{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "#Here are some reliability \+ functions. Remember that \n#r(x) = Prob(\"Failure occurs later than t ime x\")\nplot(exp(-x),x=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$\"\"!$\"\"\"F(7$$\"1mmmT&)G\\a!#<$ \"1d,(4Y_'p%*!#;7$$\"1LLL3x&)*3\"F1$\"1L)[e'F17$$\"1nm\"z%4\\Y_F1$\"1h() \\1)Hw\"fF17$$\"1MLeR-/PiF1$\"1F$3?kb&f`F17$$\"1***\\il'pisF1$\"1sb6*G +r$[F17$$\"1MLe*)>VB$)F1$\"1LjNAtG]VF17$$\"1++DJbw!Q*F1$\"1N'RI5*z8RF1 7$$\"1nm;/j$o/\"!#:$\"11n;))fY5NF17$$\"1LL3_>jU6Fao$\"15,D-Qy*=$F17$$ \"1++]i^Z]7Fao$\"10(H'GpojGF17$$\"1++](=h(e8Fao$\"1#4nbG*ypDF17$$\"1++ ]P[6j9Fao$\"1OT1>-9:BF17$$\"1L$e*[z(yb\"Fao$\"1CLOTY#e5#F17$$\"1nm;a/c q;Fao$\"16'fM\"fT\")=F17$$\"1nmm;t,mFao$\"13I<1\\8(Q\"F17$$\"1+]i!f#=$3 #Fao$\"1+a2DCLX7F17$$\"1+](=xpe=#Fao$\"1\"Rv@Y*zB6F17$$\"1nm\"H28IH#Fa o$\"1bCQyzh45F17$$\"1n;zpSS\"R#Fao$\"1pBSo>6]\"*F.7$$\"1LL3_?`(\\#Fao$ \"1O78JIyG#)F.7$$\"1M$e*)>pxg#Fao$\"1-it!yw)ptF.7$$\"1+]Pf4t.FFao$\"19 vI()Q_&p'F.7$$\"1MLe*Gst!GFao$\"1u!y8>IFao$\"1rf.&[jS)[F.7$$\"1+]i!RU07$Fao$\"1y'ezWALT%F 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>*)!#>7$$\"1+]7.\"fF&QFao$\"1;hdS)f-)fFgw7$$\"1mm;/OgbRFao$\"11h$=8&e- SFgw7$$\"1+]ilAFjSFao$\"1:2?V$>$*f#Fgw7$$\"1MLL$)*pp;%Fao$\"1g&4=O@kp \"Fgw7$$\"1ML3xe,tUFao$\"1i[4.'zV3\"Fgw7$$\"1n;HdO=yVFao$\"1g%3^$zT!)o !#?7$$\"1,++D>#[Z%Fao$\"1/(zA_od[%Ffy7$$\"1nmT&G!e&e%Fao$\"1Yi28#yfr#F fy7$$\"1MLL$)Qk%o%Fao$\"1,:8(yufr\"Ffy7$$\"1+]iSjE!z%Fao$\"1$pDs<%RS5F fy7$$\"1,]P40O\"*[Fao$\"1V@%p'\\pxj!#@7$$\"\"&F($\"1r'y?<`ms$F`[l-%'CO LOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fb[l %(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(1/(1+x),x=0..5);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$ \"\"!$\"\"\"F(7$$\"1mmmT&)G\\a!#<$\"1\"f8,OJK[*!#;7$$\"1LLL3x&)*3\"F1$ \"1A\"3)[#[s,*F17$$\"1+]i!R(*Rc\"F1$\"1eohe(Gvk)F17$$\"1nm\"H2P\"Q?F1$ \"1v'Rq:LpI)F17$$\"1LL$eRwX5$F1$\"1<\"3c(>#4j(F17$$\"1ML$3x%3yTF1$\"1x Gi8)QJ0(F17$$\"1nm\"z%4\\Y_F1$\"1xrbgi))elF17$$\"1MLeR-/PiF1$\"1m>N4zv ehF17$$\"1***\\il'pisF1$\"16#[.qPGz&F17$$\"1MLe*)>VB$)F1$\"1n7s+H\\daF 17$$\"1++DJbw!Q*F1$\"1*)QN/\\vf^F17$$\"1nm;/j$o/\"!#:$\"1!\\\\Wa)e&)[F 17$$\"1LL3_>jU6Fao$\"1HBnLr:nYF17$$\"1++]i^Z]7Fao$\"18^-]g]VWF17$$\"1+ +](=h(e8Fao$\"1OxPVN^RUF17$$\"1++]P[6j9Fao$\"1!>?*G+!*fSF17$$\"1L$e*[z (yb\"Fao$\"1Ej?z1\\4RF17$$\"1nm;a/cq;Fao$\"1$eN]MKXu$F17$$\"1nmm;t,mFao$\" 12yKG^&4O$F17$$\"1+]i!f#=$3#Fao$\"1sPi6=SVKF17$$\"1+](=xpe=#Fao$\"1!ft NDg)QJF17$$\"1nm\"H28IH#Fao$\"1$e?UbKn.$F17$$\"1n;zpSS\"R#Fao$\"148kV7 j[HF17$$\"1LL3_?`(\\#Fao$\"1TG4I*e\"fGF17$$\"1M$e*)>pxg#Fao$\"1+[(\\$f zrFF17$$\"1+]Pf4t.FFao$\"1V=wI,)**p#F17$$\"1MLe*Gst!GFao$\"1uuZpL[EEF1 7$$\"1+++DRW9HFao$\"1oS-q7kaDF17$$\"1++DJE>>IFao$\"1#o@k)=1)[#F17$$\"1 +]i!RU07$Fao$\"1])*et\\'oU#F17$$\"1++v=S2LKFao$\"18/5J%\\BO#F17$$\"1mm m\"p)=MLFao$\"11pJ\")oB2BF17$$\"1++](=]@W$Fao$\"1>kWn?;^AF17$$\"1L$e*[ $z*RNFao$\"11u)4>`E?#F17$$\"1,+]iC$pk$Fao$\"1@zO\\s&>:#F17$$\"1m;H2qcZ PFao$\"10*G_/Uj5#F17$$\"1+]7.\"fF&QFao$\"1&[@OP$og?F17$$\"1mm;/OgbRFao $\"1qo=aw\"z,#F17$$\"1+]ilAFjSFao$\"1!4<>O2](>F17$$\"1MLL$)*pp;%Fao$\" 1vwC_-PN>F17$$\"1ML3xe,tUFao$\"1@^Z@zW'*=F17$$\"1n;HdO=yVFao$\"1leg*zj $f=F17$$\"1,++D>#[Z%Fao$\"11$)zEMaE=F17$$\"1nmT&G!e&e%Fao$\"1Og[zUK!z \"F17$$\"1MLL$)Qk%o%Fao$\"1)>%yX^7f " 0 "" {MPLTEXT 1 0 142 "#Now, using the Theorem of \+ Total Probability, and a \"mixture\" modelling\n#approach we can const ruct a \"Cure-Rate\" or \"Surviving Fraction\" model" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "#If we assume a Poisson model for the nu mber of defective micro components,\n#that is, the probability of havi ng i defective components is\np[i]:=exp(-theta)*theta^i/i!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#%\"iG*&*&-%$expG6#,$%&thetaG!\"\"\" \"\")F.F'F0\"\"\"-%*factorialGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#for i=0,1,..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#Here is a plot of these probabilities for theta=3 " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "theta=3.0:\nfor i from 0 by \+ 1 while i < 21 do \n p[i]:=eval(exp(-3.0)*3.0^x/x!,x=i):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listplot([seq([i,p[i]],i=0..20)],style=point,symbol=box );" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6#777$ \"\"!$\"+Poqy\\!#67$$\"\"\"F($\"+^?h$\\\"!#57$$\"\"#F($\"+x!=/C#F17$$ \"\"$F(F57$$\"\"%F($\"+dNJ!o\"F17$$\"\"&F($\"+M\")=35F17$$\"\"'F($\"+s 1%4/&F+7$$\"\"(F($\"+XJSg@F+7$$\"\")F($\"+'z6:5)!#77$$\"\"*F($\"+KR]+F FS7$$\"#5F($FR!#87$$\"#6F($\"+F($\"+z\">pv %!#>7$$\"#?F($\"+o(y`8(!#?-%&STYLEG6#%&POINTG-%'SYMBOLG6#%$BOXG" 1 5 3 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 352 "#For the component to fail overall, we requir e that at least one of micro components fails.\n#It can be easily show n that, GIVEN that there are m micro components, each behaving \n#inde pendently and according to reliability function r(x) , then the reliab ility function\n#of the component overall is for example\nr:= x -> exp (-x^2/2): R[m](x):=(r(x))^m; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- &%\"RG6#%\"mG6#%\"xG)-%$expG6#,$*$)F*\"\"#\"\"\"#!\"\"F2F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "#Hence the Probability that the co mponent fails later than x is given by \n#the Theorem of Total Probabi lity as\nR(x):=exp(-theta*(1-r(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>-%\"RG6#%\"xG-%$expG6#,$*&%&thetaG\"\"\",&F.F.-%\"rGF&!\"\"F.F2" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#Here's an example with a sp ecific r and theta=2.0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "t heta:=2.0:\nr:= x -> exp(-x^2/2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "R:= x -> exp(-theta*(1-r(x))):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "for i from 0 by 1 while i < 21 do\n t[i]:=i/ 2: \n y[i]:=R(i/2):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(plots):listplot ([seq([t[i],y[i]],i=0..20)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'CURVESG6#777$\"\"!$\"\"\"F(7$$\"+++++]!#5$\"+@&f c!zF.7$F)$\"+yGO_XF.7$$\"+++++:!\"*$\"+W8g!f#F.7$$\"\"#F($\"+2L.uHO`8F .7$$\"+++++bF7$\"+jNN`8F.7$$\"\"'F($\"+tGN`8F.7$$\"+++++lF7$\"+MGN`8F. 7$$\"\"(F($\"+KGN`8F.7$$\"+++++vF7Fio7$$\"\")F(Fio7$$\"+++++&)F7Fio7$$ \"\"*F(Fio7$$\"+++++&*F7Fio7$$\"#5F(Fio" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "# Change the value of theta and the function r to get different plots - \+ \n#look at what happens to the level of the horizontal asymptote.\n \n #The \"surviving fraction\" are the components that NEVER fail - they \+ arise\n#because of the Poisson modelling assumption, and the assumptio n that\n#there is a positive probability of having zero defective micr ocomponents.\n#If there are zero defective microcomponents, then there is NO chance of failure\n#at any value of x. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "#We can do all of the above ca lculations using a simulation approach. We could simulate\n#the numbe r of defective micro components according to the Poisson model, simula te failure times\n#for each of them, take the time of the first failur e as the failure time of the overall\n#component. Like this" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "seed:=23:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# By changing the seed value, you can get differen t simulations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "theta:=2. 0: n:=stats[random, poisson[theta]](500):\n#n is a vector of 500 rando m numbers from the Poisson(theta) distribution" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 171 "for i from 1 by 1 while i < 501 do\nm[i]:=Inf inity:\nfor ii from 1 by 1 while ii < (n[i]+1) do \n y[ii]:=stats[ra ndom, weibull[2.0,sqrt(2.0)]](1):\n m[i]:=min(m[i],y[ii]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 " od:\n#Ignore the weibull bit for now; its just the model that gives us \n# r(x) = exp(-x^2/2.0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "#The m vector contains 500 simulated failure times for the component \+ overall." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$minG6$%)InfinityG$\"+o4(H%z!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$minG6$%)InfinityG$\"+x;_9M!#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "m[7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)Infi nityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "#That is, compone nt 1 failed after 0.79429 units of time, component 2 failed after\n#0. 34145 units of time, but component 7 did not, and will not ever fail. \+ Note that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#n[7]=0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#so there are zero defective micro components in component 7." }}}}{MARK "1 0 0" 102 }{VIEWOPTS 1 1 0 1 1 1803 }