IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 4, 1995 DECEMBER
689
Symmetric Relations in Multistate Systems Jianan Xue, Student Member IEEE Wayne State University, Detroit Kai Yang, Member IEEE Wayne State University, Detroit
Key Words - Multistate reliability, Structure function, Symmetric relation, Representativefunction.
Summary & Conclusions- This paper: 1) derivesthe upper bound for the number of critical upper (lower) vectors to level j of a monotone increasing multi-statesystem; 2) discusses the symmetric relations among the components, 3) gives several theoretical conclusions; and 4) proposes a simplified form for the structure function of the multi-state systems with symmetric relations.
1. INTRODUCTION' Acronym? CLV,, CUV, critical [lower, upper] vector at levelj MIF monotone increasing function MCS minimum cut-set MPS minimum path-set.
the system lower order MCS. The concepts of the MCS & MPS have been extended to the multi-state case. The MCS is extended to a CUVj (or minimum upper vector at level j ) ; the MPS is extended to CLVj (or maximum lower vector at level j ) [l 7, 12, 13, 151. When we apply the system state mode analysis to multi-state systems, however, we may have difficulty listing all the system CUVj & CLVj, because there can be so many of them. This can be seen in the conclusions in section 3. Thus we must find an efficientmethod for system state mode analysis for the multi-state case. This is equivalent to finding a simplified mathematical form of the multi-state system structure function. This problem can be viewed as the mathematical problem of finding the simplified form for a discrete function. It is impossible to describe a general discrete function in a simple form. We can, however, simplify some special classes of discrete functions by using some of the function properties. One subset of discrete functions is symmetric discrete functions. A symmetric relation is common in engineering systems. If two components play the same role in the system, these two components have a symmetric relation in the system. In system reliability models, parallel & series are two simple models (structures) of component behavior. If two components are in parallel (or series) in the system reliability model, the two components have symmetric relations. This paper shows how to use the symmetric relations among components to simplify the system structure function. There are some discussions about symmetric discrete function in the field of switching theory, eg, [8, 91. Symmetric discrete functions are also a special kind of Schur structure function [12, 141 or L-superadditive structure functions [13]. We introduce some of the theoretical results about symmetric discrete functions and give some new results from the viewpoint of system reliability analysis.
Reliability analysis for multi-state systems with multi-state components is a sophisticated subject in reliability; it deals with the problems of reliability analysis under multi-state models. Compared to the 2-state (binary-state)reliability models, multistate models are much more complex. Thus many theoretical problems remain to be solved in this area, eg, how to describe the structure of a multi-state system in a simple mathematical form. The structure functions of multi-state systems are discussed in [1 - 71. In the 2-state case, the system structure can be described by a Boolean function. In the multi-state case, the 2. NOTATION, NOMENCLATURE, ASSUMPTIONS system structure can be described by a discrete function, which is a mapping from a finite set to another finite set [7 - 91. Many Notation fundamental results in the 2-state s-coherent system theory [161 indicator function: S(True) = 1, S(Fa1se)=O have been extended to the multi-state case [l - 71. In extending number of components in the system the analytic methods in the 2-state case to the multi-state case, component i a major problem is how to deal with an overwhelming number state of ci of system state modes. {cI,c2,... ,c,,}: system component-set In system reliability analysis, one of the major tasks is (x1,x2,...,x n ) : state vector of c system failure-mode analysis. In the 2-state case, system failure system structure-function modes are represented by the system MCS. The dual concept of the MCS is the MPS, which corresponds to the system suca multi-state system +(x) for a symmetric system cess mode. System failure-mode analysis can be done by listing dS(x) for a partial-symmetric system highest component state; there are m t 1 states Cartesian product 'The terms, series & parallel are used in their logic-diagram sense, {0,1,. ..,ml irrespective of the schematic-diagram or physical-layout. Cartesian product of n M M x M x ...x M ( n times) 'The singular & plural of an acronym are always spelled the same. 0018-9529/95/$4.00 01995 IEEE
IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 4, 1995 DECEMBER
subset of M ma 9 (x, E SI,,): lattice exponentiation function me 9 (x, 2 j ) : simplification of X ~ J ’ ‘3
Xl+XZ+
jm
... +x,
“implies” 3 x, Ix,’ for all i=1,2, ...n * x, 2 x,’ for all i=1,2, ...n * x, Ix, ’ for all i = 1,2,. ..n; AND for at least one j , l r j l n , x, < x,’ NEITHER x d X I NOR x 2 X I x W number of subsystems C, {cz,17ct,2 ,...,c ~ , ~ subset }: of c, i=1,2 ,...,w ~ $ ~ ( x , )structure function of subsystem i ( c , , ~ , ( x , ) ) subsystem i of ( C , ~ ( X ) ) , i=1,2, ...,w CUV,,, CUV, set of + ( x ) , j=1,2 ,...,m Int(a) truncate a to an integer, a 2 0 x A x ’ x and x ’ can be deducted from each other by permutation p ; A, V [min, max] symmetric relation.
-
Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.
Nomenclature 0
MIF. 4(x) is MIF if x Ix ‘ * 4(x) 5 4(x’) Monotone increasing system. A system whose $(x) is M F .
Assumptions 1. All the multi-state systems are monotone increasing. 2. All components, and the system, have the same set of states. 3. The states are ordered in terms of goodness: state m 4 is the best, state 0 is the worst. Example 2 in section 4 illustrates some of the notation.
cw2 =
{(2,2,2)1
4(x) = 1 A ( ( X 1 : l k Z
v
(xl:IEVc;?:Z)
v
lm3:l)
(x2 l h 3 2)
v
(Xl.Zh2.1)
(Xl,lm3:2)
(x2 Zh3’l)
v
(X1,Znx3
v 2A ( x l : 2 h x Z : 2 ~ 3 . 2 ) .
1))
4
There is another basic form of discrete functions: CO tive (product-of-sums) form. The conjunctive form of a MIF corresponds to the CLV,.. Using the dual theory of multi-state systems [7], the disjunctive form and the conjunctive form of the MPF can be transformed from one to the other. So we need to discuss only the disjunctive form, and the conclusions about the disjunctive form (corresponding to the CUV,) can be easily extended to the conjunctive form (corresponding to the CLVj) by using the dual theory of multi-state systems [7]. The complexity of the structure function of a multi-state system is proportional to the number of its CUV,. Theorem 1 gives the upper bound for the number of CUV, of a monotone increasing multi-state system.
-
ZXeorem 1. Given a MIF 4 (x): M M. The upper bound of the number of CUV, is Z ( n , m ) , the number of integer solutions of:
6
xi = htl’/z(n.m + I)], 0 5 x , i m , i=1,2 ,...,n.
4
2=1
3. STRUCTURE FUNCTION AND UPPER BOUND OF THE NUMBER CUVj
Proof: See appendix. Table 1 gives some values of Z(n,m).
Let 4(x) E MIF. Define the concept of CUVj for +(x):
TABLE 1 Upper Bound of Number of CUV,: Z(n,m)
Dejinition 1. Given +(x) E MIF, and an x E M . If,
4(x)
2 j,
for any x ’
n
and
< x, 4(x’) < j ;
then, x is a CUV, of +(x).
4
Using the lattice exponentiation and the disjunction (sumof-products) form of the discrete function [7 - 91, a MIF 4(x) can be expressed in terms of CUV,:
3The theoretical conclusions in this paper do not need assumption 2 . It does, however, simplify the notation for the examples.
m
2
3
4
5
6
7
8
1 2 3 4
2 3 4 5 6
3 7 12 19 27
6 19 44 85 146
10 51 156 381 780
20 141 580 1751 4332
35 397 2120 8135 24017
70 1107 8092 38165 135954
5
Table 1 shows that when n & m are large, the number of CUV, can be very large. Thus when the size of a multi-state system increases, the complexity of the system increases rapidly.
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XUE/YANG. SYMMETRIC RELATIONS IN MULTISTATE SYSTEMS
4. SYMMETRIC RELATIONS IN MULTI-STATE SYSTEMS
a. +(XIJ~,X~,...,X,) = d,(x2,x1,x3,...,xn)
Given a multi-state system (c,d,(x)), if components c1 & c2 have the same effect on the system, then c1 & c2have a symmetric relation in the system. The formal definition of symmetric relation is: Dejinition 2. Given a multi-state system (c,d,(x)): if
-
Example 2
4
Some other theoretical conclusions about symmetric functions are in [8, 91. We now discuss how to simplify the disjunctive form of a symmetric function.
From observation #1 we conclude that the domain of the symmetric function can be represented by its subset. We construct a subset of M as follows: p
= {x: x 1 s x 2 s ... I X " } ,
and has the properties:
a, c5
that x
-b
I
N x ) = max{min{xl9x2}, min{x3,x4}, x5, X3-X4;
3 x'.
If d,(x) is a symmetric function, the domain of d,(x), M , can be represented by its subset p. The number of elements in M is (m+l)". Ref [9: theorem 5.131 shows that the number
Figure l 4is an example of a multi-state system with,
Thus: x 1 - X ~ ;
~ 4 2 ) .
Observation 1. Let d,(x) be a symmetric function. Given two vectors: x & x', if x can be obtained by some permutation on 4 x', then d,(x) = +(x').
holds for all (xl,. ..,ik,. ..,jq,... ,xn) E M , then x k & xq (or ck & cq) have a symmetric relation in the system. This relation is denoted as xk xq (or ck cq). d,(x) is partially symmetric on x k & xq. 4 Example 2 illustrates a symmetric relation.
-
b. d,(xllx2,x3,... J") = d,(xi7x3,x4,...~
11
(",'")
xg}.
. I
X5-Xg.
( m + l ) " = 66 = 46656;
If x1-x2 and ~ 2 ~ xthen 3 , x1-x3. That is, symmetric relation is an equivalent relation on the component set c. Denote the equivalent classes partitioned by as c1, 4 , ..., c,. c, is a symmetric block of c , i = 1,2,. ..,w.
-
= ( 5 ) = 462.
The domain of d, (x) is greatly simplified if p used to represent M . Given a symmetric function d,(x), construct a mapping d,~(x):
&: p
-
M, d,s(x) = d,(x) for all x E p.
represent d,('), so we name )'(S,d the representative Dejinition 3. 4(x) is a symmetric function if xk-xq holds for dS(x) all 1 s k , q r n . 4 function of +(x). Theorem 3 follows from theorem 1 (the proof is obvious). If +(x) is a symmetric function, then (c,d(x)) is a symmetric multi-state system. A symmetric function is the special neorem 3. The upper bound on the number of $s CUVj is case of the Schur structure function [12, 141 and of the L- Z,(n,m), which is the number of integer solutions of superadditive structure function [13]. m Theorem 2 is proved in [9]. neorem 2 . A function +(x) is symmetric iff it satisfies the conditions:
4A system is series iff its +(x) = mini(xi). A system is parallel iff its +(x) = maxi&).
n, = ~ n t [ l / z ( n - m + l ) ~l , ~
n ~ ~ x ~ ~ . . . < x , i1 n z .
i= I
Table 2 gives some values of Zs(n,m) and shows that the structure function of a symmetric multi-state system can be greatly simplified by using the representative functions.
IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 4,1995 DECEMBER
692
TABLE 2 The Upper Bound of CUV, of (&c): Zs(n,m) n m
2
1 2 3 4 5
1 2 2 3 3
3 1
2 3
4
5
6
7
8
1
1 3 8 12 20
1 4
1 4
1 5 3 23 73
3 5
5 6
6 8 12
1 18 32
0
1 24 49
5. STRUCTURE FUNCTIONS OF SYMMETRIC MULTI-STATE SYSTEMS Examples illustrate how to use representative functions to simplify the structure functions of symmetric multi-state systems.
Example 1 (continued) The representative function of 4(x) is:
4s (x) = 1A ( (x1 lAX2 1-3
1)
(?2.lAx3 2 ) )
2A (x1.2m2:2m3:2).
, is (0,1,2). Given an x=(1,2,0), the corresponding x r in U
c
q5(1,2,0) = r&(0,1,2) = lA(OV2) V 2A(0) = 1.
Often 4(x) is a partial symmetric function. Let xl, x2, .. x, be symmetric blocks of x. Then define:
.7
q5ps(x)is a mapping from yc
- M,
G i v e n x = ( Q 1 3 2 ) , t h e r e i s a x r = ( 0 1 2 3 )E y c , x L x r . Then:
4fx)= +(O 1 3 2) =
=
4ps(o1 2 3)
1A(oV(oA3)) V 2A(OV(3A3)) V 3A(OV(OA3))
P = cClXP2X~~~XPW = 0
p, = {xL:q 1 i x , , ~s . . .x,,~,, ~ 0 5 x L j 5 m, j = 1 , 2 ,...,n>,
v 2 v 0 = 2.
The Z(n,m) & Z, (n,m) in tables 1 & 2 show the efficiency of using #s(x) or q5ps(x)to represent 4 (x) when n & m become large.
i= 1,2)...)w. Example 3 illustrates how to use 4ps(x)to represent #(A-).
ACKNOWLEDGMENT
Example 3
Let @(XI = ~ ( x 1 , ~ 2 ~ 3 , ~and 4 : 4 )( x, ) = max(41(x1,xd, & 42 are defined by: 42(x3,x4)},M = (0,1,2,3). 61(3,0) = 1;
'#'i(oj2)
=
41(0,3) = $i(2,0)
41(1,2)
=
41(1,3) = 41(2,1) = 41(2,2)
41(2,3)
=
41(3,2) = 41(3,3) = 3;
=
We are grateful to the referees for their comments & suggestions for improving this paper.
=
$i(3,1)
=
2;
APPENDIX A . l Proof of Theorem 1 Define the vector set
41(a7b) = 0, otherwise. d ~ ( 1 ~ 2=) $z(1,3) = $z(2,1> = 42(3,1)
D ( k ) = (x: 1x1 = k; x E M } , for k=0,1, ...,nam. =
1;
D ( k ) has the properties:
693
XUE/YANG: SYMMETRIC RELATIONS IN MULTISTATE SYSTEMS
[5] H.W. Block, T.H. Savits, “A decomposition for multistate monotone systems”, J. Applied Probability, vol 19, 1982, pp 391-402. [6] B. Natvig, “Two suggestions of how to define a multistate coherent system”, Adv. Applied Probability, vol 14, 1982, pp 434-455. [7] J. Xue, “On multistate system analysis”, IEEE Trans. Reliability, vol R-34, 1985 Oct, pp 329-337. [8] M. Davio, J.-D. Deschamps, A. Thayes, Discrete and Switching Functions, 1978; McGraw-Hill. [9] S.C. Lee, Modem Switching Theory and Digital Design, 1978; Prentice-Hall. [lo] J. Xue, F. Xu, “Modular decomposition of multistate systems”, Reliability Theory and Applications: Proc. China-Japan Reliability Symp, 1987 Sep; World Scientific Publishing Co. [ll] J.C. Hudson, K.C. Kapur, “Modules in coherent multistate systems”, IEEE Trans. Reliability, vol R-32, 1983 Jun, pp 183-185. [12] A.M. Abouammoh, E. El-Neweihi, F. Proschan, “Schur structure functions”, Prob. Engineering Znformation Science, vol3, 1989, pp 581-591. [13] H.W. Block, W.S. Griffith,T.H. Savits, “L-Superadditive structure functions”, Adv. Applied Probability, vol 21, 1989, pp 919-929. [14] M. Hollander, F. Proschan, J . Sethuraman, “Functions decreasing in transposition and their applications in ranking problems”, Annals of Statistics, vol 5 , num 4, 1977, pp 722-733. [15] F. Meng, “Component-relevancy and characterization in multistate systems”, IEEE Trans. Reliability, vol 42, 1993 Sep, pp 478-483. [16] R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, 1975; Silver Spring.
n.m
1 . U D ( k ) = M, k=O
2. F~~any
& xr:
3. For any x
E ~ ( kand ) x r E ~ ( k =,)
< > x’;
D(k);
a. if 1x1 < k,thereexistsanx’ E D ( k ) suchthatx’ > x; b. if 1x1 > k,thereexistsanx” E D ( k ) ,suchthatx” < x. Notation Zk(n,m) number of vectors in D ( k ) z(n,m) maxk{Zk(n,m)}. By definition of CUVj, the number of CUVj can not exceed Z(n,m). Zk(n,m) is the number of integer solutions of xi = k , O S x i s m , i=1,2 ,...,n.
4
i=l
Consider: a. the sequence: Zk(n,m) for k=1,2 ,...,n-m; b. the polynomial: (1+u+u2+ ... +U*)”.
AUTHORS
Zk(n,m) is the coefficient of uk in this polynomial. When k = Int[ ‘/2 (men + 1)], the coefficient of uk is the maximum. Q.E. D-
Jianan Xue; Dept. of Industrial & Manufacturing Eng’g; Wayne State Univ; Detroit, Michigan 48202 USA, Internet (e-mail):
[email protected] Jianan Xue: For biography, see IEEE Trans. Reliability, vol44, 1995 Dec, p 688.
REFERENCES [l] R.E. Barlow, A.S. Wu, “Coherent systems with multistate components”, Mathematics of Operations Research, vol 4, 1978, pp 275-281. [2] E. El-Neweihi, F. Proschan, J. Sethuraman, “Multistate coherent systems”, J. Applied Probability, vol 15, 1978, pp 675-688. [3] S. Ross, “Multivalued state component systems”, Annals of Probability, V O ~7, 1979, pp 379-383. [4] W.S. Griffith, “Multistate reliability models”, J. Applied Probability, VOI17, 1980, pp 735-744.
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Dr. Kai Yang; Dept. of Industrial & Manufacturing Eng’g; Wayne State Univ; Detroit, Michigan 48202 USA. (e-mail): kyang@mie,eng,wayne,edu Kai Yang: For biography, see IEEE Trans. Reliability, vol44, 1995 Dec, 688, Manuscript received 1995 January 15. IEEE Log Number 94-14477
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