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2 Char 8 Markov Modlling INTRODUCTION. Gnral.. Par C Char 0 of hi Manual dcrib Rliabiliy Block Diagram analyi, which i rha h mo familiar mhod of analying h funcional rlaionhi of a ym from a rliabiliy andoin. Par C Char 5 dcrib analyical man of rforming calculaion uing RBD, bu h calculaion ar only valid undr crain rriciv aumion.g. indndnc of block, no quuing for rair, c.... Thr ar modlling chniqu ha may b ud a ool o ovrcom om of h rricion. Thy may b groud ino wo gnral y, h Markov modl and h imulaion or Mon Carlo modl Par D Char. Boh modl u h RBD a a man of rrning h funcional lmn of a ym and hir inrrlaionhi. Boh ar alo ochaic modlling chniqu, maning ha hy can dal wih vn having an lmn of chanc and hnc a of oibl oucom a ood o drminiic modlling, whr a ingl oucom i drivd from a dfind of circumanc. In R&M nginring ochaic modlling i ud o dcrib a ym oraion wih rc o im. Th ubym failur and rair im yically bcom h random variabl... For many yar h iz and co of comur caabl of running all bu h iml imulaion modl limid hir u di h advanag ou in Tabl. In rcn yar, howvr, incra in comuing owr and aociad co dcra hav mad Mon Carlo imulaion radily accibl, and hnc h oulariy of Markov ha dclind. Howvr i i ill uful for h analyi of mulil a ym and ho ha xhibi rong dndncy bwn comonn and i ud in commrcial AR&M modlling ool ha u a raniion diagram o calcula rliabiliy and mainainabiliy valu for comlx ym... Ohr raon for h lack of oulariy of Markov ar: Th fac ha i i no an ay ool for nginr o aly and o xlain o ohr. Mon Carlo imulaion rogram xi ha no only modl comlx ym bu alo h u of h ym in comlx oraional cnario. Pag

3 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu MARKOV ANALYSIS MONTE CARLO SIMULATION ADVANTAGES DISADVANTAGES SOURCE A. Simliic Modlling Aroach. Th modl ar iml o gnra alhough hy do rquir a mor comlicad mahmaical aroach. D. Va incra in numbr of a a h iz of h ym incra. Th ruling diagram for larg ym ar gnrally vry xniv and comlx, difficul o conruc and comuaionally xniv. A: Rfrnc D: Rfrnc Vry flxibl. Thr i virually no limi o h analyi. Can gnrally b aily xndd and dvlod a rquird. A. Rdundancy Managmn Tchniqu. Sym rconfiguraion rquird by failur i aily incororad in h modl. D. Markov modlling of rdundan rairabl ym wih auomaic faul dcion and on rair crw i flawd. Thi i bcau alhough random failur i a Markov roc, rair of mulil failur i no a Markov Proc. Th mahmaical dicrancy may b ovrcom by uing a ddicad rair crw r quimn, bu hi do no normally corrond o ral lif uor ragi. A: Rfrnc D: Rfrnc Rdundancy managmn aily handld. A. Covrag. Covrd dcd and iolad and uncovrd undcd failur of comonn ar muually xcluiv vn, no aily modlld uing claical chniqu bu radily handld by Markov mahmaic. A: Rfrnc Covrd and uncovrd failur of comonn radily handld. A. Comlx mainnanc oion can radily b modlld. D. Can only dal wih conan failur ra and conan rair ra - h lar bing unraliic in ral, oraional ym for many raon including, for xaml, changing hyical condiion and variaion in mainnanc kill. Howvr, if h MTTR i vry much horr han h MTTF, hn hi horcoming rarly inroduc riou inaccuracy in h final comud ym aramr. A: Rfrnc D: Rfrnc Comlx mainnanc oion can radily b modlld. A wid rang of diribuion including mirical diribuion can b handld. D5. Fuur a of h ym ar indndn of all a a xc h immdialy rcding on, which imli ha a rair rurn h ym o an a nw condiion. 5D: Rfrnc Can incorora diribuion ha mbrac war-ou condiion. A6. Comlx Sym. Many imlifying chniqu xi which allow h modlling of comlx ym. A6: Rfrnc Comlx ym radily handld. A7. Squncd Evn. Markov modlling aily handl h comuaion of h robabiliy of an vn ruling from a qunc of ub-vn. Thi y of roblm do no lnd ilf wll o claical chniqu. A7: Rfrnc Rfrnc Squncd vn radily handld. Tabl : Advanag and limiaion of Markov Analyi comard o Mon Carlo Simulaion Iu Pag

4 Char 8 Markov Modlling MARKOV ANALYSIS. Gnral.. Markov analyi i a comlx ubjc wih many alicaion ouid h fild of R&M nginring. Mo chnical librari will hav vral book on h ubjc. I i covrd in hi manual inc i i an analyi mhod ha can b alid o crain rliabiliy roblm. Th mhod i bad on an analyi of h raniion bwn ym a. Markov analyi i illurad by xaml in Scion of hi Char... Th bai of a Markov modl i h aumion ha h fuur i indndn of h a, givn h rn. Thi ari from h udy of Markov chain qunc of random variabl in which h fuur variabl i drmind by h rn variabl bu i indndn of h way in which h rn a aro from i rdcor. Markov analyi look a a qunc of vn and analy h ndncy of on vn o b followd by anohr. Uing hi analyi, i i oibl o gnra a nw qunc of random bu rlad vn, which aar imilar o h original... A Markov chain may b dcribd a homognou or non-homognou. A homognou chain i characrid by conan raniion im bwn a. A nonhomognou chain i characrid by raniion ra bwn h a ha ar funcion of a global clock, for xaml, lad miion im. In R&M analyi a Markov modl may b ud whr vn, uch a h failur or rair of an im can occur a any oin in im. Th modl valua h robabiliy of moving from a known a o h nx logical a, i.. from vryhing working o h fir im faild, from h fir im faild o h cond im faild and o on unil h ym ha rachd h final or oally faild a.. Sym Sa and Truh Tabl.. A SYSTEM STATE i a aricular combinaion of h a of h lmn comriing h ym. For xaml, for a ym comriing wo lmn x and y, ach lmn caabl of aking on of wo a u or down, hr ar oibl ym a: a x u b x u c x down d x down y u y down y u y down.. In gnral, if lmn can b in on of m a, h numbr of oibl ym a for an n lmn ym i m n... Th li of all oibl ym a in rm of h lmn a i calld h TRUTH TABLE for h ym; a o d abov comri a ruh abl. Each lin of h ruh abl can b idnifid wih a ym condiion, u or down or dgradd. For xaml, if lmn x and y wr in ri, hn a would b a ym u a, and b, c and d would b down a. If x and y wr in a rdundan configuraion, hn a a, b and c would rrn ym u a and d h ym down a. Pag

5 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu. Markov Analyi and Traniion Sa Diagram.. Markov analyi comu h ra a which raniion occur bwn ym a from uch aramr a h lmn failur ra and/or rair ra. Thi i hn ud o comu ym aramr uch a MTBF, rliabiliy, availabiliy, c. Th mahmaic i illurad in Scion of hi Char... Gnrally Markov analyi in rliabiliy alicaion i confind o h iuaion whr h diribuion of lmn failur and rair im i ngaiv xonnial. I i alo gnrally aumd ha wo lmn canno chang hir a imulanouly. Thu, for xaml, h lmn ym of aragrah.. canno chang from a a o a d a on im, inc hi would rquir x and y o fail imulanouly. Poibl raniion bwn ym a and h ra a which hy occur ar indicad in Traniion Sa Diagram, a hown in Scion... Di h limiaion h chniqu i of valu inc uch aumion ar frqunly mad in rliabiliy work, and i can handl iuaion whr h failur and rair im diribuion of h lmn ar no indndn a i h ca wih andby ym. Iu Pag 5

6 Char 8 Markov Modlling EXAMPLE OF MARKOV METHODS FOR ANALYSING THE RELIABILITY OF COMPLEX SYSTEMS. Inroducion.. Thi Scion dcrib, by man of an xaml, a mhod of analying h rliabiliy of comlx ym. Alhough h xaml chon i of a non-rairabl andby ym, i i rlaivly raighforward o ada h mhod o modl rairabl ym... Th analyi chniqu mloy om of h ida ud in h analyi of Markov Proc. An imoran aumion rquird by and limiaion of h mhod i ha h failur ra of h lmn comriing h ym ar conan for rairabl ym, h rair ra mu alo b conan, i.. h failur im and rair im diribuion mu b ngaiv xonnial. In ral im oraional iuaion hi may b unraliic and car mu b akn whn alying hi chniqu.. Th Analyi Mhod.. Conidr h ym hown in, comriing lmn in andby rdundancy. L h failur ra of lmn b, and h failur ra of lmn b in h oraional a and in h andby a. Figur : A Two Elmn Sandby Rdundancy Sym.. Now hi ym can occuy on of four a: and u, down, u, u, down, down, down, Th ym a may b rrnd diagrammaically a: Pag 6

7 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu,,,, Tabl : Th Sym Sa.. L h robabiliy of h ym bing in a a im Conidr h robabiliy of h ym bing in a a im Δ. Th following rlaionhi hold auming ha Δ i ufficinly mall ha h robabiliy of wo or mor raniion aking lac in h im inrval Δ i ngligibl. Δ [ ] inc h robabiliy of bing in a a Δ i h roduc of h robabiliy ha h ym wa in a a im and h robabiliy ha nihr of lmn and faild in Δ. Similarly: Th diagram can b illurad by a diagram calld h TRANSITION STATE DIAGRAM for h ym. Iu Pag 7

8 Char 8 Markov Modlling,,,, Figur : Traniion Sa Diagram No: i ii h arrow indica h dircion of raniion bwn ym a. i an aborbing a, i.. onc nrd, h ym canno lav i... Equaion can b r-arrangd a follow: Pag 8

9 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu Δ Δ Δ Δ Δ Δ Δ Δ Taking h limi a yild: 0 & & & & Whr d d &..5 Th diffrnial quaion can b olvd by aking Lalac Tranform, a hown in Andix. Th oluion ar: Iu Pag 9

10 Char 8 Markov Modlling..6 Th rliabiliy of h ym a im i h robabiliy ha h ym i in a, or : i.. R..7 Thr ar ohr mhod of analying h ym dcribd abov. On advanag of hi mhod howvr i ha givn ha h ym can b dcribd in a raniion a diagram uch a ha hown in Fig, h mhod i comlly gnral and can b incororad in a comur rogram. Thorically, a ym of any comlxiy may b analyd in hi way, alhough in racic hr will b limiaion imod by h iz of h ym and h comur rogram o analy i i.. for a ym comriing n lmn hr will b n ym a if, howvr, om of h lmn ar idnical, hi numbr may b rducd...8 A ad in h inroducion, h mhod can b adad aily o h analyi of rairabl ym. Conidr h am lmn andby rdundancy ym whr lmn and hav rair ra μ and μ rcivly. Th raniion a diagram i MTBF now: Pag 0

11 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu μ,, μ μ, μ μ, μ Figur : Traniion Sa Diagram for a Rairabl Sandby Sym No: Thr i no raniion allowd from o. Thi ari from h aumion ha a aiv failur of lmn i.. a raniion from o will no b dcd unil h lmn i rquird for oraion i.. whn lmn fail. Hnc no rair acion i akn on h aiv failur of lmn, and hnc hr can b no raniion from a o...9 Th raniional robabiliy quaion imilar o can now b u and olvd a bfor. I hould b nod ha, in hi ca, bcau w ar daling wih a rairabl ym, h um givn by rrn h availabiliy and no h rliabiliy of h ym a im. Th rliabiliy of h ym i.. i robabiliy of urvival o im R may b calculad by modifying h raniion a diagram o ha hown in Fig 5. Th raon for raing h calculaion of rliabiliy R in hi way ari from h fac ha rliabiliy i h robabiliy ha a ym will no fail in a givn riod of im. Iu Pag

12 Char 8 Markov Modlling μ,, μ,, Figur 5: Traniion Sa Diagram for a Rairabl Sandby Sym o Calcula i Rliabiliy R A gnral rul for h conrucion of raniion a diagram i ha for availabiliy calculaion, all allowabl rair raniion hould b includd, whra for rliabiliy calculaion h ym down a,.g. a in Fig 5, hould b rad a aborbing a, i.. onc nrd, h ym canno lav hm. Pag

13 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu SOLUTION OF THE DIFFERENTIAL EQUATIONS DESCRIBING THE STATE PROBABILITIES, USING LAPLACE TRANSFORMS. Inroducion.. Thi aragrah dcrib how o olv h diffrnial quaion lablld in aragrah, bu hr. Th quaion wr: & & & & whr & d d.. Th mhod of oluion adod hr i ha uing Lalac Tranform. Th bai of h mhod i no dcribd hr, xc o ay ha i convr diffrnial quaion o algbraic form by man of a ranformaion dicovrd by Lalac. Th algbraic quaion may hn b aily olvd, h invr of h ranformaion alid o obain h final oluion. Som Lalac Tranform ar givn in Tabl.. Soluion.. L L i b h Lalac Tranform of & i * and ak Lalac Tranform of quaion. Now h Lalac Tranform of & i i givn by Li i 0. If, a 0, boh lmn and ar u hn h ym will b in a aragrah.., i.. 0, * L τ τ i i dτ 0 Iu Pag

14 Char 8 Markov Modlling L L L L L L L L L L L L L L L L L L L For h objc of h xrci i o olv h quaion algbraically for h L, hn ak invr Lalac Tranform o obain oluion for h, i o in hi ca. In roducing h xrion of L i i ncary o u i ino a form uiabl for h invr ranformaion. Such form can b obaind from abl of Lalac Tranform.g. Tabl. In hi ca h uiabl form i: whr A, B,.. Thu from : A B L i c α β α, β do no involv. L * i L L i u ino h form of quaion a follow: A B Sinc hi i an idniy w hav: Pag

15 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu Iu Pag 5 B B A A Equaing cofficin of on boh id yild: A B 0 i.. B A Inring i yil h in 6 d: A A A Thrfor from 5 L * Similarly L L * Th corrc form for i mo aily obaind from h xrion: L L L L L which ilf driv from h fac ha L L *.. Th oluion now com from h quaion lablld *. Th invr Lalac Tranform of L i by dfiniion, ha of / α A i A α, and ha of / i Tabl. Hnc:

16 Char 8 Markov Modlling Pag Thu h xrion lablld 7 abov ar h oluion of h diffrnial quaion lablld in aragrah...

17 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu x n n a oiiv ingr n n! a in a a coa inh a > a a a a coh a > a a a a [ in a ] in a a a a a a Lalac Tranform of x d x.lx0 whn L i Lalac d Tranform of x d d n n x n n n n dx d x. L x0... n d 0 d 0 d i x whr i h valu of h i h d 0 drivaiv of x a 0 Tabl : Tabl of Common Lalac Tranform No: Th Lalac Tranform of x i dfind by: L 0 x d Sciali book will rovid mor comrhniv abl of ranform. Iu Pag 7

18 Char 8 Markov Modlling 5 RELATED DOCUMENTS. Alicabiliy of Markov Analyi Mhod o Rliabiliy, Mainainabiliy and Safy. by Norman B. Fuqua. Publihd in h Rliabiliy Analyi Cnr - Slcd Toic in Auranc Rlad Tchnologi START Volum 0 No.. Rairabl Rdundan Sym and h Markov Fallacy by W G Gulland and Rliabiliy Amn of Rairabl Sym I Markov Modlling Corrc? by KGL Simon and M Klly. Publihd in Safy and Rliabiliy Sociy Journal Volum No.. Briih Sandard Rliabiliy of ym, quimn and comonn. Par :99. Scion. Calculaing robabiliy of Failur of Elcronic and Elcrical Sym Markov v. FTA by Vio Faraci Jr. Publihd in h Journal of h Rliabiliy Analyi Cnr, Third Quarr 00. Pag 8