Mathematical expression for reliability, availability, maintainability and maintenance support terms.pdf

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56 56 [1]

[1] Alsmeyer, Alsmeyer, G.,G., Erneuerungstheorie. Analyse stochastischer RegenerationsschemataErneuerungstheorie. Analyse stochastischer Regenerationsschemata , Stuttgart, B.G., Stuttgart, B.G. Teubner, 1991

Teubner, 1991 [2]

[2] Ascher, Ascher, H.R., H.R., Feingold, Feingold, H.,H., Repairable System Reliability: Modeling, Inference, Misconceptions andRepairable System Reliability: Modeling, Inference, Misconceptions and Their Causes

Their Causes, New York, Marcel Dekker, 1984, New York, Marcel Dekker, 1984 [3]

[3] Asmussen, Asmussen, S.,S., Applied Prob Applied Probability and Qability and Queuesueues, Second Edition, New York, Springer-Verlag, 2003., Second Edition, New York, Springer-Verlag, 2003. [4]

[4] Aven, Aven, T.,T., Reliability and Risk AnalysisReliability and Risk Analysis, London, Elsevier Applied Science, 1992, London, Elsevier Applied Science, 1992 [5]

[5] Aven, Aven, T., T., Jensen, Jensen, U.,U., Stochastic models in Stochastic models in reliability reliability , New York, Springer-Verlag, 1999, New York, Springer-Verlag, 1999 [6]

[6] Barlow, Barlow, R.E., R.E., Proschan, Proschan, F.,F., Mathematical Theory of Reliability Mathematical Theory of Reliability , New York, Wiley, 1965. Reprinted:, New York, Wiley, 1965. Reprinted: Philadelphia, SIAM, 1996

Philadelphia, SIAM, 1996 [7]

[7] Barlow, Barlow, R.E. R.E. Proschan, Proschan, F.,F., Statistical Theory of Reliability and Life Testing: Probabilistic ModelsStatistical Theory of Reliability and Life Testing: Probabilistic Models , New, New York, Holt, Rinehart and Winston, 1975. Reprinted with corrections: Silver Spring, MD, To Begin With, York, Holt, Rinehart and Winston, 1975. Reprinted with corrections: Silver Spring, MD, To Begin With, 1981

1981 [8]

[8] Beichelt, Beichelt, F., F., Franken, Franken, P.,P., Zuverlässigkeit und Instandhaltung: mathematische MethodenZuverlässigkeit und Instandhaltung: mathematische Methoden , Berlin, VEB, Berlin, VEB Verlag Technik, 1983

Verlag Technik, 1983 [9]

[9] Birolini, Birolini, A.,A., Reliability EngineeringReliability Engineering: : Theory and PracticeTheory and Practice, Sixth Edition, Berlin, Springer-Verlag, 2010, Sixth Edition, Berlin, Springer-Verlag, 2010 [10]

[10] Cocozza-ThCocozza-Thivent, ivent, C.,C., Processus stochastiques et fiabilité des systèmesProcessus stochastiques et fiabilité des systèmes, Berlin, Springer-Verlag, 1997, Berlin, Springer-Verlag, 1997 [11]

[11] Cox, Cox, D.R.,D.R., Renewal Theory Renewal Theory , London, Methuen & Co. Ltd, 1962, London, Methuen & Co. Ltd, 1962 [12]

[12] Feller, Feller, W.,W., An An Introduction to Introduction to Probability Theory Probability Theory and Its and Its ApplicationsApplications, Volume II, Second Edition, New, Volume II, Second Edition, New York, Wiley, 1971

York, Wiley, 1971 [13]

[13] GnedenGnedenko, B.V., Belyako, B.V., Belyayev, Y.K., Soloyev, Y.K., Solovyew, A.D.,vyew, A.D., Mathematical Methods of Reliability Theory Mathematical Methods of Reliability Theory , New, New York, Academic Press, 1969

York, Academic Press, 1969 [14]

[14] Henley, Henley, E.J., KE.J., Kumamoto, umamoto, H.,H., Reliability Engineering and Risk Assessment Reliability Engineering and Risk Assessment , Englewood Cliffs, Prentice, Englewood Cliffs, Prentice Hall, 1981

Hall, 1981 [15]

[15] Heyman, Heyman, D.P., D.P., Sobel, Sobel, M.J.,M.J., Stochastic Models in Stochastic Models in Operations ResearchOperations Research, Volume I, , Volume I, Stochastic ProcessesStochastic Processes and Operating Characteristics, New York, McGraw-Hill, 1982

and Operating Characteristics, New York, McGraw-Hill, 1982 [16]

[16] Resnick, Resnick, S.,S., Adventures in Stochastic Adventures in Stochastic ProcessesProcesses, Boston, Birkhäuser, 1992 (4th printing, 2005)., Boston, Birkhäuser, 1992 (4th printing, 2005). [17]

[17] Ross, Ross, S.M.,S.M., Stochastic ProcessesStochastic Processes, Second Edition, New York, Wiley, 1996, Second Edition, New York, Wiley, 1996 [18]

[18] Villemeur, Villemeur, A.,A., Reliability, Availability, Maintainability and Safety Assessment Reliability, Availability, Maintainability and Safety Assessment , Volume 1, Chichester,, Volume 1, Chichester, Wiley, 1982

Wiley, 1982

Für eine vertiefende Behandlung der Zuverlässigkeit von Software wird auf folgende Literaturstellen Für eine vertiefende Behandlung der Zuverlässigkeit von Software wird auf folgende Literaturstellen verwiesen:

verwiesen:

BS 5760 Part 8:1998

BS 5760 Part 8:1998, Reliability of systems, equipment and components – Part 8: Guide to assessment of, Reliability of systems, equipment and components – Part 8: Guide to assessment of reliability of systems containing software.

reliability of systems containing software. IEC 61713:2000-06,

IEC 61713:2000-06, Software dependability through the software life-cycle processes – Application guideSoftware dependability through the software life-cycle processes – Application guide IEC 62628:2012,

IEC 62628:2012, Guidance on software aspects of dependability. Guidance on software aspects of dependability. _______ _________________ - ` ` ` ` , , , , ` ` ` ` , , , , , , , , , , , , , , ` ` , , , , , , ` ` , , , , , , ` ` , , , , ` ` ` ` ` ` - - ` ` - - ` ` , , , , ` ` , , , , ` ` , , ` ` , , , , ` ` -

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-CONTENTS

6.3.15 Mean Mean down down time time [191-48-10] [191-48-10] ... ... 4141 45 45 6.3.16 6.3.16 Maintainability Maintainability [191-47-01] [191-47-01] ... 42... 42 46 46 - ` ` ` ` , , , , ` ` ` ` , , , , , , , , , , , , , , ` ` , , , , , , ` ` , , , , , , ` ` , , , , ` ` ` ` ` ` - - ` ` - - ` ` , , , , ` ` , , , , ` ` , , ` ` , , , , ` ` -

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-6.3.17

6.3.17 Instantaneous Instantaneous repair repair rate rate [191-47-20] ...[191-47-20] ... .... 4343 47

47

6.3.18

6.3.18 Mean Mean repair repair time time [191-47-21] [191-47-21] ... ... 4545 48

48

6.3.19

6.3.19 Mean active Mean active corrective maintenance corrective maintenance time [191-time [191- 47-22] ...47-22] ... ... 4646 49

49

6.3.20

6.3.20 Mean Mean time time to to restoration restoration [191-47-23] ...[191-47-23] ... ... 4646 50

50

6.3.21

6.3.21 Mean Mean administrative administrative delay delay [191-47-26] ...[191-47-26] ... ... 4747 51

51

6.3.22

6.3.22 Mean Mean logistic logistic delay delay [191-47-27] ...[191-47-27] ... .. 4848 52

52

An

Annenex A x A (i(infnformorm atativeive) ) PePerfrforormamancnce e asaspepectcts as a nd nd dedescrscr iptipt orors s ... ... ... ... ... ... ... ... ... ... ... . . 5050 53

53

An

Annenex B x B (i(infnformorm atativeive) ) SuSummmmarar y oy o f f memeasasurures es rerelalateted d to to titime me to to fafailuilu re re ... ... ... ... ... ... ... ... ... . 5151 54

54

An

Annenex C x C (i(infnforormamatitiveve) ) CoCompmpararisoiso n n of of sosome me dedepependndababiliili ty ty memeasasurures es fofor r cocontntininuouoususlyly 55

55

operating

operating items ...items ... .... 5353 56

56

An

Annenex D x D (i(infnforormamatitiveve) ) SoSoftftwarwar e e dedepependndababililitit y ay a spespe ctcts s ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 5555 57

57 58 58

Figure 1

Figure 1 – Sample – Sample realization realization of a of a non-repairable non-repairable item ...item ... .... 1515 59

59

Figure 2 –

Figure 2 – Sample realization Sample realization of a repairable of a repairable item with zero item with zero time to restoratime to restora tion ...tion ... ... 1616 60

60

Figure 3 –

Figure 3 – Sample realization Sample realization of a repairable of a repairable item with non-item with non- zero time to zero time to restoration ...restoration ... .. 1717 61

61

Figure 4

Figure 4 – Comparison – Comparison of an of an enabled time enabled time for a for a COI and COI and an IOI ...an IOI ... . 1818 62

62

Figure 5

Figure 5 – S– S ample reample re alization of alization of the the item state item state ... ... 4040 63

63

Figure 6 – Plot of the up-time hazard rate function

Figure 6 – Plot of the up-time hazard rate function λ λ _U_U((t t )) ... ... 4040 64

64

Figure

Figure A.1 A.1 – – Performance Performance aspects aspects and and descriptors ...descriptors ... .. 5050 65

65 66 66

Table B.1 – Relations

Table B.1 – Relations among functional measures of among functional measures of time to failure time to failure of continuouslyof continuously 67

67

operating

operating items items ... ... 5151 68

68

Table B.2 – Summary of measures for some continuous probability distributions of time Table B.2 – Summary of measures for some continuous probability distributions of time 69

69

to failure

to failure of of continuously opcontinuously op erating items ...erating items ... .. 5252 70

70

Table C.1 – Comparison of some dependability measures of continuously operatin Table C.1 – Comparison of some dependability measures of continuously operatin gg 71

71

items

items with with constant constant failure failure raterate λ λ and restoration rate and restoration rate µ µ RR ... 53... 53

72 72 73 73 74 74 75 75 -` ` ` ` , , , , ` ` ` ` , , , , , , , , , , , , , , ` ` , , , , , , ` ` , , , , , , ` ` , , , , ` ` ` ` ` ` -` ` -` ` , , , , ` ` , , , , ` ` , , ` ` , , , , ` `

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-INTERNATIONAL ELECTROTECHNICAL COMMISSION

INTERNATIONAL ELECTROTECHNICAL COMMISSION

76 76 ____________ ____________ 77 77 78 78

MATHEMATICAL EXPRESSIONS FOR RELIABILITY,

79 79

AVAILABILITY, MAINTAINABILITY AND

80 80

MAINTENANCE SUPPORT TERMS

81 81 82 82

FOREWORD

83 83 1)

1) The International ElectroteThe International Electrotechnical Commission (IEC) is a worldwide organizchnical Commission (IEC) is a worldwide organization for standardization comprisiation for standardization comprisingng

84 84

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

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international co-opera

international co-operation on ation on a ll questions concerning standardizatioll questions concerning standardization in tn in t he electrical and electronic fields. Tohe electrical and electronic fields. To

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this end and in addition to other activities, IEC publishes International Standards, Technical Specifications, this end and in addition to other activities, IEC publishes International Standards, Technical Specifications,

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

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112 112 Publications. Publications. 113 113 8)

8) Attention is drawn to the Normative refeAttention is drawn to the Normative references cited in this publicationrences cited in this publication. Use of the referenced publications is. Use of the referenced publications is

114 114

indispensable for the correct application of

indispensable for the correct application of this publication.this publication.

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9)

9) Attention is drawn to the possAttention is drawn to the possibility that some of the elements of this IEC Pubibility that some of the elements of this IEC Publication may be the subject oflication may be the subject of

116 116

patent rights. IEC shall not be

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117 117

International Standard IEC 61703 has been prepared by IEC technical committee 56: International Standard IEC 61703 has been prepared by IEC technical committee 56: 118 118 Dependability. Dependability. 119 119

The text of this standard is based on the following documents: The text of this standard is based on the following documents: 120

120

FDIS

FDIS Report Report on on votingvoting

56/XXX/FDIS 56/XXX/RVD

121 121

Full information on the voting for the approval of this standard can be found in the report on Full information on the voting for the approval of this standard can be found in the report on 122

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voting indicated in the above table. voting indicated in the above table. 123

123

This publication has been drafted in accordance with the ISO/IEC Directives, Part This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.2. 124

124

An

Annenexexes s A, A, B, B, C C anand d D D arare e fofor r ininfoformrmatatioion n ononly.ly. 125 125 - ` ` ` ` , , , , ` ` ` ` , , , , , , , , , , , , , , ` ` , , , , , , ` ` , , , , , , ` ` , , , , ` ` ` ` ` ` - - ` ` - - ` ` , , , , ` ` , , , , ` ` , , ` ` , , , , ` ` -

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-The committee has decided that the contents of this publication will remain unchanged until The committee has decided that the contents of this publication will remain unchanged until 126

126

the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data 127

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related to the specific publication. At this date, the publication will be related to the specific publication. At this date, the publication will be 128 128 • • reconfirmed,reconfirmed, 129 129 • • withdrawn,withdrawn, 130 130 •

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The National Committees are requested to note that for this publication the stability date The National Committees are requested to note that for this publication the stability date 134 134 is .... is .... 135 135

THIS TEXT IS INCLUDED FOR THE INFORMATION OF THE NATIONAL COMMITTEES AND WILL BE THIS TEXT IS INCLUDED FOR THE INFORMATION OF THE NATIONAL COMMITTEES AND WILL BE

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DELETED AT THE PUBLICATION STAGE DELETED AT THE PUBLICATION STAGE..

137 137 138 138 139 139 -` ` ` ` , ,

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INTRODUCTION

140

IEC 60050-191, Dependability terminology , provides definitions for dependability and its 141

influencing factors, reliability, availability, maintainability and maintenance support, together with 142

definitions of other related terms commonly used in this field. Some of these terms are 143

measures of specific dependability characteristics, which can be expressed mathematically. 144

This standard, used in conjunction with IEC 60050-191, provides practical guidance essential 145

for the quantification of those performance measures. For those requiring further information, 146

for example on detailed statistical methods, reference should be made to the IEC 60605 147

series of standards. 148

Annex A provide s a diagr ammat ic explanation of the relatio nsh ips between som e basic 149

dependability terms, related random variables, probabilistic descriptors and modifiers. 150

Annex B provides a summar y of measures related to time to failure . 151

Annex C compares som e dependabili ty measu res for con tinuously o perating items. 152

Annex D explains some of the sof twa re dependability a spects. 153

The bibliography gives references for the mathematical basis of this standard, in particular, 154

the mathematical material is based on references [2], [6], [8], [9], [13], [14] and [18]; the 155

renewal theory (renewal and alternating renewal processes) may be found in [6], [8], [9], [10], 156

[11], [13], [15] and [17]; and more advanced treatment of renewal theory may be found in 157 references [1], [3], [12] and [16]. 158 159 160 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

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-MATHEMATICAL EXPRESSIONS FOR RELIABILITY,

161

AVAILABILITY, MAINTAINABILITY AND

162

MAINTENANCE SUPPORT TERMS

163 164 165 166

1 Scope

167

This International Standard provides mathematical expressions for reliability, availability, 168

maintainability and maintenance support measures defined in IEC 60050-191. The following 169

classes of items are considered separately in this standard: 170

– non-repairable items; 171

– repairable items with zer o ( or negligible) time to re sto ration; and 172

– repairable items with non- zer o tim e to re storation. 173

In order to keep the mathematical formulae as simple as possible, the following basic 174

mathematical models are used to quantify dependability measures: 175

– random var iable ( tim e to failure ) for non-repairable items; 176

– simple (ord ina ry) renewal p rocess for repairable items w ith zero time to re sto ration; 177

– simple (ordinar y) alternating renew al process for repairable items with non-zer o time to 178

restoration. 179

To facilitate location of the full definition, the IEC 60050-191 reference for each term is shown 180

[in brackets] immediately following each term, for example: 181

mean time to restoration [191-47-23] 182

The application of each dependability measure is illustrated by means of a simple example. 183

This standard is mainly applicable to hardware dependability, but many terms and their 184

definitions may be applied to items containing software. Some of the software dependability 185

aspects are explained in Annex D. 186

2 Normative references

187

The following referenced documents are applicable to this standard. For dated references, 188

only the edition cited applies. For undated references, the latest edition of the referenced 189

documents (including any amendments) applies. 190

IEC 60050-191:XXXX International Electrotechnical Vocabulary (IEV) – Part 191: 191

Dependability 192

ISO 3534-1:2006, Statistics – Vocabulary and symbols – Part 1: General statistical terms and 193

terms used in probability 194 195 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

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-3 Definitions

196

For the purpose of this International Standard, the terms and definitions given in 197

IEC 60050-191:XXXX and ISO 3534-1 apply. 198

In addition, the following terms and definitions, which do not appear in IEV-191, are used in 199

order to facilitate the presentation of mathematical expressions for other IEV-191 terms. 200

3.1 201

instantaneous restoration intensity 202

restoration intensity 203

limit, if it exists, of the quotient of the mean number of restorations (191-46-23) of a repairable 204

item (191-41-11) within time interval ( t , t +

∆

t ), and

∆

t , when

∆

t tends to zero 205 t t N t t N E t v t

∆

−

∆

+

=

+ → ∆ )] ( ) ( [ lim ) ( R R 0 206 where 207

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

208

E denotes the expectation. 209

NOTE 1 to entry: The unit of measurement of instantaneous failure rate is the unit of time to the power−1.

210

NOTE 2 to entry: Other term used for restoration intensity is “restoration frequency”.

211

[[Document to be checked for usage of this term, and harmonized with IEV-191 Ed 2]] 212

3.2 213

up-time distribution function 214

function giving, for every value of t , the probability that an up time will be less than, or equal 215

to, t 216

NOTE 1 to entry: If the up time is (strictly) positive and continuous random variable, then F U(0) = 0 and

217       − − =

∫

t x x t F 0 U U() 1 exp λ ( )d 218

where λ U(t ) is the instantaneous up-time hazard rate function.

219

NOTE 2 to entry: If the up time is exponentially distributed, then

220

F U(t ) = 1 − exp(−t /MUT)

221

where MUT is the mean up time.

222

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

223

λ U = 1/MUT

224

3.3 225

instantaneous up-time hazard rate function 226

up-time hazard rate function 227

λ U(t )

228

limit, if it exists, of the quotient of the conditional probability that the up-time will end within 229

time interval (t , t + ∆t ) and ∆t , when ∆t tends to zero, given that the up-time started at t = 0 230

and had not been finished before time t 231

NOTE 1 to entry: The instantaneous up-time hazard rate function is expressed by the formula:

232 ) ( 1 ) ( ) ( 1 ) ( ) ( 1 lim ) ( U U U U U 0 U t F t f t F t t F t F t t t − = − ∆ + − ∆ = → ∆ λ 233

where F U(t ) isup-time distribution function and f U(t ) is the probability density at the up-time.

234

NOTE 2 to entry: If the up time is exponentially distributed, then the instantaneous up-time hazard rate function is

235

constant in time and is denoted byλ U.

236 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

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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 237 power −1. 238 3.4 239

continuously operating item 240

COI 241

item for which operating time (191-42-05) is equal to its enabled time (191-42-17) 242

3.5 243

intermittently operating item 244

IOI 245

item for which operating time (191-42-05) is less than its enabled time (191-42-17) 246

4 Glossary of symbols and abbreviations

247

The symbols given below are widely used and recommended but are not mandatory. For 248

consistency of presentation, the notation in this document may differ from that used in a 249

referenced document. 250

4.1 Non-repairable items 251

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

) ( ˆ t

R Point estimate of the reliability function at time t R(t 1, t 2) Reliability for the time interval (t 1, t 2)

) | ,

(t t x t

R

+

Conditional reliability for the time interval ( t , t + x ), assuming that the item survived to time t

TTFi 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

nS(t ) Number of (non-repairable) items that are still operational at the instant of time t (n_S(0) = n)

nS(t )

−

nS(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

) ( ˆ t

λ Point estimate of the instantaneous failure rate at time t )

, (t ₁ t ₂

λ Mean failure rate for the time interval (t ₁, t ₂) 252 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

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-4.2 Repairable items with zero time to restoration 254

MOTBF Mean operating time between failures

∧

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)

) , ( ˆ 2 1 t t

R 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 ).

) (

CTTF t

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

≥

1 k F Number of failures during a given period of observation

n Number of items in the population

nF(t , t

+ ∆

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

∆

t )

nF(t 1, t 2) Number of failures observed in the time interval ( t 1, t 2)

nS(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 )

, (t ₁ t ₂

z Mean failure intensity for the time interval ( t 1, t 2)

) , ( ˆ 2 1 t t

z 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 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

(12)

-4.3 Repairable items with non-zero time to restoration 257

A Asymp totic a vaila bil ity

A(t ) Instantaneous availability (availability function), i.e. the probability of the item being in an up state at the instant of time t

) , (t ₁ t ₂

A Mean availability for the time interval (t 1, t 2)

) , ( ˆ 2 1 t t

A Point estimate of the mean availability for the time interval ( t 1, t 2)

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

GR(t ) Distribution function of the times to restoration

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)

RTi 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, i.e. the expectation of a given maintenance action time

- ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

(13)

-∧

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

∧

MTTR Mean time to restoration

∧

MTTR Point estimate of the mean time to restoration

MUT Mean up time

∧

MUT Point estimate of the mean up time

U Asymp totic u navaila bilit y

U (t ) Instantaneous unavailability (unavailability function) )

, (t ₁ t ₂

U Mean unavailability for the time interval ( t 1, t 2)

) , ( ˆ 2 1 t t

U Point estimate of the mean unavailability for the time interval ( t 1, t 2)

VRT The variance of repair time, VRT = Var [ζ ] = E [ζ 2]

−

MRT2, where ζ is a random variable representing the repair time, Var 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

corres-ponding 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

) (

CTTF t

h n Probability density function of calendar time to the nth 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

(14)

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

) (

MAT t

m Number of maintenance action times with duration greater that t (m_MAT(0)

=

m)

n Number of items in the population

nD{t } Number of items that are in a down state at the instant of time t

nF(t , t +

∆

t ) Number of failures observed during the time interval (t , t +

∆

t ), where the

time scale includes both up and down times

nF(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

nR(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 )

nS(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)

nU{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 )

, (t ₁ t ₂

z Mean failure intensity for the time interval ( t 1, t 2)

) , ( ˆ 2 1 t t

z Point estimate of the mean failure intensity for the time interval ( t 1, t 2)

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 )

(

U t

λ 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

adminis-trative 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 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

(15)

-µ R Constant restoration rate, i.e. the reciprocal of the mean time to

restoration (MTTR) when the times to restoration are exponentially distributed 258

5 Assumptions

259 5.1 General 260

In order to derive correct mathematical expressions for the measures defined in 261

IEC 60050-191, the distinction needs to be made between repairable items [191-41-11] and 262

non-repairable items [191-41-12]. The following classes of items are considered separately 263

in this standard: 264

– non-repairable items; 265

– repairable items with zer o time to restoration; 266

– repairable ite ms with non- zer o tim e to re storation. 267

In order to keep the mathematical formulae as simple as possible, the following basic 268

mathematical models are used to quantify dependability measures: 269

– random var iable ( tim e to failure) for non-repairable items; 270

– simple (ord ina ry) renewal p ro cess for repairable items w ith zero time to re sto ration; 271

– simple (ordinar y) alternating renew al process for repairable items with non-zer o time to 272

restoration. 273

The simplest mathematical model for the reliability of a non-repairable item is the random 274

variable, time to failure of the item [191-45-01]. One of the widely used reliability measures of 275

non-repairable items is the instantaneous failure rate, λ (t ) [191-45-06], also referred to as 276

the hazard rate function. It is derived fro m the distribution function of the time to failure. 277

The expression λ (t )∙

∆

t is, for small values of

∆

t , approximately equal to the conditional

278

probability of failure of an item during the time interval ( t , t +

∆

t ) given that the item has not 279

failed during the interval (0, t ]. 280

For repairable items, the basic model is a simple renewal process, when the time to 281

restoration of the item may be neglected, or a simple alternating renewal process in which the 282

time to restoration of the item is non-zero. In the latter case, the item alternates between an 283

up state and a down state, and a widely used measure of reliability of the item is the failure 284

intensity, which is equal to the renewal density. 285

The failure intensity [191-45-08] is a measure derived from the expected value of the 286

cumulative number of failures E [ N (t )] of a repairable item occurring during the time interval 287

(0, t ]. The expression z (t )∙

∆

t is, for small values of

∆

t , approximately equal to the probability of

288

failure of the item during the time interval ( t , t +

∆

t ). 289

To avoid improper use of these mathematical expressions, which could yield erroneous 290

results, the specific assumptions detailed in 5.2 and 5.3 should be observed. 291

5.2 Assumptions for non-repairable items 292

a) At any instant of time, the non-repairable item will be either in an up state or in a down 293

state (see figure 1). 294

b) Unless otherwise stated, when the item is in an up state, it is considered to be operating 295

continuously. 296

NOTE The mathematical expressions given in this sub-clause may not always be valid for IOIs.

297 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

(16)

-c) At time t = 0, the item is in an operating state, and is as good as new. Latent faults are not 298

considered, which, if present, may invalidate some mathematical expressions. 299

d) Preventive maintenance, or other planned actions that render the item incapable of 300

performing a required function are not considered. 301

e) The time to failure is a positive and continuous random variable with a probability density 302

function and finite expectation. 303 0 τ Time Down state Up state State 304 Key 305 τ Time to failure 306

Figure 1 – Sample realization of a non-repairable item

307

5.3 Assumptions for repairable items 308

a) At any instant of time, the repairable item will be either in an up state or in a down state 309

(see figures 2 and 3). 310

b) At time t = 0, the item is in an up state, and therefore, R(0) = 1, and is as good as new. 311

Latent faults are not considered. 312

c) Unless otherwise stated, when the item is in the up state, it is considered to be operating 313

continuously. 314

d) Consecutive up times of the item are statistically independent, identically distributed, 315

positive, continuous random variables with a common probability density function and 316

finite expectation. 317

e) In the case of non-zero down-time duration, the consecutive down times of the item are 318

statistically independent, identically distributed, positive, continuous random variables with 319

a common probability density function and finite expectation 320

f) The up times are statistically independent of the down times. 321

g) Preventive maintenance or other planned actions that render the item incapable of 322

performing a required function are not considered. 323

h) Unless otherwise stated, each other random variable (e.g. time to failure, repair time, 324

logistic delay, and so on) considered in the standard, is a positive and continuous random 325

variable with a probability density function and finite expectation. 326

In summary: 327

– any tra nsition from an up state to a down sta te is a failure ; 328

– any tra nsition from a down sta te to an up state is a restoration; 329

– any down sta te is a fault and, in con seq uence, the down time is equal to the time to 330

restoration; 331

– after each restoration the item is as good as new. 332 333 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

(17)

-For continuously operating items (COI) the up state coexists with the operating state, and the 334

up time is concurrent with the operating time. 335

The expressions for reliability measures of continuously operating, repairable items may not 336

be true for IOIs (see figure 4). 337

NOTE 1 Models assuming zero time to restoration are used when either the up time of the item is the only time of

338

interest in assessing the performance of the item, or the time to restoration is so short that it is negligible.

339

NOTE 2 All mathematical expressions for the reliability measures relating to the time to failure of a non-repairable

340

item may also be applied to each time to failure of a continuously operating repairable item.

341 342 0 1 2 m − 1 m m + 1 N (t ) S F,1 S F,2 S F, m− 1 S F, m S F,0= 0 Time

τ U,1 τ U,2 τ U,3 τ U, m− 1 τ U, m τ U, m + 1

Down state Up state State S F,1 S F,2 S F, m− 1 S F, m S F,0= 0 Time 343 Key 344

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

345

S F, 1,S F, 2, S F,3... Consecutive instants of failure

346

τ U,1, τ U, 2, τ U,3... Consecutive up times

347

Figure 2 – Sample realization of a repairable item with zero time to restoration

348 349 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `

(18)

-Down state Up state State

τ U,1 τ U,2 τ U,3 τ U, m τ U, m + 1

ξ R,1 ξ R,2 ξ R, m− 1 ξ R, m S F,1 S F,2 S _{R, m}− 1 S F, m S F,0=S F,0=0 S R,1 S R,2 S R,m Time 0 1 2 m − 1 m N (t ) S F,1 S F,2 S _{R, m}− 1 S F, m S F,0=S F,0=0 S R,1 S R,2 S R,m Time 0 1 2 m − 1 m N R(t ) S F,1 S F,2 S R, m− 1 S F, m S F,0=S F,0=0 S R,1 S R,2 S R,m Time m − 2 350 Key 351

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

352

N R(t ) Number of restorations during the time interval (0, t ]

353

S F, 1, S F, 2, S F,3... Consecutive instants of failure

354

S R,1, S R,2, S R,3... Consecutive instants of restoration

355

τ U,1 τ U, 2, τ U, 3 ... Consecutive up times

356

ξ R,1, ξ R,2, ξ R,3... Consecutive times to restoration

357

Figure 3 – Sample realization of a repairable item with non-zero time to restoration

358 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

(19)

-Operating time [191-42-05]

Standby time [191-42-13]

Idle time [191-42-15]

Restoration Failure

Continuously operating item (COI)

Intermittently operating item (IOI)

Enabled time

Key

359

Figure 4 – Comparison of an enabled time for a COI and an IOI

360

6 Mathematical expressions

361 6.1 Non-repairable items 362 6.1.1 General 363

All exp ressions in 6.1 are applica ble to COI s only. 364

For each measure, the following are presented: 365

a) the generic expression; 366

b) the most common expression (for exponentially distributed time to failure of the item); 367

c) a simple example of application where necessary. 368

6.1.2 Reliability [191-45-05] 369

(Symbol R(t 1, t 2), 0

≤

t 1< t 2)

370

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

371

the reliability R(0, t 2) for the time interval (0, t 2).

372

More commonly used expressions are the reliability function R(t ) = R(0, t ), with R(0) = 1, and 373

the conditional reliability R(t , t + x | t ), when no failure has occurred in time [ 0, t ]. 374 a)

∫



=

∫

∞











−

=

t t x x f x x t R( ) exp ( )d ( )d 0λ 375 where 376

λ ( x) is the instantaneous failure rate of the item; 377 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

(20)

- f ( x) is the probability density function of th e time to failure of the item, i.e. , for small values

∆

x, 378

f ( x)

⋅∆

x is approximately equal to the probability that the failure of the item will occur 379

during ( x, x +

∆

x). 380

NOTE If observed failure data are available forn non-repairable items, from a homogenous population, the

381

estimated value of R(t ) is given by

382 n t n t Rˆ()₌ S() 383 where 384

nS(t ) is the number of items that are still operational at the instant of timet (nS(0) =n).

385

The probability that the item will fail during the time interval ( t 1, t 2), 0

≤

t 1 < t 2, is given by

386

∫

=

−

2 1 d ) ( ) ( ) ( ₁ ₂ t t f t t t R t R 387

The conditional reliability, R(t , t + x | t ), is defined as the conditional probability that an 388

item can perform a required function for a given time interval ( t , t + x) provided that the 389

item is in an operating state at the beginning of the time interval. (See [9]1)_{, page 40.)}

390 ) ( ) ( d ) ( exp ) | , ( t R x t R t t t x t t R t x t

+

=













−

=

+

∫

+ _λ 391

b) When λ (t ) = λ = constant, i.e. when the (operating) time to failure is exponentially 392 distributed 393 R(t ) = exp(

−λ

t ) 394 R(t , t + x | t ) = exp(

−

λ x) 395

c) For an item with a constant failure rate of λ = 1 year −1 and a required time of operation of 396

six months, the reliability is given by 397 R(6 months) = exp(

−

1

×

12 6 ) = 0,606 5 398

6.1.3 Instantaneous failure rate [191-45-06] 399 (Symbol λ (t )) 400 a) According to definition [191-45-06]: 401 ) ( ) ( ) ( ) ( ) ( 1 lim ) ( 0 R t t f t R t t R t R t t t

=

∆

+

−

∆

=

→ ∆ λ 402

For small values of

∆

t , λ (t )

⋅∆

t is approximately equal to the conditional probability that 403

failure of the item will occur during ( t , t +

∆

t ), given that the item has survived to time t . 404

NOTE If observed failure data are available for n non-repairable items, from a homogenous population, the

405

estimated value of λ (t ) at time t is given by

406 t t n t t n t n t ∆ ∆ + − = ) ( ) ( ) ( ) ( ˆ S S S λ 407 where 408

nS(t ) is the number of items that are still operational at the instant of timet (nS(0) = n);

409

nS(t )− nS(t + ∆t ) is the number of items that fail in the time interval (t , t + ∆t ).

410

It should be noted that the estimated value of the failure density function f (t ), at time t , is given by

411 t n t t n t n t f ∆ ∆ + − = () ( ) ) ( ˆ S S 412 ________

(21)

The probability that the item will fail during the time interval ( t 1, t 2) is given by 413













−













−

=

−

∫

1

∫

2 0 0 ( )d exp ( )d exp ) ( ) (t ₁ R t ₂ t t t t t R t λ λ 414

b) When the time to failure is exponentially distributed, i.e. λ (t ) = λ for all values of t, 415 f (t ) = λ exp(

−

λ t ) 416 and 417 R(t ) = exp(

−

λ t ) 418

NOTE If observed failure data are available for n non-repairable items, from a homogenous population, with

419

constant failure rate, then the estimated value ofλ is given by

420

∑

= = n i i n 1 TTF ˆ λ 421 where 422

TTFi is the time to failure of itemi.

423

c) For 10 non-repairable items, from a homogenous population, with a constant failure rate, 424

the observed total operating time to failures of all the items is

∑

= 10 1TTF i i = 2 years. Hence 425 λ ˆ = 2 10 = 5 year -1 426

If the time to failure of a non-repairable item has a two-parameter Weibull distribution with 427

scale parameter α > 0 and shape parameter β > 0, then 428 R(t ) = exp(

−

(α t ) β ) 429 and 430 t t R t f d ) ( d ) (

=

−

= αβ (α t ) β − 1 exp(

−

(α t ) β ) 431 hence 432 λ (t ) = ) ( ) ( t R t f = αβ (α t ) β − 1 433 (See [9], page 434.) 434

For β = 2 and α = 0,5 year −1 435 λ (6 months) = 0,5

×

2

×

(0,5

×

12 6 ) = 0,25 year −1 436 λ (1 year) = 0,5

×

2

×

(0,5

×

1) = 0,5 year −1 437

6.1.4 Mean failure rate [191-45-07] 438 (Symbol λ(t ₁, t ₂),₀

≤

_t₁_{< t}₂₎ 439 a) ) ( ) ( ln 1 d ) ( 1 ) , ( 2 1 1 2 1 2 2 1 2 1 R t t R t t t t t t t t t t

−

=

−

=

∫

λ λ 440

b) When the time to failure is exponentially distributed 441 λ λ (t ₁, t ₂)

=

442 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

(22)

-for all values of t 1 and t 2.

443

c) Let t 1 = 6 months, R(t 1) = 0,8 and t 2 = 12 months, R(t 2) = 0,5, then

444 0,5 0,8 ln 6 12 1 ) 12 , 6 (

−

=

λ = ln(1,6)/6 = 6 47 , 0 = 0,078 3 month−1 445 while ( R(0) = 1) 446 0,8 1 ln 0 6 1 ) 6 , 0 (

−

=

λ = ln(1,25)/6 = 6 1 223 , 0 = 0,037 2 month−1 447

6.1.5 Mean (operating) time to failure [191-45-11] 448 MTTF (abbreviation) 449 a)

=

∫

∞

=

∫

∞ 0 0 ( )d ( )d MTTF tf t t R t t 450

NOTE If observed failure data are available for n non-repairable items, from a homogenous population, then

451 an estimate of MTTF is given by 452 n n i i

∑

= ∧ 1 TTF = MTTF 453 where 454

TTFiis the time to failure of item i.

455

b) When the time to failure is exponentially distributed, i.e. λ (t ) = λ for all values of t, 456

MTTF = λ 1 457

c) For a non-repairable item with a constant failure rate of λ = 0,5 year −1, 458

MTTF = 2 years = 17 520 h 459

If the time to failure of a non-repairable item has a two-parameter Weibull distribution with 460

a scale parameter α > 0 and shape parameter β > 0, then 461 R(t ) = exp(

−

(α t ) β ) 462 and 463 MTTF = α β Γ (1

+

1) 464 where 465 Γ ( x) =

∫

∞ − − 0 1_e _d_t t x t 466

is the complete gamma function. (See [9], page 435.) 467

For β = 2 and α = 0,5 year −1: 468 MTTF = ) 2 1 1 ( 2 5 , 0 ) 2 1 1 (

+

×

=

+

Γ Γ 469 but 470 2 / 2 / ) 1 ( ) 1 1 (

+

=

Γ

=

π

Γ 471 -` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` -` -` , , ` , , ` , ` , , `

(23)

-hence 472

MTTF =

π ≈

1,772 5 years = 21,27 months 473

6.2 Repairable items with zero time to restoration 474

6.2.1 General 475

All exp ressions in 6.2 are applicabl e to COI s. Where they are appli cable to IOIs, this is stated. 476

For each measure, the following are presented: 477

a) the generic expression; 478

b) the most common expression (for the cases when the times to failure of the item are 479

exponentially distributed); 480

c) a simple example of application where necessary. 481

6.2.2 Reliability [191-45-05] 482

(Symbol R(t 1, t 2), 0

≤

t 1 < t 2)

483

a) The reliability of an item for the time interval (t 1, t 2) may be written as (see [9], page 461

484 and [13], page 105): 485

∫

−

+

=

1 0 2 2 2 1, ) ( ) ( ) ( )d (t t R t t R t t z t t R 486 where 487

the first term, R(t 2), represents the probability of survival to time t 2, and the second term

488

represents the probability of failing at time t (t < t 1) and, after immediate restoration,

489

surviving to time t 2;

490

z (t ) is the instantaneous failure intensity (renewal density) of the item, i.e., for small values of 491

∆

t , z (t )

⋅∆

t is approximately equal to the (unconditional) probability that a failure of the item 492

occurs during (t , t +

∆

t ), and 493

R(t ) = R(0, t ) is the reliability function of the item 494

∫

∞

=

t f s s t R( ) ( )d 495

where f (t ) is the probability density function (also referred to as the failure density 496

function) of the times to failure of the item, i.e., for small values of

∆

t , f (t )

⋅∆

t is 497

approximately equal to the probability that the item fails during the time interval ( t , t +

∆

t ). 498

More precisely, it is approximately the probability that a given time to failure terminates in 499

the time interval (t , t +

∆

t ), assuming that the time to failure started at time t = 0. 500

NOTE 1 R(t 1, t 2) is also known as the interval reliability.

501

NOTE 2 If observed failure data are available forn repairable items, from a homogenous population, then an

502 estimate of R(t 1, t 2) is given by 503 n t t n t t Rˆ( , ) S(1, 2) 2 1 = 504 where 505

nS (t 1, t 2) is the number of items that were operational at the instant of time t 1and did not fail during the time

506

interval (t 1, t 2).

507

By setting t 1 = t and t 2 = t + x, one can obtain the asymptotic interval reliability (see [13],

508 page 106): 509

∫

∞ ∞

+

=

→ R t t x x R s s t MTTF ( )d 1 ) , ( lim 510 - ` ` , , ` ` , , , , , , , ` , , , ` , , , ` , , ` ` ` - ` - ` , , ` , , ` , ` , , `