Markov Model - Based Reliability

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Markov Model-Based Reliability and Safety Evaluation for AircraftMaintenance- System Optimization

Galina M . Susova Aviation Register of Interstate Aviation Committee 0 MoscowAndrei N. Petrov M.G romov Flight Research Institute ZhukovskyKey Words: Arcraft, Reliability, Flight safety, Latent failure,Maintenance effectiveness, Optimization.

SUMh.t4RY AND CONCLUSIONRedundant aircraft systems are intended for operationunder established flight m issions and maintenance programs.Systems may have latent failures and their reliability mayvary, depending on previously made scheduled checks andrepairs. System failure modes are a basis fo r assessing failureeffect on flight safety, and are determ ined by the com ponents'failure modes and their sequence of occurrence.Using a model of Markov homogeneous process it iss h o w that probabilities of the aircraft system states can bepresented as a product of two multipliers. The first of themdepends on a components failure rate and flight durationunder cond ition of serviceab ility of all the components attakc-off. It may be determined by known methods using theMarkov processes and Boolean logic. Th e other one dependson the check and repair intervals for the components andsystem states, as well as the flight phase limits. The practicalcalculation formulae are presented. The proposed methodpresents a new practical approac h for aircraft system s analysisand may be used for a wide range of complex systems withlatent failures. The analytical model may be used for solving

a number of practical tasks: redundancy optimization,determining check intervals, optimizing aircraft MinimumEquipme nt List, etc. - in order to minimize operation cost andensure flight safety. Practica l application of this technology tomodern aircraft has shown good results.1. INTRODUCTION

Aviation development over the last 15 years has shown agrowing interest in aircraft technical operating capabilities(flight safety, reliability, testability, and mainta inability). Theaviation industry became involved in the development at allstages of the life-cycle besides high aircraft performance toachieve an econom ical efficiency of operation s, a high level ofsafety, and complete system reliability. This c an be confirmedby a classical optimization interconnections chain. Indeed, agrowth of requirem ents to the co mpo sition and quality of thetasks performed by a n aircraft--a ndeconomic efficiency of itsoperations complicates the aircraft and its systems' design,

Scheduled maintenance, M arkov model, Maintenance system,

and influences the failures rates and effects. To maintain therequired reliability and safety leve ls measures should be takento enhance the system components reliability, redundancy,warning m eans, and special safety systems (both, forewarningthe critical flight modes, and predicting the hazardousconditions).These measures directly affect the aircraft life-cycle cost(through the systems costs, aircraft weight, etc.), and on theother hand, they stipulate new requirements to an aircraftmaintenance system, ground support an d test equipment. Newrequirements cause challenges within aircraft maintainabilityand testability that have to be solved. Some of thesechallenges are: new or improved ground support equipment;on-board and ground test equipment; non-destructiveinspection techmques; and the elaboration of more effectivemaintenance programs to reveal hidden failures of redundantcompone nts and p revent aircra ft system failures evident to thecrew. This helps to ensure the required reliability and safetyof the aircraft, though it considerably affects the dispatchreliability and operating cost due to increa se in labor spent fortroubleshooting complex redu ndan t systems, probable hum anerrors, and "Re-test OK" events, - and finally results in life-cycle cost increases leading to n ew ad vanced requirements forthe aircraft charac teristics.One of the key tools for op timizing operating cost, safety,and reliability is an analytical "reliability - maintenancepolicy" model whch allows the calculation of systemsreliability taking into account different kinds of maintenancechecks and their intervals. There have been a number ofdifferent models proposed in the p ast two decades, but most ofthem are no t really suitable for practical aerospaceapplications ( [ I ] .. I 41). Some of them are too simple toadeqwtely represent complex aerospace systems. Others aretoo sophisticated to be correctly used in every-dayengineering. The proposed Markov-based reliability andsafety evaluatim m ethod is based on mathematically correct,but simplified, quantitative relations between aircraft safety,reliability, and maintenance parameters and providespromising results.The following notation appears in this paper:H - system state (depend s on com ponen ts states),

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Ho- systcm state with fully serviceable components,Has,, (H,) - state to which the system transits afteroccurrence of a,P,..,p,o ailures in the specified sequence,H G FIT>- state to which the system transits afteroccurrence, in any sequence , of a se ries of a,p,. . , p p ailures,

z (z- designations for a specified sequence of failures anda random c ombination of failures respectively,r - system state index equal to the number of failures whichmoved the system from a serviceable condition to the givenstate HZor H; ,h, rate of the p-th mode of the elem ent failure,tn - flight duration,(ti-l, i) - i-th phase of flight limits,Qap ..pa (ti-l , i), QZ(ti-l til - probability of the state l%p...po(Hz)ccurrence on the i-th flight phase,Qup...pa (ti-l , i), Q; (ti-, , i> - probability of the state

H,p...po (H; ) occurrence on the i-th flight phase,"'"upper index, denotes a probability of a system stateunder condition of serviceability of all system componentsbefore next flight,I(ap...po; K ap..pa ;i (K, ,K;. 3- coefficients of the systemmaintenance policy effect on the system failures probabilities-oap .. p ;i, Qo p ..pa ; on the i-th flight phase,Oj,j-th form of maintenance actions,Tj - j-th scheduled maintenance check an d repair interval,T, - individual maintenan ce task interval that is eq ual TjC , - the overall aircraft specific ma inten anc e cost.interval at which state H, is checked and restored,

2. BASICS OF TH E MATH MODEL2 , l . Maintenance System and Mission Param etersAn aircraft operation can be realized by a series of typical

flights. Each flight includes a number of phases with specificlimits for each phase. It is shown in Figu re 1.Iv 3

v 2v 1

0 3- # 2-0 1 a 1

.-"Ati t i t 1 L l t,

The failed components of the aircraft systems may not berestored in-flight. The maintenance system provides forscheduled tasks on the system condition monitoring, andunscheduled tasks on repair of the failed system componentsaccording to the maintenance schedule established for anaircraft type.As a rule; the mainten ance sche dule involves performingseveral forms of maintenance actions 0,.= 1,2,...7J, andeach form (maintena nce check) has its own tasks scope V,,and interval T,, over the entire aircraft service life.Maintenance intervals and tasks are related by followingrules:Ti


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These transitions occur at random instants of time due tothe element failures. The trajectory (1) may be used as a basefor particulare cases. Thus, for example, the process of thesystem transition from the serviceable state Ho to a stateH,P...po (none restored Boolean model ) features acombination of possible transition sequences:Ho -+ H a -+H a p +.. -+Hap. .+ +H ap...pcrHo -+HP +HJ3a+.. -+Hpa. .p +H p a . .po

Ho -+H o -+H op +.. -+H op ..p +H op ..paOtherwise, restoration policy should be taken into account(2)..............................................................................

for calculating a math model.2.3.Reliability AssumptionsBasic system reliability assumptions are as follows:* Element failure rates are constant* For highly reliable aircraft systems the followingA= const (3)Ap t / , (12)Table 1 formulae are based on universal math model andmay be used for reliability and maintainability analysis of a

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broad range of aviation systems considering their specificscope and design, maintenanc e check and repair intervals. Table IForm ulae for probabilities Q, and coefficients K, ,K,;i

In practice, it often appears that the system unserviceablestate is defined only by the modes of a,@,..p,o elementfailures and does not depend on their occurrence sequence asshown in transits (2).The probability of this system failure m ode will beQGti-l,ti)=Q a p ...pu(ti-l,ti)+Qpa..po(ti-l,ti)+*.-

0+Qop . . .p (ti-,,t;)=QO,p . . .po(O,tn) (,9...po; i+ Q p a...po(07tdKpa..po;i+(13)(14)(15)

- -..+QOap ... a h p ..pa; i =Qo up ..pu i(07hW up ..p o i 7whereQoap...pa ;i(O,tn)= (r!)LAp...ph, d/r! ,K =;i= (1/r!)&...pu; i+ &a ...po;+ ...+Kop. . .pa; i ) -

4. RELUBILITYAND SAFETYASSESSMENTAPPLICATIONS4.1. ApproachThe basis for a design evaluation of flight safety andeffectiveness measures are aircraft system failure modes andprobabilities of failure occurrences in Werent phases offlight. Methods of calculating the measures provide forengin eerin g analysis of the element failure mode and failurecombination effects upon the system output characteristicsand, hence, upon the system serviceability as ii whole. As aresult of the analysis a set of the unserviceable system states{H,} is determined. To generate a list of possible systemfailure modes based on the engineering analysis the set of(H,} is divided to compose a group of J disjoint subsets

{H,}!, {H,}2, ... {H,}J in such a way that the systemtransition from the serviceable state to any unserviceable onebelongs to the same subset result in equal variations in thesystem output characteristics set. Transition to any subset ofthe system unserviceable states {H,}, means the systemfailure of R* mode.The proiability QR ~ti.l,t;>of the system failure occurrenceof R, mode in the i-th flight phase is determined from theformula:

QRJ ti-l,ti)= C Qz(tl-l)ti) (16)IHz l jwhere Q& ,ti ) is the probability of the unserviceable stateH, n the i-th phase of flight.In casewhen the state of the system output characteristics(and the system failure mode) depends on the failed elementcompositim, but not of their sequence, the design formula is

For particular z an d failures combinations, theprobabilities of the failure modes are found from equations(9) and (13) using Table 1 formulae.

4.2. Initial DataInitial data to calculate the probabilities of the systemfailure modes for the established maintenance policy are asfollows:1) list of the system element failure modes and failureOccurrence rates;2) list of the system failure modes, and the system state

sets which define each system failure mode;3) flight duration, and tim e limits for the phases which theflight is divided into;4) data on the system unserviceable states (system failures)rates.4.3. ProcessThe calculation procedure for the probabilities of the1)Dimensionless parameters 0i an d n, are found from2) ProbabilitiesQ, nd Q; are determined from equations3) Coefficients K,i or K;-i are calculated from the Tab le 1

system failure modes estim ation includ es the following steps:formula (12).(10) and (14).formulae using the n, values,

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4)P rob tAt ies Q z, l or Q, ,, are obtained by formula (9) or(13), respectively;5) Using formulae (16), (17) the occurrence probabilitiesQRj (ti-, ti) of the system failure modes on the i-th phase of anarbitrary selected flight a re respectively calculated.

4.5. cliclbili(vnnd Sclfit?,i4.ssess~~ientasksThe effect of each ~YYtem ailure m ode on flig ht Safety andFor esam ple. the probability of a catastrophic effect Qcan deffectiveness is ~ v a h a t e dY the engineeringanalysis.departure failure Qdshould be given in the form of

4.4. Exanrple Qc= C C aj;iQRj(kl,ti)=Qc(LTj, f i - l r f i ,tn ,aj;i) (1 s){Hz }j iTO illustrate the method fo r calcu lating the proba bility of

an aircraft system failure during typical flight let us evaluatea simple system with scheduled maintenance (Fig. 3) inaccordance with procedure recommended in par. 4. 2 an d 4.3.

Q F r _ Z , 5: bj;iQRj(fi-l,ti)=Qd(%Tj, ti-l,ti ,tn ,bj;i ) (19)

, @ 30 2

I

@ I2-

F i g . 3 . S a m p l e S y s t e m

~ H z ~ Jwhere aJs1nd bJ,lare the coefficients for the probability ofan accident or dep arture failure due to a system failureRJ.Suppose Qreqland Qreqz re the quantitative requirementsplaced to flight safety and disp atch reliability. Transfo rma tionof the relation s (lS), (19) leads us to inequ alities:Q&,Tj, fi-l,fi 9aj.i )


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maintenance tasks, and the qua litative methods to develop themaintenance intervals for those tasks which are covered bythe proposed mat hem atica l model of scheduled check tasksfrequency influence on functional system reliability, safety,and aircraft effectiveness. The methodology which was usedto develop some aircraft m aintenanc e programs combines thetask - oriented qualitative engineering analysis for PMPs and

mainteiiance tasks selection (known as RCM principles [7])with quantitative methods of rational scheduled tasks intervalevaluation using the proposed math model. It allows formalanalyzing of the influence of possible systems andcomponents failure modes on aircraf t saf~ty~ ispatchreliability, a nd econom ical efficiency.

5.2. Maintenance Intervals Optimization ProcessMaintenance intervals optim ization process includes threemain stages.1. Determination of the unreliability functions (20), (21)and cost function C, n terms of system failure modes withassociated safety and mission completion effect, andrequirements Q values for each system failure underconsideration uS%g the following data: component failuremodes and their probabilities; typical flight and its phasesduration; unknown parame ters of maintenance tasks intervalsto be optimised; scheduled and unscheduled maintenancetasks cost (man- hou rs or money values).2 . Optimization of the individual maintenance tasks

intervals Tu using L agrange's method for convex functionscase. The only problem in this step is to how im plemen t Table1formulae in case of unknown m aintenanc e intervals (meansunknown sequence of failures restoration ).A method is proposed for ranking the Tu alues using thecriteria %, = Lg (C/QU),here rank SU or the U-thmaintenance task is a function of the task cost C, nd sum ofthe possible components failures probabilities Qu which haveto be checked during this task. This assumptio n allows toestablish which m aintenance task should be performed earlierthan others, so if %


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This has bccii achieved not only by the riiaintcriaiiccprogram optimization. but due to the earlie r rectification andcliiiiiiintioii of iiiaintaiiiability rind testability deficiencies.Th e iiiost airplane coniponents (about 100% of Q ~ e s ndiicarly 90 % of the total number of the components) can beinaintaincd on-condition and operated without any lifc tinielimitations.I n this way the airplane compares favorably ibitli itspredecessors (Fig. 4). As a result of the analysis performed-N , %

arid the iniplcmentation of tlic on-condition niaintcnancc tlicspecific labor rates for all basic systems wcrc sulficieiitlyreduced (Fig. 5 ) .But tlic share of avionics and fire supprcssion systems i.11this reduction is n.=: so visible. which is mostly du e to lowcomponents reliability an d necessity of frcqucnt checks ofthese systems. Numerous false ala rm signals (Re-test OKevents) are also typical for this class of systems.

1 0 09 08 07 06 05 04 03 02 01 0n

~ ~ ~ ~( O N O . o f c o m p ~ n e n t s t y p e T ~ m ~ o - .f c o m p o n e n t sI

-. I L - 6 2 I L - 8 6 IL - 9 6 - 3 0 0F i g . 4 . T h e S h a r e o f t h e O n - c o n d i t i o n M a i n t a i n e d C o m p o n e n t s f o r t h e I L - 9 6 - 3 0 0 S y s t e m s C o m p a r e d

t o o t h e r I L A i r p l a n e s ( I L - 6 2 a n d I L - 8 6 ) .

M M H l F H , 0.63x 1 0 0 00 , 1 7 16

0110 , 0 90,080 , 0 70 , 0 60 , 0 50 , 0 40 , 0 30 , 0 20 , O l

02 2 3 8 2 8 3 2 2 7 7 6 2 1 2 6 2 4 2 9 3 5 3 02 3 , 3 1 ,

3 4 , 7 7 ,1 1 0 ,

1 1 3 , 1 1 4F i g . 5 . D i s t r i b u t i o n o f t h e P e r i o d i c M a i n t e na n c e S p e c i fi c L a b o r f o r th e IL - 96-300 S y s t e m sC o m p a r c d t o IL-86 S y s t e m s .

A T A 1 0 0Number

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6. OTENTIAL OUTLOOK BIOGRAPHIESThe future activities of niodel inwroveme nt are focused on Galina M. Susova, DR. SC.

implementation of the digital system reliability assumptionsand possible in-flight repair opportunity. Another modelenhancement is the development of Minimu m Equipmen t Listoptimization technique and ETOPS (twin engine airplanesover ocean with a probable one engine failure allowed)operations certification procedures. There is also atechnological problem of method automation realizing asystematic appro ach to safety, reliability and m aintaina bilityensuring and evaluation. The methodology can easily be usedin other industries also. It can be applied to reliability an dsafety assessment of most complicated systems (power units,ships, locomotives, etc.) with a cyclic operation, allowing aformal description of system potential failures and associatedrisks to be made.ACKNOWLEDGMENTS

The authors express their deep gratitude to all specialistsof M. Gromov Flight R esearch Institute and Ilyushin DesignBureau who gave us a helping hand in refining andimplementation of the proposed method into engineeringpractice. Dr. Valery K. Tomich also did much and indeedshall be regarded as one of the authors. Unfortunately hepassed away in 1993 and we will always keep him in ourmemories.

REFERENCES1 T .C. Sharma, B. ilbermnn, "Reliability Analysis of Redundant Aircrali

Systems w ith Possible Latent Fdlures" Proc.AM. Reliability & Maintainability2. A K. Somd., S. Pahitknr, T. C. Sharmn, "Reliability Modeling of

Systems with Latent Failures UsingMarkov Chains ",Proc. Ann. Reliability &Maintainability Symp., 1993, JM,pp 120-125.3. A K Somnni,T. . S h q h. H. guyen, "Reliability Computation ofSystems with Latent Failures and Moni tor ing" , Proc. Ann Relinbility PMaintaihbility Symp., 1994,Jan,pp 195 - 200.4. L.. Garbelhi, A Altavilla, M. Fenante, "Availability Technique8 andApproach fora Manned Re-entry Vehicle", Proc. Ann. Reliability &

Maintainability Symp., 1994,JM, pp 222-229.5. A N. Petrov, V. A Karpmko, "Experience of Development andCertification of the IL-96-300 Maintenance Program", Proc. of the ALcraftFlight Safety Intl. C o d . , 1993,Sept, p 676481.

6. "Manual for the Designers and O perators on he Civil Aviation AircraftMaintenance Program Development and Cdication (RDK-E)".M. GromovLII, GosNIIGA, 1993.

Symp., 1990,JM, pp 303-308.

7. F. S. Nowlnn, "Re hbility-Cen tercd Maintenance", 1978.8. "AirIineManufactum Maintmsnce Program Development Document"(ATA MSG-3). Revision 2, 1993.

Chief Specialist, Aviation Register, Intentate Aviation Committee,7, Krjijanovsky St., bldg 1, Moscow, 117875 RUSSIAFax: (095)125-5195E-mail: [email protected] M. Susova received her MS n Mechanics from the Moscow StateUniversity and Dr. Sc. in Engineering from the M. Gromov Flight Research

Institute in 1974. Since 1991 Mrs. Susov a is a Chief spcialist at the AviationRegister of the Interstate Aviation Committee. She delivers lectures onreliability for students at the Moscow Aviation Institute. Prior to this assignmentMrs. Susova accrued for tw o decades at the M.Gromov Flight Rwearch Instituteof Russia, working her way h mMEngineer up to the Chief of the RelinbilityLaboratory. She took part in reliability nnnlyair of a number of both civil endmilitary aircratl, testing and operating data. MIS. Susovn's resea rch intere stsinclude reliability, safety and maintainability methods for complex systems,based onMarkov models, aircraft reliability growth m-ent, data bases, andexpnt systems design. On he sub ject she participated in turning out more than25 papers and several Industry standnrds.She is a member of S A E .

Andrei N. Petrov, DR. SC.Chief of Division 4, M. Gromov L II/Flight Research InstituteZhukowky-2, Moscow Region, 140 160 RUSSIAE-mail: [email protected] a : (095) 556-5334

Andrei N. etrov entered Rursia State Flight Research Institute nnmed afterM. romov Russian abbreviation -M. romov LII) upon grnduation&om theMoscow Avidon Institute with a MS degree in &eraft MechanicalEngineering, were he worked since 1980 to present time on engineering andmanagerial positions. Since 1993 he is a Chief of the Safety, Reliability,Maintenance , nd Test Research Division in the Institute. He was closelyinvolved in development,testing and certifcation of number of both civil andmilitary ai rcraft (SU-27, Yak-42, IL-96-300, IL-114, Tu-204, etc.) with themain activity in maintenance progradmaintainability OptimiLation andcertification. He adso participated M n working group mm b r or *e projoctmanager in the development of many tochnicd and regulatory documents(specifications, natiod/indu&y stmd.rb, nd replntions) on nn aircrafttechnical opentiag capabilities (flight safety, reliability. mnintninnbility,teatability) evaludon and ccrtifcntion. Mr. etrov received his Dr.Sc. degreein AircrafWSystmuTesting and evnuation from he Flight Research Institute in1989.He has some ducational experience eaching he methodsofmaintenanceprogram development and cd ic at io n. Mr. Petrov is an author of a number ofpublications (standards,handbooks, printed pnpcrs). He is a member ofSocietyof Flight Test Engineers.

36 1997 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium

Markov Model - Based Reliability

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