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On the S-shaped theorem
Statistics & Probability North-Holland
2 January
Letters 13 (1992) l-4
1992
On the S-shaped theorem Thore Deportment
Egeland
*
of Mathematics,
University of Oslo, N-0316 Oslo, Norway
Received May 1990 Revised March 1991
Abstract: The S-shaped theorem describes the shape of the reliability function under quite general structural conditions. multivariate version of the theorem based on a new condition is provided. As opposed to the standard proof the present association. Keywords:
Reliability
function,
S-shaped
theorem.
1. Introduction
The reliability of a system depends on the structure as represented by the structure function and on the simultaneous distribution of the states of the components. Obviously, if both the structural and distributional aspects are specified, the reliability is known in principle; there may however remain severe computational problems. The goal of this paper is to discuss the shape of the reliability function based on limited information. In particular focus is on the S-shaped theorem (Barlow and Proschan, 1965, p. 214; 1981, p. 46). The S-shaped theorem is important since it describes the shape of the reliability function under quite general structural conditions. The term S-shaped was coined to indicate the appearance of a typical reliability function, an example is provided in Figure 1. A first version of the theorem was proved in Moore and Shannon (1956) for relay networks. Extensions to coherent systems were made by Birnbaum et al. (1961) and Esary and Proschan (1963) accomplished the proof which is now standard. Their proof uses a concept of dependence which is presently known as association. We
* Current address: Blindem, N-0316 0167-7152/92/$05.00
A proof of a does not use
Norwegian Computing Oslo, Norway.
Center,
consider the result of the multivariate version of the S-shaped theorem provided in Barlow and Proschan (1965). In this paper we give a new condition for this theorem to hold. The proof of this new result is based on a modification of an inequality due to Ross (1979) and (1980). A comparison of the assumptions in the Barlow and Proschan (1965) version of the result and the
h(p(8)) = w - 28”
h(p(0))
Box 114,
0 1992 - Elsevier Science Publishers
Fig. 1. Reliability B.V. All rights reserved
function.
Identical
components.
1
Volume 13, Number 1
STATISTICS & PROBABILITY
present is included and shows that the conditions are incomparable in the sense that neither implies the other. Reliability
(E, +)by: h(p)=~,GE~(~)~i,.~,, wherep= (pi,. . . , p,), p, = P( X, = l), is the vector of component reliabilities. Note that h(p) = P( C$= 1) = the reliability of the system. When p, = . . . =p, = p, we simply write h(p). Observe that h(p) is a polynomial in p of degree n or less. If (E, +) is nontrivial, then h(0) = 0 and h(1) = 1.
2. S-shaped theorem We are now in a position to state our version of the multivariate S-shaped theorem. The component reliabilities are allowed to differ as long as they depend on a common parameter, i.e., P( X, = 1) = p,( 19), 0 < 8 < 1. A typical choice may be p,(8) = P(T > t), where q is the lifelength of component i. Two lemmas will precede the proof. Theorem 2.1. Consider a binary monotone non-series structure (E, +) with n > 2 components. Let p, (8) satisfy: ~,(fl*)
asp,
i= l,..., Ifh(p(e,,)) h(p(f3)) 2
while h(p(e))>e
fore,
(lb)
o
(A2)
e(l
-e)P:(e)
aP,(e)@
the
-P,(e)).
In presence of (A2), the conclusion of Theorem 2.1 remains valid for series systems. The existence of a 6, satisfying h( p ( do)) = f3, is considered next. Assume the structure has no path or cut sets of cardinality one. The reliability function of identical components then crosses the diagonal exactly once (Esary and Proschan, 1963). If, in addition, each p, (8) crosses the diagonal exactly once, then h( p(e)) crosses the diagonal exactly once according to Barlow and Proschan (1965, p. 214). The following lemma, due to Ross (1980) modifies Lemma 4.2.3 in Barlow and Proschan (1981). Lemma 2.1. Let 0 < A < 1, and 0 < x
- [hy + (1 - h)x]
If 0 <
a > 0.
Proof. Since f(x) = xa is strictly concave in for 0 < CY< 1, it follows that f(u, + 8) - f( f ( u2 + 8) - f( u2) provided 0 -C ui < u2 and Take S =X( y - x), u1 = Xx, and u2 = x to plete the proof. 0
(2) x > 0 u,) > 6 > 0. com-
The next lemma modifies Proposition 9.5.1 in Ross (1980) by replacing a nonstrict inequality with a strict inequality. In doing so further assumptions on the structure are necessary. Lemma 2.2. Let h ( p ) be the reliability function of a binary monotone non-series structure of n > 2 independent components. If 0 < (Y < 1, then h”(p)
O
i=l,...,
n,
(3)
where pa = (Py,-..,P;). Proof. By pivotal decomposition on component e (Lemma 2.1.1 in Barlow and Proschan, 1981) and Proposition 9.5.1 of Ross (1980),
n, O
= e, then for
2 January 1992
Remark. Barlow and Proschan (1965) provides theorem with condition (Al) replaced by:
concepts
A binary monotone system (BMS) is an ordered pair (E, $J), where E = (1,. _. , n} is a nonempty set of components, and + = $(X) is a binary (O-l) nondecreasing function of the component state vector X. The function + is called the structure function of the system, and describes the state of the system; C#I = 1 if the system is functioning and + = 0 if the system is failed. Similarly, the i th entry of the component state vector, X,, is respectively 1 or 0 if the i th component is functioning or failed. The function + is multilinear in X, so for some suitable integer valued set function 6, $ may be expressed as: G(X) = C,,.s(A)II,,.X,. Assuming independent component state variables we define the reliability function of a BMS
(Al)
LETTERS
(la>
h(p”)
=p,*h(l,,
pa) + (1 -p:)h(O,,
P*>
Volume
13, Number
STATISTICS
1
& PROBABILITY
>p,*h*(l,, p> + (1 -p,“)ffyO,, P). the structure ponent e may be h(O,, p) > 0 (0
’ [&W,,
is non-series a relevant comchosen such that h(l,, p) > < 1, i # e). Apply Lemma 2.1, = h(l,, p) and x = h(O,, p) to
Utilizing pivotal decomposition clude that (3) holds. 0
again,
we con-
Proof of Theorem 2.1. Let h( p( 0,)) = 0, and 8 E (B,,, 1). Then f$j’= B for some (YE (0, 1) and condition (Al) followed by Lemma 2.2 gives h(p(O))
= G+%Y)
a WV0))
>hQ(e,))=e,*=e.
The reverse inequality
is proved
similarly.
0
3. Discussion The usual univariate version of the S-shaped theorem appears if each p,( 0) = 8. A proof of the univariate version of the S-shaped theorem with nonstrict inequalities in (1) is requested in Exercise 21, p. 347 in Ross (1980). Omitting the index, (Al) may be rewritten log p(ea)/log 8” G log p(B)/log 8. In other words, log ‘(‘I log e
The function x + log x/(1 - x), 0 < x < 1, is increasing. Hence, if p( 0) is stars/raped (Barlow and Proschan, 1981, p. 89) condition (Al) is better (worse) than (A2) if 1 > 8 > 0, (0 < 0 < a,) where 0,)
= 0,. If p,(8), i = 1,. . . , n, are reliability functions, condition (Al) holds by virtue of Lemma 2.2 while (A2) is proved in Barlow and Proschan (1981, p. 45546), see also Joag-Dev and Proschan (1990). The latter reference briefly mentions the S-shaped theorem in a context with dependent variables. In the case of dependence, the system reliability is not a function of the component reliabilities and so the question arises: In which parameter is the reliability to be S-shaped? Theorem 2.1 might be a step towards an S-shaped theorem in 8 for dependent variables. Consider for instance a case where 8 is a parameter of the simultaneous distribution of the components and P (system
functions)
for some reliability
= h ( p ( 8 )) . function
h.
Acknowledgment
a
log 0) log
8
of differentiability
e/o- 0) _p(e))
is equivalent It is a pleasure to thank Associate Professor Arne Bang Huseby and Professor B. Natvig for valuable comments.
PC’).
A condition resembling be formulated:
p(e),(l
general logical relation between the two conditions. Referring to equations (5) and (A2) it is reasonable to state that (Al) is better (easier to satisfy) than (A2) provided 1 -p(B) - (1 - 0) log p(e) /log e > 0 or equivalently if
is decreasing,
which in presence to
ep’(e)
1992
log P(B) _- log 8 l-e’“. 1 -m
PIIa.
P) + (1 -P,)h(O,,
2 January
LETTERS
(4) for condition
isdecreasing.
(A2) may References
(6)
If p(8) satisfies (Al), then so does p(8)p, p > 0. A similar closure property is not valid for (A2). The function 8 ---, B”.5 meets condition (Al) and not (A2) while the map B + 40’ complies with (A2) and contradicts (Al). Hence, there is no
Barlow, R.E. and F. Proschan (1965), Mathematrcal Theoq of Rehabibty (Wiley, New York). Barlow, R.E. and F. Proschan (1981). Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD). Birnbaum, Z.W., J.D. Esary and SC. Saunders (1961), Multicomponent systems and structures, Technometrics 3(l), 5517.
Volume
13, Number
1
STATISTICS
& PROBABILITY
Esary, J.D. and F. Proschan (1963). Coherent structures of nonidentical components, Technomerrics 5, 191-209. Joag-Dev, K. and F. Proschan (1990), A covariance inequality for coherent structures, Statist. Probab. L&t. 9(3), 215-216. Moore, E.F. and C.E. Shannon (1956), Reliable circuits using less reliable relays, J. Frunkl. Inst. 262, part I, 191-208; part II, 281-297.
LETTERS
2 January
1992
Ross, S.M. (1979) Multivalued state component systems, Ann. Probab. 7, 379-383. Ross, S.M. (1980). Introduction to Probability Models (Academic Press, New York).
2 January
Letters 13 (1992) l-4
1992
On the S-shaped theorem Thore Deportment
Egeland
*
of Mathematics,
University of Oslo, N-0316 Oslo, Norway
Received May 1990 Revised March 1991
Abstract: The S-shaped theorem describes the shape of the reliability function under quite general structural conditions. multivariate version of the theorem based on a new condition is provided. As opposed to the standard proof the present association. Keywords:
Reliability
function,
S-shaped
theorem.
1. Introduction
The reliability of a system depends on the structure as represented by the structure function and on the simultaneous distribution of the states of the components. Obviously, if both the structural and distributional aspects are specified, the reliability is known in principle; there may however remain severe computational problems. The goal of this paper is to discuss the shape of the reliability function based on limited information. In particular focus is on the S-shaped theorem (Barlow and Proschan, 1965, p. 214; 1981, p. 46). The S-shaped theorem is important since it describes the shape of the reliability function under quite general structural conditions. The term S-shaped was coined to indicate the appearance of a typical reliability function, an example is provided in Figure 1. A first version of the theorem was proved in Moore and Shannon (1956) for relay networks. Extensions to coherent systems were made by Birnbaum et al. (1961) and Esary and Proschan (1963) accomplished the proof which is now standard. Their proof uses a concept of dependence which is presently known as association. We
* Current address: Blindem, N-0316 0167-7152/92/$05.00
A proof of a does not use
Norwegian Computing Oslo, Norway.
Center,
consider the result of the multivariate version of the S-shaped theorem provided in Barlow and Proschan (1965). In this paper we give a new condition for this theorem to hold. The proof of this new result is based on a modification of an inequality due to Ross (1979) and (1980). A comparison of the assumptions in the Barlow and Proschan (1965) version of the result and the
h(p(8)) = w - 28”
h(p(0))
Box 114,
0 1992 - Elsevier Science Publishers
Fig. 1. Reliability B.V. All rights reserved
function.
Identical
components.
1
Volume 13, Number 1
STATISTICS & PROBABILITY
present is included and shows that the conditions are incomparable in the sense that neither implies the other. Reliability
(E, +)by: h(p)=~,GE~(~)~i,.~,, wherep= (pi,. . . , p,), p, = P( X, = l), is the vector of component reliabilities. Note that h(p) = P( C$= 1) = the reliability of the system. When p, = . . . =p, = p, we simply write h(p). Observe that h(p) is a polynomial in p of degree n or less. If (E, +) is nontrivial, then h(0) = 0 and h(1) = 1.
2. S-shaped theorem We are now in a position to state our version of the multivariate S-shaped theorem. The component reliabilities are allowed to differ as long as they depend on a common parameter, i.e., P( X, = 1) = p,( 19), 0 < 8 < 1. A typical choice may be p,(8) = P(T > t), where q is the lifelength of component i. Two lemmas will precede the proof. Theorem 2.1. Consider a binary monotone non-series structure (E, +) with n > 2 components. Let p, (8) satisfy: ~,(fl*)
asp,
i= l,..., Ifh(p(e,,)) h(p(f3)) 2
while h(p(e))>e
fore,
(lb)
o
(A2)
e(l
-e)P:(e)
aP,(e)@
the
-P,(e)).
In presence of (A2), the conclusion of Theorem 2.1 remains valid for series systems. The existence of a 6, satisfying h( p ( do)) = f3, is considered next. Assume the structure has no path or cut sets of cardinality one. The reliability function of identical components then crosses the diagonal exactly once (Esary and Proschan, 1963). If, in addition, each p, (8) crosses the diagonal exactly once, then h( p(e)) crosses the diagonal exactly once according to Barlow and Proschan (1965, p. 214). The following lemma, due to Ross (1980) modifies Lemma 4.2.3 in Barlow and Proschan (1981). Lemma 2.1. Let 0 < A < 1, and 0 < x
- [hy + (1 - h)x]
If 0 <
a > 0.
Proof. Since f(x) = xa is strictly concave in for 0 < CY< 1, it follows that f(u, + 8) - f( f ( u2 + 8) - f( u2) provided 0 -C ui < u2 and Take S =X( y - x), u1 = Xx, and u2 = x to plete the proof. 0
(2) x > 0 u,) > 6 > 0. com-
The next lemma modifies Proposition 9.5.1 in Ross (1980) by replacing a nonstrict inequality with a strict inequality. In doing so further assumptions on the structure are necessary. Lemma 2.2. Let h ( p ) be the reliability function of a binary monotone non-series structure of n > 2 independent components. If 0 < (Y < 1, then h”(p)
O
i=l,...,
n,
(3)
where pa = (Py,-..,P;). Proof. By pivotal decomposition on component e (Lemma 2.1.1 in Barlow and Proschan, 1981) and Proposition 9.5.1 of Ross (1980),
n, O
= e, then for
2 January 1992
Remark. Barlow and Proschan (1965) provides theorem with condition (Al) replaced by:
concepts
A binary monotone system (BMS) is an ordered pair (E, $J), where E = (1,. _. , n} is a nonempty set of components, and + = $(X) is a binary (O-l) nondecreasing function of the component state vector X. The function + is called the structure function of the system, and describes the state of the system; C#I = 1 if the system is functioning and + = 0 if the system is failed. Similarly, the i th entry of the component state vector, X,, is respectively 1 or 0 if the i th component is functioning or failed. The function + is multilinear in X, so for some suitable integer valued set function 6, $ may be expressed as: G(X) = C,,.s(A)II,,.X,. Assuming independent component state variables we define the reliability function of a BMS
(Al)
LETTERS
(la>
h(p”)
=p,*h(l,,
pa) + (1 -p:)h(O,,
P*>
Volume
13, Number
STATISTICS
1
& PROBABILITY
>p,*h*(l,, p> + (1 -p,“)ffyO,, P). the structure ponent e may be h(O,, p) > 0 (0
’ [&W,,
is non-series a relevant comchosen such that h(l,, p) > < 1, i # e). Apply Lemma 2.1, = h(l,, p) and x = h(O,, p) to
Utilizing pivotal decomposition clude that (3) holds. 0
again,
we con-
Proof of Theorem 2.1. Let h( p( 0,)) = 0, and 8 E (B,,, 1). Then f$j’= B for some (YE (0, 1) and condition (Al) followed by Lemma 2.2 gives h(p(O))
= G+%Y)
a WV0))
>hQ(e,))=e,*=e.
The reverse inequality
is proved
similarly.
0
3. Discussion The usual univariate version of the S-shaped theorem appears if each p,( 0) = 8. A proof of the univariate version of the S-shaped theorem with nonstrict inequalities in (1) is requested in Exercise 21, p. 347 in Ross (1980). Omitting the index, (Al) may be rewritten log p(ea)/log 8” G log p(B)/log 8. In other words, log ‘(‘I log e
The function x + log x/(1 - x), 0 < x < 1, is increasing. Hence, if p( 0) is stars/raped (Barlow and Proschan, 1981, p. 89) condition (Al) is better (worse) than (A2) if 1 > 8 > 0, (0 < 0 < a,) where 0,)
= 0,. If p,(8), i = 1,. . . , n, are reliability functions, condition (Al) holds by virtue of Lemma 2.2 while (A2) is proved in Barlow and Proschan (1981, p. 45546), see also Joag-Dev and Proschan (1990). The latter reference briefly mentions the S-shaped theorem in a context with dependent variables. In the case of dependence, the system reliability is not a function of the component reliabilities and so the question arises: In which parameter is the reliability to be S-shaped? Theorem 2.1 might be a step towards an S-shaped theorem in 8 for dependent variables. Consider for instance a case where 8 is a parameter of the simultaneous distribution of the components and P (system
functions)
for some reliability
= h ( p ( 8 )) . function
h.
Acknowledgment
a
log 0) log
8
of differentiability
e/o- 0) _p(e))
is equivalent It is a pleasure to thank Associate Professor Arne Bang Huseby and Professor B. Natvig for valuable comments.
PC’).
A condition resembling be formulated:
p(e),(l
general logical relation between the two conditions. Referring to equations (5) and (A2) it is reasonable to state that (Al) is better (easier to satisfy) than (A2) provided 1 -p(B) - (1 - 0) log p(e) /log e > 0 or equivalently if
is decreasing,
which in presence to
ep’(e)
1992
log P(B) _- log 8 l-e’“. 1 -m
PIIa.
P) + (1 -P,)h(O,,
2 January
LETTERS
(4) for condition
isdecreasing.
(A2) may References
(6)
If p(8) satisfies (Al), then so does p(8)p, p > 0. A similar closure property is not valid for (A2). The function 8 ---, B”.5 meets condition (Al) and not (A2) while the map B + 40’ complies with (A2) and contradicts (Al). Hence, there is no
Barlow, R.E. and F. Proschan (1965), Mathematrcal Theoq of Rehabibty (Wiley, New York). Barlow, R.E. and F. Proschan (1981). Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD). Birnbaum, Z.W., J.D. Esary and SC. Saunders (1961), Multicomponent systems and structures, Technometrics 3(l), 5517.
Volume
13, Number
1
STATISTICS
& PROBABILITY
Esary, J.D. and F. Proschan (1963). Coherent structures of nonidentical components, Technomerrics 5, 191-209. Joag-Dev, K. and F. Proschan (1990), A covariance inequality for coherent structures, Statist. Probab. L&t. 9(3), 215-216. Moore, E.F. and C.E. Shannon (1956), Reliable circuits using less reliable relays, J. Frunkl. Inst. 262, part I, 191-208; part II, 281-297.
LETTERS
2 January
1992
Ross, S.M. (1979) Multivalued state component systems, Ann. Probab. 7, 379-383. Ross, S.M. (1980). Introduction to Probability Models (Academic Press, New York).