on the use of maximum entropy concept in two-unit repairable systems

Reliability Engineering 16 (1986) 181-199

On the Use of Maximum Entropy Concept in Two-unit Repairable Systems

Yash Gupta, Avinash Dharmadhikari# and Wing Sing Chow

Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

(Received: 9 October 1985)

A BSTRA CT

This paper applies a statistical methodology based upon information theory for approximating the reliability of two-unit warm standby systems. The repair time distribution is assumed to be Weibull. Results are provided in the form of tables and graphs.

1 I N T R O D U C T I O N

The operating characteristics o f two-unit redundant systems have been studied by several authors in the past 20 years. Reddy and Grosh 1 and Srinivasan and Subramanian 2 have compiled detailed bibliographies of the literature dealing with this area. Initial at tempts in the study of two- unit systems can be found in Refs 3-5. Since then several techniques have been proposed in the study of these systems, e.g. Osaki 6 and Buzacott 7 have used the renewal theoretical approach, whereas Sriniva- san 8 and Arora 9 have used Markov and renewal processes to analyse two-unit systems. Most o f the literature provides Laplace-Stieltjes (L-S) transforms of the transient behaviour of the operating characteristics

t On leave from the University of Poona, Pune, India. 181

Reliability Engineering 0143-8174/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain

182 Yash Gupta, Avinash Dharmadhikari, Wing Sing Chow

of the systems. A standard technique for inversion of all types bf L-S transforms is not available; ~° moreover, the process of inversion is mathematically difficult. Therefore, alternative methods are desirable. In this paper, we propose the use of the maximum entropy principle (MEP) to obtain an approximate solution for the reliability of a two- unit system, when the repair times follow a Weibull distribution.

Since the early work of Shannon ~ ~ '~ 2 in communication theory, MEP has been applied successfully in a remarkable variety of fields, including reliability estimation, ~3 queueing theory and computer modelling, ~4 system simulation, ~5 production line decision making, ~6 behavioural sciences,XV-~9 stock market analysis 2° and entropy spectral analysis. 1~

The justification for the use of MEP as a general method of approximation has been provided by several authors such as Good, z2 Shore and Johnson 23 and Clough. 24 In Section 2 of this paper, we will present the model description and provide the L-S transforms of the operating characteristics of a two-unit system, as obtained by Nakagawa and Osaki. z5

2 MODEL DESCRIPTION

We consider systems with the following properties:

(l) The system consists of two units with the same set of statistical properties and a repair facility.

(2) When both units are operable, one unit is put on-line and the other is kept as warm standby.

(3) Upon failure, a unit is taken up for repair provided that the repair facility is free. Further, it is assumed that the repair facility is always open, repairs are perfect and the repair discipline is first come, first served (FCFS).

(4) The failure rates of the units are time independent and the repair time distribution is Weibull.

The L-S transforms of the operating characteristics will now be given (symbols and definitions are given in the Appendix).

The recurrence time distribution from state 1 to state 1 (H~(t)):

H 1 l(t) = Qlo(t) © Qol(t) + Q21(t) (1)

Use of maximum entropy concept 183

where

Qol(t) = 1 - exp { - r io t} (2)

fo Qxo(t) = exp {- ]? i t} dG(t) (3)

Q12 = r i l exp{- r i l t }G( t )d t (4)

O ( t ) = 1 - a ( t )

Q ~ ( t ) = ( 1 - e x p ( - r i , t ) ) d G ( t )

The first passage time distribution from state 1 to state 2 (Hx2(t)):

H12(t) = Q12(t) + Qlo © Qol © Hx2(t) (5)

The first passage time distribution from state 1 to state 0 (H~o(t)):

Hi o(t) = Q 1 o(t) + Q~I 1 © H1 o(t) (6)

In a similar fashion:

Ho2(t) = Qox © Hlz(t) (7)

and

Hoo(t) = Qol © Hxo(t)

The L-S transforms of eqns (1), (7), and (8) are

h~l(s ) = g*(s) - sg*(s + rilXs + fix)- '

h~z(s ) = r iOr i l [ l - - g*(s + r i l ) ] {Is -'}- r i l ] [S + r io(1 - - g*(s --}- r i l ) ) ]} - 1

h~o(S) = flog*( s + rio

(8)

(9)

(10)

(11)

To obtain the inversions of L-S transforms of the above equations when the repair time distribution G(t) is Weibull, the numerical methods become tedious. In the Section 3 we suggest an alternative method which uses MEP to approximate Hoo(t), H l l ( t ) and Ho2(t ).


3 THE M A X I M U M ENTROPY PRINCIPLE (MEP)

When partial information about the random variable is provided, according to MEP we should choose a probability distribution which is consistent with the given information but has otherwise maximum uncertainty associated with it. Generally, the available information is in the form of moments of an associated random variable rarely extending beyond the first two moments. 26

In other words, we wish to find a distribution that maximizes the entropy of the system, S, subject to a set of first moment constraints reflecting the known information about S. This involves solving the maximization problem:

subject to

S = - f ( x ) l n f ( x ) d x (12)

bf(x) dx = 1

and

£ xif(x) d x = m,,

where a and b are provided.

i= 1 . . . . , r ,a <_ b (13)

If, in eqn (13) r = 1 and (a,b) = (0, ~ ) eqn (12) provides a solution as

f ( t ) = m - ~ e x p ( - t / m a ) , t > O , m I >_0 (14)

On the other hand, if, in eqn (13) r = 2 and ( a , b ) = (0, oo),

f i t) = A(Ya, 72) exp (71 t - 3,2t 2) (15)

where

=

3,1 = T x ~ 2 (16)

[B'(t)] 2 3'2 = j 2 { ~ } 2 (17)

Use of maximum entropy concept

T A B L E 1

Values of 7", B(t), Bl(t) and F(t)

185

T B(t) B 1 (t) F(t)

-- 1-234 0"209 0'302 0"335 -- 0'985 0"293 0"375 0.383 - 0'387 0-580 0'544 0"499 - 0"288 0'624 0"569 0-518 -- 0"084 9 0-745 0-618 0'556

0-615 1"226 0'750 0.670 0-637 1'243 0"753 0.673 1'270 1 1.746 0"830 0"753 1"567 1.944 0"851 0-778 2-303 2'645 0'902 0-842 2'621 2-934 0"916 0'861 2'846 3-140 0'924 0.873 3-510 3'761 0'943 0'901 3-950 4.178 0.952 0"916

and B(t) is the inverse function of Mill's ratio, if(t) is the first derivative of B(t), T is the value of Mill's ratio determined by using m2(ml) -2 - 1 = F(T) from Table 1, provided 2m 2 > m 2.

For cases where 2m 2 < m2, the MEP solution does not exist for eqn (12). 27

In Section 4 we establish the relationship between Sections 2 and 3.

4 A P P R O X I M A T I O N

In order to relate the MEP with the model given in Section 2, we obtain the first two moments ofeqns (1), (7), and (8) using their L-S transforms given by eqns (9), (10), and (11), respectively:

Li 1 = #-1 + (rio)-lg*(fll) (18)

L21 = g*"(O) -- 2(flo)- 'g*'(fll) + 2(//o)-2g*(fll) (19)

1 1 L°l° = floo +/Ag*(//1) (20)

L2o = 2//o 2 + 2[p//og,(flOl-1 + g,,,(0)[g,(//01-1

+ 2{g*'(fl, ) -- g*'(0)} {/~[g*(//1)] z } -1 (21)


1 1 L~2 = ~ + flo[ 1 -- g*(fll)] (22)

LEE = g*'(fl O[fl l ( 1 -- g*(fll))]-I

+ 2(1 -- 3og*'(fll))[flofl~(1 -- g*(fll))]-

+ 2(/31)-2 + 2g*'(30[1 -- 3og*(30]{3o[1 -- g*(31)] z } -~

+ 211 -- flog*(flO]2{flo[1 -- g*(flx)]} - 2

+ g*'(fll)[fll[ 1 -- g*(fll)]]-I (23)

The exact values of Hl~( t ) given by eqn (1) can be obtained for given values of flo,/31 and G(t); however, one can see that H02(t ) and Hoo(t ) cannot be obtained in the closed form.

We assume that the repair time distribution G(t) is Weibull in nature, with parameters 0 and c. Then:

G(t) = 1 - exp ( - OtC); t > 0, > 0, c > 0 (24)

/~-1 = F (1 + ! ) / 0 , / c (25)

g*(31) = exp ( - st) da(t) (26)

g*'(fll) = ( - u ) e x p ( - f l l U ) d G ( u ) (27)

g*"(0) = F(1 + ! ) / 0 2 / c (28)

Using eqns (24)-(28) in eqns (18)-(23) the first two moments of H~l(t), Ho2(t) and Hoo(t) are obtained.

Integration of g*(flt), g*'(3,) were evaluated using Gaussian tension product formula, IMSL subroutine D M L I N (Ref. 28, p. 321).

Initially, assuming that only L]I is provided, we calculate the f irst moment approximate MEP distribution, H ] l ( t ) of H~l( t ). Similarly, H2~l(t) is an approximate MEP distribution when both L]~ and L21 are provided. For various combinations of parameters flo, ill, 0 and c, the values of H x 1(0, H l l(t) and H 2 l(t) are provided in Tables 2-4 and Figs 1-3.


TABLE 2 Comparison of Exact First Passage Time Distribution

H~l(t ) with Its Approximations when 0--0-5, f lo=0 '3 , f l ~ = l , c = l , M e a n = 2 , Variance = 4, 111 = 3.11 and V 11 = 7-209

Exact Time (t) n l ~ (t) H~ t (t) H~I (t) OlEF1 a DIFF2 b

0-1 0-003 0"031 6 0.0262 938.275 759'3 0-4 0'040 0-121 0-102 198"707 151.972 i "0 0-178 0'275 0.240 54-635 35'022 5"0 0' 825 0"799 0" 794 3" 106 3" 722

10.0 0-973 0"950 0'975 1"306 0"305 20"0 0'999 0'998 0.999 0'537 0-102 30"0 1 "000 1 '000 1 "000 0.001 0'005 45-0 1 "000 1 "000 1-000 0"000 0-000

a Absolute percentage difference between H 1 l(t) and H~ 1(0. b Absolute percentage difference between H~l(t) and H~(t) .

A s s t a t e d e a r l i e r in th i s s e c t i o n , t h e a c t u a l v a l u e s o f Ho2(t)[Hoo(t)] c a n n o t b e c o m p u t e d . T h e r e f o r e , o n l y f i rs t m o m e n t a p p r o x i m a t e M E P

d i s t r i b u t i o n H~2(t)[H~o(t)] a r e c o m p u t e d f o r t h e s a m e c o m b i n a t i o n s o f

p a r a m e t e r s a s a b o v e . T h e s e a r e p r e s e n t e d in T a b l e s 5 - 1 0 a n d F i g s 4 - 9 .

I n S e c t i o n 5 we d i s c u s s t h e a c c u r a c y o f t he r e su l t s o b t a i n e d .


H~l(t ) with Its Approximations when 0=0-6, flo =0-5, fll = 1, c = 1, M e a n = 1"667, Variance = 2.778, l~l -- 2.417, V t l = 3.653

Time H11 (t) HI 1 (t) H21 (t) DIFF1 a DIFF2 b

0.1 0.004 0-041 0.029 867.714 589.168 0.4 0.054 0.153 0.114 180.383 108.91 1.0 0.230 0.339 0.273 47.025 18-635 5.0 0.906 0.874 0.893 3.527 1.344

10.0 0'994 0-984 0.998 0.986 0-423 15.0 0.999 0.998 0.999 0.159 0.042 30-0 1.00 0-999 1-000 0.000 0.000 40-0 1.00 1.000 1.00 0.00 0.00

a Absolute percentage difference between H 11(0 and H~1(0. b Absolute percentage difference between H ll(t) and H~t(t ).

I.O0'30U~

O.90000ff

O.BO0000

0.70000C

{t)

0.601

O.qOi

O. 20:

O. O 0 0 0 0 t ~ q

0 2 q ~, ~ 10 12 ! q 16

1

Fig. 1. Comparison of exact first passage time distribution Hl l(t ) with its approximations when 0 = 0 . 5 , /30--0.3, /31--1, c = 1, m e a n = 2, variance = 4, I11 =3.11 and

V11 = 7.209.


Hll l(t) and H~l(t) when 0 = 1,/30 = 1,/31 = 1, c = 0.85, Mean = 1.088, Variance = 1-652, 111 = 1.599, Vll = 1.759

Exact Time (t) H l l ( t ) Hill( t) HZtl(t) DIFF1 a DIFF2 b

0-1 0.013 0.061 0.0474 385.25 276.89 0.4 0-121 0.221 0-182 82.29 49.499 1.0 0.399 0.468 0-413 16.34 3.232 5.0 0.974 0-956 0.977 1.801 0.339

10.0 0.999 0.998 0.999 0.103 0.086 16 1.00 1.00 1.00 0.002 0.003 20 1.00 1.00 1.00 0.00 0.00

" Absolute percentage difference between H 11(t) and H~ ~(t). b Absolute percentage difference between H 1 l(t) and H~ 1(0.

1.00000~

0.90000~

0.800006

0.700000

0.600000

O.SO0000

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0.300000

0.200000

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0.000000

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, . . . . . . . . . , . . . . . . . . . , . . . . . . . , . . . . . . . . . ) . . . . . . . r - - • , . . . . . . . . f . . . . . . . . . ,

1 2 3 q 5 8 7 8

Fig. 2. Comparison of exact first passage time distribution H 11(t) with its approximations when 0=0"6, f lo=0-5 , fll = l, c = 1, m e a n = 1.667, variance=2.778,

111 = 2-417, V i i = 3"653.

TABLE 5 Comparison of First Passage Time Distributions

lHllo2(t) and H22(t), when 0=0"5, flo =0-3 , fll = 1, c = 1, Mean = 2, Var iance=4 , 1 ~2 = 5 ' 9 9 9 , V 0 2 = 1 2 9 " 3 3 3

Time ( t ) H~2 (t) Hg2 ( t ) D I F F 0 2 ~

0-1 0.165 0.146 11.675 0.4 0.064 0.057 11.105 !.0 0-154 0.138 10.012 5.0 0-565 0.541 4.266

10.0 0.811 0.807 0-401 20.0 0.964 0.974 1.053 30-0 0.994 0-998 0.445 45.0 0.999 1.000 0-052

a Absolute percentage difference between H~2(t ) and H2z(t).

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0.00000~

0 I 2 3 q 5 5 7 8 g

Fig. 3. Comparison between exact first passage time distribution Hill(t) and H21(t) when O= 1, rio-- 1, fl~ = 1, c=0 .85 , m e a n = 1.088, va r iance= 1.652, I ~ = 1.599,

V11 = 1.759.


H~2(t) and H22(t), when 0 = 0-6, flo = 0.5, fll = 1, e = i, Mean = 1.667, Variance = 2.778,

1~2 = 1.2, Voz = 13'64

Time (t) H~o2 (t) H22 (t) DIFF02 ~

0.1 0.024 0.020 14-934 0.4 0.091 0.078 13-903 1.0 0.211 0.187 11.954 5.0 0.696 0.677 2.766

10.0 0.908 0.918 1.197 15.0 0.972 0.984 1.251 20.0 0.999 0.999 0.077 40.0 1.00 1.00 0.007

a Absolute percentage difference between H~z(t ) and Hgz(t ).

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HIE(t ) and HoZ2(t), when 0 = 1, flo = l,/31 = l, c = 0.85, Mean = 1-088, Variance = 1.652, 112 = 3"044, Voo = 7-093

Time (t) H~2(t ) H~E(t ) DIFF02 ~

0.1 0.30 0.027 15.462 0.4 0.123 0-105 13.996 1.0 0.28 0.248 11.283 5-0 0.806 0.803 0.489

10.0 0.963 0.976 1.438 16.0 0.995 0.999 0.431 20.0 0-998 0-999 0.138

a Absolute percentage difference between H~2(t ) and Hg2(t ).


H~o(t ) and Ho2o(t), when 0 = 0 . 5 , /3o=0-3, /31 = 1, c = 1, M e a n = 2 , V a r i a n c e = 4 , lolo=9.33, Voo = 63.112

Time ( t ) Hlo (t) Hgo (t) DIFFO(P

0. l 0.107 0.009 19.47 0.4 0.042 0.034 18.873 1.0 0.102 0.084 17.70 5.0 0.415 0-37 10.79

10.00 0.657 0.628 4.482 20.0 0.883 0.894 1.284 30-0 0.96 0.977 1.805 40.0 0.992 0.998 0.677

a Absolute percentage difference between H~o(t ) and H~o(t).


H~o(t) and H2o(t), when 0=0 .06 , /3o=0.5, / 3 l=1 , c = l , M e a n = l - 6 6 7 , ance = 2.778, 1~o -= 6.444, Voo = 33.012

Vari-

Time (t) H~o(t) H2o(t) DIFFO0 ~

0.1 0.154 0'013 13.267 0.4 0.060 0'053 12.666 1"0 0.144 0.127 11-508 5"0 0"539 0-511 5.274

10.0 0.788 0-782 0.808 15.0 0.902 0"911 0.958 30.0 0.990 0-997 0.616 40.0 0.998 0.999 0.178

Absolute percentage difference between H~o(t ) and HZo(t).


T A B L E 10 Compar i son o f Firs t Passage Time Dis t r ibut ions

H~o(t ) and Hgo(t), when 0 = 1, ~o = 1,/~1 = 1, c = 0.85, Mean = 1.088, Var iance = 1-652, lolo = 3.129, Voo = 9.178

Time (t) H~o(t) Hgo(t) DIFFO0 ~

0.1 0-031 0-030 3.339 0.4 0.12 0.116 3-026 1-0 0.273 0-267 2.456 5.0 0.767 0.796 0.247

10.0 0.96 0-962 0.295 16-0 0.994 0-995 0-152 20.0 0.998 0.999 0.067

a Absolu te percentage difference between H~o(t ) and H~o(t).

5 A C C U R A C Y

Since the exact distribution for the recurrence time of state 1 could be determined using eqn (1), the percentage differences between this distribution and H~l(t)[H~l(t)] for various values o f t were considered as a measure of accuracy. The values of these differences with respect to H~ 1(0 and H21(t) are given in Tables 2-4. One can observe that after a relatively short time, depending upon the combination of parameters, these differences become insignificant. Similarly, the percentage differences between H~a(t) and H~(t) become insignificant for values of t larger than 25. Therefore, we suggest that the consideration of MEP distribution on the basis of higher order moments may not contribute any further significant information. The exact solution ofHoo(t), [H02(t)] , cannot be obtained; therefore, the differences between H~o(t ) and H2oo(t)[H~2(t),Ho22(t)] were considered as a measure of accuracy. The values of these differences for various values of t are provided in Tables 8-10 [Tables 5-7]. Once again, it can be noted from these tables that these differences become insignificant for values of t larger than 25. We justify the application of the MEP to approximate Hoo(t ) and Ho2(t) on the basis of the results that the differences obtained for H~ 1(0 and its approximations become insignificant for t larger than 2.5; therefore, we assume the differences between Hoo(t), Ho2(t ) and their approximation would also become insignificant for values of t larger than 25.


6 CONCLUSION

In a two-unit system, the L-S transforms of the operating characteristics were obtained, under the assumptions (i) that failure rates are time independent and (ii) that the repair time distribution is Weibull in nature. The exact solutions of these operating characteristics were compared with the approximate solutions obtained by using MEP, for various combinations of parameter values. The values of systems parameters for various values of component parameters are summarized in Table 11. It is concluded that, for large values of t, MEP distributions

TABLE 11 Parameters of Weibull Distr ibution and Corresponding Mean and Variance of the

System

0 3o l~,

Repair time (1) (1) (1)

C Mean Variance L l x Vx 1 Loz Vo2 Loo Voo

I 1 1 4 0.9064 0'0646 1"0923 0.3534 1.7284 2-1925 5.8752 21.9694 1 0.5 2.0 1.0 1.0000 1.0000 1.6667 2.3333 3.4999 10.2499 5.0000 17.0003 0.5 0.3 1.0 1.0 2.0000 4.0000 3.1111 7.2098 5.9999 29.3333 9.3334 63.1120 0.6 0.5 1.0 1.0 1.6667 2.7778 3'4167 3.6528 4.2000 13.6400 6.4444 33.0124 1.0 1.0 2.0 2.0 0.8862 0.2146 1.1284 0-4847 1.8195 2.4674 4-6603 14.3914 1'0 1.0 1.0 0 .85 1.0879 1.6523 1"5988 1.7595 3.0443 7-0931 3.1298 0.1781

are good approximations to the original distributions of the operating characteristics of these systems. It is further concluded that MEP distributions involving their order moments (beyond second moments) do not add significant information in a system.

ACKNOWLEDGEMENTS

The authors acknowledge Professor C. R. Bector for introducing them to the field of Information Theory. The authors also acknowledge the financial support provided by the Natural Science and Engineering Research Council of Canada and the University of Manitoba Research Administration. The second author thanks the authorities of the University of Poona, India for granting him leave.


R E F E R E N C E S

1. Reddy, P. and Grosh, D. R. Standby redundancy in reliability--a review, IEEE Trans. Reliab. (sent for publication 1984).

2. Srinivasan, S. K. and Subramanian, R. Probabilistic Analysis of Redundant Systems, Springer-Verlag, New York, 1980.

3. Epstein, B. and Hosford, J. Reliability of some two-unit redundant systems, Sixth Symp. Rel. Qual. Contr. (1960), pp. 469-76.

4. Barlow, R. E. and Proschan, F. Mathematical Theory of Reliability, John Wiley, New York, 1965.

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A P P E N D I X : LIST O F SYMBOLS & D E F I N I T I O N S

a*(s)

a'(t)

a"(t)

©

DIFF1

DIFF01

G( t), t >_ 0 Hij(t ), i = 0, 1;

j = 0 , 1,2

The Laplace-Stieltjes transform of function A(t)

da(t)

dt

d ~ta'(t)

Convolution

[Hlo,(t) - H2o,(t)[ Hloi(t ) × 100, i 0, 2, t > 0

IH1 ~(t) - H~ ~(t)l = × 100, i = l , 2 , t > 0 H~ ~(t)

Repair time distribution First passage time distribution from state i to state j


Hii(t),k = 1 , 2 ; i = 0, 1; j = 0 , 1 , 2

i = (0, 1,2)

Qii(t), i = 0, 1; j--- 0, 1,2

Q121(t)

S v,j x(t) fll, i = O, l, fli > O F

Informat ion- theore t ic approximat ions to Hij(t ) when the first k moment s are specified. State of a system, indicating the number of non-operat ive units The kth m o m e n t of Hii about zero The probabili ty that after making a transit ion into state i the process next makes a transit ion into state j in (0, t) The recurrence time transit ion probabili ty for State 1 via State 2 Measure of informat ion Mean repair time State of the system at t ime t Failure rate of the system in State i G a m m a function

on the use of maximum entropy concept in two-unit repairable systems

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