AVAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKOV REWARD MODEL ABSTRACT

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1 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL M. A. El-Damcese and N. S. Temraz Deparmen of Mahemacs, Faculy of Scence, Tana Unversy, Tana, Egyp ABSTRAT Ths paper descrbes some models and measures of relably for mulsae sysems. The expeced cumulave reward for he connuous me Markov reward models are used for dervng he srucure funcon for a mulsae sysem where he sysem and s componens can have dfferen performance levels rangng from perfec funconng o complee falure. The suggesed approach presens wh respec o he non-homogeneous and homogeneous Markov reward model of wo sochasc process for compuaon of hese avalably and relably measures. A parcular case for hree levels s analyzed numercally by assumng Webull and exponenal dsrbuons for falure and repar mes. Keywords: Markov reward model, demand, mulsae sysem, avalably and relably measures.. INTRODUTION Tradonal bnary-sae relably models allow for a sysem and s componens only wo possble saes: perfec funconng up and complee falure down. However, a sysem can have a fne number of performance raes. And, many real-world sysems are composed of componens ha n her urn can have dfferen performance levels and for whch one canno formulae an all or nohng ype of falure creron. Falures of some sysem elemens lead, n hese cases, only o performance degradaon. Such sysems are called mul-sae sysems MSS []. Tradonal relably heory, whch s based on a bnary approach, has recenly been exended by allowng componens and sysems o have an arbrary fne number of saes. Accordng o he generc mul-sae sysem model [], any sysem elemen j{,,, n} can have k dfferen saes correspondng o he performance raes, represened by he se g j {g j, g j,, g jk }, where g j s he performance rae of elemen j n he sae, {,,, k}. The performance rae G j of elemen j a any nsan s a dscree-sae connuous-me sochasc process ha akes s values from g j : G j g j. The sysem srucure funcon G G,, G n produces he sochasc process correspondng o he oupu performance of he enre MSS. In pracce, a desred level of sysem performance demand also can be represened by a dscree-sae connuous-me sochasc process W. The relaon beween he MSS oupu performance and he demand represened by wo correspondng sochasc processes should be suded n order o defne relably measures for he enre MSS. For relably assessmen, MSS oupu performance and he desred performance level demand are ofen assumed o be ndependen sochasc processes. In pracce, he mos commonly used MSS relably measures are probably of falure-free operaon durng me nerval [, ] or MSS relably funcon R, MSS avalably, mean me o MSS falure, mean accumulaed performance defcency for a fxed me nerval [, ], and so on. Many echncal sysems are subjeced durng her lfeme o agng and degradaon. Afer any falure, manenance s performed by a repar eam. Manenance and repar problems have been

2 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember wdely nvesgaed n he leraure. [], [], [] survey and summarze heorecal developmens and praccal applcaons of manenance models. Agng s usually consdered as a process whch resuls n an age-relaed ncrease of he falure rae. The mos common shapes of falure raes have been observed n [], []. An neresng approach was nroduced n [], where was shown ha agng s no always manfesed by he ncreasng falure rae. Afer each correcve manenance acon or repar, he agng sysem s falure rae can be expressed as: q q, where q s an mprovemen facor ha characerzes he qualy of he overhauls q and * s he agng sysem s falure rae before repar []. If q, means ha he manenance acon s perfec sysem becomes as good as new afer repar. If q, means ha he faled sysem s reurned back o a workng sae by mnmal repar sysem says as bad as old afer repar, n whch falure rae of he sysem s nearly he same as before. The mnmal repar s usually approprae for mul-sae sysems. In such suaon, he falure paern can be descrbed by non-homogeneous Posson process NHPP. Incorporang he me-varyng falure nensy no exsng Markov model was suggesed n [] for relably modelng of hardware/sofware sysems. More deals and neresng examples one can fnd n []. Based on hs, he exended approach s suggesed, whch ncorporaes he me-varyng falure nensy of agng componen no Markov reward model ha s usng for general relably measures evaluaon of non-agng MSS []. Such unfed model wll be called as a non-homogeneous Markov reward model. Ths paper consders measures of avalably and relably for a mul-sae sysem where he sysem and s componens can have dfferen performance levels rangng from perfec funconng o complee falure. In secon a general approach s presened for he compuaon of man MSS relably measures. Ths approach s based on he applcaon of he Markov reward model. The man MSS relably measures can be found by correspondng reward marx defnons for hs model and hen by usng a sandard procedure for fndng expeced accumulaed rewards durng a me nerval [, ] as a soluon of a sysem of dfferenal equaons. In secon a general approach s presened for compung relably measures for agng MSS under correcve manenance wh mnmal repar. Ths approach s based on non-homogeneous Markov reward model, where specfc reward marx s deermned for fndng any relably measure. Ths chaper s based on [], [], and presens a model represenng demand as a connuous-me Markov chan wh hree logc levels. In secon we nroduce llusrave example n order o llusrae he approaches. *. MARKO REWARD MODEL FOR MULTI-STATE SYSTEM.. Generalzed MSS Relably Measures The MSS behavor s characerzed by s evoluon n he space of saes. The enre se of possble sysem saes can be dvded no wo dsjon subses correspondng o accepable and unaccepable sysem funconng. MSS enrance no he subse of unaccepable saes consues a falure. The MSS relably can be defned as s ably o reman n he accepable saes durng he operaon perod. The sysem sae accepably depends on he relaon beween he MSS oupu performance and he desred level of hs performance demand W ha s deermned ousde he sysem. Ofen he demand W s also a random process ha can ake dscree values from he se

3 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember w {w,, w M}. The desred relaon beween he sysem performance and he demand a any me nsan can be expressed by he accepably funcon G, W. In many praccal cases, he MSS performance should be equal o or exceed he demand. So, n such cases, he accepably funcon akes he followng form: G, W G W and he creron of sae accepably can be expressed as: G, W. A general expresson defnng MSS relably measures can be wren n he followng form: R EF G, W, where E expecaon symbol, F funconal ha deermnes correspondng ype of relably measure, and accepably funcon. Many mporan MSS relably measures can be derved from he expresson dependng on he funconal F ha may be deermned n dfferen ways. For example, may be a probably Pr G, W hroughou a specfed me nerval [, ] and he accepably funcon wll be nonnegave. In hs case, hs probably characerzes MSS avalably. I may be also an expecaon of an approprae funcon up o he me of he MSS, s nal enrance no he se of unaccepable saes, where G, W s he number of such enrances whn me nerval [, ] and so on. For a power sysem where he avalable generang capacy a me nsan s G and he correspondng load demand s W, f he accepably funcon s defned as: W G, G, W, f W G f W G A funcon, F T G, W G, W d, wll characerze an accumulaed performance defcency durng me nerval [, T]... Markov Reward Model: General Descrpon The general Markov reward model was nroduced n []. I consders he connuous-me Markov chan { X } wh a se of saes {,, k} and a ranson nensy marx A [a j ],, j,, k. I s assumed ha whle he process s n any sae durng any me un, some money r should be pad. I s also assumed ha f here s a ranson from sae o sae j he amoun r j wll be pad. The amouns r and r j are called rewards. Rewards can be negave whle represenng a loss or penaly. Such a reward process assocaed wh s saes or/and ransons s called a Markov process wh rewards. For such processes, n addon o he ranson nensy marx, a reward marx r [rj ],, j,, k should be deermned. The man problem s o fnd he oal expeced reward accumulaed up o me nsan under specfed nal condons.

4 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Le denoes he oal expeced reward accumulaed up o me a sae. The followng sysem of dfferenal equaons mus be solved under he nal condons:,,, k n order o fnd he oal expeced reward. d d r k j, j a j r j k j a, j j,, k Markov reward models are wdely used n fnancal calculaons and operaons research []. General Markov reward models for sysem dependably and performably analyss one can fnd n [], [], and []. Here he new approach s presened where he man MSS relably measures can be found by deermnaon of he correspondng reward marx. Such an dea was prmarly nroduced for a bnary-sae sysem and consan demand n []. In hs chaper, he approach s exended for mul-sae sysems and varable demand... Rewards Deermnaon for MSS Relably ompuaon MSS nsananeous pon avalably A s he probably ha he MSS a nsan s n one of he accepable saes: A Pr G, W. The MSS average avalably A s defned n [] as a mean fracon of me when he sysem resdes n he se of accepable saes durng he me nerval [, ], A A d. In order o assess A for MSS he rewards n marx r for he MSS model should be deermned n he followng manner: The rewards assocaed wh all accepable saes should be defned as one. The rewards assocaed wh all unaccepable saes should be zeroed as well as all rewards assocaed wh all ransons. The mean reward accumulaed durng nerval [, ] wll defne a me ha MSS wll be n he se of accepable saes n he case when he sae s he nal sae. Ths reward should be found as a soluon of he sysem. Afer solvng and fndng, MSS average avalably can be obaned for every nal sae,, k, A. Usually, he nal sae s assumed as he bes sae. Mean number N f of MSS falures durng me nerval [, ] measure can be reaed as he mean number of MSS enrances o he se of unaccepable saes durng me nerval [, ]. For s compuaon he rewards assocaed wh each ranson from he se of accepable saes o he se of unaccepable saes should be defned as one. All oher rewards should be zeroed. In hs case mean accumulaed reward wll defne he mean number of enrances n he unaccepable area durng me nerval [, ]: N. f

5 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Mean me o falure MTTF s he mean me up o he nsan when he MSS eners he subse of unaccepable saes for he frs me. For s compuaon he combned performancedemand model should be ransformed; all ransons ha reurn MSS from unaccepable saes should be forbdden, because for hs case all unaccepable saes should be reaed as absorbng saes. In order o assess MTTF for MSS he rewards n marx r for he ransformed performancedemand model should be deermned n he followng manner: The rewards assocaed wh all accepable saes should be defned as one. The rewards assocaed wh unaccepable absorbng saes should be zeroed as well as rewards assocaed wh ransons. In hs case mean accumulaed reward wll defne he mean me accumulaed up o he frs enrance no he subse of unaccepable saes or MTTF. Probably of MSS falure durng me nerval [, ]: The model should be ransformed as n he prevous case; all unaccepable saes should be reaed as absorbng saes, and herefore all ransons ha reurn MSS from unaccepable saes should be forbdden. Rewards assocaed wh all ransons o he absorbng saes should be defned as one. All oher rewards should be zeroed. Mean accumulaed reward wll defne for hs case he probably of MSS falure durng me nerval [, ] f he sae s he nal sae. Therefore, he MSS relably funcon can be obaned as: R, where,, k.. NON-HOMOGENEOUS MARKO REWARD MODEL FOR AGING MULTI-STATE SYSTEM UNDER MINIMAL REPAIR.. Model Descrpon The MSS oupu performance G a any nsan s a connuous-me Markov chan ha akes s values from he se g {g,, gk}, G g, where g s he MSS oupu performance n sae,,, k. For Markov MSS ranson raes nenses a j beween saes and j are defned by he correspondng sysem falure j and repar j raes. The mnmal repar s a correcve manenance acon ha brngs he agng equpmen o he condons was n jus before he falure occurrence. Agng MSS subjec o mnmal repars experences relably deeroraon wh he operang me,.e., here s a endency oward more frequen falures. In such suaons, he falure paern can be descrbed by a Posson process whose nensy funcon monooncally ncreases wh. A Posson process wh a non-consan nensy s called nonhomogeneous, snce does no have saonary ncremens []. I was shown see, for example, [] ha NHPP model can be negraed no he Markov model wh me-varyng ranson nenses a. Therefore, for agng MSS ranson nenses correspondng o falures j j of agng componens wll be funcons of me... Non-Homogeneous Markov Reward Model a j For non-homogeneous Markov model a sysem s sae a me can be descrbed by a connuous-me Markov chan wh a se of saes,, k and a ranson nensy marx A [a ],, j,, k, where each ranson nensy may be a funcon of me. For such j

6 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember model, n addon o he ranson nensy marx, a reward marx deermned []. r [rj ],, j,, k should be Le be he expeced oal reward accumulaed up o me gven he nal sae of he process a me nsan s sae. Howard dfferenal equaons [] wh me-varyng ranson nenses a j should be solved under specfed nal condons n order o fnd he oal expeced rewards: d d r k j, j a r j j k j a, j j,, k In he mos common case, MSS begns o accumulae rewards afer me nsan, herefore, he nal condons are:,,, k If for example he sae k wh he hghes performance level s defned as he nal sae, he value k should be found as a soluon of he sysem. I was shown n [] and [] ha many mporan relably measures for non-agng MSS can be found by deermnaon of rewards n a correspondng reward marx. Here hs approach s exended for agng MSS under mnmal repar. And, noce ha he approach s appled only for mnmal repar... Rewards Deermnaon for ompuaon of Dfferen Relably Measures for Agng MSS.. The relably measures can be deermned by he same manner as was ndcaed n secon. ILLUSTRATIE EXAMPLE onsder he ar-condonng sysem used n a hospal. The sysem consss of hree dencal ar condoners whch are conneced n parallel. Demand s a connuous-me Markov chan wh hree levels: peak, mddle, and low. The sae-space dagram for hs sysem s presened n fgure. There are saes. Saes from o assocaed wh he low demand perod, saes from o assocaed wh he mddle demand perod, and saes from o assocaed wh he peak demand perod. Saes,, and ndcae all componens work, he sysem performance s g g. Saes,, and ndcae wo componens work and he hrd componen g faled, he sysem performance s g g. Saes,, and ndcae ha one g g g g g g g componen only works, he sysem performance s. Saes,, and ndcae full sysem falure, he sysem performance s. If n he peak-demand perod he

7 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember requred demand level s w =, n he mddle-demand perod he requred demand level s w =, and n he low-demand perod he requred demand level s w =, hen here are sx accepable saes:,,,,, and. Saes:,,,,, and are unaccepable. Fgure : The sae-space dagram for a sysem wh hree dencal ar condoners The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh he falure of one of he hree condoners and have an nensy of λ. The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh he falure of he second condoner and have nensy of λ. The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh he falure of he hrd condoner and have nensy of λ. The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh repar of one of he hree faled condoners and have nensy of µ. The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh repar of one of he wo faled condoners and have nensy of µ. The ransons from sae o sae, from sae o sae, and from sae o sae are assocaed wh repar of he faled condoner and have nensy of µ. The ransons from sae o sae, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. The ransons from sae o sae, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. The ransons from sae o sae, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. The ransons from sae o sae

8 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. The ransons from sae o sae, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. The ransons from sae o sae, from sae o sae, from sae o sae, and from sae o sae are assocaed wh a varable demand and have nensy of λ. In order o fnd he MSS average avalably we should presen he reward marx A r n he followng form: ] [ j r A r In hs marx, rewards assocaed wh all accepable saes are defned as one and rewards assocaed wh all unaccepable saes are zeroed as well as all rewards assocaed wh all ransons. The sysem of dfferenal equaons can be wren n order o fnd he expeced oal rewards.,,, The nal condons are:.,,, Afer solvng hs sysem and fndng, MSS average avalably can be obaned as follows:, A where he -h sae s he nal sae. A

9 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember d d d d d d d d d d d d d d d d d d d d d d d d where,

10 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember In order o fnd he mean oal number of sysem falures we should presen he reward marx N r n he form. In hs marx he rewards assocaed wh each ranson from he se of accepable saes o he se of unaccepable saes should be defned as one. All oher rewards should be zeroed. ] [ j r N r The followng sysem of dfferenal equaons can be wren n order o fnd he expeced oal rewards.,,, d d d d d d d d d d d d d d d d d d d d d d d d N f

11 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Here,, are calculaed va formulas. The nal condons are:,,,. Afer solvng hs sysem and fndng, he mean oal number of sysem falures N f can be obaned as follows: Nf, where he -h sae s he nal sae. In order o calculae he mean me o falure MTTF, he nal model should be ransformed; all ransons ha reurn MSS from unaccepable saes should be forbdden and all unaccepable saes should be reaed as absorbng saes. The ransformed model s shown n fgure. Fgure : The sae-space dagram for he ransformed sysem wh hree dencal ar condoners wh absorbng saes In order o assess MTTF for MSS, he rewards n marx r for he ransformed model should be deermned n he followng manner. The rewards assocaed wh all accepable saes should be defned as one and he rewards assocaed wh unaccepable absorbng saes should be zeroed as well as all rewards assocaed wh ransons. The reward marx r for hs sysem s as follows:

12 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember ] [ j r r The followng sysem of dfferenal equaons can be wren n order o fnd he expeced oal rewards.,,,,,,,,, d d dd d d d d dd d d d d d d d d where,

13 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember The nal condons are:.,,,,,,,,, Afer solvng hs sysem and fndng, he MTTF for MSS can be obaned as, where he -h sae s he nal sae. To calculae he probably of MSS falure durng me nerval [, ] he model should be ransformed as n he prevous case: all unaccepable saes should be reaed as absorbng saes and, herefore, all ransons ha reurn MSS from unaccepable saes should be forbdden. Rewards assocaed wh all ransons o he absorbng sae should be defned as one. All oher rewards should be zeroed. The reward marx r for hs sysem s as follows: ] [ j r r Mean accumulaed reward wll defne he probably Q of MSS falure durng me nerval [, ]. The followng sysem of dfferenal equaons can be wren n order o fnd he expeced oal rewards.,,,,,,,,, d d d d d d d d d d d d d d d d d d Here,,,,,,,,, are calculaed va formulas. The nal condons are:,,.,,,,,,,

14 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Afer solvng hs sysem and fndng, MSS relably funcon can be obaned as R, where he -h sae s he nal sae. Now, we consder wo ypes of he parameers as follows: The ar condoners falure and repar raes are me-varyng As a parcular case, we assume ha he workng me and he repar me of each condoner are boh Webully dsrbued. We can hen wre: Usng MAPLE program, he MSS average avalably A agans me s llusraed n fgure wh numercal soluons based on Runge-Kua mehod..... Fgure : The average avalably A versus he me case Smlarly, he mean oal number of sysem falures N f, he MTTF for MSS, and he MSS relably funcon R agans me are llusraed n fgures,, and, respecvely.

15 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Fgure : The mean oal number of sysem falures versus he me case N f Fgure : The MTTF for MSS versus he me case Fgure : The MSS relably funcon R versus he me case

16 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember The ar condoners falure and repar raes are consan: As a parcular case, we assume ha he workng me and he repar me of each condoner are boh exponenally dsrbued. We can hen wre: Usng MAPLE program, he MSS average avalably A agans me s llusraed n fgure wh soluons based on Laplace ransform mehod. Fgure : The average avalably A versus he me case Smlarly, he mean oal number of sysem falures N f, he MTTF for MSS, and he MSS relably funcon R agans me are llusraed n fgures,, and, respecvely. Fgure : The mean oal number of sysem falures N f versus he me case

17 M. A. El-Damcese and N. S. Temraz AAILABILITY AND RELIABILITY MEASURES FOR MULTISTATE SYSTEM BY USING MARKO REWARD MODEL RT&A # ol., Sepember Fgure : The MTTF for MSS versus he me case Fgure : The MSS relably funcon R versus he me case me o MSS falure, mean accumulaed performance defcency for a fxed. ONLUSIONS. Exenson of connuous-me Markov chan o Markov reward models make hem even more useful.. A Markov reward models was developed as he bass for he generalzed compuaon of avalably and relably measures.. The mehod has been suggesed for he compuaon of MSS relably and avalably measures based on a dfferen reward marx deermnaon for he Markov reward model.. A Markov reward models s well formalzed and suable for praccal applcaon n relably engneerng.. The numercal resuls are presened n order o llusrae he suggesed model.

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