Profit analysis of a k-out-of-n trichotomous system

Reliability Engineering and System Safety 37 (1992) 39-44

1

Profit analysis of a k-out-of-n trichotomous system

Rakesh Gupta & L. R. Goel Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut-250004, India

(Received 23 March 1991; accepted 24 June 1991)

This paper studies the profit function of a mathematical model of a k-out-of-n trichotomous system. Failure times are assumed to follow negative exponential distributions with different parameters whereas the repair time distributions are taken to be general. The analysis is carried out by using the supplementary variable technique.

NOTATION AND STATES OF THE SYSTEM

Pw(t), Sw=O, 1 . . . . .

n - k + 2 pro(x, t) dx, m = n - k + l ,

n - k + 2 s

@

•j• j=O, 1 . . . . . n - - k j

Ti(x), q(x)

~t(x), q(x)

Probability that the system is in state W~ at time t

Probability that the system is in state Sm at time t and has sojourned in this state for a period of (x, x + dx) Dummy variable in Laplace transform Constant short-circuited failure rate of a component Constant open-mode failure rate of a component when j components of the system have failed due to short circuit Repair rate and corresponding pdf of repair time when system breaks down due to open-mode failure in a component, so that

0(x) exp[- fo 0 0 , ) d v ] = g ( x )

Repair rate and corresponding pdf of repair time when system breaks down due to short circuit in any of the n - k + 1 components, so that

~(x) exp[-f0 x/z(v) dv] =q(x)

Reliability Engineering and System Safety 0951-8320/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England.

The various states of the system are: Si, Operative state of the system with i = 1, 2 . . . . . (n - i) good components whereas i

n - k components have failed one by one due to short circuit

S,-k+l Failed state of the system when n - k + 1 components have failed one by one due to short circuit

S,-k+2 Failed state of the system due to the open mode failure in a component Initial operative state of the system where all the components are good in series configuration

&

39

INTRODUCTION

Trichotomous systems are ones in which components can fail in two mutually exclusive modes. Networks of relays, fuse systems for warheads, diode circuits, thyristor converters, capacitor banks, etc., are a few examples of trichotomous systems. For instance, in an electrical system having components connected in series, if a short circuit occurs in one of the components, then the short-circuited component will not operate but permit flow of current through the remaining components so that they continue to operate. However, an open-circuit failure of any of the components will cause an open-circuit failure of the system. As an example, suppose we have a number of 5 W bulbs which remain operative in satisfactory conditions at voltages ranging between 3 and 6. Obviously on using well known formulae in

40 Rakesh Gupta, L. R. Goel

physics, if these bulbs are arranged in a series network to form a trichotomous system, then the maximum and the minimum number of bulbs are n = 80 and k = 40 respectively in a situation when the system is operative at 240 V. In this case any of the bulbs may fail either in close or in open mode till the system is operative with 40 bulbs. Here it is clear that after each failure in close mode, the rate of failure of a bulb in open-mode increases due to the fact that the voltage passing through each bulb increases as the number of bulbs in the series decreases.

Various authors 1-5 have analysed the k-out-of-n system models and obtained only the pointwise and steady-state availabilities of the system by using the supplementary variable technique. The purpose of the present paper is to investigate a k-out-of-n trichotomous system model and to obtain various economics related measures of system effectiveness as follows:

(i) pointwise and steady-state availabilities of the system;

(ii) expected up-time of the system during (0, t] and in the steady state;

(iii) reliability and mean time to system failure; (iv) expected busy period of the repairman during

(0, t] and in the steady state; (v) net expected profit earned by the system in

(0, t] and in the steady state.

MODEL DESCRIPTION

Initially the system comprises n identical good components connected in series. Each component may fail in any of the two mutually exclusive modes (open and closed). The closed-mode failure in a component is defined as the failure due to a short circuit in the component. Due to short-circuit failure

in any of the components, the short-circuited component does not operate but the system still operates with the remaining ( n - 1) components. Further, due to open-mode failure in any of the components, the system breaks down (fails totally). This process goes on until we have k good components in the system. At the stage of k good components in the system, the system breaks down due to either open- or dosed-mode failure in any of the k components. A natural assumption is that the open mode failure rates increase as the number of good components in the system decreases. The repair is carried out only when the system breaks down and each repair makes the system as good as new. All the failure time distributions are taken to be negative exponential while the repair time distributions are general. A transition diagram of the system model is shown in Fig. 1.

BASIC EQUATIONS AND THEIR LT

Probabilistic considerations and limiting procedure yield the following integro-differential equations (the limits of integration, when 0 to ~, are not mentioned):

[O+ n(oc + flo)]Po(t)

=f Pn_k+l(x,t)#(x)dx + f Pn_k+2(x,t)rl(x)dx (1)

[ a + (n - i)( oc + fli) ]P~(t)

= ( n - i + l ) t r P , _ , ( t ) , i = 1 , 2 . . . . . n - k (2) a a

[~xx + ~ + [~(x)]Pn-k+l(X, t)=O (3)

+ ~ + r/(x)]P._k+2(x, t) = 0 (4)

q(x

~(x)

1 s.2 S -k -k+1

J - ooooJ

~ I n-z)p

nP° S kpn-k

[~:FAILED STATE Fig. 1. Transition diagram.

Profit analysis of a k-out-of-n trichotomous system 41

We also have the following boundary conditions:

P.-k+,(O, t) = kotP._k(t)

and

(5)

n--k Pn_k+2(0, t)= ~, (n-j)f l jPj(t) (6)

j=0

t'ondiliom

It is assumed that the system initially starts from normal state So, i.e.

Po(0) = 1, Pw(0) -- 0 = Pn-k+l(X, O) m Pn-k+2(X, 0) (7)

NOW define

P*(s) = LT[P.,(t)] = f exp(-st)Pw(t) dt

so that

LT[P'(t)] = sP*(s) - P.,(O)

Taking Laplace transforms of eqns (1)-(6), we get

Is + nor + Po)lP~(s) - f P*-k+,(x, s)tt(x) dx

-- en-k+2(X, S)TI(X) dx = 1 (8)

Is + (n - i )(a + D l p T ( s ) - (n - i + 1) .P* , ( s ) = 0 (9)

0 -~X P*-k+I(X' S) + [$ + ll(x)]P*_k+l(X, s) = 0 (10)

0 OX P*-k+2(X, S) 4" [S 4" rl(x)lP*.-k+2(x, s) = 0 (11)

P*-k÷I(0, S) = kocP*_k(S) (12) n--k

P*-k+2(0, S) = 2 (n - j )a/PT(s ) (13) j=0

CALCULATION OF P*(s)

Integrating (10) and using (12), we get

P~-k+l(x, s) = kocP*~_k(s ) exp - s x I~(V) dv (14)

so that

f e*~-k+t(X, dx = kocq*(s)e*~_k(S) (15) S)~(X)

Also from (14),

" f = P . - k ÷ l ( x , s ) dx P.-k+I(S) *

=kffP*_k(s) LT exp[ - f [ /~ (v ) d r ]

= kocP*_k(s)[1 -- q*(s)]/s (16)

Integrating (11) and using (13), one gets

[ fo ] * -- --SX - - Pn_k+2(X, S) -- exp q(v) dv

n-k x ~, ( n - j ) p j P t ( s ) (17)

j=0

so that

f n-k P,-k+2(x, s)TI(X) dx =g*(s) ~ (n - j ) p j P ; ( s ) (18) j=O

Also from (16),

, -f Pn_k+2(S) - - * Pn-k+l(X, S) dx

n-k = s - l [ 1 - g * ( s ) ] ~ (n - j )p j eT( s ) (19)

j=O

Using (15) and (18) in (8)

[s + n(oc + Po)le~(s) - kaq*(s)P*_k(s) n--k

- g * ( s ) ~ (n-j)pjPt(s)= 1 (20) i=O

From (9),

P*(s) =P~(s)/A,(s), i= 1, 2 , . . . , n - k (21)

where

Ai(s) = (-I s + (n - r)(tr + fl,)

Using (21) in eqn (20) we get P~(s) as

kaq*(s) e~(s) = s + n(oc + flo) A,_k(s)

/ %k (n -- i)fl A . . . ] - 1 - I n f l o + ~ - - - g Ls)

i = , A i ( $ ) ) J \ (22)

Finally, using (21) in (16) and (19), one gets

P*-k+l(s) = :111 - q*(s)]kaP~(s)/An_k(s) (23)

P'n-k+2(s) --$-1[ 1 __ g , ( s ) ] [ n p o 4- ~ k (n~_/)_~Pi]p~(s) i=t /tiLs) 3

(24)

ANALYSIS OF CHARACTERISTICS

Using the results in (21)-(24), the expressions giving important measures of system effectiveness are:

(i) The reliability of the system R(t) in terms of its Laplace transform is

R*(s) = LT[R(t)I = 1 + ~ 1/Ai(s) [s + n(a + flo)] -1 i=1

(25)

and the mean time to system failure (MTSF) is given


by

where

MTSF = lim R*(s) $---*0

= 1 + ~ l /A i tr+flo)n i= l

(26)

lLi (n - r)(oc + ~r) Ai = r= l ( ; -~ ~ 1 ) ~

(ii) Pointwise availability A(t) of the system in terms of its Laplace transform is given by

A*(s) = L T [ A ( t ) I = 1 + 1/A~(s P~(s) (27) t. i=1

the steady-state availability of the system

A(~) = lim sA*(s) $-'*0

n-k ] = 1 + ~ 1/A, Po (28)

i=1 -J

as such becomes

ti, = f xg(x) dx

i = 1, 2 . . . . . (n - k)

where in terms of

ep = f xq(x) dx,

and i

D, = ~ [(n - r)(a~ + ~r)] -1, r= l

we have

--1 Po = 1 + nfloW + ktr(dp + Dn-k)An-k

n-k ] - 1 + X (n - - i)fli(q I + Di)A; 1

i=1

(iii) The expected up time of the system during (0, t] is given by

so that

(iv) The during (0, t] is

It.p(t) = A(v ) do

It*,p(s) = A*(s)/s (29)

expected busy period of the repairman

I~b(t) = B(u) du

so that

I*7,(s) = B*(s)/s (30)

where

(v) The net expected profit incurred in (0, t] is

P(t) = Total revenue in (0, t] - Expected cost of repair in (0, t]

= Ko~up(t) - Kll~b(t) (31)

and the expected profit per unit time in the steady state is given by

P = lim P(t) / t t----~oo

= Ko lim Izup(t)lt - K1 lim I~b(t)/t t---*oo t----, oo

= KoA(Oo) - K , B ( ~ ) (32)

where Ko is revenues per unit up time, K1 is cost of repair per unit of time and

B(~) = lim sB*(s) s'-'-~0

n--k ] = nflo~ + kocdP/An_k + X (n - i)flil¥/Ai Po

i=1

(33)

P A R T I C U L A R C A S E

When repair time distributions are also exponential with parameters/~ and r/, i.e.

q(x) = l~e - ~ , g(x) = tie -"x

then for n = 3 and k = 2 we have

negative

(i, 3 ° ] s + 2(tr + ill) Is + 3(a~ + flo)] - l

(34)

MTSF = [1 -t 3a~ 2 ( d ? ~ 1 ) ] / 3 ( c ~ ' "+" ~0) (35 )

A(~) = [1 -~ 3tr 2(a~ ~ ill) ]Po (36)

B ( o o ) = a [ f l ° + tr tr fll

(ii)

(i i i)

(iv)

where

[ 3a, 1 tl ]] -1 po = 1+ ÷ (a~+ ill) ( ~ + ~ + # r / / j

On taking the inverse Laplace transform of R*(s) given in (34), the reliability of the system is given by

.(cl + 5a~ + 2fll)e c: - (c2 + 5re + 2fll)e c2t R(t) (38)

C 1 -- C 2

where cl and c 2 are the roots of the equation

B*(s) = P*-k+,(s) + P*-k+E(S) s2 + (5re + 3flo + 2flOs + 6(tr + flo)(tr + ill)

Profit analysis of a k-out-of-n trichotomous system 43

STUDY OF SYSTEM B E H A V I O U R T H R O U G H GRAPHS

For a more concrete study of system behaviour, we plot the various measures of system effectiveness given in the particular case. In Fig. 2, curves represent the graph for the reliability of the system R(t) in respect to mission time t for ¢ = 0.5, 1-0 and 1.5 taking flo = 0.5 and fl~ = 0.7. It is observed that the reliability of the system decreases uniformly as the time passes. This decrease is rapid initially and stabilises as t increases. Further, it is also noted that there is a uniform decrease in the reliability of the system as the short-circuited failure rate of the component, a~, increases.

In Fig. 3, dotted and smooth curves respectively represent the graphs for A(~) and B(=) w.r.t . /~ taking a~ = 1-0, 1.2 and 1.4, whereas, fl0 = 0.5, fll = 0.7 and T/= 50. Here the steady-state availability of the system increases uniformly as the repair rate of the system '/z' increases. This availability also increases with the decrease in failure rate of a component due to a short circuit. For the study of B(oo) curves, we noted that the trend is the reverse to that observed in the case of A ( ' ) .

Finally Fig. 4 shows the variations in steady-state profit w.r.t. /~ and o( when Ko=150 , K1=50 , fl0 = 0.5, fll = 0.7 and r /= 50 are kept fixed. Different curves have been sketched for different values of a~(=l-0, 1.2, 1-4). Here, an obvious conclusion is that

100

90

8O

70 m <

,.J " ' 60

LI.I

W

uJ 0 .

30

20

10

\ I ~=o.%R:o.7

\

I I o.1 o'.2 0.'3 o.~ 0.'5 0.'6 0:7 o:8 1:

~.CC=O 5

"~. C{=1.0

"° ~,(:[=1"5

o.; 1.5

B(~) A(~) • 60 [.90 7

q

/ / / /

/ . s / / / /

/ / / " / , ; /

/ / l • /

..... C{=I.O

/

/ i

or o "

I E:°'5,P':°'7,n:5° I • 30 -.60 F ,' ,' ;

. \ , , , , i 11 / ~ • i i • \ \ \

i " / / . • , \

, , , : "~. ~-~ ~. a:14

~ , ~ " ~ , (E=1.2 -'40 ';- ~ - " ~ . f f : l .0 II0

Fig. 3. Steady-state beha~our of availability and the probability that the repair man is busy w.r.t. /~ and o~.

130

120

110

100

90

8O I--

0 70

60

50

l ~o=O.S ~=o-7, ~:so I , ~=lso K~5o

3O

0 2 3 4 5 6 7 9 1

i=1'0 =1.2

Fig. 2. Behaviour of percentage reliability w.r.t, time for Fig. 4. Behaviour of profit function w.r.t. /~ for different different values of a~. values of a~.


the steady-state profit of the system increases as the repair rate increases or failure rate of a component ' ~ ' decreases.

REFERENCES

1. Dhillon, B. S., A k-out of N three state devices system with common cause failures. Microelectron. Reliab., 18 (1978) 447-8.

2. Chung, W. K., A k-out of N redundant system with

common cause failures. IEEE Trans. Reliab., R-29(4) (1980) 344.

3. Chung, W. K., An availability calculation for k-out of N redundant system with common-cause failures and replacement. M icroelectron. Reliab., 20 (1980) 517.

4. Chung, W. K., An availability analysis of a k-out of N:G redundant system with dependent failure rates and common cause failures. Microelectron. Reliab., 28 (1988) 391-3.

5. Mokhles, N. A. & Salch, E. H., Reliability analysis of a k-out of N reparable system with dependent units. Microelectron. Reliab., 28 (1988) 535-9.

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Profit analysis of a k-out-of-n trichotomous system