availability and reliability of system with.pdf

IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 2, JUNE 2001 151

Availability and Reliability of System WithDependent Components and Time-Varying Failure

and Repair RatesTieling Zhang and Michio Horigome, Member, IEEE

Abstract—Summary & Conclusions—System reliability de-pends not only on the reliabilities of components in the systembut also on their interactions, viz., the dependencies among them.Generally, in a system, not only -independent failures but also-dependent failures among components can occur; thus there

are many studies where the -dependencies among componentsare taken into account in system reliability & availability analysis,but in which the failure & repair rates were assumed constant.Whereas, from a practical viewpoint, the constant failure rateassumption for components has been, and is repeatedly challengedby knowledgeable reliability practitioners. Therefore, there areother studies which handled the problem of time-varying failurerates, among which all concerned repairable systems did notinvolve -dependent failures. In most cases, however, to combine-dependent failures and time-varying failure & repair rates in

system reliability & availability analysis is the most appropriatefor real systems. But it is very difficult to obtain the analyticsolution and, in most cases, the closed-form solution for systemreliability & availability does not exist, so that numerical orsimulation methods must be used.

This paper studies one kind of systems that endure environ-mental shocks, and where one or more components can failsimultaneously due to a cumulative shock-damage process. Anapproach for reliability & availability analysis of such kind ofrepairable systems is presented, where failure & repair rates ofcomponents can be varying with time. One type of special vehicleswith such mechanical systems illustrates system reliability &availability solutions.

Index Terms—Dependent failure, Markov chain, repairablesystem, system availability, system reliability, time-varying failurerate.

ACRONYMS1

MLE maximum likelihood estimationNHPP nonhomogeneous Poisson process

NOTATION

system availability

Manuscript received November 14, 1998; revised August 17, 2000.Responsible Editor: W. KuoT. Zhang is with the Dept. of Electronic Control Eng., Tokyo Univ. of

Mercantile Marine, 2-1-6 Etchujima, Koto-ku, Tokyo 135-8533, Japan (e-mail:[email protected]).

M. Horigome is with the Hiroshima National College of Marine Technology,Touyochou 4272-1, Toyotagun, Hiroshima Ken 725-0200, Japan (e-mail:[email protected]).

Publisher Item Identifier S 0018-9529(01)09548-3.

1The singular & plural of an acronym are always spelled the same.

system availability with constant failure & repairratescombination of out offailure-state set:system state:

: working state: failure states

system-state probability vectortime-derivative ofPr system is in stateatsystem reliabilityreliability of componenttime (or moved mile)total moved distance of a vehicle in teststate-transition rate matrix of systemoperation-state set:variate of system state atfailure rate of system from state 0 to stateatrepair rate atstate-transition matrix of linear time-varying system

I. INTRODUCTION

A LARGE complex system consists of many differentfunctional subsystems in which each subsystem can be

divided into sub-subsystems, elements, or components. Thereliabilities of elements or components are in the basis ofsystem reliability. System reliability also depends on interac-tions among components, viz., the-dependence among them.The often-made assumption of-independence for componentsis an approximation, to some extent, on certain conditions, andsome systems can not be modeled with the-independenceassumption among components, e.g., share-load redundantsystems. However, because mechanisms of-dependenceamong components are very complicated, and carrying outquantitative description about the degree of such-dependenceis very difficult, some assumptions must be made.

In recent years, the studies on-dependent failure theorieshave been widely developed in many engineering fields, suchas nuclear industry, oil & chemical industry, rocket launching,and mechanical industry [1]; these are applied to analyze systemsafety. The main elements in research are the common-causefailures in redundant systems. Therefore, techniques for ana-lyzing common-cause failures have been developed, and thedata library of common-cause failures was established [2]–[4].

0018-9529/01$10.00 © 2001 IEEE

152 IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 2, JUNE 2001

A. -Dependent Failures

References [5]–[9] describe the parallel redundant systems.Reference [10] describes reliability bounds of a series system

with -dependent component failures.Reference [11] presents the MCS approach for extending the

cut-set method to systems involving-dependencies.Reference [12] gives the preventive replacement policies for

systems with -dependent components.Reference [13] analyzes the availability of a 2-component

system where both components can fail simultaneously due toa shock-damage process that is Poisson with parameter.

Reference [14] proposes a multiple-state Markov model ofthe system with -dependent components, in which the systemis a homogeneous continuous-time Markov process with dis-crete states. It investigates the condition upon which the transi-tion-rate matrix of the system has the form of a modified Kro-necker sum of transition-rate matrices of its units, and intro-duces an algorithm for determining the transition-rate matrix ofthe system, based on the Kronecker algebra.

Reference [15] presents a multiple-state stochastic descrip-tion of the -dependent failures, and analyzes the ‘relationshipbetween objective variability and subjective uncertainty’, andthe ‘influence of both features on creating the effect of-de-pendent failures’. It gives a unified method in which both the‘state-of-knowledge correlation approach as a model describingsubjective parameter uncertainty’ and the ‘generalized binomialdistribution function model of real-world variability’ are inte-grated into a quantitative multiple-state stochastic descriptionof systems.

Reference [16] presents the Clayton model for-dependentcomponents in system reliability analysis. A general expressionof system reliability is derived as a function of the componentreliabilities and the -dependency parameter. An advantage ofthis model is that it describes the-dependency in terms of 1parameter, which has a practical interpretation. Estimation ofthis -dependency parameter is the key to applying this model.

Reference [17] takes-dependent failures with different com-ponents as different states of the system.

Reference [18] summarizes developments of techniques fordealing with -dependent failures.

Thus many reliability practitioners have been searching for amore practical and more accessible method for dealing with the-dependent failure problems in real systems. However, in all

studies (of which we are aware) concerning-dependent fail-ures, the failure & repair rates are assumed constant, which be-comes unacceptable by knowledgeable practitioners (in recentyears) in some situations, especially for mechanical systems.

B. System Reliability With Time-Varying Failure Rates

Reference [19] proposes parametric estimation of time-de-pendent failure rates for probabilistic risk assessment, in whichfailures of components are assumed to follow the Poissonprocess. And it proposes many parametric models for thetime-dependent failure rate .

Reference [20] studies the availability & reliability of re-pairable systems with time-varying failure rates in which the-dependencies in the system are not involved.

Reference [21] examines the NHPP models in the practicalanalysis of maintenance failure data and briefly reviews NHPPmodels.

C. General Discussion

Up to now, when analyzing system reliability involving time-varying failure & repair rates, the-dependencies among com-ponents were not considered; on the other hand, when con-sidering problems of -dependence, the failure & repair rateswere regarded as constants. However, to analyze system reli-ability by combining -dependencies among components, andtime-varying failure & repair rates is the most appropriate formodeling real systems. But it is difficult to handle them mathe-matically, and closed-form solutions of availability & reliabilityare not obtained analytically in general. Herein numerical solu-tions or simulating approaches must be applied. Whereas, withthe increase in the number of components in a system, the ma-nipulating time in calculation is enormous and the calculationrequires very large computer memory. To solve such problems,the preferable method is to apply large, specialized software andto model the real system by simplifying its structures in detail.

This paper presents the method for analyzing availability& reliability of repairable systems with-dependent failuresamong components; failure & repair rates of components arefunctions of time. The main contents of this paper are dividedinto three parts:

1) state description of repairable system with-dependentfailures,

2) solutions of system availability & reliability with time-varying failure & repair rates,

3) one kind of special vehicles illustrates system availability& reliability solutions.

II. DESCRIPTION OFSYSTEM STATES

Assumptions

1) System components have 2 discrete states: failed and op-erational.

2) Failure & repair rates of components are functions oftime.

3) State transition occurs instantaneously; the probability fortwo or more states to come into being state-transitions inan interval of time is .

4) A set including components failed is one failure state ofthe system regardless of which one is active or passive infailure mechanism.

5) A system with components has total of states.Regarding Assumptions 4 & 5, [22] considered-depen-

dencies among components, and split component-failures intosingle-failure and joint-failure. This method is complicated& tedious, and can only be suitable to systems with fewcomponents because of the enormous state numbers increasingwith increment of components.

Because the-dependencies among components are consid-ered, there can be an occurrence of compo-nents failing simultaneously, then the system fails. The systemis put into repair after failure. After repair, it is put into work

ZHANG AND HORIGOME: AVAILABILITY & RELIABILITY OF SYSTEM 153

again. The system is always in the failure or operational state.System states are defined as:

0: components of a system are all operational; thus thesystem is operational.

: The system is in 1 of the following 4 failure types,according to the amount of-dependence, which are:

1) Exactly 1 component fails; there are failurestates. These failure states are labeled: .

2) 2 out of components fail at the same time; there is atotal of failure states. These failure states are labeled:

, respectively.3) out of components fail simultaneously; there is a total

of failure states. Similarly, these failure states are la-beled:

, respectively,continuing with case that out of components arefailed.

4) All components are failed; there is only one failure state,.

Then . im-plies that all system components operational.

implies that the system is in state(at least 1 compo-nent is failed).

is a stochastic process whose state space is.If failure & repair rates are constant, this stochastic process

becomes a homogeneous Markov process. Thus the homoge-neous Markov chain can be applied to solving such a problem.Otherwise, it is not easy to solve this kind of problem.

III. A VAILABILITY & RELIABILITY OF SYSTEM WITH

TIME-VARYING FAILURE & REPAIR RATES

Assumptions

A0. The system has 3 different components.A1. The system operates under a random environment.

Whenever at least 1 component fails, the system fails.When the system is failed, no other components canfail.

Fig. 1. System state-transition diagram.

A2. At least 2 components can fail simultaneously due to acumulative shock-damage process caused by environ-ment with .

A3. Whenever all components fail simultaneously, they areall replaced by new ones, so that there is a state tran-sition from this state to the system operative state with

; see the system state-transition diagram in Fig.1.

A4. There are 3 repair facilities with the same operationalproficiency; the failure & repair rates in Fig. 1 are allfunctions of time.

Equation (1) is derived from Fig. 1. (Please see the equationat the bottom of the page.) The initial condition of (1) is:

Similarly, this procedure can be applied to a large system withcomponents. There is no difficulty in obtaining the state-transi-tion diagram like Fig. 1 in accordance with failure mechanismsin the system, and repair and/or replacement policies prescribedfor components. Then,

(3)

with initial condition

(1)

(2)


is state-transition rate matrix with order.The and are similar each other; the character-

istics of can be induced from (2).From the state-transition rate matrix, we know that there exist

coupling terms in this group of differential equations, (3), andit contains time-varying-factor ones, that make the solution of.these equations complicated & difficult. Therefore, a numericalsolution is necessary.

If factors in (3) are regarded as constants, then linear constant-factor equations are obtained through Laplace transform of (3).The solution of is:

(4)

is the Laplace transform of is the Laplace oper-ator,

is a polynomial with up to power , and it can beproved that there are negative real roots. These rootscan be obtained with mathematical software. Take the inverseLaplace transform of (4); the result is:

(5)

& are the negative real roots of

From (5), the system transitory availability is

(6)

the steady-state availability is

(7)

Taking all failure states in the state-transition diagram as ab-sorbing states, then

(8)

all failure states containing component. Ifand in (3) are not constant, then (3) becomes a groupof linear time-varying homogeneous equations; its solution is[23]–[25]:

is solution vector of (3), is a nonsingular ma-trix, which is the state-transition matrix of a linear time-varyingsystem and depends only on ; it is a unique solution ofthe matrix differential equation

(9)

with initial condition .The state-transition matrix of a linear time-varying

system, and the state-transition matrix of a linearconstant-factor system are similar, to some extent, in their formand some characteristics, but they are really different from eachother. is a function only of is afunction of and . can be easily given in the matrixexponential form:

(10)

However, in general, can not be obtained in the form of(10) because is time-variant. But, if satisfies thecondition of Lemma 1, then can be written as a matrixexponential:

(11)

Lemma 1: The sufficient condition that the solution,, of matrix differential equation of linear time-varying

system, (9) with initial condition can be ex-pressed as a matrix exponential is: andare commutative:

(12)

It can be proved that if is a matrix with constant ele-ments or a diagonal matrix, then (12) is satisfied. Otherwise,only can be obtained directly by solving (9). That is,when integrating (9) in , then (13) can be obtained:

Hence,

(13)


TABLE IFAILURE DATA OF 1-COMPONENT

FAILURES OF THE3 MOVE-DRIVE SYSTEMS.

To solve numerically, (13) is extended into a series:

The final solution is:

(14)

IV. EXAMPLE

Three special vehicles of the same kind are tested. The purposeis to examine the performance of the move-drive system of thevehicles. The move-drive system consists of 8 LWTSS (loadwheel and torsional shaft suspensions) which can be treated asthe same kind. If one LWTSS is regarded as a component, thenthe move-drive system has 8 components. These 8 componentsare a series system in logic. The test-mile for each vehicle is11 000 km. Because the vehicles often run under severe/harshconditions, the move-drive system endures severe shocks in-duced by the environment so that one or more units can fail si-multaneously. Under such conditions, one system hastotal states. Because there are so many system states, they oughtto be simplified. Since the components are same, integrate thesestates by the following definition:

State0 —components are all in operation,state1 —1 out of components is failed,state2 —2 out of components are failed, and so on. Based

on these considerations, this system has states.Tables I & II show the test data on LWTSS, where repair

data are not involved. First, to estimate the distribution type and

TABLE IINUMBER OF COMPONENTS FAILURES ( ) FOR THE 3

MOVE-DRIVE SYSTEMS.

TABLE IIIFAILURE -RATE FUNCTIONS AND AVERAGE FAILURE RATES.

Fig. 2. System state-transition diagram of move-drive system.

parameters followed by the data in Table I where the data fromdifferent vehicles are differentiated into three groups #1, #2, #3.These data fit 2-parameter Weibull distributions. Estimates ofparameters and failure rate functions for 1 component failingare obtained by MLE, as shown in Table III.

In addition to the data of 1 component failures, the failuredata for 2 or more components failing together are much fewer.Hence, the failure rate functions can be estimated as follows.For example, the average failure rate of 2 components failingsimultaneously is . Therefore, the average failurerate from state 0 to state 2 in Fig. 2 is

km

Similarly, are obtained; and are alsoassigned to km by considering the real situation.

It is not unreasonable to assume that the failure rate functionshave the form . Use the following equation to calculateone of the parameters when the other is given:

Because there are fewer data, use the average failure rates in thereal calculation.


Fig. 3. System reliability vs moved distance.

In this example, there are at most 3 repair facilities for eachvehicle. Let them have the same proficiency in operation andlet the repair rate for a failed component by 1 repair personbe independent of time: the repair rate is constant. Fig.2 showsthe resulting system state-transition diagram. Equation (3) ap-plies, and Eq. (15) (shown at the bottom of the next page). Insert

into (14) to calculate elements of the system state-tran-sition matrix ; then can be derived. In this case,the element in row #1 and column #1 of is obtainedas in (19).

By inserting into (19), is obtained. Conse-quently, (please see the equation at the bottom of the next page).

Figs.3 and 4 show the calculated results of system availability& reliability at various moved miles for km. If failure& repair rates are all constant, it is convenient to obtain systemavailability & reliability, for example, with the averagefailure & repair rates. We often attempt doing so because of thesimplistic mathematics.

In Fig.3, system reliability decreases with increase of. Fig.4shows the relationship of and vs . As the distance

increases, decreases and quickly reduces to itssteady value from the beginning. Therefore, it is clear that thereis a great difference between and . It can be easilyestimated that if the system availability is evaluated by ,then system availability is under-estimated in the former testmile, and over-estimated in the latter test-mile. If failure ratesare larger, such a difference is greater. Fig. 4. System availability vs moved distance.

(15)


APPENDIX

A. Proof of Lemma 1

From (9) and its initial condition , the result is,

Let have the simplistic form,

(18)

If is the solution of (9), it will besatisfied when inserting (18) into it. Then, the lhs & rhs of (9)are, respectively,

Therefore, if

i.e., and are commutative, then

is the solution of (9). Otherwise, can not be expressedas a matrix exponential.

B. Expression of

(Please see the equation at the bottom of the next page.)

ACKNOWLEDGMENT

We are grateful to Professor Way Kuo and the referees fortheir constructive comments & suggestions for improving thispaper.

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(16)

(17)


(19)

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Tieling Zhang received his B.E. (1985), M.E. (1988), and his Ph.D. (1998) invehicle engineering from Beijing Institute of Technology. From 1988 to 1994,he worked as a senior lecturer at Beijing Institute of Printing. Since 1997 Oc-tober, he has carried out his research on system reliability and maintainabilityin Tokyo University of Mercantile Marine aided by the Ministry of Education,Science, Sports and Culture of Japan. He completed two research projects bycooperating with other scholars, which were supported by the National Foun-dation of Natural Science of China. His research interests include simulation,vibration, and stability of dynamic systems, reliability and maintainability ofcomplex systems, and application of neural networks.

Michio Horigome (born in Japan, 1940) received the B.E. from Tokyo Univer-sity of Mercantile Marine and Ph.D. (1984) in engineering from the Universityof Tokyo. From 1985 to 2000 March, he was a Professor of electronic controlengineering at Tokyo University of Mercantile Marine. Since 2000 April, hehas served as the Head of Hiroshima National College of Marine Technology.He was a VC of Marine Engineering Society of Japan from 1996 to 1998; aCo-Chair of ICRMS’1999 Shanghai, China; and now is Chair of Reliability En-gineering Association of Japan and Chair of Reliability & Maintainability Sym-posium sponsored by the Union of Japanese Scientists and Engineers (JUSE).His research interests include Bayes inference, and system reliability & main-tainability. Dr. Horigome is a Member of IEEE.

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