Applications of semi-markov processes in reliability

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1 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I rbk rczk Nl Ur d old Alco o m-mrko roc rlbl Kword m-mrko roc rlbl rdom lr r cold db m wh rr Abrc Th bc do d horm rom h m-mrko roc hor r dcd h r Th m- Mrko roc hor llow o corc h modl o h rlbl m olo wh h m rm Alco o m-mrko roc rlbl r codrd Sm-Mrko modl o h cold db m wh rr m-mrko roc h rlbl modl o h oro wh rrbo d m-mrko roc lr r r rd h r Irodco Th m-mrko roc wr rodcd ddl d lmo mlol b W Smh d Tkc Th l dlom o m-mrko roc hor wr rood b Clr [3] Korolk & Trb [3] mo & Or [4] W wold l ol m- Mrko roc wh or cobl c Th m-mrko roc r cocd o h Mrko rwl roc Th m-mrko roc hor llow o corc m modl o h rlbl m olo hroh h m rm Do o m-mrko roc wh dcr c S b dcr or cobl c d l [ N { } So h r h rdom rbl dd ϑ o o robblc c Ω wh l o S d rcl A wo-dmol rdom c { ϑ } clld Mrko rwl ch or ll S N : { ϑ ϑ } ϑ { ϑ } { o ϑ } { } o hold rom h bo do ollow h Mrko rwl ch homoo wo-dmol Mrko ch ch h h ro robbl do o dd o h cod como I o oc h rdom c { : } homoo o-dmol Mrko ch wh h ro robbl {! } lm 3 Th mr [ : S] 4 I clld Mrko rwl krl Boh Mrko rwl krl d h l drbo d h Mrko rwl ch Th c llow o corc m-mrko roc τ τ ϑ ϑ ϑ τ { τ : N A ochc roc { X : } ollow rlo } b h - 6 -

2 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I X or τ τ 5 [ clld m-mrko roc o S rd b h Mrko rwl ch rld o h krl d h l drbo Sc h rcor o h m-mrko roc k h co l o h hl-rl [ τ τ d rh-coo co rom l X τ ollow h h c { X : } τ Mrko ch wh h ro robbl mr [ : S] 6 τ clld mbddd Mrko ch m-mrko roc { X : } Th co Th c { X : } { τ τ X τ X } τ 7 cml robbl drbo o rdom rbl T h clld hold m o h wll b rom w h 8 Th co { τ X } τ τ 9 S cml robbl drbo o rdom rbl T h clld w m o h Th w m T m h m b wh w do o kow h ccor N : dd b A ochc roc { } N or τ τ [ clld co roc o h m-mrko roc { X : } Th m-mrko roc { X : } d o b rlr or ll { N < } X h h mbr o ch o rod Er Mrko roc { X : } wh h dcr c S d h rh-coo rcor k co l o h hl-rl wh h r mr o h ro r I m h h roc { : } Α [ : S] < h m- < Mrko roc wh h krl [ : S ] Whr or d I h rlbl modl h rmr d chrcrc o m-mrko roc r rrd h rlbl chrcrc d rmr o h m 3 Tro robbl o m-mrko roc Th ro robbl o h m-mrko roc r rodcd ollow: { X X } S Al h Mrko ror o h m-mrko roc h m mom rl w ob Mrko rwl o or h ro robbl [4] [] δ [ ] d S k S U lc-sl rormo w ob h m o lr o δ [ ] 3 S whr h rorm d k S k k k k r kow whl h rorm - 6 -

3 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I k d k d whr π [ π S] h or drbo o h mbddd Mrko ch h m o o r o mrc w ob h ollow o [ I ] 4 whr [ : S] [ : S] δ [ : S] I m c h ro robbl d h robbl { X } S 5 roch co l or lr lm lm 6 To orml h ror horm w h o rodc rdom rbl { N : X } m τ 7 Th do h m o r rrl A mbr { < X τ } 8 h robbl h h ch h l wll oor or lr ch h A coclo o horm rd b Korolk d Trb [3] w h obd ollow horm Thorm { X : } b m-mrko roc wh dcr c S d coo krl [ : S] I h mbddd Mrko τ co o o rcrr cl C ch h or ch S C d < E T < S h ch { X : } lm π E T lm π E T S 9 π π S π S S 4 r m rom h o h b A Th rdom rbl whr τ A A A m{ N : X τ A} do h m o r rrl o m-mrko roc h o A Th co Φ { X } A A h cml drbo o h rdom rbl A h do h r m rom h o h b A Thorm [4] [3] or h rlr m-mrko roc ch h A { A < X } A h drbo Φ A r ror d h r h olo o h m o o Φ A A A Φ k S ka d k Al lc-sl rormo w ob h m o lr o φ φ 3 A A wh kow rorm φ A dφ A ka k r mr orm w o k - 6 -

4 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I 4 I b whr I [ δ : ] [ ' : A] r h r mrc d b A [ : ] A A T T : 5 Sm-Mrko modl o h cold db m wh rr Th roblm wll kow rlbl hor Brlow & roch [] Th modl rd hr om modco o h modl h w codrd b Brod & oo [] 5 Dcro d mo A m co o o or como dcl d-b como d wch r r h o-colm mrc o rorm Th orml olo o h o I b To ol h o w comr rorm or ml MATHEMATICA Ob h r lc rorm mch mor comlcd I ll mlr o d h cd l d h cod mom o h rdom rbl A A I h cod mom o h w m T A r o d commol bodd d A A h h cd l o h rdom rbl o o I T whr A A r h olo 5 [ δ : ] ' [ : ] I A T [ E : ] [ E T : ] T T A d h cod mom o h m o lr r h olo o o I A B whr [ δ : ] ' [ : ] I A 6 r Drm o h m Wh h or como l h r moo b h wch mmdl Th ld como rrd Thr l rr cl Th rr ll ror h como h como rr m hr rwl Th m l wh h or como l d como h w oor ld o rrd or wh h or l d h wch l W m h h m o lr o h or como r rrd b h dd co o o- rdom rbl ς wh drbo b robbl d co d W o h h lh o h rr rod o h como r rrd b h dcl co o h o- rdom rbl γ wh cml drbo co CD γ U b rdom rbl h br drbo b k U k k k k < < whr U wh wch ld h mom o h or como lr d U wh h wch work h mom W o h h whol ld m rlcd b h w dcl m Th rlc m o- rdom rbl η wh CD H η T [ E A : ] [ b : ] T A B b E T E T E k k k ka

5 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I ς γ ζ z z 3 3 z 5 U ς γ < ζ z z 4 4 [ ] d d 3 4 r lbl olo o h db m Moror w m h h ll rdom rbl mod bo r dd 5 Corco o h m-mrko modl To dcrb rlbl olo o h m w h o d h d h rwl krl W rodc h ollow : - h m ld - h ld como rrd r ord - boh or como d r r τ τ τ do h o h ch d { Y : } b rdom roc wh h c S { } whch k co l o h hl-rl [ τ τ d rh-coo Th rlzo o h roc how r Th roc o m-mrko bc h codo o do o d or ll o h ch o h roc corc w rdom roc ollow w τ d τ τ do h o h m como lr or h o whol m rwl Th rdom roc { X : } dd b o X X Y τ or [ τ τ h m-mrko roc To h m-mrko roc modl w m d l drbo d ll lm o krl or w ob η H d U ς γ < ζ d U ς U ς W m h h l I m h l drbo [ ] Hc h m-mrko modl corcd 53 Th rlbl chrcrc Th rdom rbl A h do h r m rom h o h b A or d A {} or modl rr h m o lr o h m Th co Φ 7 h rlbl co o h codrd cold db m wh rr Sm o lr o 3 or h lc- Sl rorm o h co Φ h c φ φ φ φ Th olo φ φ

6 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I Hc w ob h lc rorm o h rlbl co φ 9 Th ro robbl mr o h mbddd Mrko ch h m-mrko roc X : { } 3 Whr U γ < ζ U d U orml 9 w ob h CD o h w m o T H Hc E T E η E T E ς E T E 3 ς Th o 5 h c h orm Th olo E E E E E ς E ς E ς E ς E ς 3 W wll l horm o clcl h lm robbl drbo o h Now h m o lr o π π π π π π π π π π π Sc h or drbo o h mbddd Mrko ch π π π U orml 9 w ob h lm drbo o m-mrko roc E η E η E ς E ς 3 E η E η 54 Coclo E ς E ς E ς E ς E ς E ς Th co E do h m m o lr E whr E ς E ς d oc h h cold db drm cr h mm o lr m Th lm lbl coc o h m E ς E ς A E η E ς E ς

7 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I 6 Sm-Mrko roc h rlbl modl o h oro wh rrbo Sm-Mrko roc h rlbl modl o ml- oro w codrd b rbk [8] d [] M oro co o om lmr k whch r rlzd r Dro o h ch k rlzo md o b o rdom rbl Ech lmr oro m b rrbd or ld Th rrbo cr h m o oro d h robbl o lr wll 6 Dcro d mo So h h oro co o whch ollow r W m h dro o -h o rdom rbl ; wh cml robbl drbo d K whr do robbl d co dd Tm o lr o h oro o h -h como h o rdom rbl η wh ol drbo 6 Sm-Mrko modl To corc rlbl modl o oro w h o r rom do o h roc do -h rlbl o -h o h oro whr do rrbo d do cc - lr -cc o h oro - l or coc w mr h Udr h bo mo ochc roc dcrb o h orll oro rlbl c m-mrko roc { X : } wh c o S { } d low rh how r 3 - ; K η - Th oro o ch m b rrbd W m h o mor h o c rrbo o ch o h oro m occr Tm o c o oro rrbo o -h o rdom rbl ζ ; K wh ol drbo ; K ζ Th rrbo dr h robbl o h oro l W o h m o lr o h rrbd oro o h -h h o rdom rbl ν h h h ol drbo wh rmr ν ; K W m h h oro cclcl W m h rdom rbl η ν ζ r mll dd r 3 Tro rh or - cclc oro To ob m-mrko modl w h o d ll o lm o m- Mrko krl [ : S] { X τ τ τ X } τ r w d ro robbl rom h o h or m o rr h or - η ζ whr z D D { z : z } d d dz z

8 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I Sc w h d d z d or w ob dz z w Th m o o rom { w : w } h o rom l ro D Th Jcob o h m J w ζ η ζ ζ Sc w or [ ] d w z d d dz D w J w d d dw η η < ζ η < w d dw d [ ] d I o -h rrbo h hd h ro robbl o or m l h or l o whr D { z : ζ ν D E z z z ζ ζ d d dz d d } E { : } d d oc h E d d [ ] d ll w ob d I h m w w d [ d] d [ ] d To d h rl rl or h ro D w l ch o coord: w z Hc

9 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I d d dz d d E z D < ζ ζ ν ν whr } : { z z z D < } : { E Hc ] [ d d d dw w w Smlr w w ob d ζ η ] [ d d d ζ ζ ν ζ Thror h m-mrko rlbl modl o oro h b corcd 63 Two- cclcl oro W wll codr 3 W wll rclr c o h modl m A ro mr or h m- Mrko modl o h - cclc oro rlbl c k h ollow orm whr d ] [ 3 d ] [ 5 d d ] [ 4 d ] [ 5 d ] [ ] [ d d d ] [ 3 5 d d d dw w w ] [ ] 4 d d d ] [ 4 5 d d d dw w w 55 U

10 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I Th modl llow o ob om rlbl chrcrc o h oro Th rdom rbl 5 do h r m rom o 5 or modl m m o lr o h oro Th lc-sl rorm or h cml drbo co o h rdom rbl w wll ob rom mr o 5 I h c w h } {5 } { A A d ' A b φ φ φ φ ' A rom h olo o o 4 w ob lc- Sl rorm o h cml drbo co o h rdom rbl 5 do m o lr o h oro 5 b φ b Th lc rorm o h rlbl co b h orml 5 φ 64 Eml Eml W o h Th lc-sl rorm o h co r: 3 5 4

11 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I 5 Th co how o r 5 4 or ; ; ; ; ; 4 ; ; l 33 d wh hl o MATHEMATICA comr rorm w ob h d co d h rlbl co r lc rorm Th d co b h orml φ Th co how r 4 Th rlbl co r 5 Th rlbl co o - cclc oro M m o lr w c d ol h mr o I A ' T whr E T E T T E T3 E T4 E E E E 5 5 rom h o w ob h m m o lr E Eml Now w m h dl dl I m h h dro o h r drmd d h r l or I h c h lm o r: or or r 4 Th d co h m o lr o - cclc oro - 7 -

12 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I or or 5 or or or or 4 or or 5 or or 4 Th lc -Sl rorm o h co r: Th m m o lr w c d ol h mr o whr

13 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I 4 5 E T E T E T 3 E T 4 d d 4 d d or h m rmr ; ; ; ; ; 4 ; ; d h m m o lr o h oro E I ro c h m m o lr E Coclo I m h or h drmd dro o h m m o lr o h oro ll rr h or oll drbd dro o h wh h m co To rlbl o h m oro w c l m-mrko roc Corco o h m-mrko modl co d krl o h roc A w o bld h krl or h m-mrko modl o h m oro rd h r rom Sm-Mrko modl w c ob m r rmr d chrcrc or l rlbl o h oro rom rd ml w coclo h or h drmd dro o h m m o lr o h oro ll rr h or oll drbd dro o h wh h m co 7 Sm-Mrko roc lr r Th rlbl co wh m-mrko lr r w codrd b Kooc k & Kooc k [] Kooc k [] d b rbk [4] [6] [9] So h h lr r { : } h m-mrko roc wh h dcr c S { : J} J { m} or J { } < < wh h krl [ : J ] d h l drbo [ : J] W d codol rlbl co E d J I [6] rod h or h rlr m-mrko roc { : } h codol rlbl co J dd b 7 h m o o [ ] d J Al h lc rormo w ob h m o lr o J whr d d - 7 -

14 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I d I mr oo w h ] [ H I whr ] [ I O M M M M M H Th codol m m o lr w ob rom h orml J lm µ Th codol m m o lr h orm N µ µ 7 Alr rdom roc lr r Am h h lr r m-mrko roc wh h c } { S d h krl whr r h cml robbl drbo co wh o or So h l o o h co boll coo wh rc o h b mr ] [ b l robbl drbo o h roc Th ochc roc clld h lr rdom roc I h c h mrc rom h o r ] [ I whr d d H Th olo o k h orm [ ] [ ] Th lc rorm o h codol rlbl co Eml 3 Am h d d whr Γ Γ

15 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I So h l Hc h l drbo [ ] d h lc rorm o h codol rlbl co Now h o k h orm o or 3 5 w h U h MATHEMATICA comr rorm w ob h rlbl co h r lc rorm [899co ] 375 [4668co ] r 6 how h rlbl co r 6 Th rlbl co rom ml Th corrod d co ' how r r 7 Th d co rom ml 8 Coclo Th m-mrko roc hor co or dcro o h rlbl m olo hroh h m Th robblc chrcrc o m-mrko roc r rrd h rlbl coc o h m I A rr h b o l d l h rdom rbl A d h r m rom h o h b A do h m o lr o h m Thorm o m- Mrko roc hor llow o d h rlbl chrcrc lk h drbo o h m o lr h rlbl co h m m o lr h lbl coc o h m d m ohr W hold rmmbr h m- Mrko roc mh b ld modl o h rl m rlbl olo ol h bc ror o h m-mrko roc do r d b h rl m rc [] Brlow E & roh 975 Scl hor o rlbl d l Hol hr d Wo Ic Nw York [] Brod S M & oo I A 973 Embddd ochc roc hor K Nko Dmk

16 rbk Alco o m-mrko roc rlbl - TA # Dcmbr - Scl I [3] Clr E 969 Mrko rwl hor Ad Al robbno 3-87 [4] rbk Sm-Mrko modl o rlbl d oro IBS AN 3 Wrzw [5] rbk & Ko owrock K 999 Amoc rlbl o ml m wh m-mrko o como S d lbl AA Blkm ordm 37-3 [6] rbk 3 Th rlbl o h obc wh m-mrko lr r Ald Mhmc d Como Elr [7] rbk & J w k J 3 Som roblm o h ror m modll ITE [8] rbk 5 lbl modl o h ml oro Adc S d lbl Ko owrock d Tlor & rc ro odo ISBN [9] rbk 6 dom lr r roc Sbmd or blco Ald Mhmc d Como [] rbk 6 Sm-Mrko Modl o Oro Irol Jorl o Mrl & Srcrl lbl V [] Kooc k I & Kooc k B 98 O m rlbl dr rdom lod o lm Alco Mhmc XVI 5-5 [] Korolk V S & Trb A Sm-Mrko roc d hr lco Nko Dmk K [3] mo N & Or Sm-Mrko roc d lbl Boo Brkhr

Example: Two Stochastic Process u~u[0,1]

Example: Two Stochastic Process u~u[0,1] Co o Slo o Coco S Sh EE I Gholo h@h. ll Sochc Slo Dc Slo l h PLL c Mo o coco w h o c o Ic o Co B P o Go E A o o Po o Th h h o q o ol o oc o lco q ccc lco l Bc El: Uo Dbo Ucol Sl Ab bo col l G col G col