Partial and catastrophic failure analysis of a complex system

Mkroelectron. Reliab., Vol. 19, pp. 349-350 O Pergamon Press Ltd. 1979. Printed in Great Britain

002(~2714/79/08014)349 $02.00/0

PARTIAL A N D CATASTROPHIC FAILURE ANALYSIS OF A COMPLEX SYSTEM BALBIR S. DHILLON Business Administration, Central Thermal Services, Ontario Hydro, 700 University Avenue, Toronto, Ontario M5G 1X6, Canada (Received for publication 11 M a y 1979)

Abstraet--This paper presents a generalized mathematical Markov model to analyse partial and catastrophic failures. The state probability equations are developed.

INTRODUCTION

Related literature on the subject may be found in references [ 1 4 ] . This paper presents a mathematical model to representa complex system subject to both partial and catastrophic failures• The complex system may be composed of several sub-systems. Each noncritical sub-system may fail in its partial or catastrophic failure mode. The catastrophic failure mode of each sub-system could be assigned with the order of criticality. Sub-systems of a system may be divided into two categories: (a) sub-systems with no partial failure mode; (b) sub-systems with partial failure mode. It is assumed that the sub-systems with no partial failure state are more critical than the ones with partial failure state. Finally, it is emphasized again that a criticality factor may be assigned to sub-systems with and without partial failure state which will be useful to obtain the appropriate system failure probability•

The associated system of differential equations with the mathematical model shown in Fig. 1 are:

P'~(t) - Po(t)2 = 0

(2)

P'k(t) + yk-iPk(t) = Po(t)Ctk-i

(3)

l, k = 2

f

-2, k = 4

for/=

3, k = 6

l, (n 1)

P'.(t) = Po(t)fi.-, + P . - l(t)y.-i.

(4)

n=5

for i =

n=7

ASSUMPTIONS

(a) Sub-systems fail independently. (b) Partial and other failure rates are constant.

n = 21 + 1,

(5)

g

2 = ~ 2,

(6)

i=1 NOTATION

where the prime denotes differentiation with respect is the last system state to time t. At P0(0)= 1, other initial condition probis the number of sub-systems subject to partial failures is the number of sub-systems which are not subject abilities are equal to zero. Solutions to differential to partial failures equations (1)-(4) are: k partial failure state in question Po(t) is the failure free probability at time t Po(t) = e at, (7) P l(t) probability of failure of critical sub-systems (i.e. subsystems without partial failure state) at time t where Pk(t) kth partial failure state probability at time t l l (k = 2,4,6,8 ..... (n - l)) A = 2 + ~ cq + ~ fl,, P l ( t ) = 2(1 - e - a t ) / A , (8) p,(t) nth system catastrophic failure mode failure probi=l i=1 ability at time t, More clearly, n represents the catastrophic failure state of a sub-system (i.e. Pk(t) = Ctk-i(e - a' - e -~k Q/(Yk-i - A) (9) n = 3,5,7,9,...) flj constant catastrophic failure rate of the ith subI, k = 2 system (sub-systems with partial failures) -2, k=4 2i constant failure rate of the ith critical sub-system (sub-systems without partial failures) f o r / = 3, k = 6 cq constant partial failure rate of the ith sub-system y~ constant partial to catastrophic failure state failure rate of the ith sub-system. l, (n 1) 349 n l g

i

350

BALBIR S. DHILLON

Sub- systems with par t"ial failures a

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Fig. I. System transition diagram. One star denotes sub-system partial failure state, whereas two stars denote sub-system catastrophic failure state.

P,(t)

Y,-iTk-i 7k-i2

k

ft,-i(1-e'at)

+

A

~k i)'.-i e-.~k / )'k-i()'k

~!n_ i 3(k_ i e

i--

Similarly, the system failure probability, Psf(t), is given by the following expression:

A)

n

-At

A(yk- i - A)

(10)

Psf(t) = Px(t) + Z Pi(t),

i # even number.

(12)

i=3

REFERENCES f o r i = f 23i int == 53 4n=7 ]

'

.

The system reliability, SR, is given by equation (11 ): n

1

SR = Po(t) + Z Pi(t), i=2

i ¢ odd number

(11)

1. B. S. Dhillon, A note on a four-state system, Microelectron. Reliab. 15 (1976). 2. B. V. Gnedenko, Y. K. Belyayev and A. D. Solovyev. Mathematical Methods of Reliability, Academic Press, New York (1969). 3. B. S. Dhillon, The analysis of the reliability of multistate device networks, Ph.D dissertation (1975). Available from the National Library of Canada, Ottawa. 4. B. A. Kozlov and 1. A. Ushakov, Reliability Handbook, Holt, Rinehart & Winston, New York (1970).