# CHAPTER 38 MARKOV MODELLING CONTENTS

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1 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu CHAPTER 8 MARKOV MODELLING CONTENTS INTRODUCTION MARKOV ANALYSIS EXAMPLE OF MARKOV METHODS FOR ANALYSING THE RELIABILITY OF COMPLEX SYSTEMS 6 SOLUTION OF THE DIFFERENTIAL EQUATIONS DESCRIBING THE STATE PROBABILITIES, USING LAPLACE TRANSFORMS 5 rlad documn Error! Bookmark no dfind. Iu Pag

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

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 a.co ] 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