Safety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis

Transcription

1 Safy and Rlably of Embddd Sysms (Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss

2 Safy and Rlably of Embddd Sysms Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons of Falurs ovr Tm Rlably Modlng Exampls of Dsrbuon Funcons Th xponnal dsrbuon Th Wbull dsrbuon Th Posson dsrbuon Musa's Excuon Tm Modl Drmnaon of Modl Paramrs Slcon of Modls Basd on Falur Obsrvaon 2

3 Dfnon Rlably Par of h qualy wh rgard o h bhavor of an ny durng or afr gvn m nrvals undr gvn applcaon condons (ranslad from DIN 44) Th propry of an ny o fulfll s rlably rqurmns durng or afr a gvn m span undr gvn applcaon condons (ranslad from DIN ISO 9 Tl 4) A masur for h capably of an m undr consdraon o rman funconal, xprssd by h probably ha h dmandd funcon s xcud whou falur undr gvn condons durng a gvn m span (Broln) 3 Safy and Rlably of Embddd Sysms

4 Safy and Rlably of Embddd Sysms Hardwar- vs. Sofwar Rlably Hardwar Rlably (ypcal assumpons) Falurs ar a rsul of physcal dgradaon Whn h fauly componn s subsud, h rlably bcoms h nal valu of hs componn Th rlably of h sysm dos no xcd h nal valu of h sysm rlably hrough h subsuon of componns wh nw componns Hardwar rlably s drmnd by farly consan paramrs T 2 T 3 4

5 Safy and Rlably of Embddd Sysms Hardwar- vs. Sofwar Rlably Sofwar Rlably (ypcal assumpons) Falurs ar a rsul of dsgn rrors ha ar conand n h produc from h sar and appar accdnally Afr faul corrcon h sysm rlably xcds s nal valu (undr h assumpon ha no addonal fauls ar nroducd) Fauls ha ar nroducd durng dbuggng dcras rlably Rlably paramrs ar assumd o vary T T 2 T 3 5

6 Safy and Rlably of Embddd Sysms Tool Asssd Rlably Modlng How rlabl s my sysm now? How rlabl wll b a h plannd rlas da? How many falurs wll hav occurrd by hn?... 6

7 Tool Asssd Rlably Modlng Us of modls Whch modls do xs? How can I fnd ou, whch modl fs my purposs bs? How can I dfn h modl paramrs n ordr o g dpndabl rlably prdcons? 7 Safy and Rlably of Embddd Sysms

8 Safy and Rlably of Embddd Sysms Dscrpon of falur ovr m Falur ms Tm nrvals bwn falurs Toal numbr of falurs a a pon n m Falurs whn a gvn m nrval Falur Tms Ausfallzpunk T T2 T3 T - T T' T2' T3' T' ' Znrvall zwschn Ausfälln Inrvals Bwn Falurs Numbr of Falurs Anzahl Ausfäll 8

9 Modlng of Rlably Lfm T Larg numbr of smlar sysms undr consdraon Smulanous sar of h sysms a m = Obsrvd m of h frs falur of ach sysm s h so-calld lfm T of hs sysm Plo of h fracon of fald sysms ovr s h so-calld mprcal dsrbuon funcon of h lfm (or mprcal lf dsrbuon) 9 Safy and Rlably of Embddd Sysms

10 Safy and Rlably of Embddd Sysms Modlng of Rlably If h numbr of sysms bcoms largr (approxmas nfny), h mprcal lf dsrbuon approxmas h lf dsrbuon F() Hr, lfm T s a random varabl and F() s h probably ha an arbrary sysm s no opraonal a F() = P{T } F() s h probably ha lfm T s lss or qual o, manng ha a sysm has alrady fald by. W us h followng assumpons: F( = ) =,.. a nw sysm s nac, and lm F() =,.. vry sysm fals somms Falur Tms of Sysms T (h) n / N,,2,3,4,5,6,7,8,9, F(),94,74,264,37,457,585,668,765,879,94

11 Safy and Rlably of Embddd Sysms Modlng of Rlably Lf dsrbuon F(),,8 n/n F(),632,6,4,2, = 2834 h

12 Safy and Rlably of Embddd Sysms Modlng of Rlably Rlably funcon R() F() gvs h probably ha a m a las on falur has occurrd; hus R() = - F() s h probably ha a m no falur has occurrd y Probably dnsy f() Th probably dnsy f() dscrbs h modfcaon of h probably ha a sysm fals ovr m () = d F() d 2

13 Safy and Rlably of Embddd Sysms Modlng of Rlably MTBF, MTTF A rlvan masur for rlably s h Man Tm To Falur (MTTF) or Man Tm Bwn Falur (MTBF) Th MTTF rsp. MTBF dfns h man valu of h lfm rsp. h man valu for h m nrval bwn wo succssv falurs I s drmnd by calculang h followng ngral: T = E(T) = f() d Falur ra Th falur ra s h rlav boundary valu of fald ns a m n a m nrval ha approxmas zro, rfrrng o h ns sll funconal a h bgnnng of h m nrval () df() / d () = = = R() R() - dr() / d R() 3

14 Safy and Rlably of Embddd Sysms Modlng of Rlably Th condonal probably ha a sysm ha oprad falur fr unl also survvs h prod s R( + ) R() Thus, h probably ha h produc fals whn s R( + ) - F( + ) F () ( - F( + )) - = - = = R() - F() - F() F( + ) F() - F() 4

15 Safy and Rlably of Embddd Sysms Modlng of Rlably As h gvn probably for shor m nrvals s proporonal o, w dvd h rm by and drmn h boundary valu whn approxmas lm F( + ) F() - F() = R() lm F( + ) F() = f() R() = () Thus h probably ha a sysm, ha s opraonal a m fals whn h (shor) m nrval, s approxmaly () 5

16 Safy and Rlably of Embddd Sysms Modlng of Rlably R() and falur ra,,8 R(),6,4,2, = 2834 h h () -5 /h 4, 3,53 3, 2,,, h 6

17 Safy and Rlably of Embddd Sysms Exampl for h Dsrbuon Funcon Assumpon: For h gvn daa (abl p. ) lfm s xponnally dsrbud: F() = - Th paramr (falur ra) has o b drmnd basd on falur obsrvaons n ordr o achv an opmal adusmn of h funcon, accordng o a prdrmnd crron. Th Maxmum-Lklhood-Mhod provds h followng paramr for h xponnal dsrbuon: = N N T = =,353 / h Rlably: R() = F() = - 7

18 Safy and Rlably of Embddd Sysms Exampl for h Dsrbuon Funcon Th falur ra s consan ovr m () df() / d - dr() / d () = = = = - = R() R() R() A consan falur ra causs an xponnal dsrbuon of h lfm Drmnaon of h MTTF T = E(T) = () d = - d = - d = (- ) = If lfm s xponnally dsrbud, h MTTF s h rcprocal of h falur ra and hus consan

19 Safy and Rlably of Embddd Sysms Th Exponnal Dsrbuon Lf dsrbuon: F() = - - Dnsy funcon: () = - Rlably funcon: R() = - F() = - Falur ra: () = MTTF: T = 9

20 Safy and Rlably of Embddd Sysms Th Wbull Dsrbuon Lf Dsrbuon : F() = -() ;, > or: F() = ;, >, d. h. = Dnsy: df() () = = ( ) - d -() Rlably: R() = (-) - Falur ra: () () = = ( ) R() - 2

21 Safy and Rlably of Embddd Sysms Th Wbull Dsrbuon Falur ra of h Wbull dsrbuon dpndng on h form paramr 5 () =5 =3 4 =2 3 = 2 =,5 = =,5,5,,5 2, 2

22 Th Posson Dsrbuon Assumpons Th probably of mor han on falur whn h (shor) m nrval can b gnord. Thus, falurs occur rlavly nfrqunly Th probably of a falur whn, rspcvly whn [, + ], s (s dfnon of falur ra). Th probably s proporonal o h lngh of h m nrval P x () s h probably, ha whn m nrval [, ] x falurs occur 22 Safy and Rlably of Embddd Sysms

23 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon No falurs Th probably ha whn m nrval [, +] no falurs occur s drmnd by mulplyng h probably ha unl m no falurs hav occurrd (P ()) and h probably ha whn [, +] no falurs occur (- ): For owards on rcvs: P P P P P lm P P d P d P P () =, snc nw sysms (=) ar always opraonal by dfnon. For a consan valu of and P () = h dffrnal quaon has h soluon: P 23

24 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon Th probably ha a nw sysm shows no falurs unl s R() P R Usng h dfnons for F() and f(), w g: df F R and f d 24

25 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon Falurs Th probably ha whn m nrval [, +] x falurs occur can b drmnd as follows: Px P P x falurs bwn and... 2 Px 2 P falurs bwn and Px P falur bwn and Px P no falur bwn and 25

26 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon Du o h prcondon h probably o obsrv mor han on falur n s zro. Thrfor w g: x Px Px P falur bwn and P P x x x x P P no falur bwn and P P Px Px Px P x Px 26

27 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon Wh approxmang zro: lm P x P dp x x d P x P x Th followng rm for P x () s a soluon for hs dffrnal quaon P x x! (Posson Dsrbuon) x 27

28 Safy and Rlably of Embddd Sysms 28 Th Posson Dsrbuon Ths can b shown vry asly Th probably P X () provds h corrc valu P () also for h cas ha w rad sparaly bfor P P x x x x d x d d dp x x x x x x x X!!!! P d dp x X!

29 Safy and Rlably of Embddd Sysms Th Posson Dsrbuon P X () fulflls h boundary condons for =,.. P () = and P X () =, for x. Furhrmor h sum of h probabls of all x for vry mus b,.. P x x x! x x x x! x x! x x! Th spcfd sum on h lf hand sd of h quaon s h powr srs of h xponnal funcon on h rgh hand sd. Th Posson Dsrbuon hus fulflls h prcondons. If s consan, h man valu s ()=. Ths s calld a homognous Posson Procss. If s a funcon of m, h man valu s d and P x x! Ths s calld a non-homognous Posson Procss (NHPP) x 29

30 Safy and Rlably of Embddd Sysms Falur Tms and Tms bwn Falurs Th m of falur s T Th m nrval bwn falur ( - ) and falur s T T = T, T = = M() s h numbr of falurs a M T 3

31 Safy and Rlably of Embddd Sysms Falur Tms and Tms bwn Falurs Th probably for falurs unl m s P P M Th probably for a las falurs a s P M!! P T 3

32 Musa's Excuon Tm Modl A sofwar sysm fals du o rrors n h sofwar randomly a, 2,... ( hr rfrs o xcuon m,.. CPU-sconds) I s assumd ha h numbr of falurs obsrvd n s lnarly proporonal o h numbr of fauls conand n h sofwar a hs m () s h oal numbr of falurs for ms () s a lmd funcon of Th numbr of falurs s a monoonc ncrasng funcon of A = no falurs hav bn obsrvd y: ()= Afr vry long xcuon m ( ) h valu () s qual o a. a s h oal numbr of falurs n nfn m. (Thr ar also modls whr nfn numbrs of falurs ar assumd o happn) 32 Safy and Rlably of Embddd Sysms

33 Safy and Rlably of Embddd Sysms Musa's Excuon Tm Modl Modl dvlopmn Th numbr of falurs obsrvd n a m nrval s proporonal o and o h numbr of rrors no y dcd ba Wh w g: d d ba b ' Wh ()= and ()=a w g: ba b b a Th falur ra s: b ' ab 33

34 Safy and Rlably of Embddd Sysms Musa's Elmnary Excuon Modl Th curv for h accumulad numbr of falurs () approxmas asympocally h xpcd oal numbr of falurs a 34

35 Safy and Rlably of Embddd Sysms Musa's Elmnary Excuon Modl Th curv for h falur ra () for = sars a h nal falur ra = ab and approxmas asympocally h valu. Th nal falur ra s proporonal o h xpcd numbr of falurs a, wh h consan of proporonaly b b a a b a ab a b a b a b a a and ab ab 35

36 Safy and Rlably of Embddd Sysms Musa's Excuon Tm Modl a ab a a ba ab 36

37 Safy and Rlably of Embddd Sysms 37 Musa's Excuon Tm Modl If s h prsn falur ra and a arg z s dfnd, addonal falurs wll occur unl hs arg s rachd Th addonal m unl hs arg s rachd s z z z a a a z z z z a a a ln ln ln ln ln

38 Safy and Rlably of Embddd Sysms 38 If s nsrd no h gnral quaon of h Posson dsrbuon, w g: Musa's Excuon Tm Modl a a a a a a a a a a P T!!!

39 Safy and Rlably of Embddd Sysms Exampls of Modlng For a program wh an xpcd oal numbr of 3 falurs wh an nal falur ra of,/cpu-scond, modls ar o b gnrad Wha s h probably ha a a parcular xcuon m a las a cran numbr of falurs wll hav occurrd? Formula for P[T ] for, 2 and 3 falurs,9,8 Probably Wahrschnlchk,7 T,6,5 T2,4,3 T3,2, Tm Zpunk (xcuon (Ausführungsz: m: CPU-Sc.) CPU-Sk.) 39

40 Safy and Rlably of Embddd Sysms Exampls of Modlng Wha wll b h numbr of falurs w.r.. xcuon m? Formula for () 25 2 Falurs Ausfäll Excuon Ausführungsz m (CPU-Sc.) (CPU-Sk.) 4

41 Safy and Rlably of Embddd Sysms Exampls of Modlng How wll h falur ra dvlop dpndng on h xcuon m? Formula for (),,9,8,7 Ausfallra Falur ra ( (/CPU-Sc.) / CPU-Sk.),6,5,4,3,2, Ausführungsz Excuon m (CPU-Sc.) (CPU-Sk.) 4

42 Safy and Rlably of Embddd Sysms Drmnaon of Modl Paramrs Las squars Targ: Dfn paramrs n such a way ha h sum of h squars of h dvaons bwn h calculad and h obsrvd valus bcoms mnmal. If F rfrs o h valu of h mprcal dsrbuon funcon a pon, h followng rm s o b mnmzd: n 2 Maxmum-Lklhood-Mhod n 2 F F Targ: Choos paramrs n such a way ha h probably s maxmzd o produc a "smlar" obsrvaon o h prsn obsrvaon. Th probably dnsy has o b known 42

43 Drmnaon of Modl Paramrs Las squars Safy and Rlably of Embddd Sysms Targ: Dfn paramrs n such a way ha h sum of h squars of h dvaons bwn h calculad and h obsrvd valus bcoms mnmal. If F rfrs o h valu of h mprcal dsrbuon funcon a pon, h followng rm s o b mnmzd: n n 2 F F For h xponnal dsrbuon w g: n n 2 xp xp 2 n 2 F F F 2 43

44 Safy and Rlably of Embddd Sysms 44 Drmnaon of Modl Paramrs Las squars Th valu ha mnmzs hs rm s o b drmnd Th valu s calculad by drmnng h zro pon n n F d d d d 2 xp 2 xp 2! ˆ ˆ F ˆ

45 Drmnaon of Modl Paramrs Las squars Safy and Rlably of Embddd Sysms Somms numrcal mhod mus b usd for hs ask. A Nwonan raon provds h followng rsuls for h Exponnal Dsrbuon wh: and: n f n f df n n d n xp n xp 2 2 d f 2 F F 2 For h falur ms of h abl on pag h sarch for zro pons accordng o h Nwonan raon provds a valu ˆ 3, * -5 /h for h xponnal dsrbuon 2 45

46 Drmnaon of Modl Paramrs Maxmum-Lklhood-Mhod Safy and Rlably of Embddd Sysms Targ: Choos paramrs n such a way ha h probably s maxmzd o produc a "smlar" obsrvaon o h prsn obsrvaon Prcondon: Probably dnsy has o b known Lklhood funcon F Produc of h dnss a h obsrvd falur ms Th valu s proporonal o h probably o obsrv falur ms ha do no xcd h dvaon w.r.. h prsn obsrvaon I s a funcon of h dsrbuon funcon's paramrs ha ar o b drmnd Exampl: Th paramr of h xponnal dsrbuon s o b drmnd wh h Maxmum-Lklhood-Mhod, f 46

47 Safy and Rlably of Embddd Sysms 47 Drmnaon of Modl Paramrs Maxmum-Lklhood-Mhod Wh n obsrvd falur ms,..., n w g h Lklhood Funcon: Du o h monooncy of h logarhmc funcon, L und ln L hav dncal maxma In ordr o calcula h valu ha maxmzs h Lklhood Funcon, h drvaon accordng o mus b drmnd n n n n n n n f f f L ,...,,,...,, n n n L ln,...,, ln n n n d L d,,..., ln ˆ

48 Drmnaon of Modl Paramrs Maxmum-Lklhood-Mhod Safy and Rlably of Embddd Sysms ˆ s h zro pon. For h xponnal dsrbuon w g: n n! ˆ n n 48

49 Modl Slcon basd on Falur Obsrvaons Safy and Rlably of Embddd Sysms U-Plo-Mhod Prqunal-Lklhood-Mhod Holdou-Evaluaon 49

50 Modl Slcon basd on Falur Obsrvaons U-Plo U-Plo Graphc mhod ha ss whhr a dsrbuon funcon can b accpd wh rgard o h prsn obsrvaon Addonally, sascal ss (.g. Kolmogoroff-Smrnov) mgh b usd If a random varabl T s dscrbd by h dsrbuon F(), h F( ) of h random varabl ar qually dsrbud ovr h nrval [,] 5 Safy and Rlably of Embddd Sysms

51 Modl Slcon basd on Falur Obsrvaons U-Plo Th n valus U ar chard n a U-Plo as follows Th valus U ar usd as y-valus n such a way ha h valu U wh h poson s arbud o h x-valu / n If h valus U ar approxmaly qually dsrbud, h appld pons ar locad "nar by" h funcon y = x, for x 5 Safy and Rlably of Embddd Sysms

52 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons U-Plo Exampl T (h) n / N,,2,3,4,5,6,7,8,9, F(),94,74,264,37,457,585,668,765,879,94 Th valus prsnd n h abl for F() ar h U accordng o h dfnon sad abov 52

53 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons U-Plo U-Plo of h daa U,,9,8,7,6,5,4,3,2,,,,,2,3,4,5,6,7,8,9, 53

54 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod Th Prqunal-Lklhood-Mhod compars h suably of wo dsrbuon funcons undr consdraon wh rgard o a gvn falur obsrvaon I s basd on h followng approach Th falur nrval s a ralzaon of a random varabl wh h dsrbuon F () and h dnsy f () F () and f () ar unknown Th dnss of h dsrbuon funcons A and B ( rsp. ) can b drmnd basd on h falur nrvals,..., - A If h dsrbuon A s mor suabl han h dsrbuon B, can b xpcd ha h valu fˆ s grar han h valu B fˆ fˆ A fˆ Th quon wll b grar han B ˆ f A ˆ f B 54

55 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod f() ^ f B () f () ^f A () ^f A ( ) ^ f B ( ) 55

56 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod If hs analyss s don for vry obsrvd falur m nrval w g h socalld Prqunal-Lklhood-Rao concrnng h dsrbuons A and B PLR AB If A s mor appropra han B wh rgard o h prsn falur daa, h PLR shows a rsng ndncy Exampl s fˆ fˆ A B W compar h xponnal dsrbuon and h normal dsrbuon basd on h daa from h abl on pag usng h Prqunal-Lklhood-Mhod 56

57 Safy and Rlably of Embddd Sysms 57 Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod Th paramrs of h dsrbuons ar drmnd usng a Maxmum-Lklhood-Approach. For h xponnal dsrbuon, w g: For h normal dsrbuon w g: Th paramrs accordng o h Maxmum-Lklhood-Mhod ar: ˆ k k / 2 2 2,, f 2 2 ˆ ; ˆ k k k k

58 Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod Th followng abl shows h dnss of h xponnal dsrbuon and h normal dsrbuon for h arrval m nrvals basd on h falur ms T o T - from h abl on pag Th calculaon sars wh = 4. In addon h logarhm of h quon of h dnss and h logarhm of h PLR s conand n h abl Th rsng of h PLR undrlns ha h assumpon of xponnally dsrbud arrval ms for h prsn daa maks mor sns han h assumpon of normally dsrbud arrval ms 58 Safy and Rlably of Embddd Sysms

59 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod T (h) (h) f Exp / -6 74,9 84,8 32,5 52,4 3,3 3,8 7,3 f Norm / -9, , , log (f Exp / f Norm) log (PLR ) 4,83 -,46 5,63 -,29,49 8,67,49 4,83 4,37, 9,7,2 8,87 9,36 59

60 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Prqunal-Lklhood-Mhod PLR of h daa 2 log(plr )

61 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Holdou Evaluaon Approach Only pars of h falur daa ar usd for modl calbraon. Th rmanng daa ar usd o udg h prdcon qualy of h calbrad modl If an xponnal dsrbuon and a Wbull dsrbuon ar calbrad o h frs 6 falur ms (abl p. ) usng a Las-Squars-Algorhm, w g h followng rsuls: Exponnal dsrbuon: Wbull dsrbuon: F F xp xp 3,89292*,5525* 5 / h, / h 6

62 Safy and Rlably of Embddd Sysms Modl Slcon basd on Falur Obsrvaons Holdou Evaluaon Th Wbull dsrbuon has as xpcd - a br adusmn o h falur ms T o T 6. Th sum of h dvaon squars for h frs 6 falur ms s,459 compard o,79 n h xponnal dsrbuon Th prdcon qualy of h Wbull dsrbuon s howvr wors han ha of h xponnal dsrbuon. Th sum of h dvaon squars for h falur ms T 7 o T s,446 for h Wbull dsrbuon; for h xponnal dsrbuon s only,2. W mgh prfr o us h xponnal dsrbuon n ordr o avod ovr-calbraon 62

63 Sochasc Rlably Analyss Summary Safy and Rlably of Embddd Sysms Sofwar Rlably can b adqualy masurd and prdcd usng appropra modls Th us of sochasc rlably modls rqurs som knowldg w.r.. h undrlyng mahmacs Appropra ools ar a prcondon for h succssful us of rlably modls 63