5 CTBER 984 TABLE 5.3..-: VALUES F <J>R z (t'i_ a ) MST CMMNLY USED IN MAINTAINABILITY ANALYSIS l- <p r z (ti_ a ) 0.80 0.85 0.90 0.95 0.99 0.846.036.8.645.36 Fllwing is an example f maintainability analysis f a system which has a lgnrmal distributin f repair times. 5.3... GRUND ELECTRNIC SYSTEM MAINTAINABILITY ANALYSIS EXAMPLE Giv the active repair times data f Table 5.3...- 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
5 CTBER 984 TABLE 5.3...-: TIME T REPAIR DATA N A GRUND ELECTRNIC SYSTEM Gup N. Times t repair t, hr Frequcy f bservatin n 3 4 5 6 7 8 9 0 0. 0.3 0.5 0.6 0.7 0.8.0..3.5.0. 4 3 4 4 K ' '' '.' ' v 3 4 5 6 7 8 9 0 3 4 5 6 7 8.5.7 3.0 3.3 4.0 4.5 4.7 5.0 5.4 5.5 7.0 7.5 8.8 9.0 0.3.0 / -. - N'= 9 4.5 5-45
5 CTBER 984. Pbability Dsity Functin f q(t) T determine the lgnrmal^. pdf f the times-t-repair giv 5.3...-, 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 5.3...-, 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 5.3...-, 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 5.3...- was prepared. Table 5.3...-, t' and at', are btained as fllws: x "jt'j 30.30439 46 N' Fm r 0.65879 5-46
i» -C» C C -~J -»J -P> b c c c c i i-* ---J cn c i i I I c c r- C I» C c C -Pi c c v ^J H A c» -p» C C c c c i > c C -vl -p» -P» -P» cn c c 00 C c c c c c c n i ' r - v i c i -» J - P i v c n r c e n r r - p > i c c i cn c - p i c n c r i ' c n c c n c n - v j c n -Pi -"J c i -P=- c - P» - p i c r r r r r i» i» -Pi -~i -vi C 00 C C -Pi C 00 e e 0D -Pi C c C i-» -Pi -vl C c cn C -Pi i» i > v C i C C -Pi C -Pi C -vi -~i -Pi I c c -vf i» -Pi -Pi cn v 00 i I» -Pi -Pi 00 -Pi c C -vl C -»J C C» -Pi v II M I» Cj. ri II M II i r\5 3 C C_J. M i 'i 'i '-Pii ' c c i-» i» e I» I» C cn c c c C "vl I -Pi c I» C cn C C c C i» -vj.p».pi i» N M l IB -Pi c -Pi -Pi C c cn cn -Pi c -vl -Pi i» -Pi -Pi c c c c c c cn cn C c cn C c -Pi i» cn cn i -vi cn c c cn C -Pi "~i ll C C -Pi c <J II M c e n - p i - p i - p i c r r r r r c r r I r cn -Pi -vi -~i c cn C C C c cn c cn C C -Pi C cn e c -Pi C c cn -Pi I» 00 e cn cn C -Pi i» -Pi n -«j C C cn cn C -Pi e C i > cn c -~i 00 C C I i» i» C cn C i» n cn cn C c C C cn cn c cn <» cn C i» -Pi cn C -~i -~j -Pi C cn C -«j U> c c II 0 -Pi C
5 CTBER 984 and fm Eq. (5.80) f = 75.8437-46 (0.65879)' 56TI r <7 t ' =.U435 Cnsequtly, the lgnrmal pdf represting the dataintable 5.3...- is r g(t) = g(t) = t a t. Vn t(l.435) im ^ A' - 0.65879\ -?{.435 J where t' = lg e t. The plt f this pdf is giv in Figure 5.3...- in terms f the straight times in hurs. See Table 5.3...-3 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 -» ( * - 0.65879 \ 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 5.3...-, the values f g(t') were btained fm Table 5.3...-3. 5-48
5CTBER 984 TABLE 5.3...-3: The pbability dsity f Time t Repair Data (Fm Table 5.3...- based n the straight times t repair and the natural lgarithm f the times t repair used t plt Figures 5.3...- and 5.3...-, respectively.*) Time t restre, t hurs 0.0 0. 0. 0.3 0.5 0.7.0.4.8.0.4 3.0 3.4 4.0 4.4 5.0 6.0 7.0 8.0 9.0 0.0 0.0 30.0 y 40.0 80.0 Pbability dsity, g(t) 0.00398 0.0480 0.55 0.950 0.34300 0.33770 0.30060 0.454 0.9849 0.789 0.4638 0.039 0.0960 0.073 0.0695 0.04976 0.03559 0.065 0.0985 0.0534 0.006 0.0099 0.00058 Pbability dsity g(t') = gdg e t) 7.95 x 0" 5 0.0048 0.0450 0.08853 0.750 0.3636 0.30060 0.34334 0.3578 0.35784 0.3530 0.338 0.3483 0.899 0.758 0.4880 0.35 0.8373 0.5884 0.3804 0.06 0.0397 0.0733 0.00888 0.0035 *At the mde, * = 0.5584, V At the median, t =.93 g(t) = 0.34470 and g( ) = 0.8530 and g(?) = 0.947. g(t') = 0.35800. 5-49
3 r- -I I a: c m D T 7<i > c «0 t 3 T 4 5 Mde = t =0.5584 Median = t=.93 Mean = t = 3.595 lirs. Time t restre, t, hurs FIGURE 5,3,,,.; plqt F THE LGNRMAL PDF F THE TIMES^T-RESTQRE DATA GIVEN IN TABLE 5.3...-3 IN TERMS F THE STRAIGHT t's
+-> 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 5.3...-3 IN TERMS F THE LGARITHMS F t, R LG e t = t' 00.0
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.65879 + / (.435) ) r _ t = 3.595 hr. 3. Median Time t Repair The median f the times-t-repair (5.76) t = et' t = e -65879 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
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 5.3...- we previusly calculated t"' = 0.65879 a t - =.435 The quantified M(t) is shwn in Figure 5.3...-3. The values were btained by inserting values fr, t' = lg e t, int the expressin,,/+-\ - f - 0.65879 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\ -.9957-0.65879 _, Z(lg nq e 0) = 743 " - 097 Fm Appdix A we find that fr z =.097.097 M(0) = C <f>{l) (dz) = -0.08 = 0.98 r 98.% -00 6. 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
3»- r- I i a: a m B T 7* i J < J 00 00 Time t xbktdre\,\t} i i i! i i : : i JL_i_i_: i_ r i T i j Jrbiitg I I I FIGURE 5.3...-3: PLT F THE MAINTAINABILITY FUNCTIN FR THE TIMES-T-REPAIR DATA F EXAMPLE
and MIL-HDBK-338 5 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 0.99. Cnsequtly, the z(t'i_ a ) values which wuld be used mst cmmnly wuld be thse previusly giv in Table 5.3..-. 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 X65879 +.8 (.435) = antilg e (.08737) = 8.06 hrs. Fr 95% M maxct t.95 = antilg e [Q5879 +.645 (.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 5.3...-3 and the previusly calculated value fr M(0) Table,tn\ q(0) = 0.0099 _ 0.0099 uuu; " -M(0) -0.98 " 0.08 = 0. repairs/hr. 5-55