assuming that the down time started at time t = 0.
1066 1067
NOTE If observed down times are available for n repairable items, from a homogenous population, then an estimate of MDT is given by n
∧
MDT =
1068
total down time k D
∑ (down time) i
= i =1
k D
1069
where
1070
total down time is the aggregate down time of all n items during a given time period;
1071
k D is the total number of the down times of the items in the given time period;
1072
(down times) i is the total down time of the i th item during the given time period.
1073
b) If the down times are exponentially distributed with a parameter µ D , i.e. g D (t ) = µ D exp( − µ D t )
1074 1075
then
MDT =
1076 1077 1078
NOTE According to the assumptions in 5.3 (any fault is the result of a failure, and no preventive maintenance), any down time is equal to the time to restoration, i.e.
1079 1080
MDT = MTTR and, for exponentially distributed down times
1081 1082
1 µ D
µ D = µ R
c) For a repairable item with µ D = 100 year −
1
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MDT =
1084
6.3.16 Maintainability [191-47-01]
1085
(Symbol M ( t 1 , t 2 ), 0 ≤ t 1 < t 2 )
1 = 0,01 years = 87,6 h 100
1086
The expressions in this sub-clause also apply to IOIs.
1087 1088
a) The probability that a given maintenance action on an item can be completed in the time interval ( t 1 , t 2 ), assuming that the maintenance action started at time t = 0, is given by M (t 1, t 2 ) =
1089
t 2
∫t g 1
MA (t ) dt
1090
where g MA (t ) is the probability density function of the time to complete a given
1091
maintenance action for the item, i.e., for small values of ∆ t , g MA (t ) ⋅ ∆t is approximately
1092 1093
equal to the probability of completing a given maintenance action during the time interval ( t , t + ∆ t ) , assuming that the maintenance action started at time t = 0.
1094
In practical applications, the maintainability function, M ( t ) , defined as
1095
1096 1097 1098 1099
M ( t ) = M ( 0, t ) =
MA ( x ) d x
is used, with M ( 0) = 0. This is equal to the probability that a given maintenance action will be completed by time t , assuming that the action started at time t = 0, i.e. M ( t ) is the distribution function of that time. The maintainability, M ( t 1 , t 2 ), and the maintainability function, M ( t ) , are related as follows: M ( t 1 , t 2 ) = M ( t 2 ) − M ( t 1 )
1100 1101 1102
t
∫0 g
NOTE 1 follows:
The maintainability function, M ( t ) , and the mean maintenance action time, MMAT, are related as
MMAT =
1103
∞
∞
∫ 0 (1 − M (t )) dt = ∫ 0 t g
MA (t ) dt
1104 1105
NOTE 2
1106 1107
where G ACM ( t ) is the distribution function of the active corrective maintenance time and MACMT is the mean active corrective maintenance time.
1108 1109
If the repair is considered as the maintenance action, then
1110
where G( t ) is the distribution function of the repair time and MRT is the mean repair time.
1111
Similarly, when the maintenance action consists of all actions attributed to the time to restoration, then
If the maintenance action time is equal to the entire active corrective maintenance time, then: M ( t ) = G ACM (t ) , MMAT = MACMT
M (t ) = G (t ) , MMAT = MRT
1112
M ( t ) = G R ( t ) , MMAT = MTTR
1113
where GR ( t ) is the distribution function of the time to restoration and MTTR is the mean time to restoration.
1114 1115
NOTE 3 If observed data for m maintenence action times of given type are available, from a homogenous population, the estimated value of M (t ) is given by: m − mMAT (t ) ˆ M (t ) = m
1116 1117 1118
where mMAT (t ) is the number of maintenance action times with duration greater that t , i.e. not finished up to
1119
b) If a given maintenance action times are exponentially distributed with a parameter µ MA , i.e.
1120
g MA ( t ) = µ MA exp( − µ MA t )
1121
time t , and mMAT (0) = m.
then
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M ( t 1 , t 2 ) = exp( − µ MA t 1 ) − exp( − µ MA t 2 )
1123
M ( t ) = 1 − exp( − µ MA t )
1124
and
1125 1126 1127
MMAT =
c) For a repairable item with µ MA = 1 000 year − , and t 2 − t 1 = 16 h, then [simplify by using hours, rather than years, as the unit] 1
M ( 16) = M ( 0, 16) = 1 − exp( − 16 ×
1128 1129
1134
M ( 2, 18) = exp( − 2 ×
1 000 1 000 ) − exp( − 18 × ) = 0,667 758 8 760 8 760
M ( 4, 20) = exp( − 4 ×
1 000
and
1132
1133
1 000 ) = 0,839 021[too many sig. figures] 8 760
also:
1130 1131
1 µ MA
8 760
) − exp( − 20 ×
1 000 8 760
) = 0,531 453
6.3.17 Instantaneous repair rate [191-47-20] (Symbol µ ( t ))
1135
The expressions in this sub-clause also apply to IOIs.
1136
a) By definition,
1 G(t + ∆t ) − G(t ) g (t ) = 1 − G(t ) 1 − G(t ) ∆t → 0 ∆t
µ (t ) = lim
1137 1138
where
1139 1140 1141 1142
g ( t ) is the probability density function of the repair time of an item (excluding technical, logistic and administrative delays), i.e., for small values of ∆ t , g ( t ) ⋅∆ t is approximately equal to the probability that the repair started at time t = 0 will be completed during ( t , t + ∆ t );
1143 1144
G ( t ) is the distribution function of the repair time of the item, i.e. G ( t ) is the probability that the repair, started at time t = 0, with G (0) = 0, will be completed by time t :
G(t ) = 1 − exp −
1145
t
t
∫ 0 µ ( x) d x = ∫ 0 g ( x) d x
1146 1147 1148
For small values of ∆ t , µ ( t ) ⋅∆ t is approximately equal to the conditional probability that the repair will be completed in the time interval ( t , t + ∆ t ) , given that the repair started at time t = 0 and has not been completed by the instant of time t .
1149 1150
NOTE If observed repair data are available for n repairable items, from a homogenous population, the estimated value of µ (t ) at time t is given by ˆ (t ) = nR (t ) − nR (t + ∆t ) µ nR (t )∆t
1151 1152 1153
where
1154
n R (t ) − n R ( t + ∆ t ) is the number of items with repair completed in the time interval (t , t + ∆ t ) .
1155
It should be noted that the estimated value of the repair density function g ( t ) , at time t , is given by
n R (t ) is the number of items that are still under repair at the instant of time t ( n R (0) = n );
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1157 1158 1159 1160 1161 1162 1163 1164
nR (t ) − nR (t + ∆t )
ˆ (t ) = g
1156
n∆t
b) When the time to failure is exponentially distributed, g ( t ) = µ exp( − µ t )
and G ( t ) = 1 − exp( − µ t )
hence µ ( t ) = µ
for all values of t . In this case:
1165
1 µ
MRT =
1166
where MRT is the mean repair time.
1167 1168
NOTE If observed repair data are available for n repairable items, from a homogenous population, with constant repair rate, then the estimated v alue of µ is given by n
ˆ= µ
1169
n
∑RTi i =1
1170
where
1171
RT i is the repair time of item i .
1172 1173
c) For 10 repairable items, from a homogenous population, with a constant repair rate, the
ˆ = µ
1174 1175 1176
g (t ) =
10 1 = 2 h− 5
1 t σ 2π
exp −
(ln t − m) 2 2σ 2
and G(t ) =
1179 1180
10
If the repair time of a repairable item has a lognormal distribution with scale parameter m and shape parameter σ > 0, then (see Table B.2)
1177 1178
∑ i =1RTi = 5 hours. Hence
observed total repair time of all the items is
t
∫0
g ( x ) d x
hence µ (t ) =
1181
g (t )
1 − G(t )
1182
EXAMPLE
1183 1184 1185 1186
Assuming that an item has a mean repair time (M RT) of 1,5 hour and re pair time var iance 2 (VRT) of 0,16 hour , in order to compute the repair rate, the parameters m and σ of the lognormal distribution of repair times have first to be determined. According to the results given in Table B.2,
1187
1188
MRT = exp m +
σ 2
2
VRT = exp (2m + 2σ 2 ) − exp (2m + σ 2 ) = MRT 2 ⋅ (exp (σ 2 ) − 1)
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Solving the above equations yields:
1 MRT 4 , m = ln 2 VRT + MRT 2
1190
VRT + MRT 2 , σ = ln 2 MRT 2
2
1191
giving m = 0,371 117 and σ = 0,068 697, and hence the repair rates are:
0,000 881 1 = 0,000 881 h − 0,999 912
1192
µ ( 1 hour) =
1193
µ ( 1,5 hours) =
1194
µ ( 2 hours) =
1195
0,015 735 1 = 0,015 784 h − 0,996 894 0,065 206 1 = 0,066 662 h − 0,978 159
6.3.18 Mean repair time [191-47-21]
1196
MRT (abbreviation)
1197
The expressions in this sub-clause also apply to IOIs.
1198
a)
MRT =
∞
∞
∫0 t g (t ) dt = ∫0 (1 − G(t )) dt
1199 1200
where g ( t ) is the probability density function of the repair time of an item, and G ( t ) is the distribution function of the repair time of the item.
1201
NOTE 1
From the definition of the repair time MRT = MACMT − MTD
1202 1203
where
1204
MTD
1205
MACMT is the mean active corrective maintenance time.
1206 1207
NOTE 2 If observed repair times are available for n repairable items, from a homogenous population, then an estimate of MRT is given by
is the mean technical delay;
n
∧
MRT =
1208
∑ (repair time) i
total repair time = i =1 k
k
1209 1210
where
1211
k is the total number of repair times of the items during the given time period;
1212
(repair time) i is the total repair time of the i th item during the given time period.
total repair time is the aggregate repair time of all n items during a given time period;
1213
b) If the repair times are exponentially distributed with a parameter µ , i.e. g (t ) = µ exp( − µ t )
1214 1215
then
1216
MRT =
1 µ
c) For a repairable item with µ = 1 000 year − : 1
1217 1218
MRT =
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , ,
1 = 0,001 years = 8,76 h 1 000
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6.3.19 Mean active corrective maintenance time [191-47-22]
1220
MACMT (abbreviation)
1221
The expressions in this sub-clause also apply to IOIs.
1222
a)
∞
∫0
MACMT =
(1 − G ACM (t )) dt =
∞
∫0 t g
ACM (t ) dt
1223
where
1224
g ACM (t ) is the probability density function of the active corrective maintenance times of an
1225 1226
item (including technical delay and repair time, but excluding logistic and administrative delays), i.e., for small values of ∆ t , g ACM (t ) ⋅ ∆t is approximately equal to the probability
1227 1228
that the active corrective maintenance of the item is completed in the time interval ( t , t + ∆ t ) , assuming that the active corrective maintenance started at time t = 0;
1229 1230 1231
G AC M( t ) is the distribution function of the active corrective maintenance time of the item, i.e. G ACM ( t ) is the probability that the active corrective maintenance, started at time t = 0, will be completed by time t : G ACM (t ) =
1232
t
∫0 g ACM ( x) d x
1233
NOTE 1 By definition of the active corrective maintenance time:
1234
MACMT = MRT + MTD
1235
where MTD is the mean technical delay.
1236 1237
NOTE 2 If observed active corrective maintenance times are available for n repairable items, from a homogenous population, then an estimate of MACMT is given by n
∧
MACMT =
1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248
total active corrective maintenance time k ACM
∑ (active corrective maintenance time)i =
i =1
k ACM
where total active corrective maintenance time is the aggregate active corrective maintenance time of all n items during a given time period; ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
k ACM is the total number of active corrective maintenance actions on the items during the given time p eriod;
(active corrective maintenance time)i is the total active corrective maintenance time of the i th item during the given time period.
b) If the active corrective maintenance times are exponentially distributed with parameter µ AC M, i.e. g ACM (t ) = µ ACM exp( − µ ACMt )
then
MACMT =
1249
1 µ ACM
1250 1251
c) For a repairable item with the mean technical delay MTD = 5 h and the mean repair time MRT = 9 h:
1252
MACMT = 5 + 9 = 14 h
1253 1254 1255
6.3.20 Mean time to restoration [191-47-23] MTTR (abbreviation) The expressions in this sub-clause also apply to IOIs.
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a)
MTTR =
∞
∫0 t g (t ) dt R
1257
where g R (t ) is the probability density function of the times to restoration of the item, i.e.,
1258
for small values of ∆ t , g R (t ) ⋅ ∆t is approximately equal to the probability that the item is
1259 1260
restored from a fault to an up state in the time interval ( t , t + ∆ t ) , assuming that a failure resulting in a fault occurred at time t = 0.
1261 1262
NOTE 1 The mean time to restoration (of a faulty item), MTTR, may be written as the sum of the expected values of its constituent times:
1263
MTTR = MFDT + MAD + MLD + MACMT
1264
= MFDT + MAD + MLD + MTD + MRT
1265
where
1266
MFDT is the mean fault d etection time;
1267
MAD is the mean administrative delay;
1268
MLD is the mean logistic delay;
1269
MACMT is the mean active corrective maintenance time given by
1270
MACMT = MTD + MRT
1271
where
1272
MTD is the mean technical delay;
1273
MRT is the mean repair time.
1274 1275
NOTE 2 If observed times to restoration are available for n repairable items, from a homogeneous population, then an estimate of MTTR is given by n
∧
MTTR =
1276 1277
1280 1281 1282 1283
i =1
k R
k R is the total number of times to restoration of the items during the given time period;
(time to restoration) i is the total time to restoration of the i th item during the given time period.
b) If the times to restoration are exponentially distributed, i.e. g R (t ) = µ R exp( − µ R t )
where µ R is the constant restoration rate, then:
MTTR =
1 µ R
c) For a repairable item with a restoration rate of µ R = 100 year −
1286
1287
=
total time to restoration is the aggregate time to restoration of all n items during a given time period; ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
1284 1285
k R
∑ (time to restoration)i
where
1278 1279
total time to restoration
MTTR =
1
1 = 0,01 years = 87,6 h 100
6.3.21 Mean administrative delay [191-47-26]
1288
MAD (abbreviation)
1289
The expressions in this sub-clause also apply to IOIs.
1290
a)
MAD =
∞
∫0 t g
AD (t ) dt
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where g AD (t ) is the probability density function of the administrative delay during a time to
1292
restoration of a faulty item, i.e., for small values of ∆ t , g AD (t ) ⋅ ∆t is approximately equal to
1293 1294
the probability that the delay ends in the time interval ( t , t + ∆ t ) , assuming that the delay started at time t = 0.
1295 1296
NOTE If observed administrative delays are available for n repairable items, from a homogenous population, then an estimate of MAD is given by n
∧
MAD =
1297
total administrative delay k AD
∑ (administrative delay)i = i =1
k AD
1298 1299
where
1300
k AD is the total number of administrative delays during the given time period;
1301
(administrative delay)i is the total administrative delay of the i th item during the given time period.
1302
b) If the administrative delays are exponentially distributed with parameter µ AD , i.e.
total administrative delay is the aggregate administrative delay of all n items during the given time period;
g AD (t ) = µ AD exp( − µ AD t )
1303 1304
then
MAD =
1305
1 µ AD
c) For a repairable item with µ AD = 1 000 year − : 1
1306 1307
MAD =
1308
1 = 0,001 years = 8,76 h 1 000
6.3.22 Mean logistic delay [191-47-27]
1309
MLD (abbreviation)
1310
The expressions in this sub-clause also apply to IOIs.
1311
a)
1312 1313 1314 1315 1316 1317
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
MLD =
∞
∫0 t g
LD (t ) dt
where g LD (t ) is the probability density function of the logistic delay during a maintenance time of a faulty item, i.e., for small values of ∆ t , g LD (t ) ⋅ ∆t is approximately equal to the probability that the delay ends in the time interval ( t , t + ∆ t ) , assuming that the delay started at time t = 0. NOTE If observed logistic delays are available for n repairable items, from a homogenous population, then an estimate of MLD is given by n
∧
MLD =
1318
total logistic delay k LD
∑ (logistic delay)i = i =1
k LD
1319
where
1320
total logistic delay is the aggregate logistic delay of all n items during a given time period;
1321
k LD is the total number of logistic delays during the given time period;
1322
(logistic delay) i is the total logistic delay of the i th item during the given time period.
1323
b) If the logistic delays are exponentially distributed with parameter µ LD , i.e.
1324
g LD (t ) = µ LD exp( − µ LD t )
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61703/Ed2/CD © IEC (E) 49 1325
then
1326
MLD = c) For a repairable item with µ LD = 1 000 year −
1327 1328
MLD =
1 µ LD
1
1 = 0,001 years = 8,76 h 1 000
1329
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
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Annex A (informative)
1330 1331 1332 1333 1334
Performance aspects and descriptors Performance aspects
Random variables
Probabilistic descriptors
Modifiers
Ava ila bil it y
Tim e t o f ail ure
Dis tri bu tio n function
True
Time between failures
Probability density function
Predicted
Reliability
Number of failures in interval ( t 1, t 2 ) Reliability function Time to restoration
Maintenance support
Estimated
Preventive maintenance time
Extrapolated
Instantaneous Failure rate
Asymptotic
Renewal function Up time Maintainability
Renewal density
Down time Expectation Variance Standard deviation p -fractile
1335 1336 1337
NOTE A mathematical operation on a random variable results in a basic measure. The addition of a modifier to a basic measure results in a specific measure.
1338
Figure A.1 – Performance aspects and descriptors
1339
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Annex B (informative)
1340 1341 1342 1343
Summary of measures related to time to failure
1344 1345 1346
Table B.1 – Relations among functional measures of time to failure of continuously operating items Relation to other measures
Functional measure
F ( t )
f ( t )
R ( t )
F ( t )
–
d F (t ) dt
1 − F (t )
f ( t )
R ( t )
λ ( t )
t
∫0
f ( x) d x
−
1 − R (t )
1 − exp −
t
∫
–
∫0 λ ( x) d x
∞ t
t
1 1 − F (t )
∫ −
–
∫0 λ ( x) d x
exp −
d F (t ) dt
f (t ) f ( x ) d x
d R(t ) dt
λ (t ) exp −
λ ( t )
t
∫0 λ ( x) d x
∞ t
f ( x ) d x
1 R(t )
d R(t ) dt
–
NOTE Similar relationships hold among functional measures of any random variable, for example time to first failure, up time, down time, time to restoration, corrective maintenance time, repair time.
1347 1348
` , , ` , ` , , ` , , ` ` ` ` ` , , ` , , , ` , , , ` , , , , , , , ` ` , , ` ` -
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52
61703/Ed2/CD © IEC (E)
1349 1350 1351 1352 1353
Table B.2 – Summary of measures for some continuous probability distributions of time to failure of continuously operating items
Distribution
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
Exponential
Weibull
Gamma
Range
Probability density function f ( t )
Reliability (survival) function R ( t )
Failure rate ( t )
λ exp (−λ t )
exp ( −λ t )
λ
λ > 0 t ≥ 0
α > 0, β > 0
(
β α (α t ) β −1 exp − (α t ) β
t ≥ 0
α (α t ) β −1
α > 0, β > 0 t ≥ 0
Γ ( β )
λ (λ t ) k −1
k ∈ {1, 2, ...}
(k − 1) !
t ≥ 0
Rayleigh
Lognormal
exp ( −α t )
exp ( −λ t )
)
β α (α t ) β −1
kt exp −
t ≥ 0
(ln t − m)2 exp − 2σ 2 t σ 2 π
Γ ( x ) is the complete gamma function defined as Γ( x ) =
∞
∫t f (u) du
exp ( −λ t )
k −1
∑ i =0
k t 2 2
k ∈ {1, 2, ...}
−∞ < m < +∞ σ > 0, t > 0
(
exp − (α t ) β
1
(λ t ) i i!
k t 2 exp − 2
∞
∫t f (u) du
∞
∫0 t x −1e−t dt , x > 0.
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Variance
1
1
λ
λ 2
Γ (1 +
1 β
1 2 1 Γ (1 + ) − Γ 2 (1 + ) β β α 2
)
α
λ > 0
Erlang
)
Expected value MTTF
f (t )
β
β
R(t )
α
α 2
f (t )
k
k
R(t )
λ
λ
f (t )
2 π 1 − 4 k
π
kt
R(t )
2
2k
exp m +
σ 2
exp (2m + 2σ 2 )
− exp (2m + σ 2 )
2
E DIN EN 61703:2013-11
53
61703/Ed2/CD © IEC (E)
Annex C (informative)
1355 1356 1357 1358
Comparison of some dependability measures for continuously operating items
1359 1360 1361
Table C.1 – Comparison of some dependability measures of continuously operating items with constant failure rate and restoration rate R Measure
zero ( µ R → ∞ )
exp(−λ t )
exp(−λ t )
exp(−λ t )
exp(−λ t 2 )
exp(−λ ×( t 2 − t 1))
µ R λ exp[− (λ + µ R ) t 1] exp[−λ × (t 2 − t 1)] + λ + µ R λ + µ R
Reliability function R( t ) = R (0, t ) Reliability R( t 1, t 2 ) Mean time to failure MTTF Mean operating time between failures MOTBF
Asym pto tic fail ure int ensi ty z ( ∞ ) Mean failure intensity z (t 1, t 2 )
1
1
1
λ
λ
1
1
λ
λ
λ exp(−λ t )
λ
0
λ
exp(−λ t 1) − exp(−λ t 2 ) t 2 − t 1
λ
exp(−λ t )
1
Mean availability A(t 1, t 2 )
exp(−λ t 1) − exp(−λ t 2 ) λ × (t 2 − t 1)
1
Asym ptot ic avai lab ilit y A
0
1
1 − exp(−λ t )
0
Instantaneous availability A( t )
Instantaneous unavailability U ( t )
Mean unavailability U (t 1, t 2 )
1−
exp(−λ t 1) − exp(−λ t 2 ) λ × (t 2 − t 1)
0
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non-zero (0 < µ R <
λ –
Instantaneous failure intensity z ( t ) ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
Repairable item with time to restoration equal to
Non-repairable item ( µ R = 0)
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λµ R
+
λ + µ R
λ 2 λ + µ R
∞)
exp[− (λ + µ R )t ]
λµ R λ + µ R
λ µ R λ + µ R
+
λ 2
exp[− (λ + µ R )t 1] − exp[− (λ + µ R )t 2 ]
(λ + µ R )2
t 2 − t 1
µ R λ + µ R µ R λ + µ R
+
+
λ
(λ + µ R )2
λ
(λ + µ R )
exp[− (λ + µ R )t ]
exp[− (λ + µ R )t 1] − exp[− (λ + µ R )t 2 ] t 2 − t 1 µ R λ + µ R
λ λ + µ R λ λ + µ R
−
λ
(λ + µ R )2
(1 − exp[− (λ + µ R ) t ] ) exp− (λ + µ R )t 1] − exp[− (λ + µ R )t 2 ] t 2 − t 1
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Measure
Asym ptot ic unav aila bili ty U
Non-repairable item ( µ R = 0) 1
Repairable item with time to restoration equal to zero ( µ R → ∞) 0
non-zero (0 < µ R <
∞)
λ λ + µ R
1362
` , , ` , ` , , ` , , ` ` ` ` ` , , ` , , , ` , , , ` , , , , , , , ` ` , , ` ` -
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Annex D (informative)
1363 1364 1365 1366
Software dependability aspects
1367 1368 1369
Items containing software are prone to design failure due to the activation of a latent design fault resulting from human error during software development.
1370 1371
In test or operation, these latent faults may be activated in response to certain trigger circumstances.
1372 1373 1374
Such failures tend to be transient (i.e. the item becomes operable again if the trigger is removed) and systematic (i.e. the item will fail in a similar manner if th e trigger is encountered again or if deliberately reproduced).
1375 1376
Corrective maintenance of software requires a modification to remove the fault. This can usually be performed away from the installation after normal operation has been resumed.
1377 1378 1379
An imp ortant attribu te of ite ms containing sof tware is the ability to recover to an upstate following the failure of the item due to a software fault. This is measured by the time to recover, which comprises three aspects:
1380
a) restart time (for example time to reload the software);
1381 1382
b) recording time (for example time to take memory dumps, complete incident reports and record other evidence);
1383 1384
c) restoration time for the software (for example time to restore files to a clean state, reestablish communication links).
1385
Availability with regar d to softw are is a combina tio n of its reliability a nd the ability t o recover .
1386 1387
Some modes of failures will have a zero time to recovery, for example giving wrong results for a calculation, and do not affect availability.
1388 1389 1390 1391
Maintainability of the software is measured by the time to effect a modification. This time is a combination of time to diagnose (subdivided into time to localize and time to identif y the fault), active maintenance time (to devise the modification) and re-testing time (time to re-run earlier tests to ensure that the modification is successful).
1392 1393 1394 1395 1396
` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` ` ` , , ` , , ` , ` , , ` -
Maintenance of software requires the movement of information rather than of people and parts, and logistic delays are due to queues within the support system. The overall failure intensity (rate of occurrence of failures) for items containing software is the sum of the failure intensity of the hardware plus the failure intensities resulting from processes of activation of all individual latent design faults.
1397
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Bibliography
1398 1399 [1]
Alsmeyer, G., Erneuerungstheorie. Analyse stochastischer Regenerationsschemata, Stuttgart, B.G. Teubner, 1991.
[2]
Ascher, H.R., Feingold, H., Repairable System Reliability: Modeling, Inference, Misconceptions and Their Causes, New York, Marcel Dekker, 1984.
[3]
Asmussen, S., Applied Probabili ty and Queues, Second Edition, New York, SpringerVerlag, 2003.
[4]
Aven, T., Reliability and Risk Analysis, London, Elsevier Applied Science, 1992.
[5]
Aven, T., Jensen, U., Stochastic models in reliability , New York, Springer-Verlag, 1999.
[6]
Barlow, R.E., Proschan, F., Mathematical Theory of Reliability, New York, Wiley, 1965. Reprinted: Philadelphia, SIAM, 1996.
[7]
Barlow, R.E. Proschan, F., Statistical Theory of Reliability and Life Testing: Probabilistic Models, New York, Holt, Rinehart and Winston, 1975. Reprinted with corrections: Silver Spring, MD, To Begin With, 1981.
[8]
Beichelt, F., Franken, P., Zuverlässigkeit Methoden, Berlin, VEB Verlag Technik, 1983.
[9]
Birolini, A., Reliability Engineering: Theory and Practice , Sixth Edition, Berlin, Springer-Verlag, 2010.
[10]
Cocozza-Thivent, C., Processus stochastiques et fiabilité des systèmes , Berlin, Springer-Verlag, 1997.
[11]
Cox, D.R., Renewal Theory, London, Methuen & Co. Ltd, 1962.
[12]
Feller, W., An Introduction to Probability Theory and Its Applications, Volume II , Second Edition, New York, Wiley, 1971.
[13]
Gnedenko, B.V., Belyayev, Y.K., Solovyew, A.D., Mathematical Methods of Reliability Theory, New York, Academic Press, 1969.
[14]
Henley, E.J., Kumamoto, H., Reliability Englewood Cliffs, Prentice Hall, 1981.
[15]
Heyman, D.P., Sobel, M.J., Stochastic Models in Operations Research, Volume I, Stochastic Processes and Operating Characteristics, New York, McGraw-Hill, 1982.
[16]
Resnick, S., Adventures in Stochastic Processes, Boston, Birkhäuser, 1992 (4th printing, 2005).
[17]
Ross, S.M., Stochastic Processes, Second Edition, New York, Wiley, 1996.
[18]
Villemeur, A., Reliability, Availability, Maintainability and Safety Assessment, Volume 1, Chichester, Wiley, 1982
und
Instandhaltung:
Engineering
and
mathematische
Risk
Assessment,
1400 1401
Further information on the treatment of software aspects of dependability is available from the following references:
1402 1403
BS 5760: Part 8:1998, Reliability of systems, equipment and components. Part 8. Guide to assessment of reliability of systems containing software.
--``,,``,,,,,,,`,,,`,,,`,,```-`-`,,`,,`,`,,`---
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