Chap : lably ad valably Modls lably = prob{s s fully fucog [,]} Suppos from [,] m prod, w masur ou of N compos, of whch N : # of compos oprag corrcly a m N f : # of compos whch hav fald a m rlably of h compo N N Nf Nf N N N d dnf d N d dnf d or N d d Physcal mag: saaous ra a whch compos ar falg 4
How may ufalg compos ar hr a m? N Z * N Falg ra of a sgl compo dn d f N N N f d N f +d - N f Z N N dn f d N d d d d f +d m.3 x p.8 Z s also calld h hazard ra. 5
For lcroc compos, Z s rlaoshp wh rspc o m s a bahub curv. Z Hgh falur ra du o fauly dsg, maufacurg or assmbly. Wak compos ar rmovd durg h bur- prod. Bur- prod Ifa moraly phas Usful m phas Falur ra ca b assumd o b a cosa durg h usful m prod, say War-ou phas du o agg m 6
d d d d Expoal Falur Law.g. =. hr -, wha s a = hrs? s: -.* For hardwar compos, xpoal falur law s frquly assumd. For sofwar compos, h rlably may grow as h sofwar s dsg fauls ar rmovd durg h sg/dbuggg phas. 7
I gral, w ca assum Z = - Wbull = - ds. Ths ylds Z Us hs o modl sofwar falur ra.g. = - - = < > =,, lably mprovs as a fuco of 8
Formal Dfo of : L x b a.. rprsg h lf of a sysm ad l F b h cumulav dsrbuo fuco CDF of x. Th, pr x F For a compo obyg h xpoal falur law f x dx x dx 9
Ma Tm o Falur MTTF: Th xpcd m ha a sysm wll opra bfor h frs falur occurs Dscr cas MTTF N frs falg m Idcal sysms Q: wha s h rlably of a sysm obyg h xpoal falur law a = MTTF? s:.3678... MTTF Couous cas MTTF falur m. g. E T d d d MTTF d f d d N df d d d as d.78... N d d uv' d uv u' vd 3
MTT Ma Tm o par If also assum a fald sysm obys Expoal par Law, h MTT, whr s h rpar ra laoshp bw MTBF Ma m bw falur, MTT & MTTF: m MTTF MTT MTTF MTT Tm of frs falur Tm of d falur MTBF = MTT + MTTF If MTTF >> MTT Th MTBF MTTF 3
valably Isaaous or po valably = prob {h sysm s fucog a m } gardlss of h # of ms may hav fald & b rpard durg [, ] m Sady-Sa valably = MTTF MTTF+MTT For a sysm whou rpar, = 3
ssum xpoal falur & rpar law O F S also pag 67, x chapr 4 Tm doma: P o = - P o + P F P F = P o - P F wh al sa P o = & P F = Laplac doma: P o SP o S - = - P o S + P F S SP F S - = P o S - P F S P F 33
P o S Smlarly P F S P o = P F = = S + S + + S S S + + Physcal mag: - -+ + S S + + -+ Ivrs LT o rur o m doma S F a S LT B Ivrs LT LF = fs /S /S!/S + /S-a P o = prob {h sysm s fucog a m } 34
Q: uavalably? s: P F Q:? Sll - Q3: Sady-sa avalably? s: = + MTTF MTTF MTT.g. =. & =...9.999... 35
Modlg: Srs-Paralll lably Block Dagrams srs-paralll block dagram rprss h logcal srucur of a sysm wh rgard o how h rlabls of s compos affc h sysm rlably. Compos ar combd o blocks srs paralll or k-of- cofguraos 36
37. Sral sysm: ach lm of h sysm s rqurd o fuco corrcly for h sysm o fuco corrcly. 3 srs...... srs B. Paralll sysm: oly o of svral lms mus b opraoal for h sysm o b opraoal. ssumpos: dpd radom varabls prfc covrag so up o - falurs ca b olrad. paralll paralll x F F F :
C. Combao of srs & paralll sysms.g. Compur Irfac Dsplay Bus Compur Irfac Dsplay Bus srs Paralll 3 4 4 4 j, paralll j Numrcal x: =.9 h sysm = [--.9 ] 4 =.96 v.s. o-rduda =.9 4 =.656, Whr j, s for jh compo of h u 38
.g. 3 3 paralll = - - srs, - srs, - srs, 3 = - - - 3-3 Q: Prov h followg horm: plcao a h compo lvl s mor ffcv ha rplcao a h sysm lvl mprovg sysm rlably usg h sam # of compos. s br ha s: Show ha ssum =/ for ach compo sysm 9 6 sysm 7 6 39
D. k-ou-of-.g. TM Trpl Modul dudacy s a -ou-of-3 sysm. TM = * * 3 + - 3 + 3 - + 3 - wh = = 3 = -ou-of-3 = 3-3 I gral,.g. k ou of all ar fucog fald & ar fucog 3 k 3 3 3! 3!! 3 3 3 3 3 ou of 3 4
Q: Is TM > sysm wh a sgl compo? L 3-3 =. sysm TM - 3 +.5 = =. or.5 sgl TM = wh h rlably of a Mor sgl modul s. or.5 ralsc rgo I fac, wh <.5, > TM.5. Q: wha s h MTTF of a k-ou-of- sysm wh ach sgl modul follows h xpoal falur law wh a falur ra of?.g.!!! sysm k MTTF sysm d... k -ou-of-3: MTTF= 3 -ou-of-5: MTTF= 5 4 3 5 6 77 6 4
Q: wha s h rlably & MTTF of h followg srucur? Fg..5, p36 lso a paralll sysm P P p =.764/day -ou-of- m m m 3 m =.39/day -ou-of-3 paralll sysm oo sysm = [ - - -.764 ][ - - -.39 3 ] = 6 -.93-3 -.4-6 -.667 +3 -.86 + -.43 - -.57 MTTF = sysm d = 6.93-3 - 6 3 +.4.667.86.43 call ha par ra Falur ra of compo = Equaos & 3 obad abov ca also b usd o compu h sysm avalably by rplacg wh + - = 6.9.57 4
43 Spcfcally, For sady sa avalably Us = + o quaos 6 5 4 & abov k of ou k paralll srs 4 5 6 ssumg...