MIL-HDBK-338 15 OCTOBER 1984

TABLE 5.3.2.1-1: VALUES OF <J>OR z (t'i_a) MOST COMMONLY USED IN

MAINTAINABILITY ANALYSIS

l-o

0.80

0.85

0.90

0.95

0.99

<p or z (ti_a)

0.8416

1.036

1.282

1.645

2.326

Following is an example of maintainability analysis of a system which has a lognormal distribution of repair times.

5.3.2.1.1 GROUND ELECTRONIC SYSTEM MAINTAINABILITY ANALYSIS EXAMPLE

Given the active repair times data of Table 5.3.2.1.1-1 on a ground electronic system find the following:

1. The probability density function, g(t) 2. The MTTR of the system 3. The median time to repair the system 4. The maintainability function 5. The maintainability for a 20 hour mission 6. The time within which 90% and 95% of the maintenance actions

are completed. 7. The repair rate, u(t), at 20 hours.

5-44

MIL-HDBK-338 15 OCTOBER 1984

TABLE 5.3.2.1.1-1: TIME TO REPAIR DATA ON A GROUND ELECTRONIC SYSTEM

Group No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20 21

22

23

24

25

26

27

28 N'= 29

Times to repair t, hr

0.2

0.3

0.5

0.6

0.7

0.8

1.0

1.1

1.3

1.5

2.0

2.2

2.5

2.7

3.0

3.3

4.0

4.5 4.7

5.0 5.4

5.5

7.0

7.5

8.8 9.0

10.3

22.0

24.5

Frequency of observation n

1

1

4

2

3

2

4

1

1

4

2

1

1

1

2

2

2

K '• ''•'.' '• v

/ -. -

5-45

MIL-HDBK-338 15 OCTOBER 1984

1. Probability Density Function of q(t)

To determine the lognormal^. pdf of the times-to-repair given in Table 5.3.2.1.1-1, the values of 7' a n d a f , should be calculated from

V

ni t'

j=l

(5.79)

ni

where ni is the number of identical observations given in the third column of Tabte 5.3.2.1.1-1, N' is the number of different-in-value observed times-to-repair, or number of data groups, which for this problem is N = 29, given in the second column of Table 5.3.2.1.1-1, and N is the total number of observed times-to-repair,

N'

i=l J

which, for this example, is 46.

And

rf

N 9 _ 9 .ix (t;-)z - N(t')z

N-l

N' nj(t'j)2 - N(t')2

N-l

lh

(5.80)

To facilitate the calculations, Table 5.3.2.1.1-2 was prepared. Table 5.3.2.1.1-2, t' and at', are obtained as follows:

From

x "jt'j

N'

30.30439 46

or

0.65879

5-46

ro ro i—» -C» ro O

CO

oo

en i

-P=-

II M II i—• r\5

3 CO C_J.

M -pi

II M I—»

C j . r i ­

ll CO O

CO o -Pi co <J2

II M

ro II

en

CO

O

oo -~J -»J

oo en o

en

en

en

-pi

en

O

- p i

en

-P>

b co co

co o

ro ro

en

co co ro ro ro r-o

ro ro

ro o

O O

en co i—•

i-* O O o O O O

O oo ---J cn en co ro

o o o o CO

CO co ro

vO -vl

I—» O CO ^ J H A -p» -P» en

O -P»

en c» cn

en O co

e n -p»

e n O

co Co e n co

o CO 0 0

CO co CO en

oo oo

en co co

c n o i — ' r o - v i c o i o - » J - P i v o c n o r o c o e n r o r o - p > i — •

-Pi O en

ro en ro

o co en

O O O

i o ro ro co

i

o I

o I

o

ro en o - p i c n o o c o r o i — ' c n c o c n c n - v j c n co co o i—• cn co

co en en ro en en i—> co o O en o co co co

co en - P » - p i c o r o r o r o r o r o i—» i—» o O O

O -Pi -"J co en

O O O O O i—• ro ro en eo en i-» -Pi

en en 1—• -Pi

-Pi eo oo CO O

oo ro -vl oo O

-~i ro CO en en

o en co oo CO

-vi 0D en en -vi

CO o cn i—1

-~i

00 -Pi CO CO -Pi

en CO o ro co

CO co -Pi CO en

ro en ro ro en

CO ro i — »

oo I — •

-Pi ro en -Pi en

ro O en CO en

CO oo en en en

00 CO vo en co

en ro i — >

en -vf

-pi oo o -pi en

i — »

en -Pi -Pi O

O cn oo oo -pi

o o vO o 00

o o o o o

o -pi CO -»J CO

I—»

ro -vl ro 1—»

ro en O CO -Pi

-Pi 00 o -pi en

-Pi -Pi CO en en

en co O ro vo

r o r o r o i — ' i — ' i — ' r o - P i i — ' -pi ro co ro -Pi i—»

co co ro ro ro ro ro ro ro O i-» O O i—» o eo co co co Co i—» ro en o i—» -vl -pi -Pi

I—» I—» O CO CO "vl I—• -Pi -vj .p» .pi en N M lO IB ro en o i—»

O -Pi

en

en oo en -Pi O

cn O co co -Pi

en -Pi en en

en O -Pi

o ro en

oo co

co CO oo -pi

I—» CO CO CO

ro ro

co ro en

en ro co

oo oo -Pi cn

co oo cn CO

o

cn ro co co

ro en ro co cn

O co cn co

O O O o o

-pi -pi cn ro CO

O

o o

o ro

cn cn

i ro

-vi

ro cn O

ro cn o o co co CO -Pi "~i -pi

O c o e n - p i - p i - p i c o r o r o r o r o r o c o r o r o O O O O O O O I—• r—•

ro CO i — »

cn i—»

cn cn -Pi en -Pi

-Pi eo on CO O

oo ro -«j

no o

-vi ro CO cn cn

O en CO oo CO

-~i co cn cn -~i

co O cn i — »

-~j

oo -Pi CO CO -Pi

cn CO o ro CO

CO co -Pi co cn

ro cn ro ro en

oo -Pi eo en ro

CO en o CO o

-pi I—» CO CO o

CO 00 cn en en

co eo co cn CO

cn ro i — >

cn -«j

co cn O co o

cn en -~i cn O

o cn 00 <» -pi

o o CO o oo

o o o o o

o CO CO en oo

CO oo I — •

cn U>

en ro i — »

oo oo

CO ro i — »

CO O

-Pi -Pi CO en en

en cO O ro cO

0O - P i CO

MIL-HDBK-338 15 OCTOBER 1984

and from Eq. (5.80)

of =

or

75.84371 - 46 (0.65879)' 56TI

<7t' = 1.U435

Consequently, the lognormal pdf representing the datainTable 5.3.2.1.1-1 is

g( t ) =

or

g( t ) =

t a t . V2n

1

•iM

t(l.11435) ^

1 A ' - 0.65879\ -?{ 1.11435 J

where t ' = loge t . The p lot of th is pdf is given in Figure 5.3.2.1.1-1 in terms of the straight times in hours. See Table 5.3.2.1.1-3 for the g( t ) values used.

The pdf of the loge t or of the t ' s is

g ( t ' ) to11 V2ff

i(*&) = t g( t )

or

g ( t ' ) = (l.H435)V2ff - » ( *

- 0.65879 \ 2

1U1435 )

This pdf is that of a normal distribution which is what one should expect since if t follows a lognormal distribution, loget should be normally distributed. This is shown plotted in Figure 5.3.2.1.1-2, the values of g(t') were obtained from Table 5.3.2.1.1-3.

5-48

MIL-HDBK-338 15 OCTOBER 1984

TABLE 5.3.2.1.1-3: The probability density of Time to Repair Data (From Table 5.3.2.1.1-1 based on the straight times to repair and the natural logarithm of the times to repair used to plot Figures 5.3.2.1.1-1 and 5.3.2.1.1-2, respectively.*)

Time to restore , t hours

0.02

0.1

0.2

0.3

0.5

0.7

1.0

1.4

1.8

2.0

2.4

3.0

3.4

4.0

4.4

5.0

6.0

7.0

8.0

9.0

10.0

20.0

30.0 y

40.0

80.0

*At the mode, * = 0.5584, V

At the median, t = 1.932

Probability density,

g(t)

0.00398

0.10480

0.22552

0.29510

0.34300

0.33770

0.30060

0.24524

0.19849

0.17892

0.14638

0.11039

0.09260

0.07232

0.06195

0.04976

0.03559

0.02625

0.01985

0.01534

0.01206

0.00199

0.00058 —

—

g(t) = 0.34470 and

g(¥) = 0.18530 and

Probability density

g ( t ' ) = gdog e t )

7.95 x 10"5

0.01048

0.04510

0.08853

0.17150

0.23636

0.30060

0.34334

0.35728

0.35784

0.35130

0.33118

0.31483

0.28929

0.27258

0.24880

0.21351

0.18373

0.15884

0.13804

0.12061

0.03971

0.01733

0.00888

0.00135

g ( ? ) = 0.19247.

g ( t ' ) = 0.35800.

5-49

0 t 1 12 3 T 4 5

Mode = t =0.5584 Median = t = 1.932 Mean = t = 3 .595 l i r s .

Time t o r e s t o r e , t , hours

FIGURE 5 , 3 , 2 , 1 , 1 . 1 ; pLQT OF THE LOGNORMAL PDF OF THE TIMES^TO-RESTQRE DATA GIVEN IN TABLE 5 . 3 . 2 . 1 . 1 - 3 IN TERMS OF THE STRAIGHT t ' s

en o 3 o r-- I I o a: co o m DO TO 7<i

>— co «o oo oo oo

+-> bfi

c -a

O

(1.01 0 .1

FIGURE 5 .2 .2 .1 .1 -2 :

1.0 l o .o Time to res to re , t , l iours

PLOT OF THE LOGNORMAL PDF OF THE TIMES-TO-RESTORE DATA GIVEN IN TABLE 5 . 3 . 2 . 1 . 1 - 3 IN TERMS OF THE LOGARITHMS OF t , OR LOGet = t '

100 .0

en 3

O •-" c i i— —) i o ^ GO O m Da

I i— CO l O CO OO OO

MIL-HDBK-338 15 OCTOBER 1984

2. MTTR (Mean Time to Repair) of the System

The mean time to repair of the system, t, is obtained from Eq. (5.73).

t s e(t' + 1/2 ( o t , ) 2 )

f a JO.65879 + 1/2 (1.11435)2)

or _ t = 3.595 hr.

3. Median Time to Repair

The median of the times-to-repair the system, t, is obtained from Eq. (5.76)

t = et'

or

t = e°-65879

t = 1.932 hr.

This means that in a large sample of t's half of the t's will havej/alues smaller than t, and the other half will have values greater than t. In other words, 503. of the repair times will be < t.

4. Maintainability Function M(t)

The maintainability of a unit can be evaluated as follows, using Eq. (5.62):

ti t'i z(t'i) M(tl) = f g(t) dt = f g(t') d f = f 4>{z) dz (5.81)

where t' = loget, (5.81a)

t'i - f z(t'i) = aV (5.81b)

and t' and °\* are given by Eq. (5.79) and (5.80), respectively.

By means of the transformations shown in Eqs. (5.81a) and (5.81b), the lognormal distribution of the pdf of repair times, g(t), is transformed to the standard normal distribution <j> (z) which enables the use of standard normal distribution tables (Table A-l, Appendix A).

5-52

MIL-HDBK-338 15 OCTOBER 1984

The maintainability function for the system, M(t), from (5.81) is:

z(t') M(t) = f 0(z) dz

where

t' - 7' z(t') a*, i

t1 = loge t

From the data in Table 5.3.2.1.1-1 we previously calculated

t"' = 0.65879

at- = 1.11435

The quantified M(t) is shown in Figure 5.3.2.1.1-3. The values were obtained by inserting values for, t' = loget, into the expression,

,/+-\ - f - 0.65879 Z{1 ' ~ 1.11435

solving for z(t'), and reading the value of M(t) directly from the standard normal tables in Appendix A (Table A-l).

5. Maintainability for a 20 Hour Mission

z(loge20) M(20) = f 0(z) dz

where loge 20 = 2.9957 and 7/inn on\ - 2.9957 - 0.65879 _ , n Q „

Z(loge 20) = 1711431 " 2 - 0 9 7 2

From Appendix A we find that for z = 2.0972

2.0972 M(20) = C <f>{l) (dZ) = 1-0.018 = 0.982 or 98.2%

-00

6. The time within which 90% and 95% of the Maintenance Actions are Completed (Mma„ct)

This is the time tj.afor which the maintainability is 1-a, or

r 1 _ a *'!-« z(t'l-a) M(ti_a) = P(t< tsti.aj - J g(t) dt B r g(tI) dt, = r 0 ( z ) dZj (5>82)

0 -oo ^-oo

5-53

Time t o xbktdre\,\t} Jrbiitg I

i i i ! i i : | :

i r i T i j |

JL_i_i_:__i_

I I

en 3 o »-• o r-

—I i o a: oo a m OB TO 7*

i •— OJ <o OJ 00 00

FIGURE 5 .3 .2 .1 .1 -3 : PLOT OF THE MAINTAINABILITY FUNCTION FOR THE TIMES-TO-REPAIR DATA OF EXAMPLE 2

MIL-HDBK-338 15 OCTOBER 1984

and

*(t'i.J 1 1-a l

l-a1 ( (5.83)

at'

The commonly used maintainability, or (1-a), values are 0.80, 0.85, 0.90, 0.95, and 0.99. Consequently, the z(t'i_a) values which would be used most commonly would be those previously given in Table 5.3.2.1-1. Using Eq. (5.83) the time t*i_a would then be calculated from

t'i_a= t' + z(t'i-a) • av

or

t l - a = antnoge( t ' i_a) = antilogeC't' + z( t ' i_ a ) • o t J (5.84)

Thus, for 90% Mmax ., from the previously obtained value of t' and ot

to.go = antiloge t' + z (t'o.go)cr f

= antiloge |X65879 + 1.282 (1.11435)1

= antiloge (2.08737)

= 8.06 hrs.

For 95% M m a X c t

tO.95 = antiloge [Q5879 + 1.645 (1.11435)1

= antiloge (2.491896) = 12.08 hrs.

7. Repair Rate at t = 20 hours

Using Eq. (5.63) and substituting the values for g(20) from Table 5.3.2.1.1-3 and the previously calculated value for M(20)

,ton\ q(20) = 0.00199 _ 0.00199 uuu;" 1-M(20) 1-0.982 " 0.018

= 0.11 repairs/hr.

5-55