MIL-HDBK OCTOBER 1984 MAINTAINABILITY ANALYSIS
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1 5 CTBER 984 TABLE : VALUES F <J>R z (t'i_ a ) MST CMMNLY USED IN MAINTAINABILITY ANALYSIS l- <p r z (ti_ a ) Fllwing is an example f maintainability analysis f a system which has a lgnrmal distributin f repair times GRUND ELECTRNIC SYSTEM MAINTAINABILITY ANALYSIS EXAMPLE Giv the active repair times data f Table n a gund electnic system find the fllwing:. The pbability dsity functin, g(t). The MTTR f the system 3. The median time t repair the system 4. The maintainability functin 5. The maintainability fr a 0 hur missin 6. The time within which 90% and 95% f the maintance actins are cmpleted. 7. The repair rate, u(t), at 0 hurs. 5-44
2 5 CTBER 984 TABLE : TIME T REPAIR DATA N A GRUND ELECTRNIC SYSTEM Gup N. Times t repair t, hr Frequcy f bservatin n K ' '' '.' ' v / -. - N'=
3 5 CTBER 984. Pbability Dsity Functin f q(t) T determine the lgnrmal^. pdf f the times-t-repair giv , the values f 7' andaf, shuld be calculated fm in Table V j=l ni t' ni (5.79) where ni is the number f idtical bservatins giv in the third clumn f Tabte , N' is the number f differt-in-value bserved times-t-repair, r number f data gups, which fr this pblem is N = 9, giv in the secnd clumn f Table , and N is the ttal number f bserved times-t-repair, N' i=l J which, fr this example, is 46. And r f N 9 _ 9.i x (t;-) z - N(t') z N-l N' nj(t'j) - N(t') N-l lh (5.80) T facilitate the calculatins, Table was prepared. Table , t' and at', are btained as fllws: x "jt'j N' Fm r
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5 5 CTBER 984 and fm Eq. (5.80) f = ( )' 56TI r <7 t ' =.U435 Cnsequtly, the lgnrmal pdf represting the dataintable is r g(t) = g(t) = t a t. Vn t(l.435) im ^ A' \ -?{.435 J where t' = lg e t. The plt f this pdf is giv in Figure in terms f the straight times in hurs. See Table fr the g(t) values used. The pdf f the lg e t r f the t's is g(t') t Vff i(*&) = t g(t) r g(t') = (l.h435)vff -» ( * \ U435 ) This pdf is that f a nrmal distributin which is what ne shuld expect since if t fllws a lgnrmal distributin, lg e t shuld be nrmally distributed. This is shwn pltted in Figure , the values f g(t') were btained fm Table
6 5CTBER 984 TABLE : The pbability dsity f Time t Repair Data (Fm Table based n the straight times t repair and the natural lgarithm f the times t repair used t plt Figures and , respectively.*) Time t restre, t hurs y Pbability dsity, g(t) Pbability dsity g(t') = gdg e t) 7.95 x 0" *At the mde, * = , V At the median, t =.93 g(t) = and g( ) = and g(?) = g(t') =
7 3 r- -I I a: c m D T 7<i > c «0 t 3 T 4 5 Mde = t = Median = t=.93 Mean = t = lirs. Time t restre, t, hurs FIGURE 5,3,,,.; plqt F THE LGNRMAL PDF F THE TIMES^T-RESTQRE DATA GIVEN IN TABLE IN TERMS F THE STRAIGHT t's
8 +-> bfi c -a 3 -" ci i ) i ^ G m Da I i C l C (.0 0. FIGURE 5...-:.0 l. Time t restre, t, liurs PLT F THE LGNRMAL PDF F THE TIMES-T-RESTRE DATA GIVEN IN TABLE IN TERMS F THE LGARITHMS F t, R LG e t = t' 00.0
9 5 CTBER 984. MTTR (Mean Time t Repair) f the System The mean time t repair f the system, t, is btained fm Eq. (5.73). t s e(t' + / ( t,) ) f a J / (.435) ) r _ t = hr. 3. Median Time t Repair The median f the times-t-repair (5.76) t = et' t = e the system, t, is btained fm Eq. r t =.93 hr. This means that in a large sample f t's half f the t's will havej/alues smaller than t, and the ther half will have values greater than t. In ther wrds, 503. f the repair times will be < t. 4. Maintainability Functin M(t) The maintainability f a unit can be evaluated as fllws, using Eq. (5.6): ti t'i z(t'i) M(tl) = f g(t) dt = f g(t') df = f 4>{z) dz (5.8) where t' = lg e t, (5.8a) t'i - f z(t'i) = a V (5.8b) and t' and \* are giv by Eq. (5.79) and (5.80), respectively. By means f the transfrmatins shwn in Eqs. (5.8a) and (5.8b), the lgnrmal distributin f the pdf f repair times, g(t), is transfrmed t the standard nrmal distributin <j> (z) which ables the use f standard nrmal distributin tables (Table A-l, Appdix A). 5-5
10 5 CTBER 984 The maintainability functin fr the system, M(t), fm (5.8) is: where z(t') M(t) = f 0(z) dz z(t') t = lg e t t' - 7' a*, i Fm the data in Table we previusly calculated t"' = a t - =.435 The quantified M(t) is shwn in Figure The values were btained by inserting values fr, t' = lg e t, int the expressin,,/+-\ - f Z{ ' ~.435 slving fr z(t'), and reading the value f M(t) directly fm the standard nrmal tables in Appdix A (Table A-l). 5. Maintainability fr a 0 Hur Missin z(lg e 0) M(0) = f 0(z) dz where lg e 0 =.9957 and 7/inn n\ _, Z(lg nq e 0) = 743 " Fm Appdix A we find that fr z = M(0) = C <f>{l) (dz) = = 0.98 r 98.% The time within which 90% and 95% f the Maintance Actins are Cmpleted (M m a ct ) This is the time tj.afr which the maintainability is -a, r M(ti_ r _a a ) = P(t< tsti. *'!-«z(t'l-a) a j - J g(t) dt B r g(ti) dt, = r 0(z) dzj (5>8) 0 - ^- 5-53
11 3»- r- I i a: a m B T 7* i J < J Time t xbktdre\,\t} i i i! i i : : i JL_i_i_: i_ r i T i j Jrbiitg I I I FIGURE : PLT F THE MAINTAINABILITY FUNCTIN FR THE TIMES-T-REPAIR DATA F EXAMPLE
12 and MIL-HDBK CTBER 984 -a l *(t'i.j l-a ( (5.83) at' The cmmnly used maintainability, r (-a), values are 0.80, 0.85, 0.90, 0.95, and Cnsequtly, the z(t'i_ a ) values which wuld be used mst cmmnly wuld be thse previusly giv in Table Using Eq. (5.83) the time t*i_ a wuld th be calculated fm r t'i_ a = t' + z(t'i- a ) a v tl-a = antng e (t'i_ a ) = antilgec't' + z(t'i_ a ) t J (5.84) Thus, fr 90% M max., fm the previusly btained value f t' and t t.g = antilg e t' + z (t'.g)cr f = antilg e X (.435) = antilg e (.08737) = 8.06 hrs. Fr 95% M maxct t.95 = antilg e [Q (.435) = antilg e (.49896) =.08 hrs. 7. Repair Rate at t = 0 hurs Using Eq. (5.63) and substituting the values fr g(0) fm and the previusly calculated value fr M(0) Table,tn\ q(0) = _ uuu; " -M(0) " 0.08 = 0. repairs/hr. 5-55
, which yields. where z1. and z2
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