uniform stochastic ordering on a system of components ... - CiteSeerX

Sankhy¯ a : The Indian Journal of Statistics 1999, Volume 61, Series A, Pt. 2, pp. 218-228

UNIFORM STOCHASTIC ORDERING ON A SYSTEM OF COMPONENTS WITH DEPENDENT LIFETIMES INDUCED BY A COMMON ENVIRONMENT By CHUNSHENG MA∗ The University of Sydney, Australia SUMMARY. A system of n non-renewable components sharing a common random environment is considered. In terms of the uniform stochastic orderings, it is shown how the random environment can affect the number of components functioning and the lifetime of a k-out-of-n system.

1.

Introduction

This paper is concerned with a system of n non-renewable components sharing a common random environment; specifically, the component lifetimes T1 , · · · , Tn are supposed to be conditionally independent, given the “environment” of the system which is a certain random variable, say Λ. The systems of this type with dependent component lifetimes have been recently introduced and developed. When the components are identical, Shaked (1977) introduced the so-called positive dependence by mixture and obtained bounds for the reliability of some kout-of-n systems consisting of identical components. Lindley and Singpurwalla (1986), Nayak (1987), Roy and Mukherjee (1988), Whitmore and Lee (1991) assumed that the joint lifetimes are distributed as a mixture of n independent exponential laws. For other mixture models see for instance, Hougaard (1986, 1987), Crowder (1989), Youngren (1991), Egeland (1992) and Al-Mutairi (1996). Currit and Singpurwalla (1988) and Woodham and Richards (1997) compare the reliability function of a redundant k-out-of-n system operating within the laboratory where all component lifetimes are independent and identically distributed according to a gamma distribution, with the reliability function of the same system operating subject to environmental effects. Lefevre and Malice (1989) and Gupta and Richards (1992) investigate the influence of the environment on the Paper received. April 1997; revised October 1998. AMS (1991) subject classification. 62K10, 62N05, 90B25. Key words and phrases. k-out-of-n system, likelihood ratio ordering, order statistics, stochastic ordering, uniform stochastic orderings. ∗ Supported by the Award OPRS-UPRA-9302659.

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performance of the system when the joint lifetimes of the components are taken as a mixture of n independent exponential laws using different notions of stochastic majorizations. Under the general setting, this paper will show the influence of the environment on the performance of the system in terms of the usual stochastic ordering and the uniform stochastic orderings. These orderings are recalled below. For references see Pfanzagl (1964), Keilson and Sumita (1982), Ross (1983) and Lynch, Mimmack and Proschan (1987), amongst others. For other related work see a recent paper of Garren and Richards (1998), where comparisons are made when the length of the time interval is sufficiently small. Throughout this paper “increasing” and “decreasing” are used in the nonstrict sense. Denote the distribution function and survival function of Wi by FWi and F¯Wi respectively, i = 1, 2. (i) W1 is said to be stochastically smaller than W2 , written W1 ≤st W2 , if F¯W1 (t) ≤ F¯W2 (t) for all t; or equivalently, if Eψ(W1 ) ≤ Eψ(W2 ) for all increasing functions ψ(·) for which the integrals are well-defined. (ii) W1 is uniformly smaller than W2 in the positive direction, written W1 ≤(+) W2 , if F¯W2 (t) is increasing in t. F¯W1 (t) In the case of absolutely continuous distributions, this is equivalent to the hazf 1 (t) ard rate of FW1 (t), F¯W , being uniformly greater than the hazard rate of W (t) 1

f

(t)

2 FW2 (t), F¯W . Thus this ordering is alternatively called the hazard (failure) W2 (t) rate ordering. (iii) W1 is uniformly smaller than W2 in the negative direction, written W1 ≤(−) W2 , if FW2 (t) is increasing in t. FW1 (t) In the absolutely continuous case, this ordering is also interpreted by Joag-Dev, Kochar and Proschan (1995) as the survival rate ordering. (iv) W1 is smaller than W2 in the sense of likelihood ratio, written W1 ≤lr W2 , if, when W1 and W2 are absolutely continuous random variables with densities fW1 (t) and fW2 (t), fW2 (t) is increasing in t, fW1 (t) or, when W1 and W2 are discrete random variables,

P (W2 = t) is increasing in t. P (W1 = t) It is well-known that W1 ≤lr W2 ⇒ W1 ≤(+) W2 (W1 ≤(−) W2 ) ⇒ W1 ≤st W2 .

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Several characterizations of uniform stochastic orderings are derived by Cap´era`a (1988), Shanthikumar and Yao (1991), and Joag-Dev, Kochar and Proschan (1995). Our purpose of this paper is to show the influence of the environment on the performance of the system under the general setting. The analysis is carried out by using the partial orderings mentioned above. We will examine how a partial ordering on the environment Λ can be translated into a partial ordering on the number of components functioning and how it affects the lifetime of a k-out-of-n system. The main results are presented in Section 2, and the proofs are given in Section 3. 2.

Main Results

Suppose that the effect of the common operating environment is represented by a random variable Λ which has the distribution function GΛ and survival ¯ Λ = 1 − GΛ , and that given Λ = λ, T1 , · · · , Tn are conditionally indefunction G pendent with survival functions F¯1 (λt), · · · , F¯n (λt). Without loss of generality, let Λ ≥ 0, Ti ≥ 0(i = 1, · · · , n). The joint survival function F¯ (t1 , · · · , tn ) of the vector (T1 , · · · , Tn ) is then written as F¯ (t1 , · · · , tn ) =

n ∞Y

Z 0

F¯i (ti λ)dGΛ (λ).

. . . (2.1)

i=1

This model may be alternatively derived as an accelerated life model. To this end, let Ti = Λ−1 Xi , i = 1, · · · , n, where Λ, X1 , · · · , Xn are independent, and Xi has distribution Fi (t). So Z ∞ P (T1 ≥ t1 , · · · , Tn ≥ tn ) = P (Λ−1 X1 ≥ t1 , · · · , Λ−1 Xn ≥ tn | Λ = λ)dGΛ (λ) 0 Z ∞ = P (λ−1 X1 ≥ t1 , · · · , λ−1 Xn ≥ tn )dGΛ (λ) 0

Z = 0

n ∞Y

F¯i (ti λ)dGΛ (λ),

i=1

which coincides with (2.1). If GΛ is supported on (1, ∞), then the components are operating in a harsh environment; and if GΛ is supported on (0, 1), then they are operating in a kinder, gentler environment. Clearly, the choice of the function GΛ (·) depends on the actual operating conditions. We are interested in the influence of GΛ (·) on the performance process of the system.

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2.1 Influence of the environment on the number of components functioning. Denote by N (t, Λ) the number of components working at time t, t ≥ 0, under the environmental effects measured by Λ. In particular, N (·, c) represents the number of components working under a constant environment degenerated at c > 0. Then for t ≥ 0, Z ∞ P (N (t, λ) = j)dGΛ (λ) P (N (t, Λ) = j) = 0 Z ∞ = P (N (λt, 1) = j)dGΛ (λ), j = 0, 1, · · · , n, 0

where N (t, λ) =

n X

Ii (t; λ),

i=1

and I1 (t; λ), · · · , In (t; λ) are independent Bernoulli random variables with P (Ii (t; λ) = 1) = F¯i (λt), P (Ii (t; λ) = 0) = Fi (λt), i = 1, · · · , n. The following theorem shows how a partial ordering on the different environments Λ1 and Λ2 can be translated into a partial ordering on N (t, Λ1 ) and N (t, Λ2 ), the numbers of components functioning at the same time t ≥ 0. Theorem 1. Let Λ1 and Λ2 be distinct random environments. (i) If Λ1 ≤st Λ2 , then N (t, Λ1 )≥st N (t, Λ2 ), t ≥ 0. (ii) If Λ1 ≤(−) Λ2 , then N (t, Λ1 )≥(+) N (t, Λ2 ), t ≥ 0. (iii) If Λ1 ≤(+) Λ2 , then N (t, Λ1 )≥(−) N (t, Λ2 ), t ≥ 0. (iv) If Λ1 ≤lr Λ2 , then N (t, Λ1 )≥lr N (t, Λ2 ), t ≥ 0. Under some special settings, part (i) of the above theorem is derived by Lefevre and Malice (1989) and Gupta and Richards (1992, Section 3). Now consider the influence of the same environment Λ on the number of components functioning at the different times, 0 ≤ t < t0 . Obviously, N (t, Λ)≥st N (t0 , Λ), 0 ≤ t < t0 . Theorem 2. (i) If Λ ≤(−) aΛ for all constants a ≥ 1, then N (t, Λ)≥(+) N (t0 , Λ), 0 ≤ t < t0 . (ii) If Λ ≤(+) aΛ for all constants a ≥ 1, then N (t, Λ)≥(−) N (t0 , Λ), 0 ≤ t < t0 . For the discussion concerning the condition that Λ ≤(+) aΛ, or Λ ≤(−) aΛ holds for all constants a ≥ 1, see Keilson and Sumita (1982). It is easy to see that the following statements are equivalent:

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(i) Λ ≤(+) aΛ holds for all constants a ≥ 1; ¯ Λ ( λ ) is T P2 in (λ, θ); (ii) G θ

¯ Λ (eλ ) is log-concave. The last statement is true, if for example, G ¯ Λ (λ) (iii) G is log-concave or equivalently, if GΛ is IFR (increasing failure rate). Corollary 1. If GΛ is IFR, then N (t, Λ)≥(−) N (t0 , Λ), 0 ≤ t < t0 .

Corollary 2. If Λ has a log-concave density function, then N (t, Λ)≥(+) N (t0 , Λ)andN (t, Λ)≥(−) N (t0 , Λ), 0 ≤ t < t0 . ¯ Λ are log-concave when the corCorollary 2 follows, since both GΛ and G responding density is log-concave; see Keilson and Sumita (1982, Theorem 3.5 (c)). 2.2 Influence of the environment on the lifetime of a k-out-of-n system. A k-out-of-n system is a system of n components which functions if and only if at least k components function. For a k-out-of-n system formed from n components with lifetimes distributed as T1 , · · · , Tn , denote its lifetime by Zk|n (Λ) under the environment Λ. Specifically, Zk|n (c) stands for the lifetime under a constant environment degenerated at c > 0. A useful relationship between the reliability function of Zk|n (Λ) and that of Zk|n (1) is given by Z P (Zk|n (Λ) ≥ t) =

∞

P (Zk|n (1) ≥ λt)dGΛ (λ), t ≥ 0.

. . . (2.2)

0

Another important fact is P (Zk|n (Λ) ≥ t) = P (N (t, Λ) ≥ k), t ≥ 0, k = 1, · · · , n, which yields Z P (N (t, Λ) ≥ k) =

∞

P (Zk|n (1) ≥ λt)dGΛ (λ), t ≥ 0, k = 1, · · · , n.

. . . (2.3)

0

Clearly, Z1|n (Λ) ≥st Z2|n (Λ) ≥st · · · ≥st Zn|n (Λ). It is proved by Boland, El-Neweihi and Proschan (1994) that Z1|n (1) ≥(+) Z2|n (1) ≥(+) · · · ≥(+) Zn|n (1),

. . . (2.4)

in the absolutely continuous case. (2.4) holds actually in general, as shown by Boland, Shaked and Shanthikumar (1998). Similarly, it follows from Esary and

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Proschan (1963, p188) that for k = 1, · · · , n − 1, P (Zk|n (1) ≤ t) = P (Zk+1|n (1) ≤ t)

P(

n P

{1 − Ii (t; 1)} ≥ n − k + 1)

i=1 n P

P(

{1 − Ii (t; 1)} ≥ n − k)

i=1

is an increasing function of t; that is, Z1|n (1) ≥(−) Z2|n (1) ≥(−) · · · ≥(−) Zn|n (1).

. . . (2.5)

For the likelihood ratio ordering of k-out-of-n systems see Ma (1998) and references therein. Under a random environment Λ, one has Theorem 3. (i) If Λ ≤(+) aΛ for all constants a ≥ 1, then Z1|n (Λ) ≥(−) Z2|n (Λ) ≥(−) · · · ≥(−) Zn|n (Λ). (ii) If Λ ≤(−) aΛ for all constants a ≥ 1, then Z1|n (Λ) ≥(+) Z2|n (Λ) ≥(+) · · · ≥(+) Zn|n (Λ). Corollary 1. If GΛ is IFR, then Z1|n (Λ) ≥(−) Z2|n (Λ) ≥(−) · · · ≥(−) Zn|n (Λ). Corollary 2. If Λ has a log-concave density function, then Z1|n (Λ) ≥(+) Z2|n (Λ) ≥(+) · · · ≥(+) Zn|n (Λ), and Z1|n (Λ) ≥(−) Z2|n (Λ) ≥(−) · · · ≥(−) Zn|n (Λ). Theorem 3 is indeed an equivalent form of Theorem 2 due to the fact (2.3). It has an important interpretation in terms of order statistics. For (dependent) lifetimes T1 , · · · , Tn with joint survival function (2.1), let Tk:n represent the k-th order statistic (in increasing order). Then Tk:n = Zn−k+1|n (Λ), k = 1, · · · , n. Now Theorem 3 says that if GΛ ( λθ ) is T P2 in (λ, θ), then T1:n ≤(+) T2:n ≤(+) · · · ≤(+) Tn:n ; ¯ Λ ( λ ) is T P2 in (λ, θ), then and if G θ T1:n ≤(−) T2:n ≤(−) · · · ≤(−) Tn:n . The following theorem shows how a partial ordering on Λ can be translated into a partial ordering on the lifetime of a k-out-of-n system.

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Theorem 4. Let Λ1 and Λ2 be distinct random environments. (i) If Λ1 ≤st Λ2 , then Zk|n (Λ1 ) ≥st Zk|n (Λ2 ), k = 1, · · · , n. (ii) If Λ1 ≤(−) Λ2 , and Zk|n (1) ≤(+) aZk|n (1) holds for all constants a ≥ 1, then Zk|n (Λ1 ) ≥(+) Zk|n (Λ2 ). (iii) If Λ1 ≤(+) Λ2 , and Zk|n (1) ≤(−) aZk|n (1) holds for all constants a ≥ 1, then Zk|n (Λ1 ) ≥(−) Zk|n (Λ2 ). As an immediate consequence of Barlow and Proschan (1981, p108, Theorem 5.8), one has Corollary. Let F1 = · · · = Fn = F . If Λ1 ≤(−) Λ2 and F is IFR, then Zk|n (Λ1 ) ≥(+) Zk|n (Λ2 ), k = 1, · · · , n. The following result is a generalization of Gupta and Richards (1992, Theorem 3.4). Theorem 5. Assume that Λ1 and Λ2 are distinct random environments, F1 (t) = · · · = Fn (t) = F (t) and F (t) is absolutely continuous with density f (t). If f (et ) is strictly log-concave, and GΛ2 (λ)−GΛ1 (λ) has at most one sign change as λ varies on (0, ∞), then P (Zk|n (Λ2 ) ≥ t) − P (Zk|n (Λ1 ) ≥ t) has at most one sign change as t varies on (0, ∞). 3.

The Proofs

Proof of theorem 1. (i) Obviously, P (N (t, Λ1 ) ≥ 0) = P (N (t, Λ2 ) ≥ 0) = 1. For k = 1, · · · , n, since P (Zk|n (1) ≥ λt) is a decreasing function of λ, from (2.3) and Λ1 ≤st Λ2 one has, Z ∞ P (N (t, Λ1 ) ≥ k) = P (Zk|n (1) ≥ λt)dGΛ1 (λ) Z0 ∞ ≥ P (Zk|n (1) ≥ λt)dGΛ2 (λ) = P (N (t, Λ2 ) ≥ k). 0

Thus N (t, Λ1 )≥st N (t, Λ2 ), t ≥ 0. (ii) Since Λ1 ≤(−) Λ2 implies Λ1 ≤st Λ2 and thus from part (i), P (N (t, Λ1 ) ≥ 0) P (N (t, Λ2 ) ≥ 0) ≤ , P (N (t, Λ1 ) ≥ 1) P (N (t, Λ2 ) ≥ 1)

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it suffices to show that for k = 1, · · · , n − 1, P (N (t, Λ2 ) ≥ k) P (N (t, Λ1 ) ≥ k) ≤ , P (N (t, Λ1 ) ≥ k + 1) P (N (t, Λ2 ) ≥ k + 1) or equivalently, by (2.3), R∞ R∞ P (Zk|n (1) ≥ λt)dGΛ1 (λ) P (Zk|n (1) ≥ λt)dGΛ2 (λ) R ∞0 ≤ R ∞0 . P (Z (1) ≥ λt)dG (λ) P (Zk+1|n (1) ≥ λt)dGΛ2 (λ) Λ1 k+1|n 0 0 Now notice that Λ1 ≤(−) Λ2 is equivalent to that R∞ R∞ α(λ)dGΛ1 (λ) α(λ)dGΛ2 (λ) 0 R∞ ≤ R0∞ β(λ)dGΛ1 (λ) β(λ)dGΛ2 (λ) 0 0

. . . (3.1)

. . . (3.2)

holds for all functions α and β, integrable with respect to GΛ1 and GΛ2 , such that α β is increasing, β is nonnegative and decreasing. This result may be proved in a similar way to that of Cap´era`a (1988, Corollary) or Joag-Dev, Kochar and Proschan (1995, Theorem 3.1). In particular, for each k ∈ {1, · · · , n − 1}, taking α(λ) = P (Zk|n (1) ≥ λt), β(λ) = P (Zk+1|n (1) ≥ λt), it yields (3.1) since in this case β(λ) ≥ 0 is decreasing, and α(λ) β(λ) is increasing by (2.4). (iii) For each k ∈ {0, 1, · · · , n − 1}, P (N (λt, 1) ≤ k + 1) = P (Zk+1|n (1) ≤ λt) is obviously increasing in λ, and by (2.5), P (Zk|n (1) ≤ λt) P (N (λt, 1) ≤ k) = P (N (λt, 1) ≤ k + 1) P (Zk+1|n (1) ≤ λt) is also an increasing function of λ. So it follows from Cap´era`a (1988, Corollary) that for k = 0, 1, · · · , n − 1 R∞ R∞ P (N (λt, 1) ≤ k)dGΛ1 (λ) P (N (λt, 1) ≤ k)dGΛ2 (λ) 0 R∞ ≤ R ∞0 ; P (N (λt, 1) ≤ k + 1)dGΛ1 (λ) P (N (λt, 1) ≤ k + 1)dGΛ2 (λ) 0 0 namely, N (t, Λ1 )≥(−) N (t, Λ2 ). (iv) By definition of the ≤lr ordering, one has to show that for k = 0, 1, · · · , n − 1, P (N (t, Λ1 ) = k) P (N (t, Λ1 ) = k + 1) ≤ , P (N (t, Λ2 ) = k) P (N (t, Λ2 ) = k + 1) or equivalently, Z Z {P (N (λ1 t, 1) = k)P (N (λ2 t, 1) = k + 1) −P (N (λ1 t, 1) = k + 1)P (N (λ2 t, 1) = k)}gΛ1 (λ1 )gΛ2 (λ2 )dλ1 dλ2 ≤ 0.

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Using the basic composition formula (Karlin, 1968, p17)), the above integral is P (N (λ1 t, 1) = k) P (N (λ1 t, 1) = k + 1) λ1