AD-A129 553 IFR (INCREASING FAILURE RATE) FOR REPAIRABLE SYSTEMS /U) CITY COLL NEW YORK DEFY OF MATHEMATICSN R CHAGANTY APR 83 CUNY-MB4 AFOSR-TR-63-0U1UNCLASIFIED FOSR-8-0024F/ 12/ 2

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MICROCOPY RESOLUTION TEST CHARTNA71ONAL BUREAU OF STANDARDS-1963-A

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AFOSR-TR. 83- 0511

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IFR Results for Repairable Systems

by N.R. Chaganty

April 1983

City College, CUNY Report No. MB4AFOSR Technical Report No. 82-04

AFOSR Grant No. 82-0024 DTVSELErT

JUN 21 iw

A I

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AFOSR-TR. 8 3-0 f-) - A27 534. TITLE (and Subtitle) S. TYPE OF REPORT B PERIOD COVERED

IFR RESULTS FOR REPAIRABLE SYSTEMS TECHNICAL

S. PERFORMING OG. REPORT NUMBER

-__CUNY Report No. MB47. AUTHOR(s) 1. CONTRACT OWGRANT NUMBER(s)

N.R. Chaganty AFOSR-82--0024

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The City College - CUNY PE61102F; 2304/A5New York NY 10031

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1. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if neceasary md identify by block number)

IFR distributions; IFRA distributions; NBU distributions; repairable systems;k out of n systems; time to first failure; Markov processes.

20. ABSTRACT (Continue an rrve side f necessary and identity by block nimiber)

Consider a k-out-of-n system with Independent repairable components.

Assume that the repair and failure distributions are exponential with

parameters n,...,Un and nI...,). It is shown that if A 1-j1 - A for

all 1, then the distribution of system life is in1.

DO :0:S 147 EDITION OF IV S O14O7fT3 8 UNCLASSIFIED

83 06 ra 2 SECURITY CLASSIFICATION OF TMIS PAGE (Ph"m be 0eod

ABSTRACT

Considers a k-out-of-n system with independent repairable

components. Asumethat the repair and failure distributions

are exponential with parameters i and f l'''''Xn ]

respectively. In this paper we showsthat if AXi-p. = A for

all i then the life distribution of the system is Increasing

Failure Rate.

Thi ter o

"

e.'.

* t ~i

AIR ? OVFI 1E O~l, SOTJ:1V. R'F SFARgt% (AFSC

qpprov o t ...Distr t * j **MATTHEWJ .Chief, Technical Inforuiation Division

A .-

1. Introduction. A k-out-of-n system is one which functions if

and only if at least k of the n components function. An

n-out-of-n system is known as a series system and an

1-out-of-n system is known as a parallel system. Consider a

k-out-of-n system with independent repairable components,

each of which has exponentially distributed failure times and

repair times. Assume that the exponential failure time distributions

have parameters X,...I Xn and the exponential repair times

have parameters PiV-''' Pn" The study of the life distribution

of these systems has received the attention of many authors.

Birnbaum-Esary-Marshall [2] have shown that if the components

are irrepairabe (pi 0, 1 5 i ! n), then the life distribution

of the k-out-of-n system is Increasing Failure Rate on the

Average (IFRA). This is true even for general coherent systems.

In the special case where X Is are all equal to X and pi's are

w equal to p, the number of components not working at any time

point forms a birth and death process. The results on birth

and death processes of Keilson F71, Derman-Ross-Schechner [61,

Brown and Chaganty (4) all imply that the life distribution of

the k-out-of-n system is Increasing Failure Rate (IFR). More

generally without any restrictions on the Xi's and pi's, it was

shown by Ross [9], the life distribution of a coherent system

is New Better than Used (NBU), from which it follows that the

life distribution of the k-out-of-n system is NBU. The theorem

of Ross (9) is actually a special case of the result in section

6.6 of Brown and Chaganty !4]. The question whether the

conclusion of Ross r91 can be strengthened to IFRA for a

k-out-of-n system remains open. The case k=n=2 failed to

yield any counter example. In this paper we show that for

a k-out-of-n system if A1 - i is constant over i, the

life distribution of the system is IFR. The motivation and

importance for the study of the life distribution of the

k-out-of-n system comes from the results of Esary and Proschan

r5. They have shown that the distribution function of the

parallel and series systems provide lower and upper bounds

respectively for the distribution function of an arbitrary

coherent system. The Laplace transform and moments of

the parallel system, without any restrictions on A 's and 0i's,

and related results were derived in Brown [3].

2. Definitions and Preliminaries. A nonnegative random

variable T with distribution F is said to be IFR if F(x+t)/F(t)

is nonincreasing in -= < t < -, for each x a 0, where F = 1-F.

If the density f exists, this is equivalent to saying that

r(x) - f(x)/i(x) is nondecreasing in x a 0. The function r(x)

is known as the failure rate. The random variable T is said

to be IFRA if r(x)Jl/x is nonincreasing in x k 0 and T is NBU

if F(x+y) s F(x) F(y) for all x,y k 0. It is well known that

IFR . IFRA -0 NBU

and none of the reverse implications are true. The properties

and usefulness of theme three classes of life distribUtions were

well explained in Barlow and Proschan Ell.

QI

Consider a coherent system with n components. Let Xi(t)=O

if component i is functioning at time t and 1 if it is under

repair, i=l,... ,n. The vector X(t)=(Xi(t),... Xn (t)) determines

the state of the system at time t. To start with we assume that

all the components are working, that is, X(O) = (0,0,...,0) with

probability 1. Under the asumptions of exponential failure

and repair distributions with parameters 0 < Xi < a, 0< < '

i=1,... ,n, the stochastic process X = (X(t), t z 01 forms a

markov process with state space S, consisting of 2n vectors of

O's and l's. Let (Ti, i k 0) be the successive times of transition

for the process X and Y {Y., i a 01 be the successive states

visited by X. If Y. - , = (xl,...,xn ) c S, then the

sojourn interval in X, [Ti, Ti+l), is exponentially distributed

with parameter Z(xiui + (i-xi)A i) and further Y forms a markov

chain with state space S and transition matrix Q, of dimension

2n 2 n2 x 2n given by

ON!5, Y) = £(xiyj + (lYxi))) if xi=0, Yi=l, xj=yj for j i

+i (if XiOl, y1 -O, xj-yj for j i=£(x-iPi + (1-xi) TTi!

- 0 otherwise.

Let the function f: S N (0,1,2,...en) be defined by

f(x) - Exi . The function f partitions the state space S into

(n+l) parts A0,...,An, where A - (15 - L, I k 0. Letn

6(0) - E Ai.i-i

UnCar the assumption Ai-j i is constant over i, the quantityIt n n

6(j) T i + E Xi is equal to E(xivi + (l-xi)l i ) for all

x A. This means that the probability of transition from

a state x E A to Aj, j # L, t k 0, is the same for all x c A inJ

the embedded discrete time markov chain Y. Thus by Theorem

6.3.2, page 124, of Kemeny and Snell (8) the lumped process

f(Y) = ff(Yih i a 01 is again a markov chain. Hence the

process f(X) = ff(X(t)): t 2 0) is a markov process with

state space N and successive states visited given by f(Y)

at times {Ti , i k 0). Also [T i , Ti is exponentially

distributed with parameter 6(1) if f(Yi)=I. To compute the

transition matrix of f(Y) we proceed as follows. Let M(L)be the cardinality of At. We can partition the matrix Q as

Qo ,l . (n

_Oo ni O nn

where Qij -(Q(x,y)), x C Ai. y e Aj. is of order M(i) x M(J),

i,j £ N. Let i. e N, 1 < I S m,, m k 1. Then the joint

probabilities of the lumped process f(Y) are given by

P-f(YO) M i , t - 1,..., m - Qi1 0ki 2 mmli eii (2.3)

where ei is a column vector of M(i m) one's. Note that Q ij-0

if Ii-JIk 2. Thus relation (2.3) shows that f(Y) is a birth

and death process. The following lemma will be useful to

compute the transition probabilities.

Lemma 2.1. Let Qjj be defined as in (2.2). Then

m! E E E NiA Al~il<i2<. .. <i t 1 "2 m

Q0112... m-l,mem = 6(0)6(1) ... 6(m-l) (2.4)

for m a 1.

Proof: From (2.3) it follows that the L-H-S of (2.4) is the

probability that m components fail in succession. These m

components can be choosen in (n) ways. For each selectionM

(,.O..,i m ) there are ml arrangements of the components and

each arrangement has probability (A. A )/6(0)...6(m-l)

and hence the R-H-S.

Let P = (pij) be the transition matrix of the markov

chain f(Y). Using (2.3) and Lemma 2.1 we can easily verify

that for 1 • m f n-i,

(m+l) E E E A A I A i14 1 2 M+l

Pmj 1(<)= E E3L 41 , if j-m+l (2.5)a(m} E E E Al .. A i

I - Pm m+1 ,if J-m-i

- 0 , otherwise,

and Pol - Pn n-I .

The preceding discussion also shows that the markov

process f(X) is a birth and death process with birth rates

{pr m+8S(m), Otcn-1} and death rates prm m-I 6 (m) , lsnmn).

If the coherent system under consideration is a k-out-of-n

system then the life length of the system is just the first

passage time to the set fk+l, ..., n) for the process f(X).

Thus the life distribution of the k-out-of-n system is

IFR follows from the result in section 6.3 of Brown and

Chaganty [4]. Summarizing the above discussion we have

the following main result of this paper.

Theorem 2.2. Consider a k-out-of-n system with independent

repairable components. Assume that the life distributions

are exponential with parameters ll,...Xn and repair

distributions also exponential with parameters 1 '...# ln"

If A i-1Ii = A for all i, then the distribution of the system

life is IFR.

Remark 2.3. If A=X 2= 0.. MXn and Vl=P2= ... Pn the above

Theorem 2.2 shows that the life distribution of the k-out-of-n

system is IFR. This result was mentioned earlier in the introduction.

REFERENCES

[1] Barlow, R. E. and Proschan, F. (1975). Statistical Theoryof Reliability and Life Testing. Holt, Rhinehart andWinston, Now York.

C2) Birnbaum, Z. W., Esary, J. D., and Marshall, A. W. (1966)."Stochastic characterization of wearout for componentsand systems," Ann. Math. Statist., 37. 816-25.

[3) Brown, M. (1975). "The first passage time distribution fora parallel exponential system with repair," Reliabilityand Fault Tree Analysis, edited by Barlow, R. E.,Fussell, J., and Singpurwalla, N. Conference volumepublished by SIAM, Philadelphia.

[4) Brown, M and Chaganty, N.R. (1980). "On the first passagetime distribution for a class of markov chains,"to appear in Ann. of Prob.

[5) Esary, J. D. and Proschan, F. (1970). "A reliabilitybound for systems of maintained interdependentcomponents," JASA, 65, 329-339.

[6) Derman, C., Ross, S. M., and Schechner, Z. (1979)."A note on first passage times in birth and deathand negative diffusion processes," unpublishedmanuscript.

[7) Keilson, J. (1979). Markov Chain Models - Rarity andExponentiality. Springer-Verlag, New York.

[8) Kemeny, J. G. and Snell, J. L. (1960). Finite MarkovChains. D. Van Nostrand Co., Inc.. Princeton, N.J.

[9] Ross, S. M. (1976). "On time to first failure inmulticomponent exponential reliability systems,"Journal of Stochastic Processes and their Appln.,4, 167-173.

J