Comparisons and bounds for expected lifetimes of reliability systems

European Journal of Operational Research 207 (2010) 309–317

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Comparisons and bounds for expected lifetimes of reliability systems Jorge Navarro a,*,1, Tomasz Rychlik b a b

Facultad de Matematicas, Universidad de Murcia, 30100 Murcia, Spain ´ , Poland Institute of Mathematics, Polish Academy of Sciences, Chopina 12, 87100 Torun

a r t i c l e

i n f o

Article history: Received 19 September 2009 Accepted 3 May 2010 Available online 12 May 2010 Keywords: Coherent system Mixed system Signature Mean residual life Convex order Gini index

a b s t r a c t Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The evaluation of coherent system lifetimes is a central subject of reliability theory. The reliability and expected lifetime calculations of a multi-component system are important issues which has been addressed widely in the literature. A general approach was presented in Barlow and Proschan (1975, 1976, 1996) and in the references therein. More recent overviews can be seen in Cocozza-Thivent and Roussignol (1995) and Cocozza-Thivent (1997). Further results under some preventive maintenance conditions were given in Cocozza-Thivent (2007), Papageorgiou and Kokolakis (2010), Zhu et al. (2010) and in some references therein. In general, it is not easy to compute the reliability or the expected lifetime (called in this context mean time to failure, MTTF) of a coherent system, especially when the component distributions are not precisely known. This is a common situation in practice. Hence, in this case, it is important to have bounds to approximate the behavior of a system. The notion of system signature is a useful tool for computing system characteristics. The signature of a coherent system with independent and identically distributed (IID) component lifetimes was defined by Samaniego (1985) as the vector s = (s1, s2, . . . , sn) with si being the probability that the system fails with the ith

* Corresponding author. E-mail addresses: [email protected] (J. Navarro), [email protected] (T. Rychlik). 1 Partially supported by Ministerio de Ciencia y Tecnología under Grant MTM200612834 and Fundación Séneca under Grant 08627/PI/08. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.05.001

component failure, i = 1, 2, . . . , n. A comprehensive survey of signatures can be found in Samaniego (2007) which includes applications of signatures to network reliability as well. The signatures are also used in analysis of mixed systems which are stochastic mixtures of coherent systems. Recently, Navarro et al. (2007a) introduced the concept of minimal signature and gave alternative expressions to compute the reliability functions and expected lifetimes of coherent systems with exchangeable component lifetimes. The signatures were used in Navarro and Rychlik (2007) (see also Navarro et al., 2007b) to obtain bounds for the reliability functions and expected lifetimes of coherent systems with exchangeable component lifetimes. The signatures were also applied for obtaining stochastic comparisons of coherent systems with different structures (see Navarro et al., 2008b) and determining the behavior of the ageing functions of the system when the time increases (see Navarro and Shaked, 2006; Navarro and Hernandez, 2008). In the reliability context, the order statistics represent the lifetimes of k-out-of-n systems (i.e., systems which work if do so at least k of their n components). Bounds for distributions and expectations of order statistics were obtained, e.g., in Papadatos (1997) and Rychlik (2001). Bounds for the variance of coherent and mixed systems were presented in Jasinski et al. (2009). In this paper, we study comparisons and bounds for the expected lifetimes of coherent or mixed systems with independent components having unknown distributions based on the components expected lifetimes and on their domination polynomials which only depend on the system structure functions. In the IID case, the domination polynomial can be computed from the minimal signature. In Section 2 we introduce the notation and

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we obtain some comparison results for the IID case. The comparisons obtained in this case allow to verify which systems improve when the component lifetimes become more dispersed in the sense of convex ordering. In Section 3 we obtain bounds for the expected lifetimes of systems with IID components based on their domination polynomials and the mean of component lifetime. Some of these results are based on the comparisons given in Section 2. In particular, more refined inequalities are obtained based on a concentration parameter connected to the Gini dispersion index. Similar results are derived (Section 4) for coherent systems with independent non-necessarily identically distributed (INID) components with use of the generalized domination polynomial. Some refinements are established in terms of expected lifetimes of series systems of smaller sizes. Some other are based on evaluations for equivalent systems with IID components, and these are further simplified in the case of weak Schur-concavity and Schurconvexity of the system generalized domination polynomial. 2. Notation and preliminary comparison results In this section we introduce the notation for the IID case and we give some preliminary comparison results which are used in Section 3. These comparison results can also be used to study if a system improves when its components are more dispersed. Let T = w(X1, X2, . . . , Xn) be the lifetime of a coherent system w with independent and identically distributed (IID) nonnegative component lifetimes X1, X2, . . . , Xn with common continuous reliability function FðtÞ ¼ PrðX i > tÞ and mean l = E(Xi), i = 1, 2, . . . , n. Here and later on, writing an expectation or a conditional distribution, we tacitly assume that they exist. The definition and basic properties of coherent systems can be found in Barlow and Proschan (1975). The order statistics associated to X1, X2, . . . , Xn are denoted by X1:n, X2:n, . . . , Xn:n, and F 1:n ; F 2:n ; . . . ; F n:n stand for the respective reliability functions. The reliability function F T ðtÞ ¼ PrðT > tÞ of T can be written as

F T ðtÞ ¼

n X

si F i:n ðtÞ;

ð2:1Þ

i¼1

of sizes k = 1, 2, . . . , n. If the component lifetimes are IID, by (2.1) the reliability function F T of a mixed system lifetime T satisfies (2.1) and (2.2) for some coefficients si and ai, i = 1, . . . , n. Accordingly, the respective signature and minimal signatures s ¼ swP ¼ ðs1 ; s2 ; . . . ; sn Þ and a ¼ awP ¼ ða1 ; a2 ; . . . ; an Þ, and the domination P polynomial of the mixed system pðxÞ ¼ pwP ðxÞ ¼ ni¼1 ai xi are well defined, and share the above mentioned properties of the corresponding notions for the standard coherent systems. Every coherent system is obviously a mixed system. If X is a nonnegative random variable with continuous reliability function F and mean l, then

l¼

Z

1

FðxÞdx ¼

0

Z

1

F 1 ðuÞdu;

ð2:3Þ

0

where F 1 ðuÞ ¼ supfx : FðxÞ P ug. Hence

hF ðuÞ ¼ F 1 ðuÞ=l;

0 < u < 1;

ð2:4Þ

(with hF(u) = 0 if either u P 1 or u 6 0) is a decreasing probability density function with the support [0, 1]. It is easy to see that this density does not depend on scale parameters, that is, X and aX determine the same density for all a > 0. Moreover, the reliability function HF associated to hF satisfies HF ðxÞ ¼ LF ð1  xÞ, where LF(x) is the Lorenz curve associated to F (see, e.g., Shaked and Shanthikumar, 2007, p. 119). Throughout the paper, for a distribution function F, ZF will represent a random variable having probability density function hF. By (2.1) and (2.2), the lifetime expectation E(T) of a coherent or mixed system can be calculated as

EðTÞ ¼

n X

si EðX i:n Þ ¼

i¼1

n X

ai EðX 1:i Þ:

i¼1

In the following proposition, using the density given in (2.4), we obtain an alternative expression for E(T). Proposition 2.1. If T is the lifetime of a coherent or mixed system with domination polynomial p, and IID component lifetimes having common continuous distribution function F, then

EðTÞ ¼ lEðp0 ðZ F ÞÞ:

ð2:5Þ

and, equivalently,

F T ðtÞ ¼

n X

Proof. By (2.2), we have

ai F i ðtÞ ¼ pðFðtÞÞ;

ð2:2Þ

i¼1

EðTÞ ¼

where s = sw = (s1, s2, . . . , sn) is the signature of the system w, a = aw = (a1, a2, . . . , an) is the respective minimal signature and p(x) = pw(x) is the respective domination polynomial (see Agrawal and Barlow, 1984; Samaniego, 1985, 2007; Navarro et al., 2007a, 2008b). The vectors s and a and the polynomial p(x) only depend on the structure function w of the system. It is well known that the reliability function of Xi:n can be written as

F i:n ðtÞ ¼

n X j¼niþ1

ð1Þjnþi1

   n j1 j F ðtÞ; j ni

(see David and Nagaraja, 2003, p. 46). So s = aAn for a triangular (non-singular) matrix An. Therefore, we can obtain s from a and vice versa. It is easy to see that p(x) is a polynomial of degree n, strictly increasing in (0, 1), and satisfying p(0) = 0 and p(1) = 1. The mixed systems are stochastic mixtures of coherent systems (see Boland and Samaniego, 2004). The lifetime T of a mixed system wP with component lifetimes X1, X2, . . . , Xn is the lifetime of randomly chosen among coherent system wi, i = 1, 2, . . . , m, with Pm probabilities pi P 0, i = 1, 2, . . . , m, such that i¼1 pi ¼ 1. By (2.1) and (2.2), every mixed system of size n, can be represented as either a convex mixture of at most n k-out-of-n systems, k = 1, 2, . . . ,n, or a linear combination of at most n series systems

Z

1

F T ðtÞdt ¼

0

¼

Z

Z

1

0

pðFðtÞÞdt ¼ 

Z

1

tp0 ðFðtÞÞdðFðtÞÞ

0

1

p0 ðxÞF 1 ðxÞdx:

0

Hence

EðTÞ ¼ l

Z

1

p0 ðxÞhF ðxÞdx

0

and (2.5) holds. h The formula obtained in the preceding proposition can be used for comparing systems with the same structure function. To this end, we shall use the following stochastic order. Its basic properties can be found in Shaked and Shanthikumar (2007, pp. 109–139). Definition 2.2. If X and Y are two random variables, X is said to be smaller than Y in the convex order (denoted as X CX Y) if E(/ (X)) 6 E(/(Y)) for all convex functions / such that the expectations exist. If X CX Y, then E(X) = E(Y) and Var(X) 6 Var(Y) (see Shaked and Shanthikumar, 2007, p. 110). So it can be treated as a comparison of dispersion of random variables and their distributions. The

J. Navarro, T. Rychlik / European Journal of Operational Research 207 (2010) 309–317

convex ordering is characterized by the following property. If E(X) = E(Y), then X CX Y holds if and only if

Z t

1

F X ðxÞdx 6

Z

1

F Y ðxÞdx

ð2:6Þ

t

for all real arguments t (see Shaked and Shanthikumar, 2007, p. 110). Moreover, from Theorem 3.A.5 in Shaked and Shanthikumar (2007, p. 112), we get the following result. Lemma 2.3. If X and Y are two nonnegative random variables with respective continuous distribution functions F and G and equal means, then X CX Y holds if and only if ZG ST ZF, where ST denotes the usual stochastic order. If E(X) = E(Y), then ZG ST ZF is also equivalent to the Lorenz ordering X Lorenz Y of the original variables (see Theorem 3.A.10 in Shaked and Shanthikumar (2007, p. 119)). The preceding lemma, representation (2.5), and the convex order can be used for comparing expected lifetimes of coherent or mixed systems with the same structure function and IID component lifetimes having the same mean (but different distributions). The result can be stated as follows. Proposition 2.4. Let T1 = w(X1, X2, . . . , Xn) and T2 = w(Y1, Y2, . . . , Yn) be the lifetimes of two identical coherent or mixed systems with IID component lifetimes having common continuous reliability functions F and G, respectively, and common mean l = E(X1) = E(Y1). Let p be the common domination polynomial. Then we have the following properties. (i) If p is convex on (0, 1) and X1 CX Y1, then E(T1) P E(T2). (ii) If p0 is convex on (0, 1) and ZF CX ZG, then E(T1) 6 E(T2). Concavity of the respective functions results in reversing the inequalities for the expectations. Proof. The proof of (i) can be obtained from Theorem 3.A.7 in Shaked and Shanthikumar (2007, p. 116). The proof of (ii) is an immediate consequence of (2.5) and the definition of the convex order. h Property (i) of the preceding proposition says that the systems with convex (concave) domination polynomials improve (in the expected lifetimes) when the components are less (more) dispersed. For example, the domination polynomial of a parallel system with lifetime Xn:n can be written as pn:n(x) = 1  (1  x)n, which is a concave function. Hence we deduce an apparent property of the parallel systems that they improve with the increase of component dispersion (in the sense of the convex ordering). However, the domination polynomial p1:n(x) = xn of a series system with lifetime X1:n is a convex function and so the series systems get worse with the dispersion increase. These are well known results which can be obtained from Theorem 7.6 in Barlow and Proschan (1975, p. 122). The next example shows that this property can be applied to other systems different from the series and parallel ones. In Property (ii), it is not clear under which conditions for F and G, ZF 6CX ZG holds. Note that if E(ZF) = E(ZG), then ZF 6CX ZG is equivalent (see Theorem 3.A.10 in Shaked and Shanthikumar (2007, p. 119)) to the ordering of the Lorenz curves obtained from ZF and ZG. We can also use Lemma 2.3 to get that if E(ZF) = E(ZG), then ZF 6CX ZG is equivalent to Z HG 6ST Z HF . Example 2.5. Consider a system with 5 IID components and lifetime given by T1 = min(X1, X2, X3, max(X4, X5)). Its reliability function is F T ðtÞ ¼ pðFÞ, where F is the common reliability function of the components and its domination polynomial p(x) = 2x4  x5 is

311

a convex function in (0, 1). That means that T1 get worse with the increase of dispersion. If an identical system with the lifetime T2 = min(Y1, Y2, Y3, max(Y4, Y5)) has also IID components such that X1 CX Y1 (implying E(X1) = E(Y1) and Var(X1) 6 Var(Y1)), then E(T1) P E(T2). It is easy to see that the property is reversed for the dual systems with lifetimes T D 1 ¼ maxðX 1 ; X 2 ; X 3 ; minðX 4 ; X 5 ÞÞ and T D 2 ¼ maxðY 1 ; Y 2 ; Y 3 ; minðY 4 ; Y 5 ÞÞ, that is, if X1 CX Y1, then D EðT D 1 Þ 6 EðT 2 Þ. From Proposition 2.4, if a system has a convex domination polynomial and IID components, it works good when the component lifetimes are not dispersed. Therefore, the maximal lifetime is attained in the limit when the component lifetimes have distributions which tend to a degenerate distribution concentrated at some l. Equivalently, by (2.5), this holds when the random variable ZF is maximal in the ST-order, that is, when the distribution of ZF converges to a uniform distribution on the interval (0, 1). Then, for a general distribution function F, we obtain the following bound

EðTÞ ¼ lEðp0 ðZ F ÞÞ 6 lEðp0 ðUÞÞ ¼ l:

ð2:7Þ

Analogously, the system works worst if the distribution of ZF converges to a degenerate distribution at 0. Indeed,

EðTÞ ¼ lEðp0 ðZ F ÞÞ P lEðp0 ð0ÞÞ ¼ lp0 ð0Þ ¼ la1 :

ð2:8Þ

If the system has a concave domination polynomial, then the inequalities are reversed. Moreover, bounds (2.7) and (2.8) are attainable in the limit when ZF has the convex reliability function Hc ðxÞ ¼ 1  xc ; 0 < x < 1, for c 2 (0, 1). Note that ZF has a limiting uniform distribution when c ? 1 and a degenerate one at 0 when c ? 0. Note that by (2.5), Hc defines uniquely a continuous Pareto  1=ðc1Þ , for x > cl, component lifetime reliability function F c ðxÞ ¼ cxl up to a scale parameter l. In order to study the case when the system has a convex (or concave) derivative domination polynomial, we need to consider the convex ordering in random variables with decreasing densities in (0, 1) (or, equivalently, convex reliability functions). To this end we use the following proposition. Proposition 2.6. If Z is a random variable with support included in [0, 1], mean E(Z) = a 6 1/2 and convex reliability function H, then ZL CX Z CX ZU, where ZL and ZU have reliability functions

HL ðxÞ ¼ 1  x=ð2aÞ;

0 6 x 6 2a;

ð2:9Þ

with HL ðxÞ ¼ 1 for x < 0 and HL ðxÞ ¼ 0 for x > 2a, and

HU ðxÞ ¼ 2að1  xÞ;

06x61

ð2:10Þ

with HU ðxÞ ¼ 1 for x < 0 and HU ðxÞ ¼ 0 for x > 1, respectively. Proof. The proof of ZL CX Z can be obtained directly from Theorem 3.A.46 in Shaked and Shanthikumar (2007, p. 135). It is easy to see that

EðZ U Þ ¼

Z

1

HU ðxÞdx ¼ a:

0

Hence, by (2.6), Z CX ZU if

Z t

1

HðxÞdx 6

Z

1

HU ðxÞdx

t

holds for all t 2 (0, 1), where H is the reliability function of Z. The proof is trivial in the case a = 1/2, and so we assume a < 1/2. We also exclude from considerations the case that H coincides with (2.10) on the whole unit interval [0, 1]. By the moment condition,

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the difference HU ðxÞ  HðxÞ changes the signs at least once in (0, 1). By convexity of HðxÞ, and linearity of HU ðxÞ, the number of sign changes does not exceed 2. Consider the function

AU ðtÞ ¼

Z

1

½HU ðxÞ  HðxÞdx:

t

Note that AU(0) = 0 and AU(1) = 0. Then, we have HU ð0Þ < Hð0Þ ¼ 1, and HU ð1Þ ¼ Hð1Þ ¼ 0. Therefore, there exists xU 2 (0, 1) such that HU ðxÞ 6 HðxÞ for x 2 (0, xU) and HU ðxÞ P HðxÞ for x 2 (xU, 1). Hence AU(t) is increasing in (0, xU) and decreasing in (xU, 1). So AU(x) P 0 for x 2 (0, 1), and Z CX ZU holds. h

mean so that the bound in (3.1) is attained in the limit by the sequences. The lower bounds are obtained in a similar way. h For some particular systems, the above bounds may be specified more precisely. Corollary 3.2. Let the assumptions of Proposition 3.1 hold. (i) If the domination polynomial p is convex on (0, 1), then

la1 6 EðTÞ 6 l:

ð3:4Þ

If p is concave, the inequalities are reversed. (ii) If the derivative of the domination polynomial p0 is convex on (0, 1), then

Therefore, for a coherent or mixed system with convex derivative of the domination polynomial, Propositions 2.4 and 2.6 imply that the greater expected system lifetime is obtained in the limit when ZF converges in distribution to ZU. Analogously, for a coherent or mixed system with concave derivative of the domination polynomial, the greater expected lifetime is obtained when ZF converges in distribution to ZL. The bounds obtained from these results and some other ones are presented in the following section.

l inf

x2ð0;1

pðxÞ 6 EðTÞ 6 l maxð1; a1 Þ: x

If p0 is concave, then

l minð1; a1 Þ 6 EðTÞ 6 l sup x2ð0;1

3. Bounds for expected lifetimes of systems with IID components

Proposition 3.1. Let T = w(X1, X2, . . . , Xn) be the lifetime of a coherent or mixed system with IID component lifetimes having common continuous reliability function F and mean l = E(Xi). Let pðxÞ ¼ Pn i i¼1 ai x be its domination polynomial. Then the following bounds x2ð0;1

pðxÞ pðxÞ 6 EðTÞ 6 l sup x x x2ð0;1

are sharp. Proof. By strict increase of the domination polynomial, we have pðFðtÞÞ ¼ 0 iff FðtÞ ¼ 0. Therefore the relations

EðTÞ ¼

Z

Z

1

pðFðtÞÞdt ¼

0

pðFðtÞÞ 6 sup FðtÞ 0
Z

f0
pðFðtÞÞ FðtÞdt FðtÞ

FðtÞdt ¼ sup x2ð0;1

f0
pðxÞ x

Z

1

FðtÞdt ð3:1Þ

0

f0
pðFðtÞÞ pðxÞ FðtÞdt ¼ x FðtÞ

Z

ð3:6Þ

Proof. Note that p(x)/x is the slope of the line passing through the points (0, 0) = (0, p(0)) and (x, p(x)). Also, p is increasing from p(0) = 0 to p(1) = 1. Accordingly, if p is convex, then from the analysis of the graph of p, it follows that supx 2(0,1]p(x)/x = p(1)/1 = 1, and infx2(0, 1]p(x)/x = p0 (0) = a1. Therefore, (3.4) holds. If p is concave then supx2(0,1]p(x)/x = p0 (0) = a1 and infx2(0,1] p(x)/x = p(1)/1 = a1. If p0 is convex, then it is either decreasing or increasing or decreasing-increasing. So p is either concave or convex or first concave and then convex. The upper bound in (3.5) is deduced as follows. It is easily seen that in the last case p(x)/x is first decreasing and then increasing. So supx2(0,1]p(x)/x = max(p0 (0), p(1)/1) = max(a1, 1). In the former cases it is a1 and 1, respectively. We cannot improve the lower bound of Proposition 3.1. If p0 is concave, then p0 is either decreasing or increasing or increasing–decreasing, and so p itself is either concave or convex or convex–concave. Similarly, in the last case p(x)/x is increasing– decreasing, and its infimum is either a1 or 1. Combining it with the first two cases, we get infx2(0,1]p(x)/x = min(a1, 1). The upper bound remains unchanged. h If the assumptions of Corollary 3.2(ii) hold, and moreover parameter aF = E(ZF) is known, we can obtain more refined bounds than those of (3.5) and (3.6). Proposition 3.3. Under assumptions of Proposition 3.1, with p0 being convex on (0, 1) and aF = E(ZF) < 1/2, we obtain

provide the upper bound. Equation

Z

pðxÞ : x

All the above bounds are sharp.

In this section we obtain bounds for the means and the mean residual lifetime functions of coherent or mixed systems with IID components. We use the notation introduced in Section 2. First we present bounds for E(T).

l inf

ð3:5Þ

FðtÞdt

ð3:2Þ

f0
l

pð2aF Þ 6 EðTÞ 6 l½ð1  2aF Þa1 þ 2aF : 2aF

ð3:7Þ

for some 0 < x < 1, and the moment condition determine the twopoint distribution

If p0 is concave, then the inequalities are reversed. In both the cases, the evaluations are sharp.

PrðX i ¼ 0Þ ¼ 1  x ¼ 1  PrðX i ¼ l=xÞ;

Parameter aF can be treated as a measure of concentration of distribution function F. Indeed,

Pn

i ¼ 1; 2; . . . ; n:

ð3:3Þ

Function pðxÞ=x ¼ i¼1 ai xi1 ; 0 < x < 1, is continuous with the limits limx&0p(x)/x = p0 (0) = a1 = nsn and p(1) = 1 at the ends. If supx2ð0;1 pðxÞ is attained at some 0 < x < 1, then we get the equality in x (3.1) for the distribution (3.3). If the supremum amounts to p0 (0) = a1, it is attained in the limit by sequences of the two-point distributions (3.3) with x tending to 0. If the supremum is attained at x = 1, then it is equal to p(1) = 1 and the upper bound l is attained by a component distribution concentrated at l. We finally note that the above degenerate or two-point discrete distributions can be approximated by some sequences of continuous ones with the same

aF ¼ EðZ F Þ ¼

1 2l

Z

1

F 2 ðtÞdt ¼

0

EðX 1:2 Þ EðX 1:2 Þ ¼ ; 2l EðX 1:2 Þ þ EðX 2:2 Þ

ð3:8Þ

where X1:2 and X2:2 are the order statistics from F. It is maximal and equal to 1/2 iff E(X1:2) = E(X2:2) = l, that is, if X1 is degenerate. Also observe that

aF ¼

Z 0

1

HF ðuÞdu ¼

Z 0

1

LF ðuÞdu ¼

1  cF ; 2

where cF is the well-known Gini dispersion index of F.

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Proof of Proposition 3.3. Suppose that p0 is convex. By (2.5) and Proposition 2.6, we have

lEðp0 ðZ L ÞÞ 6 EðTÞ ¼ lEðp0 ðZ F ÞÞ 6 lEðp0 ðZ U ÞÞ;

ð3:9Þ

where ZL and ZU have reliability functions given by (2.9) and (2.10), respectively. Since

Eðp0 ðZ L ÞÞ ¼

Z

2aF

0

p0 ðzÞ pð2aÞ dz ¼ ; 2aF 2a

and

Eðp0 ðZ U ÞÞ ¼ ð1  2aF Þp0 ð0Þ þ 2aF

Z

1

p0 ðzÞdz ¼ ð1  2aF Þa1 þ 2aF ;

0

bounds (3.7) follow. For the concave derivative, the statement is the immediate consequence of the reversed inequalities in (3.9). For the density of (2.9), the inverse transformation to (2.4) determines the two-point distribution supported on 0 and 2laF with probabilities 1  2aF and 2aF, respectively. Similarly, simple absolute continuous approximations of (2.10) correspond with the ð12aF Þl aF l distribution supported on 21 with respective weights e and e 1  e and e, as e ? 0. Sharpness of the evaluations can be deduced from the fact that the two-point distribution can be properly approximated by continuous ones. h Note that Corollary 3.2(ii) can be easily concluded from (3.7). Also, the claim of Corollary 3.2(i) immediately follows from (2.7) and (2.8). We now use the above results for obtaining bounds on the mean residual life of the system E(T  tjT > t) as follows. By (2.2), the reliability function of (T  tjT > t) can be written as

F T ðxjtÞ ¼

F T ðx þ tÞ F T ðtÞ

Pn

¼

i¼1 ai F

i

ðx þ tÞ

F T ðtÞ

¼

n X ai F i ðtÞ i¼1

F T ðtÞ

F i ðxjtÞ

¼ pt ðFðxjtÞÞ; where FðxjtÞ ¼ Fðx þ tÞ=FðtÞ is the common reliability function of the component residual lifetime (Xi  tjXi > t), i = 1, 2, . . . , n, and

pt ðxÞ ¼

n X

ai ðtÞxi ;

ð3:10Þ

i¼1

with ai ðtÞ ¼ ai F i ðtÞ=F T ðtÞ; i ¼ 1; 2; . . . ; n. This representation was obtained in Navarro et al. (2008a). The vector of coefficients a(t) = (a1(t), a2(t), . . . , an(t)) is called as the dynamic minimal signature of T and pt(x) is the dynamic domination polynomial. Using this representation we immediately obtain the following bounds for the mean residual lifetime function of the system. Proposition 3.4. Let T = w(X1, X2, . . . , Xn) is the lifetime of a coherent or mixed system with IID component lifetimes having common continuous reliability functions F and common mean residual life function m(t) = E(Xi  tjXi > t). Let (3.10) stand for the respective dynamic domination polynomial. Then, for every fixed t > 0, we have the following sharp bounds

mðtÞ inf

x2ð0;1

pt ðxÞ p ðxÞ 6 EðT  tjT > tÞ 6 mðtÞ sup t : x x x2ð0;1

In particular, (i) if pt(x) is convex (concave, resp.) on (0, 1), then

mðtÞa1 ðtÞ 6 EðT  tjT > tÞ 6 mðtÞ ðP resp:Þ; (ii) if p0t ðxÞ is convex on (0, 1), then

mðtÞ inf

x2ð0;1

pt ðxÞ 6 EðT  tjT > tÞ 6 mðtÞ maxð1; a1 ðtÞÞ; x

(iii) if p0t ðxÞ is concave on (0, 1), then

mðtÞ minð1; a1 ðtÞÞ 6 EðT  tjT > tÞ 6 mðtÞ sup x2ð0;1

pt ðxÞ : x

Below we apply the above results in some illustrative examples. Furthermore, in Table 1, we provide numerical lower and upper bounds obtained from Proposition 3.1 for the expected values of all the coherent systems composed of 2–4 components with common mean 1. If the components have mean l, then the bounds are multiplied by factor l. Furthermore, in Table 2, we give more refined bounds obtained from Proposition 3.3 for the expected values of coherent systems having convex or concave derivatives of their domination polynomial, composed of 2–4 components with common mean 1 and Gini index 1/2 (i.e. aF = 1/4) which coincides with the Gini index of exponential distributions. We also give the expected lifetimes for systems with IID exponential components having common mean 1. Again, if the components have mean l, then the bounds are multiplied by factor l. Note that from (3.8), l and the Gini index determine the expected lifetimes of parallel and

Table 1 Minimal signatures a and lower (L) and upper (U) bounds for the expected lifetimes E(T) of coherent systems with 1–4 IID components with common mean 1. N

T = w(X1, X2, . . . , Xn)

a

L

U

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

X(1:1) = X1 X1:2 = min(X1X2) X2:2 = max(X1X2) X1:3 = min(X1, X2, X3) min(X1, max(X2, X3)) X2:3(2-out-of-3) max(X1, min(X2, X3)) X3:3 = max(X1, X2, X3) X1:4 = min(X1, X2, X3, X4) max(min(X1, X2, X3), min(X2, X3, X4)) min(X2:3, X4) min(X1,max(X2, X3),max(X2, X4)) min(X1, max(X2, X3, X4)) X2:4(2-out-of-4) max(min(X1, X2), min(X1, X3, X4), min(X2, X3, X4)) max(min(X1, X2), min(X3, X4)) max(min(X1, X2), min(X1, X3), min(X2, X3, X4)) max(min(X1, X2), min(X2, X3), min(X3, X4)) max(min(X1, max(X2, X3, X4)), min(X2, X3, X4)) min(max(X1, X2), max(X1, X3), max(X2, X3, X4)) min(max(X1, X2), max(X3, X4)) min(max(X1, X2), max(X1, X3, X4), max(X2, X3, X4)) X3:4 (3-out-of-4) max(X1, min(X2, X3, X4)) max(X1, min(X2, X3), min(X2, X4)) max(X2:3, X4) max(X1, X2, min(X3, X4)) X4:4 = max(X1, X2, X3, X4)

(1) (0, 1) (2, 1) (0, 0, 1) (0, 2, 1) (0, 3, 2) (1, 1, 1) (3, 3, 1) (0, 0, 0, 1) (0, 0, 2, 1) (0, 0, 3, 2) (0, 1, 1, 1) (0, 3, 3, 1) (0, 0, 4, 3) (0, 1, 2, 2) (0, 2, 0, 1) (0, 2, 0, 1) (0, 3, 2, 0) (0, 3, 2, 0) (0, 4, 4, 1) (0, 4, 4, 1) (0, 5, 6, 2) (0, 6, 8, 3) (1, 0, 1, 1) (1, 2, 3, 1) (1, 3, 5, 2) (2, 0, 2, 1) (4, 6, 4, 1)

1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

1 1 2 1 1 1.125 1.25 3 1 1 1 1 1 1.0535 1.0671 1.0887 1.0887 1.125 1.125 1.1852 1.1852 1.2722 1.3796 1.1481 1.3849 1.5282 2 4

Table 2 Lower (L) and upper (U) bounds obtained from Proposition 3.3 for the expected lifetimes E(T) of coherent systems with 1–4 IID components given in Table 1 with common mean 1 and Gini index 0.5. We also provide the exact values for E(T) in the case of IID exponential components with common mean 1 (which have Gini index equals 0.5). N

p0 (x)

L

E(T)

U

1 2 3 4 5 6 7 8 9 16–17 18–19 20–21 28

Linear Linear Linear Convex Concave Concave Concave Convex Convex Concave Concave Concave Convex

1 0.5 1.5 0.25 0.5 0.5 1 1.75 0.125 0.5 0.5 0.5 1.875

1 0.5 1.5 0.3333 0.6667 0.8333 1.1667 1.8333 0.25 0.75 0.8333 0.9167 2.0833

1 0.5 1.5 0.5 0.75 1 1.25 2 0.5 0.875 1 1.125 2.5

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series systems with two IID components through E(X1:2) = l(1  cF) and E(X2:2) = l(1 + cF). The bounds for the coherent systems with 5 IID components can be obtained by using the signatures given in Navarro and Rubio (2010, in press). Example 3.5. Suppose that T is the lifetime of a mixed system with two IID components and signature (s1, s2) = (s1, 1  s1) for some s1 2 [0, 1]. Its reliability function is

The maximum value of the upper bound is 1, and it is obtained when c ? 1. Therefore

0 6 EðX 2:3 Þ 6 l hold for all Pareto type II distributions with a finite mean. The reliability function of (X2:3  tjX2:3 > t) has the form

F 2:3 ðxjtÞ ¼

2

F T ðtÞ ¼ s1 F 1:2 ðtÞ þ s2 F 2:2 ðtÞ ¼ 2s2 FðtÞ þ ðs1  s2 ÞF ðtÞ: It has the domination polynomial p(x) = 2s2x + (s1  s2)x2. Evidently, p is convex whenever s1 P 1/2, and concave otherwise. From Corollary 3.2(i), we get

2s2 l 6 EðTÞ 6 l

3  2FðtÞ

F 2 ðxjtÞ 

2FðtÞ 3  2FðtÞ

F 3 ðxjtÞ:

Its dynamical polynomial

pt ðxÞ ¼

3 3  2FðtÞ

x2 

2FðtÞ 3  2FðtÞ

x3 :

has a concave derivative for all t. Consequently, by Proposition 3.4,

iff s1 P 1/2, and the reversed inequalities otherwise. The reliability function of (T  tjT > t) can be written as

F T ðxjtÞ ¼

3

2s2 FðtÞ ðs1  s2 ÞF 2 ðtÞ 2 FðxjtÞ þ F ðxjtÞ F T ðtÞ F T ðtÞ

( 0 6 EðX 2:3  tjX 2:3 > tÞ 6

9=8 mðtÞ; 3FðtÞ2F 2 ðtÞ

if FðtÞ P 3=4;

mðtÞ;

if FðtÞ 6 3=4:

It follows in particular that at the time exceeding the first quartile of F, the 2-out-of-3 system works worse than a single component.

Accordingly, its dynamical polynomial is

pt ðxÞ ¼

2s2 ðs1  s2 ÞFðtÞ xþ x2 : 2s2 þ ðs1  s2 ÞFðtÞ 2s2 þ ðs1  s2 ÞFðtÞ

It is convex (concave, respectively) whenever s1 P 1/2 (s1 6 1/2, respectively). Hence, from Proposition 3.4(i), we conclude

2s2 mðtÞ 6 EðT  tjT > tÞ 6 mðtÞ 2s2 þ ðs1  s2 ÞFðtÞ with m(t) = E(Xi  tjXi > t) when s1 P 1/2, and the reversed inequalities for s1 6 1/2. In particular, taking s1 = 0, we obtain

mðtÞ 6 EðX 2:2  tjX 2:2 > tÞ 6

2 2  FðtÞ

0 6 EðX 1:2  tjX 1:2 > tÞ 6 mðtÞ for the series system. The inequalities above imply the well-known relations X1:2 MRL Xi MRL X2:2, where MRL denotes the mean residual life order. Example 3.6. Let T = X2:3 be the lifetime of a 2-out-of-3 system with IID components. Its domination polynomial p(x) = 3x2  2x3 (see Navarro et al., 2007a) has a concave derivative. By Proposition 3.1, we have

pðxÞ 9 ¼ l: 0 6 EðX 2:3 Þ 6 l sup x 8 x2ð0;1 The upper bound coincides with one obtained in Papadatos (1997). If we know the concentration parameter aF ¼ EðX 1 FðX 1 ÞÞ=l, this bound can be improved as follows:

pð2aF Þ ¼ lð6aF  8a2F Þ; 2aF

for 0 6 aF 6 1/2. The general upper bound 9/8 is obtained for aF = 3/8. In particular, if the components have a Pareto type II distribution with common reliability function given by

 FðtÞ ¼ 1 þ

t ðc  1Þl

c

;

t P 0;

with c > 1 and l = E(Xi), then aF = (c  1)/(4c  2) and

0 6 EðX 2:3 Þ 6

ð4c  1Þðc  1Þ ð2c  1Þ2

l:

l 6 EðTÞ 6 l sup x2ð0;1

pðxÞ 5 ¼ l: x 4

Furthermore, its dynamic domination polynomial

pt ðxÞ ¼

1 1 þ FðtÞ  F 2 ðtÞ

xþ

FðtÞ 1 þ FðtÞ  F 2 ðtÞ

x2 

F 2 ðtÞ 1 þ FðtÞ  F 2 ðtÞ

x3

mðtÞ

for the parallel system. Analogously, taking s1 = 1, we get the trivial bounds

0 6 EðX 2:3 Þ 6 l

Example 3.7. Suppose that T = max(X1, min(X2, X3)) is the lifetime of a coherent system with IID components. It has the domination polynomial p(x) = x + x2  x3 (see Navarro et al., 2007a) with a concave derivative. From Proposition 3.1

has a concave derivative for all t. Therefore, ( 5=4 mðtÞ; mðtÞ 2 6 EðT  tjT > tÞ 6 1þFðtÞF ðtÞ 2 1 þ FðtÞ  F ðtÞ mðtÞ;

if FðtÞ P 1=2; if FðtÞ 6 1=2:

Note that m(t) and E(T  tjT > t) are ordered in the latter case. This means that the system gets worse than its components if their age exceeds the median of F. This confirms the fact that T and X1 are not necessarily hazard rate ordered (see Fig. 2 in Navarro et al., 2008b) because otherwise it would imply that they are MRL ordered.

4. Bounds for expected lifetimes of systems with INID components In this section we study the lifetime T = w(X1, X2, . . . , Xn) of a coherent (or mixed) system with independent non-necessarily identically distributed (INID) component lifetimes X1, X2, . . . , Xn which have continuous reliability functions F i ðtÞ ¼ PrðX i > tÞ and means li = E(Xi), i = 1, 2, . . . , n. Unfortunately, in this case representations (2.1) and (2.2) are not necessarily true (see Navarro et al., 2008b). From the results in Esary and Proschan (1963) or from the minimal path set representation given in Barlow and Proschan (1975, p. 12) (see also Navarro et al., 2007a), the reliability function F T ðtÞ ¼ PrðT > tÞ of T can be written as

F T ðtÞ ¼

X

ai1 ;i2 ;...;in F i11 ðtÞF i22 ðtÞ    F inn ðtÞ

16i1 þi2 þþin 6n

¼ pðF 1 ðtÞ; F 2 ðtÞ;    ; F n ðtÞÞ;

ð4:1Þ

where ai1 ;i2 ;...;in 2 R and p(x1, x2, . . ., xn) is a polynomial of degree not greater than n called the generalized domination polynomial. This

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polynomial was called the structure reliability function of the system by Esary and Proschan (1963). Note that p only depends on the structure function of the system and p(x) = p(x, x, . . . , x) is the standard domination polynomial of the same system with IID component lifetimes. Also note that p(0, 0, . . . , 0) = 0, p(1, 1, . . . , 1) = 1, and p(x1, x2, . . . , xn) is strictly increasing in (0, 1)n for any coordinate xi. A similar representation holds for the mixed systems with INID components. For example, the reliability of the system with lifetime T = min(X1, max(X2, X3)) has the form

F T ðtÞ ¼ F 1 ðtÞF 2 ðtÞ þ F 1 ðtÞF 3 ðtÞ  F 1 ðtÞF 2 ðtÞF 3 ðtÞ

The upper bound is derived in a similar way. In this case we assume that F j ðtÞ ¼ 0 implies F i ðtÞ ¼ 0 for all i – j and hence pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ ¼ 0. Therefore, by (4.1),

EðTÞ ¼

Z

pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ F j ðtÞ

fF j ðtÞ>0g

6 lj sup

xj 2ð0;1

F j ðtÞdt

pð1; . . . ; 1; xj ; 1; . . . ; 1Þ : xj

We merely mention that if the supremum is attained at some 0 < xj < 1, then the equality holds when

PrðX j ¼ 0Þ ¼ 1  xj ¼ 1  PrðX j ¼ lj =xj Þ;

¼ pðF 1 ðtÞ; F 2 ðtÞ; F 3 ðtÞÞ;

PrðX i ¼ lj =xj Þ ¼ 1 i – j:

where p(x, y, z) = xy + xz  xyz. We use the representation presented above for establishing bounds for the expected lifetime of the system as follows.

Further details of the sharpness proof are left to the reader. h

Proposition 4.1. Let T = w(X1, X2, . . . , Xn) denote the lifetime of a coherent or mixed system with INID component lifetimes having continuous reliability functions F i and means li = E(Xi), i = 1, 2, . . . , n. Let p(x1, x2, . . . , xn) be its generalized domination polynomial. Then, for each j = 1, 2, . . . , n, the lower bound

For example, for T = min(X1, max(X2, X3)), the proposition implies 0 6 E(T) 6 l1. For the 2-out-of-3 system it gives two trivial bounds 0 6 E(X2:3) 6 1. Nevertheless, we note that the bounds are optimal under the general assumptions and cannot be improved without imposing more restrictive conditions. Some nontrivial evaluations are described in the following generalization of Proposition 4.1.

EðTÞ P lj inf

xj 2ð0;1

pð0; . . . ; 0; xj ; 0; . . . ; 0Þ xj

ð4:2Þ

is sharp. Moreover, if for arbitrary t > 0 condition F j ðtÞ ¼ 0 implies F i ðtÞ ¼ 0; i ¼ 1; 2; . . . ; n, then we have the upper sharp bound

EðTÞ 6 lj sup

xj 2ð0;1

pð1; . . . ; 1; xj ; 1; . . . ; 1Þ : xj

ð4:3Þ

The condition for the upper bound is satisfied if Xj has an unbounded support. If tj is a finite right-end of its support, then the assumption is satisfied when all the other Xi, i – j, fail almost surely by time tj. Proof of Proposition 4.1. Due to (4.1),

EðTÞ ¼

Z

F T ðtÞdt

F j ðtÞ

Z

inf

x1 ;x2 ;...;xn 2½0;1;xj – 0

¼ lj inf

xj 2ð0;1

ð4:5Þ

sup xi 2ð0;1;i2P;xi ¼1;iRP

pðx1 ; x2 ; . . . ; xn Þ Q i2P xi

The proof is analogous to that of the previous proposition. For instance, the lower bound (4.5) is derived as follows

F j ðtÞdt

pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞdt

fF j ðtÞ¼0g

P

pðx1 ; x2 ; . . . ; xn Þ Q i2P xi

is the sharp bound.

pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ

fF j ðtÞ>0g

inf

xi 2ð0;1;i2P;xi ¼0;iRP

where lP = E(mini2PXi) is the expected lifetime of the series system whose components have independent lifetimes Xi, for i 2 P. Moreover, if for any t > 0, condition F i ðtÞ ¼ 0 for some i 2 P implies F j ðtÞ ¼ 0 for all j – i, then

1

Z

þ

EðTÞ P lP

EðTÞ 6 lP

0

¼

Proposition 4.2. Let T = w(X1, X2, . . . , Xn) denote the lifetime of a coherent or mixed system with a generalized domination polynomial p(x1, x2, . . . , xn), composed of components with INID lifetimes having continuous reliability functions F i and means li = E(Xi). Then for every P # {1, 2, . . . , n} we have the optimal bound

pðx1 ; x2 ; . . . ; xn Þ xj

pð0; . . . ; 0; xj ; 0; . . . ; 0Þ : xj

Z

EðTÞ ¼

Z fmin F j ðtÞ>0g i2P

F j ðtÞdt

þ

fF j ðtÞ>0g

Z

pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ Y F i ðtÞdt Q i2P F i ðtÞ i2P pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞdt

fmin F j ðtÞ¼0g i2P

ð4:4Þ

Suppose that the infimum is attained at some xj, with 0 < xj < 1. Then the inequality in (4.4) becomes equality for xj. Hence F j ðtÞ > 0 implies F j ðtÞ ¼ xj and F i ðtÞ ¼ 0 for all i – j. Analogously, F j ðtÞ ¼ 0 implies pðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ ¼ 0. These conditions together with E(Xj) = lj are satisfied when Xj has the two-point distribution on 0 and lj/xj with probabilities 1  xj and xj, respectively, and the other components have their lifetimes concentrated at 0. If the infimum is equal pð0;...;0;xj ;0;...;0Þ to limxj &0 , then taking the above mentioned distributions xj with xj & 0, we approximate the third line of (4.4) by the sum of the second line with an arbitrarily desired accuracy. When the infimum amounts to p(0, . . . , 0, 1, 0, . . . , 0), the equality holds if Xj is concentrated at lj, and the other lifetimes at 0. We finally take sequences of continuous approximations of the discrete distributions constructed above so that (4.1) actually defines the expectation of the system lifetime, and it is arbitrarily close to the bound (4.2).

P

inf

xi 2ð0;1;i2P;xi 2½0;1; iRP

¼ lP

inf

xi 2ð0;1;i2P;xi ¼0; iRP

pðx1 ; x2 ; . . . ; xn Þ Q i2P xi

Z

Y

F i ðtÞdt

fmin F j ðtÞ>0g i2P i2P

pðx1 ; x2 ; . . . ; xn Þ Q : i2P xi

The proofs of the other statements of the proposition are omitted. Applying the proposition with varying subsets of indices P # {1, 2, 3} for the system with lifetime T = min(X1, max(X2, X3)), we obtain

maxðEðminðX 1 ; X 2 Þ; EðminðX 1 ; X 3 ÞÞ 6 EðTÞ 6 EðX 1 Þ; and the last argument of the maximum can be obviously dropped. For the 2-out-of-3 system yields

EðX 2:3 Þ P max EðminðX i X j ÞÞ: 16i
In both the examples the evaluations are easily deduced from the analysis of system structure. The above bounds can be refined under

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an additional information, for instance, if the component lifetimes are stochastically ordered.

where lY = E(Y) is the expectation of a random variable with reliability (4.7).

Proposition 4.3. Assume that a coherent or mixed system is composed of independent components whose lifetime continuous reliability functions satisfy F 1 ðtÞ 6 F 2 ðtÞ 6    6 F n ðtÞ; t > 0. Let p(x1, x2, . . . , xn) be its generalized domination polynomial. Then for every P # {1, 2, . . . , n} we have the sharp bounds

Usually it is not easy to calculate the expectation lY since it depends on the component reliability functions through the mean reliability function GðtÞ given in (4.7). We present below some evaluations based on the notions of weak Schur-concavity and convexity. These notions were introduced for bivariate copulas by Durante and Papini (2007). We say that a real function p defined on a convex set in Rn is weakly Schur-concave if

lP

inf

0
pðx1 ; x2 ; . . . ; xn Þ pðx1 ; x2 ; . . . ; xn Þ Q Q 6 EðTÞ 6 lP sup ; 0
pðx1 ; x2 ; . . . ; xn Þ 6 pðx; x; . . . ; xÞ

ð4:8Þ

where the latter is valid under the assumption that for every t > 0 relation F i ðtÞ ¼ 0, with i being the smallest element of P, implies F j ðtÞ ¼ 0 for every 1 6 j < i.

where x is the arithmetic mean of x1, x2, . . . , xn. Of course, all the Schur-concave functions are weakly Schur-concave. In particular, a generalized polynomial p is weakly Schur-concave on [0, 1]n iff

Obviously, both the extremes are calculated with respect to xi, i 2 P, only. If j R P lies between two consecutive elements i < k of P, we put xj = xi and xk when we determine the infimum and supremum, respectively. In particular, Proposition 4.3 implies the following inequalities

mp ðx1 ; x2 ; . . . ; xn Þ 6 x;

max EðminðX i X j ÞÞ 6 EðX 2:3 Þ   9 6 min 2EðX 2 Þ; EðX 3 Þ; 3EðminðX 2 ; X 3 ÞÞ 8

16i
for the 2-out-of-3 system when the components lifetimes satisfy X1 ST X2 ST X3. Another method of determining bounds for the expected lifetimes of coherent or mixed systems with INID components consists in considering equivalent systems (i.e., equal in law) with IID components and using the bounds of the preceding section. The idea is based on the following construction. Let p: [0, 1]n ´ [0, 1] be the generalized domination polynomial of a system. We recall that it is increasing in each coordinate, and p(0,0,. . .,0) = 0, and p(1, 1, . . . , 1) = 1. Let p(x) = p(x, x, . . . , x) stand for the standard domination polynomial associated with p. It is strictly increasing from 0 at 0 to 1 at 1, and so its inverse p1:[0, 1] ´ [0, 1] is well defined. Now we need the old concept of mean function which can be traced to de Finetti (1931). The mean function of a real valued function g : D # Rn # R is a function mg : Rn # R such that

gðx1 ; x2 ; . . . ; xn Þ ¼ gðz; z; . . . ; zÞ for all (x1, x2, . . . , xn) 2 D, where z = mp(x1, x2, . . . , xn). Hence the mean function mp:[0, 1]n ´ [0, 1] associated with p is mp(x1, x2, . . . , xn) = p1(p(x1, x2, . . . , xn)). If F 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞ are lifetime reliability functions, then

GðtÞ ¼ mp ðF 1 ðtÞ; F 2 ðtÞ; . . . ; F n ðtÞÞ

ð4:7Þ

is a reliability function called mean reliability function in Navarro and Spizzichino (in press). It is easy to check that the system whose independent components have lifetimes X1, X2, . . . , Xn with reliability functions F 1 ; F 2 ; . . . ; F n , respectively, and the system with identical structure and independent components that have lifetimes Y1, Y2, . . . , Yn with the common reliability function G have the same lifetime distribution. Accordingly, the following proposition is an immediate consequence of Proposition 3.1.

ð4:9Þ

and, in consequence,

lY 6 l ¼

n 1X l: n i¼1 i

ð4:10Þ

For the weakly Schur-convex functions we have the reversed inequalities in Eqs. (4.8), (4.9), and (4.10). The following examples illustrate applications of these notions combined with Proposition 4.4 to get bounds for the expected lifetimes of systems with INID components. Example 4.5. Consider the two-components parallel system with the reliability function F 2:2 ðtÞ ¼ F 1 ðtÞ þ F 2 ðtÞ  F 1 ðtÞF 2 ðtÞ, domination polynomials p(x,y) = x ffi+ y  xy and p(x) = 2x  x2. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp ðx; yÞ ¼ 1  1  x  y þ xy. From Table 1, we get

lY 6 EðX 2:2 Þ 6 2lY ; where

lY ¼

Z

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  1  F 1 ðtÞ  F 2 ðtÞ þ F 1 ðtÞF 2 ðtÞ dt:

1

0

It is easy to verify that p(x, y) is weakly Schur-convex which implies mp(x, y) P (x + y)/2. Hence GðtÞ ¼ mp ðF 1 ðtÞ; F 2 ðtÞÞ P ½F 1 ðtÞ þ F 2 ðtÞ=2 and lY P (l1 + l2)/2. Finally, we obtain

EðX 2:2 Þ P

l1 þ l2 2

:

In this case, using Proposition 4.1, we have the trivial bounds E(X2:2) P li, i = 1, 2. Hence, we can deduce the bound E(X2:2) P max(l1, l2), which is a better bound than the bound (l1 + l2)/2 obtained from Proposition 4.4. Example 4.6. Take the n-components series system with reliability function F 1:n ðtÞ ¼ F 1 ðtÞF 2 ðtÞ    F n ðtÞ. Here p(x1, x2, . . . , xn) = x1x2    xn and p(x) = xn. Then, using Proposition 4.1, we have E(X1:n) 6 li for all i = 1, 2, . . . , n, and so

EðX 1:n Þ 6 minðl1 ; l2 ; . . . ; ln Þ:

ð4:11Þ 1/n

Proposition 4.4. Let T = w(X1, X2, . . . , Xn) be the lifetime of a coherent or mixed system with INID component lifetimes having continuous reliability functions F 1 ; F 2 ; . . . ; F n . Let p(x1, x2, . . . , xn) and p(x) denote the respective generalized and associated standard domination polynomials. Then the following bounds are sharp

pðxÞ pðxÞ 6 EðTÞ 6 lY sup ; lY inf x2ð0;1Þ x x x2ð0;1Þ

Since mp(x1, x2, . . . , xn) = (x1x2    xn) , using Proposition 4.4 and Table 1, we obtain E(X1:n) 6 lY, where

lY ¼

Z

1

ðF 1 ðtÞF 2 ðtÞ    F n ðtÞÞ1=n dt:

0

P Due to weak Schur-concavity of p, we have lY 6 1n ni¼1 li , and ultiP n mately EðX 1:n Þ 6 1n i¼1 li . This is an evaluation inferior to (4.11), though.

J. Navarro, T. Rychlik / European Journal of Operational Research 207 (2010) 309–317

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