ANSI ASQ D61165-1997 Application of Markov Techniques (T86E).pdf

1、STD-ASQ DbLLb5-ENGL 1797 E 075750b 0002320 575 m ANSI/IEC/ASQ D61165-1997 Application of Markov techniques Approved as an American National Standard by: American Society for Quality - STD.ASQ DbL1bS-ENGL 1797 075950b 0002321 401 H ANSI/IEC/ASQ D61165-1997 AMERICAN NATIONAL STANDARD Application of Ma

2、rkov Techniques Approved as an American National Standard by: American Society for Quality An American National Standard Approved on September 16, 1997 American National Standards: An American National Standard implies a consensus of those substantially concerned with its scope and provisions. An Am

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6、titute. - STD-ASB DbLLb5-ENGL 1997 M 075950b 0002322 348 M O 1995 by IEC Copyright Protection Notice for the ANSI/IEC/ASQ D61165- 1997 Standard. This standard is subject to copyright claims of IEC, ANSI and ASQ. Not for resale. No part of this publication may be reproduced in any form, including an

7、electronic retrieval system, without the prior written permission of ASQ. All requests pertaining to the ANSI/IEC/ASQ D61165-1997 standard should be submitted to ASQ. Note: As used in the document, the term “International Standard” refers to the American National Standard adoption of this and other

8、International Standards. ASQ Mission: To facilitate continuous improvement and increase customer satisfac- tion by identifying, communicating, and promoting the use of quality principles, con- cepts, and technologies; and thereby be recognized throughout the world as the leading authority on, and ch

9、ampion for, quality. 1098 76543 2 1 Printed in the United States of America Printed on acid-free paper Published by: American Societv for Qualitv Quality Press 61 1 East Wisconsin Avenue Milwaukee, Wisconsin 53201 -3005 800-248-1 946 Web site hltp:/www.asq.org STD-ASQ DbLLbS-ENGL 1997 m 075750b 0002

10、323 289 m ANSI/IEC/ASQ D61165-1997 Contents Page Foreword . 1 Introduction 2 Clause 1 2 3 4 5 6 7 a 9 10 11 12 Scope . 3 Normative references 3 Definitions . 3 Symbols . 4 4.1 State-transition diagram 4 4.2 Dependability measures 4 4.3 Example 5 General 5 Assumptions 6 Development of Markov diagrams

11、 . 7 7.1 Precautions 7 7.2 Rules 7 7.3 Examples . 8 Evaluation of state-transition diagrams . 11 8.1 General . 11 8.2 Evaluation of reliability 12 8.3 Evaluation of availability and maintainability 12 Simplifications and approximations 12 Collapsed state-transition diagram . 13 Reliability and avail

12、ability expressions for system configurations . 15 Presentation of results . 15 iii STIIOASQ DbLLbS-ENGL 1777 075750b 0002324 110 = ANSI/IEC/ASQ D61165-I 997 Annexes A Example: Numerical evaluation of some dependability measures of a two-unit active redundant system 17 B Tables of reliability and av

13、ailability expressions for basic system configurations, . 20 C Bibliography 22 iv STD=ASd DbLlb5-ENGL L797 W 07C750b 0002325 057 m Six Months Rule ANSIA EC/ASQ D61165-1997 Report on Voting Application of Markov Techniques 56(C0)164 Foreword 56 (CO) 1 76 1 ) The IEC (International Electrotechnical Co

14、mmission) is a world-wide organization for standardization comprising all national electrotechnical committees (IEC National Committees). The object of the IEC is to promote international cooperation on all questions concerning standardization in the electrical and electronic fields. To this end and

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18、ed in the form of standards, tech- nical reports or guides and they are accepted by the National Committees in that sense. 4) In order to promote international unification, IEC National Committees undertake to apply IEC International Standards transparently to the maximum extent possible in their na

19、tional and regional standards. Any divergence between the IEC Standard and the corresponding national or regional standard shall be clearly indicated in the latter. International Standard IEC 1165 has been prepared by IEC Technical Committee No. 56: Dependability. The text of this standard is based

20、on the following documents: Full information on the voting for the approval of this standard can be found in the report on voting indi- cated in the above table. Annexes A, B and C are for information only. 1 ANSIA EC/ASQ D61165-1997 Introduction Several distinct analytical methods of dependability

21、analysis are available, of which Markov analysis is one. IEC 60300-3-1 gives an overview of available methods and their general characteristics. The relative merits of various methods and their individual or combined applicability in evaluating the dependability of a given system or component, shoul

22、d be examined by the analyst prior to deciding on the use of Markov analysis. For each method, consideration should also be given to the results pro- duced, the data required to perform the analysis, the complexity of analysis, and other identified factors. 2 ANSI/IEC/ASQ 061165-1997 Application of

23、Markov Techniques 1 Scope This International Standard provides guidance on the application of Markov techniques to dependability analysis. 2 Normative references The following normative documents contain provisions which, through reference in this text, constitute provisions of this standard. At the

24、 time of publication, the editions indicated were valid. All normative documents are subject to revision, and parties to agreements based on this standard are encouraged to investigate the possibility of applying the most recent editions of the normative documents indicated below. Members of IEC and

25、 IS0 maintain registers of currently valid International Standards. IEC 60050( 191 ): 1990, International Electrotechnical Vocabulary (If V), chapter 19 1: Dependability and quality of service. IEC 61 078: (1 991 ), Analysis techniques for dependability-Reliability block diagram method 3 Definitions

26、 For the purposes of this International Standard the terms and definitions of IEC 60050(191) apply. In addition, the following terms and definitions are used: 3.1 unit: A component or set of components, which function as a single entity. NOTE-AS such, the unit can exist in only two states: functiona

27、l or failed (see 3.3 and 3.4). For convenience, the term unit state will be used to denote the state of a unit. 3.2 system state: A system state is a particular combination of unit states. NOTE-Several system states may be combined into one state. 3.3 functional state: A system (or unit) state in wh

28、ich the system (or unit) performs the required function. 3.4 failed state: A system (or unit) state in which the system (or unit) does not perform the required function. NOTE-A system can have several distinguishable failed states. 3.5 transition: A change from one state to another, usually as a res

29、ult of failure or restoration. NOTE-A transition may be also caused by other events such as human errors, external events, reconfiguration of software, etc. 3.6 transition probability: The probability of transition between one state and another state. 3 STD.ASQ DbIILb5-ENGL II797 075950b 0002328 8bb

30、 ANSIA EC/ASQ D61165-1997 3.7 initial state: The system state at time t = O. NOTE-Following a system failure, the system may be restored to the initial state. Generally, a system starts its operation at t = O from the complete functional state in which all units of the system are functioning and tra

31、nsits towards the final system state, which is a failed state, via other system functional states having progressively fewer functioning units. 3.8 absorbing state: A state from which, once entered, transitions are not possible. NOTE-Once in an absorbing state, the system will stay there until in ef

32、fect it is replaced, in its entirety, by a fully functional system. 3.9 restorable system: A system containing units which can fail and then be restored to their func- tional state, without necessarily causing system failure. NOTES 1 This corresponds to transitions in the state diagram in the direct

33、ion towards the initial state. For this to be possible, the units concerned will invariably operate in redundant configurations. 2 For a restorable system, dependability measures such as reliability, MTTFF, and availability are calculated 3.1 O non-restorable system: A system, the state transition d

34、iagram of which contains only transitions in the direction towards the final system failure state. NOTE-For a non-restorable system, reliability measures such as reliability and MTF are calculated 4 Symbols and abbreviations 4.1 Symbols for state-transition diagrams 4.1.1 state symbol: A state is re

35、presented by a circle or a rectangle. 4.1.2 state description: The state description is placed inside the state symbol and may take the form of words or alphanumeric characters defining those combinations of failed and functioning units which characterize the state. 4.1.3 state label: A state label

36、is a number in a circle, placed adjacent to the state symbol, or in the absence of a state description, within the symbol itself. NOTE-The state can often be adequately represented by a circle with the state number. 4.1.4 transition arrow: The transition arrow indicates the direction of a transition

37、 (as a result of failure or restoration). 4.1.5 rates: Restoration rates and/or failure rates are written on the transition arrow. 4.2 Other symbols and abbreviations Symbols for dependability measures follow those of IEC 50(191), where available. In this standard, the following symbols are used: Sy

38、mboVab brevia tion Term IEC 50(191)No R(t) reliability NOTE-191-12-01 used the general symbol (t,t,) MllF mean time to failure 191 -1 2-07 4 MUT MDT Pi( t) At mean time to first failure mean operating time between failures mean time to restoration failure rate restoration rate instantaneous availabi

39、lity asymtotic availability mean up time mean down time probability of finding the system in state “i” at time t a small time interval NOTE-191-13-02 uses (1) for repair rate NOTE-191-1345 uses A for asyrntotic availability 191 -1 2-06 191 -1 2-09 191 -1 3-08 191 -1 2-02 191-1 1-01 191-11-11 191-11-

40、12 4.3 Example An example of a state-transition diagram, applicable to a one-unit system, is shown in figure 1, h( t)At functional failed state state Figure l-State-transition diagram of a non-restorable one-unit system h(t)At is the probability of a transition between states O and 1 in the small ti

41、me interval At. Usually how- ever, terms like h(t)At are replaced simply by h. The reason for this is that in this standard h(t) is constant with respect to time (see clause 6) and transition arrows are, by convention, labeled using transition rates rather than transition probabilities. Hence the di

42、agram in figure 1 is frequently drawn in the form shown in figure 2. Figure 2-State-transition diagram (simplified) of a non-restorable one-unit system 5 General A Markov analysis makes use of a state-transition diagram which is a pictorial representation of the dependability performance of a system

43、. It models the dependability aspects of the systems behaviour with time. In this standard, a system is regarded as a number of units, each of which can exist in only one of two states: failed or functional. The system as a whole, however, can exist in many different states, each being determined by

44、 the particular combination of failed and functioning units. Thus, as a unit fails or is repaired, the system “moves” from one state to the next. This kind of model is generally called a discrete-state, continuous time model. However, because of the way in which the model is presented, the associate

45、d methodology is also a special type of “state space” analysis. 5 STDIASQ DbLLb5-ENGL 1997 075950b 0002330 4L4 = ANSI/IEC/ASQ D61165-I 997 State space analysis is especially suited to the dependability assessment of systems with redundancy, or to systems where system failure depends on sequential ev

46、ents, or to systems for which the mainte nance strategies are complex, for example priority restoration, queuing problems, and resource restric- tions. The analyst should ensure that the model adequately reflects the operation of the real system with respect to maintenance strategies and policies. P

47、rovided the limitations described in clause 6 (below) can be accepted, one of the major advantages of Markov analysis methods is that maintenance strategies, for example restoration priorities, can easily be modelled. Moreover, the order in which multiple failures occur can be represented in the mod

48、el. It should be noted that other dependability analysis techniques, for example fault tree analysis and reliability block diagram methods, do not allow complex maintenance strategies to be taken into account. Although state space analysis, from a theoretical viewpoint, is flexible and versatile, sp

49、ecial precautions are necessary to deal with the difficulties of practical applications. The main problem is that the number of system states and possible transitions increases rapidly with the number of units in the system. The larger the number of states and transitions, the more likely is it that there will be errors and misrepresen- tations. To reduce this risk, it is advisable that certain rules be followed in designing the diagram. Also, the numerical techniques used for the evaluation of the diagram may be complex and may require spe- cial computer programs and/or assis