A Markov system dynamics (MSD) based availability and reliability analysis of a process industry

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1 CDQM, Volume, Number, 009, pp. 9-8 COMMUNICATIONS IN DEENDABILITY AND QUALITY MANAGEMENT An Inernaonal Journal UDC 9.7:.0/.0 A Markov sysem dynamcs MSD based avalably and relably analyss of a process ndusry Meesala Srnvasa Rao and V. N. A. Nakan Relably engneerng cenre, Indan Insue of Technology, Kharagpur 70, Wes Bengal, Inda E mal: msrsrnvasa@gmal.com Relably engneerng cenre, Indan Insue of Technology, Kharagpur 70, Wes Bengal, Inda E mal: nakan@hjl.kgp.erne.n acceped June, 009 Summary Sysem relably/avalably s consdered as an mporan performance ndex. erformance of engneerng sysems can be assessed by varous echnques. Ths paper proposes a mehod o compue relably and long run avalably of he man pars of a Jue processng plan. Ths plan consss of egh sub sysems workng n a seres. Two subsysems, namely Morah machne and packagng devce, are suppored by sandby uns wh perfec swch over devces and he remanng sx subsysems are prone o falure. Relably, avalably and MTBF of he seral process n he Jue processng plan have been compued for varous choces of falure and repar raes of sub sysems of hs plan usng a novel Markov sysem dynamcs MSD smulaon approach. The proposed approach and analyss for relably, long run avalably and mean me before falure of he Jue processng plan can help n ncreasng s producvy and performance. Indusral mplcaons of he resuls have also been brefly dscussed. Key words: Sysem relably/avalably, modelng, Markov processes, Markov sysem dynamcs smulaon approach, producvy and performance, ndusral mplcaons.. INTRODUCTION Avalably s he mos mporan ermnology used for evaluaon of he effecveness of any ndusral plan, where mos of he machnes are reparable sysems. Avalably s an mporan 9

2 characersc of a reparable sysem. I s a measure of sysem performance. I s a combned measure of relably and mananably. For a reparable sysem, s also a beer measure of sysem performance han relably. The relably of a sysem s he probably of falure free operaon and s generally a monooncally decreasng funcon of me. The mananably of a sysem s he probably ha he faled sysem s resored o operable condon n a specfed me. A sysem s avalable f s n a usable sae. Avalably s defned as he fracon of me a sysem performs s specfed funcons under saed condons of use and manenance. And also avalably, whch defnes connuy of servce, s he proporon of me a sysem s n a usable sae. I s an mporan arbue of fness for use. I s herefore of hgh sgnfcance n equpmen replacemen analyss and s producvy analyss. Relably and avalably engneers are ofen called upon o make decsons as o wheher o mprove a ceran componen or componens n order o acheve beer resuls. In recen years relably, avalably and mananably RAM conceps have been wdely appled n producon plans as a ool for plan manenance as well as for performance monorng. Relably and mananably conceps are manly applcable a he desgn sage of a machnery or plan layou, whle he avalably concep s mosly applcable afer commssonng he plan or afer a seady sae of producon s reached. Among several avalable mehods, Markov mehod s wdely used for relably/avalably analyss. A Markov chan analyss looks a a sequence of evens, defned as ransons beween saes, and calculaes he relave probably of encounerng hese evens n boh he shor run and he long run. A Markov chan s useful for analyzng dependen random evens ha s, evens whose lkelhood depends on wha happened las [,]. Markov chan models provde accurae long run avalably and falure characerzaon calculaons, and can somemes be solved analycally, bu are hard o formulae and nvolve hgh compuaonal effor parcularly as he number of saes grows large. The soluon procedure of hese models s also mahemacally nensve. And also n he leraure several research papers and books have been publshed ha dscuss varous facs of relably/avalably echnology [,,,,7]. In hese papers, auhors used eher Laplace ransforms mehod or Lagrange s mehod o solve Chapman Kolmogorov dfferenal equaons assocaed wh a parcular problem. I has been observed ha hese mehods nvolve complex compuaons and s very dffcul o calculae relably/ avalably of he sysem by hese mehods. The exsng leraure shows ha f he falure me and/or he repar me dsrbuons are no exponenal, he analycal expresson for he avalably becomes dffcul. Therefore, a smulaon echnque s requred for esmang he avalably [8, 9, 0, ]. And also many sudes have been performed o mprove and opmse he avalably of a sysem hrough dfferen mehods and echnques, e.g. [,, ]. Many researchers have been searchng for alernae mehodologes for more praccal and realsc avalably analyss. Smulaon has been used as a powerful ool for modelng and analyss of sysem avalably. I s used o represen he dynamc behavor of sysems n he mos realsc sense. The presen work proposes a hybrd approach called as Markov Sysem Dynamcs MSD approach whch combnes he Markov approach and sysem dynamcs smulaon for relably/avalably analyss and o sudy he dynamc behavor of mananable sysems n a Jue processng plan. To llusrae he proposed mehodology, Sr Seea Rama Jue Twne Mlls v Ld., Rajam, Andhra radesh Inda has been consdered as a case sudy n hs paper. The remanng par of he paper s srucured as follows. Secon gves he modelng ams and approach. Secon gves a Markov sysem dynamcs approach o sysem avalably/relably assessmen. Secon gves he relably and avalably assessmen of a Jue processng plan wh resuls and analyss and secon gves he concludng commens. 0

3 . MODELING AIMS AND AROACH Smulaon has been used as an approxmaon ool o remedy he lmaons of analycal Markov chans. I has been proved by he auhors Srnvasa and Nakan [,, 7, 8, 9] ha he saonary, connuous me Markov models are algebracally equvalen o lnear sysem dynamcs models. For he enre dervaon, he neresed readers can refer he quoed papers. From hs, a sysem dynamcs represenaon of Markov models opens up he possbly of numercal soluon and of avodng he edum of analycal soluon. Anoher advanage of sysem dynamcs modelng s ha s easy o expermen wh alernave values of parameers. Hence sensvy analyss can be performed easly durng relably/avalably esmaon and predcon. Fnally, he seady sae soluons for hese problems can be obaned easly only by vsual nspecon of he flow dagrams and by makng use of he fac ha he ne flow no a level s zero n he seady sae. Ths has movaed us o propose a novel hybrd approach called as Markov Sysem Dynamcs MSD approach whch combnes he Markov approach wh sysem dynamcs smulaon approach o overcome some of he lmaons of Markov process n a smple and effcen way for me dependen relably and avalably analyss. The auhors beleve ha hs echnque wll provde he mos realsc me dependen relably and avalably analyss.. A MARKOV SYSTEM DYNAMICS MSD AROACH TO SYSTEM AVAILABILITY/RELIABILITY ASSESSMENT The procedure by Srnvasa and Nakan [,, 7, 8, 9] o develop he avalably/relably sudy usng connuous me sysem dynamcs smulaon has he followng fve seps.. Sep : Sysem s Saes Descrpon The frs sep of he sudy s he denfcaon of he saes of he sysem ha means he selecon/deermnaon of he sysem s or componens whn he sysem funconal or up and down saes blocks, and how hey relae o each oher. A sae ranson dagram provdes he oupu of a sysem as he oucome of a jon even defned by he npus o he sysem and s varous saes. Sae ransons correspondng o dfferen subsysems are combned ogeher o form a sae ranson dagram represenng he sae ranson characerscs of he combned sysem [0]. As a resul of hs sep, we wll oban a sae ranson dagram of he sysem ha conans he relaons among s componens saes and her relably feaures.. Sep : Daa Collecon Before sarng o buld he smulaon model n sep, we need o know he desgn, he complee axonomy of componens of he sysem, and we wll ry o fnd ou full relably and mananably nformaon of each em []. Once he number of saes and her neracons are denfed, s requred o defne from one sae o anoher wo caegores of daa: falure raes, and repar/ resoraon and prevenve manenance mes and dependences. In erms of componens falure rae and repar dae daa nformaon, here are several sources o fnd hs nformaon: publc daa books and daabanks, performance daa from he acual plan, exper judgmens, and laboraory esng. Once he daa for each funconal block componen s gahered, MTBF and MTTR can be calculaed for each funconal block of he sysem aendng o her confguraon and probably rules.

4 . Sep : Buldng he Markov Sysem Dynamcs Smulaon Model In hs sep, he sae ranson dagram of he sysem wll be convered no equvalen sock and flow dagram accordng o he sysem dynamcs approach. The resulng model s called as Markov sysem dynamcs model. Accordng o hs approach, he saes n he sae ranson dagram are equal o he sock varables and he sae ransons are equal o he rae varables.. Sep : Markov Sysem Dynamcs Smulaon In smulang he Markov sysem dynamcs model, several avalable sysem dynamcs sofware packages can be used. In hs work Sella sofware s used for developng varous scenaros and o oban he requred smulaon replcaons and he resuls.. Sep : Resuls and Analyss Ths sep wll nclude he presenaon of resul for he avalably and relably parameers correspondng o he funcons of our neres n he dfferen confguraons. These resuls wll laer requre her dscusson when compared wh avalably and relably requremens ha may be esablshed for he funcons provded by he sysem. Ths sep mples explanng he resuls obaned wh he smulaon, and he facors ha may lead o hose resuls, bu also provdng possble acons o mprove sysem s avalably or relably o mee sysem s funconal requremens. Anoher mporan aspec of he sudy ha has o be nroduced a hs me s he sensvy analyss. Once npu parameers may no be very accurae somemes, he nfluence ha parameers have on he fnal resuls, especally hose more mporan and unceran for he sudy, mus be explored. Ths proposed approach has been dscussed and llusraed wh a case sudy n deal by akng he case of a Jue processng plan as follows.. RELIABILITY AND AVAILABILITY ASSESSMENT OF A JUTE ROCESSING LANT USING MARKOV SYSTEM DYNAMICS MSD AROACH The approach of sysem dynamcs was creaed and developed n he lae 90s by a group of researchers led by Forreser a he Massachuses Insue of Technology MIT, Cambrdge, MA []. I s a mehodology for modelng and redesgnng manufacurng, busness, and smlar sysems ha are par man, par machne [,,, ]. I bulds on nformaon feedback heory, whch provdes symbols for mappng sysems n erms of dagrams and equaons, and a programmng language for conducng compuer smulaons. Hence, n hs paper, sysem dynamcs SD s seleced as he smulaon analyss mehod. Ths SD modelng s carred ou a an aggregae level, whch s more approprae for supporng managemen decson makng han convenonal quanave smulaon. In hs work, nally he Markov analyss procedure s presened hrough he use of a Jue processng plan aken as a case sudy o derve and calculae s relably and avalably. Thereafer he same sysem s modeled by he proposed MSD approach as descrbed n he prevous secon. Ths modelng and analyss seps descrbe he problem, elaborae on he MSD model, and evaluae he smulaon analyss as follows.

5 . Sep : Sysem Analyss Wh Sysem s Saes Descrpon and Assumpons In hs sysem Jue processng plan he followng procedural seps and assumpons are used o analyze he sysem hrough Markov analyss. The sysem, noaons and assumpons A Jue manufacurng plan, dscussed n hs paper consss of egh sub sysems ou of whch Morah machne and packagng devce are suppored by sandby uns wh perfec swch over devces. The mahemacal modelng s carred ou for he sx sub sysems ha are prone o falure: Sofener Machne subsysem S: The funcon of he sofener s o bend and unbend he jue fber whle passng hrough heavy flued rollers. Ths machne s used o sofenng of he jue and has a seres of pars of rollers. The rollers are sprally flued and are placed n he machne as op and boom n one par. The emulson s appled abou he / rd dsance of he machne. Ths bendng and unbendng process also helps even spreadng of he ol and waer and furher down breakng down sffness and warness, dry gum ec,. Sofener machne componens are Flued roller, Bevel gears, bushes, Elecrcal moor, Machne shafs/ pnon gears, Bel pulleys, Emulson ppes. Major falure componens of sofener are Flued rollers, Bevel gears and bushes, pnon gears. Mnor falure componens of sofener are Emulson ppes, Bel pulleys. These all are conneced n seres. Breaker card subsysem B: Jue afer passng hrough sofener wll be send o breaker card. In breaker card he jue wll be send hrough nake rollers and wll be delvered hrough delvery rollers. Roll forms wll also be aached o breaker card machne so ha he oupu from breaker wll be n form of rolls. In he enre process of jue preparaon breaker card s he frs machne n whch he combng process s done. The slver s passed hrough wo ses of rollers whch are surrounded by pns. When he slver s passed hrough hese se of rollers combng operaon s done and he qualy of slver s ncreased. Major falure componens of Breaker card are ns, Rollers and Suds. These all are conneced n seres. Fnsher card subsysem C: Jue afer passng hrough he breaker card wll be nex brough o fnsher card and a hs furher combng process wll be done. The funcon of he Fnsher card s o furher make he slver regular and unform n lengh and wegh. The slver also ges aenuaed. The doublng of Breaker slvers s also done n hs machne by whch he dfferen quales of jue from Breaker card delvery are properly blended n he course of cardng by Fnsher. The cardng process s compleed by Fnsher card. The essenal pars of Fnsher card are Arm lnked bracke, Res bracke, Frcon rollers, Frcon roller bracke, Roll drawng drum, Spur gears.. These all are conneced n seres. Drawng Machne subsysem D: In he drawng machnes furher aenuaon, blendng, unformy and regulary of he slver s acheved. In he fnsher drawng he slver s crmped and delvered n slver cans and made suable for spnnng no yarn. The objecve and doublng s an aemp o equalze he slver n regard o hckness, qualy, color ec., and ulmaely o reduce he wegh of slver unl can be formed no rove on he rovng frame or spun no yarn n he slver spnnng frame. In shor, he hck carded slver s reduced o a suable sze for subsequen operaon. All hs s done n drawng

6 frames. The essenal pars of he drawng machne are Rubber pressng rollers, cam shaf, Drawng rollers, delvery rollers. These all are conneced n seres. Spnnng Machne subsysem : Spnnng s a very mporan sage n fber converson. All he spnnng s carred hrough bobbn holders and he bobbns are resed n bobbn carrer. Spnnng s he las process of yarn converson. Spnnng machne consss of large number of small pars and a flyer whch flls he bobbn carrer wh jue. The essenal pars of spnnng machne are Top rollers, Flued rollers, Rack shaf and Bobbn carrers. These all are conneced n seres. Twsng and Wndng Machne subsysem W: Twsng Machne consss of number of bobbn carrers. In hese bobbn carrers he bobbns are placed and are feed o a same nake roller and he roller wss he yarns. And accordng o hs he ply number depends. If wo are placed n and wsed, s called as wo ply. And hese are sendng o reelng machne o form he fnal producs. The funcon of he wsng machne s o ws he jue. Sngle yarns are wsed ogeher o make a hread or wne whch s requred for specfc purpose such as sewng, carpe makng, selvedges n weavng ec,. Tws wll brough grea unformy o ply yarn he exensbly of he wne, qualy rao, ensle srengh ec., are all dependen on ws n a ply yarns. The essenal pars of wsng machne are pressng roller, yarn gude, Rack shaf, man cylnder. These all are conneced n seres. A flow dagram of hs Jue manufacurng plan s presened n Fgure. In addon o he noaons used for sub sysems,.e. S; B; C; D; and W, we have also used he followng noaons: * W - ndcaes ha sub sysem W s workng n reduced sae. *, =,,.7, represens, respecvely, he consan falure raes of sub sysems S; ; B;C;D; W - and W. *,, =,, ; represens, respecvely, he consan repar raes of sub sysems S;; B;C;D; ; W..* We have also defned =,,..., as he probably ha he sysem s n j h sae a me j. Dash represens dervave wh respec o. * The symbols s; b; c; d; p and w represen he faled sae of he sub sysems S; B; C;D; and W, respecvely. In he presen analyss, s assumed ha: * Repar and falure raes are ndependen of each oher and her un s per day. * There are no smulaneous falures among he sub sysems.

7 * Sub sysem W fals hrough reduced sae only. * Repared componens funcon lke he new componens. * The swchover devces used for sandby sysems are perfec. Based on he above noaons and assumpons, he ranson dagram of he sysem s gven n Fgure. Ths fgure s used as he bass for applyng he mnemonc rule based on Markov analyss ha helps us n consrucng he dfferenal equaons governng he Jue processng plan. Mahemacal formulaon of he sysem To deermne relably and long run avalably of a manufacurng plan, he mahemacal formulaon of he model s carred ou usng mnemonc rule based on he Markov analyss for sx sub sysems. We have analysed wo saes of he sysem, namely ransen sae and seady sae. Transen sae: robably consderaons, usng mnemonc rule based on he Markov analyss, gves he followng sysem of frs order dfferenal equaons assocaed wh he ranson dagram of he sysem as shown n Fgure a me are as follows: = = 7 7 ] [ ] [

8 Dvng boh sdes by, we ge: 7 ] [ = Takng 0, we ge: 7 ' 7 ' Q Q = = Smlarly: ' Q =,,...,, ' = =,...,, 7 ' 7 = = 7 ' = Where: 7 = = Q Q wh nal condons: 0 = j, If j= and 0, oherwse. The sysem of dfferenal equaons o wh nal condons gven n has been solved by Markov sysem dynamcs smulaon approach. The numercal compuaons have been carred ou sarng from me, =0 o =0 days for dfferen choces of repar and falure raes of he sub sysems. The relably R of he sysem can be compued by: R = 7 Seady sae: In process ndusres, managemen s generally neresed n he long run avalably of he sysem. We need he seady sae probables of he sysem n order o calculae s long run avalably. Seady sae probables of he sysem are obaned by mposng he followng resrcons: as d d,, 0 In hs sae equaons o reduce o he followng sysem of equaons. Here, we have used he symbol j for ;,,...,, = j j

9 Q = 7 8 Q = =,..., 0, = =, =,,..., 7 = 7 Solvng hese equaons recursvely, we ge: = Z where: Z 7 = ; = ;, =,,...; = ; Z, =,,... 7 = Now, usng he normalzng condon: = = we ge: [ = Z ] The long run avalably of he sysem A can now be calculaed usng: A = = Z = The long run avalably of he sysem as gven above has been solved by sysem dynamcs smulaon approach. The numercal compuaons have been carred ou for dfferen choces of repar and falure raes of he sub sysems.. Sep : Daa Collecon The proposed MSD mehodology sars afer denfcaon of he sysem saes as menoned n he prevous secon. The remanng sages of radonal Markov analyss are hghly mahemacally nensve, whereas he MSD approach s very smple. Moreover, MSD approach s capable of modelng hose sysems whose falure raes and repar raes have decreasng, ncreasng, or consan rend wh me. I s worh rememberng ha Markov analyss s possble only f falure rae and repar raes remans consan. In he presen sudy, he requred daa for falure and repar raes of all subsysems n he Jue processng plan has been colleced by he auhors from manenance log books of he plan and from conducng nervews wh expers whn he plan n varous vss. Ths daa regardng he falure and repar raes has been used n buldng he proposed Markov sysem dynamcs smulaon model and n s furher analyss as descrbed n he followng seps. 7

10 . Sep : Buldng he Markov sysem dynamcs smulaon model The nex sep n he modelng process s o conver he rae dagram of sysem n o he rae and level dagrams. The rae dagram Fgure of he Jue processng plan s now convered no a comprehensve Markov sysem dynamc model. Ths s presened n Fgure. In he model depced n Fgure, he possble saes of he sx subsysems n he Jue processng plan are ndcaed wh level varables SBW bar CD, where =,,. and he sae ransons are ndcaed wh rae varables R jk, where j,k =,,.., j k. lke R, R ec,. wh he correspondng ranson raes LAMDA.e., n he rae dagram, where =,,.7, whch ndcaes falure raes and also wh he correspondng ranson raes MU.e., n he rae dagram where =,,., whch ndcaes repar raes of he subsysems as per he noaons gven n earler secons. The nal value of sysem relably as ndcaed a he level varable SBW bar CD s assumed as uny. The level of sysem relably wll be decreased by he rae of falures of he subsysems and wll be recovered wh he repars of he subsysems before her falures. These are measured as probably densy funcons pdfs of he subsysems whch are ndcaed as he rae varables and hese are nfluenced by he respecve auxlary varables,.e., falure rae and repar raes of he respecve subsysems along he enre msson or operang me. Addonally, he level of falure sae of he sysem.e., level varable SBwCD s ncreased by hese rae varables of he subsysems, leadng o he declnng relably of he sysem. Fgure depcs he basc srucure of he Jue processng plan relably/avalably model wh he correspondng sock/flow dagram.. Sep : Markov sysem dynamcs smulaon The nex sage of MSD approach s o smulae he comprehensve MSD model of he sysem Jue processng plan. The sae probables of he sysem have been calculaed for me dependen relably and avalably analyss n boh seady sae and ransen saes and also he requred smulaon can be performed o sudy he dynamc behavor of he sysem as follows. The followng algorhm explans he proposed Markov Sysem dynamcs smulaon procedural seps. roposed algorhm: Sep: The values of ranson raes LAMDA.e., n he rae dagram, where =,,.7, and MU.e., n he rae dagram where =,,. and he me nerval d are aken as npus. Also he oal me T s aken as npu,.e. he me for whch he sysem has o be smulaed. Sep: Inally se S equal o one. Sep: Se SBW bar CD equal o zero for =,,... Sep: A condonal loop s formed wh he condon, <T. Sep: In each execuon of he loop, he me s ncreased by d,.e. = d. So, he loop wll connue ll me T wh each sep aken a me dfference d. Sep: As assumed SBW bar CD nally has probably uny, and he sysem fals when becomes zero. Run a condonal loop as long as he condon.e. he probably of SBW bar CD >0 s sasfed. Sep7: Whn he loop all he rae varables are calculaed. The probables of he saes of all subsysems are calculaed frs and he ouflow raes are calculaed accordng o he logc as follows. SBW bar CD = SBW bar CD -d R jk - R kj * d a where =,,, j,k =,,.., j k Sep8: Then all he requred values are dsplayed o sudy he dynamc behavor. 8

11 .. The Model Expermenaon By mplemenng hs algorhm, he model expermenaon can be performed as follows. The smulaon of he proposed model confrms ha he operang sae relably/avalably of he Jue processng plan decreases wh me In hs sudy, accordng o he proposed Markov sysem dynamcs approach, hs model has been smulaed for dfferen choces of repar and falure raes of he sub sysems n he comprehensve MSD model as shown n Fgure by usng he proposed algorhm and performed he requred smulaon runs. The relably of he sysem based on dfferen value combnaons of falure and repar raes s presened n Tables -0. The las row of hese ables gves he MTBF n days for he respecve falure and repar raes of sub sysems. MTBF has been compued wh he help of numercal mehods. The long run avalably of he sysem has been calculaed recursvely for varous value combnaons of repar and falure raes and s presened n Tables-. In hs way, by usng he proposed mehod, he requred sensvy analyss has been performed easly durng relably and avalably analyss. Wh hs analyss, we can fnd ou he crcal subsysem whch effecs more when compared o oher subsysems n he oal relably and avalably of he sysem lke he one whch nvesgaed n hs work as descrbed wh he followng expermens. Numercal llusraons In hs secon, we calculae he relably for ransen sae of he sysem based on emprcal values of falure and repar raes of sub sysems by conducng he followng expermens whn he proposed MSD model by usng he proposed algorhm as follows. Seady sae analyss The long run avalably of he sysem as defned n equaon has been calculaed recursvely for varous value combnaons of repar and falure raes. The effec of change n falure and repar raes of some mporan subsysems on he long run avalably of he sysem s presened n Tables-. Fgure. A comprehensve Markov sysem dynamcs model 9

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16 Expermen : Effec of falure raes of subsysems Sofener and Spnnng machnes on long run avalably: Effec of falure raes of subsysems Sofener and Spnnng machnes on long run avalably of he sysem s suded by varyng her values as = 0.008,0.009,0.0707,0.0,0.0 and = 0.009,0.0,0.0,0.087,0.0. The falure and repar raes of oher subsysems have been aken as: =0.0898, =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =8.7, =., =0.97, =9.0. The long run avalably of he sysem s calculaed usng hs daa and he resuls are shown n Table. Ths able shows ha ncrease n falure rae of spnnng machne has approxmaely 0.9 per cen negave mpac on he long run avalably of he sysem n comparson wh he falure rae of sofener ha has only 0.8 per cen negave mpac on he avalably. Expermen : Effec of falure raes of subsysems Sofener and Breaker card on long run avalably: Here we have used fxed falure and repar raes of all he subsysems oher han sofener and breaker card as: =0.087, =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =8.7, =., =0.97, =9.0. The values of falure raes of subsysems sofener and breaker card have been consdered as =0.008, 0.009, , 0.0, 0.0 and =0.008, 0.0, , 0.0, Agan, he long run avalably of he sysem s calculaed usng hs daa and he resuls are shown n Table. Ths able shows ha ncrease n falure rae of breaker card has approxmaely. per cen negave mpac on he long run avalably of he sysem n comparson wh he falure rae of sofener ha has only 0.8 per cen negave mpac on he avalably. Expermen : Effec of falure rae and repar rae of sofener on long run avalably: We have also calculaed he long run avalably of he sysem afer varyng he falure and repar raes of sofener. Followng daa has been used and resuls are shown n Table. Fve levels each of falure and repar raes of sofener have been consdered as: = 0:008, 0.009, , 0.0, 0.0 and =.0,.,.0,.,.0. These raes for oher sub sysems have been aken as: =0.087, =0.0898, =0.0, =0.09, =0.09, 7 =0.0, =.0, =8.7, =., =0.97, =9.0. Table reveals ha ncrease n falure rae of sofener decreases long run avalably of he sysem and ncrease n s repar rae ncreases. The avalably decreases by 0.70 per cen o 0. per cen wh an ncrease n he falure rae of sofener from0.008 o 0.0 and ncreases by 0.09 per cen o 0. per cen wh an ncrease n repar rae of sofener from.0 o.0.

17 Expermen : Effec of repar rae of spnnng machne and breaker card on long run avalably: Here, we have vared he repar raes of spnnng machne and breaker card as: =.0,.0,.0,.0, 7.0 and = 8., 8., 8.7, 8.8, 8.9. Oher falure and repar raes of he sub sysems have been aken as: =0.0707, =0.087, =0.0898, =0.0, =0.09, =0.09, 7 =0.0, =., =., =0.97, =9.0. Usng hese values, long run avalably of he sysem s calculaed and has been abulaed n Table. Ths able shows ha he repar rae of breaker card does no affec he avalably. However, avalably ncreases approxmaely 0.70 percen whle he repar rae of spnnng machne ncreases from.0 o 7.0. Transen sae analyss The relably of he seral processes n a Jue processng plan has been compued for one year by akng value combnaons of he repar and falure raes of he sub sysems. These value combnaons have been aken from Sr Seea Rama Jue Twne Mlls v Ld., Rajam, Andhra radesh Inda. I may be menoned here ha hese combnaons are no exhausve and we have only consdered he man subsysems n hs sudy. The relably of he sysem based on dfferen value combnaons of falure and repar raes s presened n Tables -0. The las row of hese ables gves he MTBF n days for he respecve falure and repar raes of sub sysems. MTBF has been compued wh he help of numercal mehods. Expermen : Effec of falure rae of sofener on he relably of he sysem: Effec of falure rae of sub sysem sofener on relably of he sysem s suded by varyng s value as: = 0:008, 0.009, , 0.0, 0.0. The falure and repar raes of oher sub sysem has been aken as =0.087, =0.0898, =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =8.7, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table. Ths able shows he effec of falure rae of separaor on he relably of he sysem. The values of have been consdered wh he nerval of days. The relably of he sysem decreases by approxmaely 0. per cen wh he ncrease of me. However, decreases by approxmaely 0.9 per cen wh he ncrease n he rae of falure of separaor from o 0.0 and MTBF decreases by approxmaely 0.7 per cen. Expermen : Effec of falure rae of spnnng machne on he relably of he sysem: Effec of falure rae of sub sysem spnnng machne on relably of he sysem s suded by varyng s value as: = 0.009, 0.0, 0.0, 0.087, 0.0. The falure and repar raes of oher sub sysems have been aken as =0.0707, =0.0898, =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =8.7, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table. Ths able shows he effec of falure rae of spnnng machne on he relably of he sysem. The values of have been consdered wh he nerval of days. The relably of he sysem decreases by approxmaely 0.0 per cen wh he ncrease n me from 0 o 0 days bu relably decreases by approxmaely 0.9 per cen wh he ncrease n he rae of falure of spnnng machne from o 0.0 and MTBF also decreases by approxmaely 0.8 per cen. Expermen 7: Effec of falure rae of breaker card on he relably of he sysem: Effec of falure rae of sub sysem breaker card on relably of he sysem s suded by varyng s value as: =0.008, 0.0, , 0.0, The falure and repar raes of oher sub sysems have been aken as =0.0707, =0.087, =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =8.7, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table7. Ths able shows he effec of falure rae of separaor on he relably of he sysem. The values of have been consdered wh he nerval of days. The ncrease n decreasng he relably of he sysem from 0.09 per cen o 0. per cen wh he

18 ncrease of me. However, he ncrease n decreasng he relably of he sysem from 0.79 per cen o. per cen wh he ncrease n he rae of falure of breaker card from o 0.07 and MTBF decreases by approxmaely.0 per cen. Expermen 8: Effec of repar rae of sofener on he relably of he sysem: Effec of repar rae of sub sysem sofener on relably of he sysem s suded by varyng s value as: =.0,.,.0,.,.0. The falure and repar raes of oher sub sysems have been aken as =0.0707, =0.087, = , =0.0, =0.09, =0.09, 7 =0.0, =.0, =8.7, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table8. Ths able shows he effec of repar rae of sofener on he relably of he sysem.. The relably of he sysem decreases by approxmaely 0.0 per cen wh he ncrease n me from 0 o 0 days. However, ncreases from 0.79 per cen o 0.9 per cen wh he ncrease n he repar rae of sofener from.0 o.0 and MTBF ncreases by approxmaely 0.9 per cen. Expermen 9: Effec of repar rae of spnnng machne on he relably of he sysem: Effec of repar rae of sub sysem spnnng machne on relably of he sysem s suded by varyng s value as: =.0,.0,.0,.0, 7.0. The falure and repar raes of oher sub sysems have been aken as =0.0707, =0.087, = , =0.0, =0.09, =0.09, 7 =0.0, =., =8.7, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table9. Ths able shows he effec of repar rae of spnnng machne on he relably of he sysem. The relably of he sysem decreases by approxmaely 0. per cen wh he ncrease n me from 0 o 0 days. However, ncreases by approxmaely 0.70 per cen wh he ncrease n he repar rae of spnnng machne from.0 o 7.0 and MTBF ncreases by approxmaely 0.8 per cen. Expermen 0: Effec of repar rae of breaker card on he relably of he sysem: Effec of repar rae of sub sysem breaker card on relably of he sysem s suded by varyng s value as: = 8., 8., 8.7, 8.8, 8.9. The falure and repar raes of oher sub sysems have been aken as =0.0707, =0.087, = , =0.0, =0.09, =0.09, 7 =0.0, =., =.0, =., =0.97, =9.0. The relably of he sysem s calculaed usng hs daa and he resuls are shown n Table0. Ths able shows he effec of repar rae of breaker card on he relably of he sysem. The relably of he sysem decreases by approxmaely 0.0 per cen wh he ncrease n me from 0 o 0 days. However, does no change wh he ncrease n he repar rae of separaor from 8. o 8.9 and MTBF also does no change.. Sep : Resuls and Analyss Analyss of relably, long run avalably and mean me before falure of Jue manufacurng plan can help n ncreasng s producvy and performance. Some of he earler research workers, who red o fnd ou relably of he processng sysems by usng Laplace ransformaon mehod or Lagrange s mehod, consderng he long run avalably of process ndusres only and he effecs of falure and repar raes of sub sysems on relably of he sysem could no be esablshed. Ths s because hese mehods are no easly applcable when he sysems are complex. The proposed Markov sysem dynamcs MSD mehod can be appled o complex sysems ha may be governed by a large sysem of dfferenal equaons as well. Usng hs mehod, we can easly sudy he effecs of falure and repar raes of dfferen sub sysems on relably/avalably of he complee sysem. Dealed sudy of Tables -0 reveal ha sub sysem spnnng machne has maxmum effec on he long run avalably and relably of complee sysem. Oher sub sysems are almos equally effecve. Hence, s suggesed ha managemen should ake umos care of hs

19 sub sysem o ncrease overall performance and producvy of he Jue manufacurng plan and o ge more profs.. CONCLUSIONS In hs paper, a hybrd approach called as Markov Sysem Dynamcs MSD approach whch combnes he Markov approach wh sysem dynamcs smulaon approach has been proposed for relably /avalably analyss and o sudy he dynamc behavor of sysems. The proposed framework s llusraed for a Jue processng plan. The resuls of he smulaon have denfed he crcal subsysem spnnng machne whch has maxmum effec on he long run avalably and relably of complee sysem. Oher sub sysems are almos equally effecve. Hence, s suggesed ha managemen should ake umos care of hs sub sysem o ncrease overall performance and producvy of he Jue processng plan and o ge more profs. Ths analyss conforms ha he Markov Sysem Dynamcs MSD approach s an alernave approach for relably and avalably analyss of hs ype of complex sysems. The procedure for he developmen of he MSD approach for hs sysem s explaned and he model s run o observe all of s saes. The proposed mehodology s applcable for all ypes of falure raes and repar raes and s much smpler compared o radonal approaches. Furher, hs mehodology can be used for sudyng varous scenaros havng manageral mplcaons of sysem relably and avalably analyss. I s mporan o noe ha relably/ avalably declnng of componen or sysem has o be observed carefully n order o acheve he desred resuls. Managers mus beware of he exsng nerdependences whn he componen or sysem. Accordngly, he model can be used as a smulaon ool. Based on smulaon analyses managers can learn how o deal wh such a comprehensve approach lke he one nvesgaed n hs paper. And also, he dfferen pares,.e., engneers and machne operaors, can jonly work wh he model n order o undersand he dynamc behavor of sysems. REFERENCES [] Derman C. 9, On opmal replacemen rules when changes of saes are Markovan, Mahemacal Opmzaon Technques, Berkley, CA: Unversy of Calforna ress. [] Cafaro G, Cors F, Vacca F. 98, Mul sae markov models and srucural properes of he ranson rae marx,. IEEE Transacons on Relably, : [] Dhllon B.S and Naesan, J. 98, Sochasc analyss of oudoor power sysem n flucuang envronmen, Mcroelecroncs Relably, : [] Dayal B and Sngh J.99, Relably analyss of a sysem n a flucuang envronmen, Mcroelecroncs Relably, : 0-. [] Goel,. and Sngh, J. 99a, Avalably analyss of buer ol manufacurng sysem n a dary plan, roceedngs of Inernaonal Conference on Operaonal Research for a Beer Tomorrow, 09-. [] Mahajan,. and Sngh, J. 999, Relably of uensls manufacurng plan a case sudy, Opsearch, : 0-7. [7] Goel,. and Sngh, J. 99b, Relably analyss of a sandby complex sysem havng mperfec swch over devce, Mcroelecroncs Relably, : 8-8. [8] Broln, A. 997, Qualy and Relably of Techncal Sysems Theory, racce. Managemen, nd ed., Sprnger Verlag, Berln. 7

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