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On likelihood ratio ordering of parallel system with two exponential components
Operations Research Letters 43 (2015) 195–198
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On likelihood ratio ordering of parallel system with two exponential components Jiantian Wang a,∗ , Henry Laniado b a
Department of Mathematics, Kean University, Union, NJ, 07083, USA
b
Departamento de Estadística, Universidad Carlos III de Madrid, 28911, Leganés, Spain
article
info
Article history: Received 8 August 2014 Received in revised form 31 January 2015 Accepted 31 January 2015 Available online 7 February 2015
abstract This paper considers stochastic comparison for parallel systems with two exponential components. For a given such system, we identify a region, such that, if the hazard rate pair of another parallel system lies in that region, then there exists likelihood ratio ordering between the two systems. The new results presented in this paper extend most existing ones in the literature. Published by Elsevier B.V.
Keywords: Parallel system Allocation policy Stochastic order Likelihood rate order
1. Introduction Parallel systems are commonly used to improve the reliability of a device. For economical reason, parallel systems with only two components are quite popular. To compare the lifetimes of two such systems is fundamental in engineering reliability theory. For systems made up of components with general distributed lifetimes, the stochastic comparison for those systems is quite elusive, since the distribution theory becomes quite complicated. For this reason, in this paper, we confine ourselves to the parallel systems with components whose lifetimes follow exponential distributions. Pledger and Proschan [11] were the first ones to compare stochastically for such systems. Since then, many researchers have worked in this field, including Kochar and Rojo [7], Dykstra et al. [3], Khaledi and Kochar [5,6], Kochar and Xu [8], Joo and Mi [4], Da et al. [2], and Zhao and Balakrishnan [14]. Let T (λ1 , . . . , λn ) be the lifetime of a parallel system with n exponential components whose hazard rates are λ1 , . . . , λn , respectively. For technical reason, we just focus on the case of n = 2 in the present paper. For the systems T (λ1 , λ2 ) and T (γ1 , γ2 ), by symmetry, we assume λ1 ≤ λ2 and γ1 ≤ γ2 . Denote by ≻ as majorization order, ≥st , ≥hr , ≥rh , and ≥lr as the usual stochastic
∗
Corresponding author. E-mail address: [email protected] (J. Wang).
http://dx.doi.org/10.1016/j.orl.2015.01.012 0167-6377/Published by Elsevier B.V.
order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order, respectively. The formal definitions of these orders will be given in the next section. So far, several stochastic comparison results for the lifetimes T (λ1 , λ2 ) and T (γ1 , γ2 ) have been established. For instance, Pledger and Proschan [11] showed that if (λ1 , λ2 ) ≻ (γ1 , γ2 ), T (λ1 , λ2 ) ≥st T (γ1 , γ2 ). Boland et al. [1] strengthened it as T (λ1 , λ2 ) ≥hr T (γ1 , γ2 ) and T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). Dykstra et al. [3] further enhanced it as T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). Joo and Mi [4] revealed that when λ1 ≤ γ1 ≤ γ2 ≤ λ2 and λ1 + λ2 ≤ γ1 + γ2 , T (λ1 , λ2 ) ≥hr T (γ1 , γ2 ). Zhao and Balakrishnan [14] improved this result from hazard rate order to likelihood ratio order. Yan et al. [13] showed, when λ1 ≤ γ1 ≤ λ2 ≤ γ2 and λ1 + γ2 ≤ γ1 + λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). Misra and Misra [9] proved that, when (λ1 , λ2 ) weakly majorizes (γ1 , γ2 ), then T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). For a given point (λ1 , λ2 ), denote Ωλ1 ,λ2 as the region such that, if (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , (γ1 , γ2 ) is weakly majorized by (λ1 , λ2 ). The result of Misra and Misra [9] thus can be stated as: when (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). A natural question now is: can we extend the result of Misra and Misra [9] to likelihood ratio order? Or, in what a subregion of Ωλ1 ,λ2 that T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds? This paper intends to answer this question. In this paper, we reveal a subregion of Ωλ1 ,λ2 , denote it as Θλ1 ,λ2 , such that, for any (γ1 , γ2 ) ∈ Θλ1 ,λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). The picture below ¯ = (λ1 + λ2 )/2. shows the regions Ωλ1 ,λ2 and Θλ1 ,λ2 , where λ
196
J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
Lemma 3.2. For x > 0, the function f (x) = 1−xe−x is increasing and convex. The following lemma is a classical result of Dykstra et al. [3]: Lemma 3.3 (Theorem 3.1, Dykstra et al. [3]). If (λ1 , λ2 ) ≻ (γ1 , γ2 ),
The region Ωλ1 ,λ2 is formed by the inequalities y ≥ x ≥ λ1 , and x + y ≥ λ1 + λ2 , and the region Θλ1 ,λ2 is formed by the ¯ , or, x + y ≥ λ1 + λ2 , and y ≤ λ2 + inequalities y ≥ x ≥ λ 1 ( x − λ ) . Notice (λ , λ 1 1 2 ) ≻ (γ1 , γ2 ) is equivalent to that the 2 point (γ1 , γ2 ) is on the line segment connecting the point (λ1 , λ2 ) ¯ λ) ¯ . Thus, Dykstra’s result can be stated as: for any point and (λ, (γ1 , γ2 ) on that line segment, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). The result of Zhao and Balakrishnan [14] can be stated as: for any point (γ1 , γ2 ) ¯ λ) ¯ , and in the triangle formed by the three points (λ1 , λ2 ), (λ, (λ2 , λ2 ), T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ), and the result of Yan et al. [13] can be stated as: for any point (γ1 , γ2 ) lies in the triangle with vertexes (λ1 , λ2 ), (λ2 , λ2 ), and (λ2 , 2λ2 −λ1 ), T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). As we can see, the result established in this paper extends the results of Dykstra et al. [3], Zhao and Balakrishnan [14], improves part of the result of Da et al. [2] from hazard ratio ordering to likelihood ratio ordering, and overlaps that of Yan et al. [13]. 2. Definitions and notations
Proof of Theorem 3.1. At first, by using Theorem 1. C. 33 in ¯ T (λ1 , λ2 ) Shaked and Shanthikumar [12], we have, when γ1 ≥ λ, ≥lr T (γ1 , γ2 ). Now we want to show T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) for all point (γ1 , γ2 ) in the triangle region formed by the lines x + y = λ1 + λ2 , y = x, and y = λ2 + 21 (x − λ1 ). By the result of Misra and Misra [9], T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ) on that region. From Theorem 1.C.4(b) of Shaked and Shanthikumar [12], it is enough to show that the ratio of the reversed hazard rate functions is increasing for t > 0. Denote the reversed hazard rate function of T (λ1 , λ2 ) as r(λ1 ,λ2 ) (t ). We have,
λ1 e−λ1 t (1 − e−λ2 t ) + λ2 e−λ2 t (1 − e−λ1 t ) (1 − e−λ1 t )(1 − e−λ2 t ) −λ1 t λ2 e−λ2 t λ1 e + . = 1 − e−λ1 t 1 − e−λ2 t
r(λ1 ,λ2 ) (t ) =
sgn
For our convenience, we denote A = B if the signs of A and B are the same. We have,
ψ(t ) =
To state the results, we introduce some notations and concepts first. Let X be a nonnegative continuous random variable with distribution function FX (t ), survival function F¯X (t ) = 1 − FX (t ), and density function fX (t ). The hazard function and the reversed hazard function of X are defined as λX = fX /F¯X and rX = fX /FX , respectively. For two nonnegative continuous random variables X and Y , X is said to be smaller than Y in the usual stochastic order (denoted by X ≤st Y ), if F¯X (t ) ≤ F¯Y (t ). X is said to be smaller than Y in hazard rate order (denoted by X ≤hr Y ), if λX (t ) ≥ λY (t ). X is said to be smaller than Y in reversed hazard rate order (denoted by X ≤rh Y ), if rX (t ) ≤ rY (t ). X is said to be smaller than Y in likelihood ratio order (denoted by X ≤lr Y ), if the ratio fY (t )/fX (t ) is increasing in t. It is well known that likelihood ratio order implies both hazard rate order and reversed hazard rate order, while these two orders imply usual stochastic order. Given two vectors a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ), let a(1) ≤ a(2) ≤ · · · ≤ a(n) and b(1) ≤ b(2) ≤ · · · ≤ b(n) be the increasing arrangements of the components of the two vectors, then the vector a is said to majorize the vector b (denoted by a ≻ b) n n k k if and only if, i=1 a(i) = i=1 b(i) , and i=1 a(i) ≤ i=1 b(i) , for k = 1, . . . , n − 1. If for k = 1, . . . , n, i=1 a(i) ≤ i=1 b(i) , then the vector a is said to weakly majorize the vector b (denoted by
k
k
w
a ≻ b). For some extensive and comprehensive discussions on the theory of these orders and their applications, one can see Müller and Stoyan [10], or, Shaked and Shanthikumar [12]. 3. Main result and proof We state the main result as the following theorem. Theorem 3.1. For any (γ1 , γ2 ) ∈ Θλ1 ,λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). To prove the theorem, the following lemmas will be used. The proof of the first lemma is easy and thus is omitted.
then, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ).
r(λ1 ,λ2 ) (t ) r(γ1 ,γ2 ) (t )
λ1 e−λ1 t
=
1−e−λ1 t γ1 e−γ1 t 1−e−γ1 t
+ +
λ2 e−λ2 t 1−e−λ2 t γ2 e−γ2 t 1−e−γ2 t
def
=
ϕ(λ1 , λ2 ; t ) , ϕ(γ1 , γ2 ; t )
so, sgn
ψ ′ (t ) = ϕt′ (λ1 , λ2 ; t )ϕ(γ1 , γ2 ; t ) − ϕ(λ1 , λ2 ; t )ϕt′ (γ1 , γ2 ; t ) ′ ϕ ′ (γ1 , γ2 ; t ) sgn ϕt (λ1 , λ2 ; t ) = − t , ϕ(λ1 , λ2 ; t ) ϕ(γ1 , γ2 ; t ) where
ϕt′ (λ1 , λ2 ; t ) ϕ(λ1 , λ2 ; t ) =−
λ1
e−λ1 t (1
λ21 e−λ1 t (1 − e−λ2 t )2 + λ22 e−λ2 t (1 − e−λ1 t )2 , − e−λ1 t )(1 − e−λ2 t )2 + λ2 e−λ2 t (1 − e−λ2 t )(1 − e−λ1 t )2
and similarly to ϕt′ (γ1 , γ2 ; t )/ϕ(γ1 , γ2 ; t ). Consider the function
Ψ (x1 , x2 )
=
x21 e−x1 (1 − e−x2 )2 + x22 e−x2 (1 − e−x1 )2 x1 e−x1 (1 − e−x1 )(1 − e−x2 )2 + x2 e−x2 (1 − e−x2 )(1 − e−x1 )2 0 < x1 ≤ x2 .
,
To show ψ ′ (t ) ≥ 0 for all (γ1 , γ2 ) in that triangle region, we just need to verify that the function Ψ (x1 , x2 ) is increasing along the direction v = (1, α) with 0 ≤ α ≤ 1/2. Denote the numerator part of Ψ (x1 , x2 ) as N, and the denominator part as D. We have,
∂Ψ (1, α) ∂ x1 ∂ x2 ∂N ∂D ∂N ∂D sgn = D−N , D−N (1, α). ∂ x1 ∂ x1 ∂ x2 ∂ x2
▽v Ψ =
∂Ψ
,
Some calculations lead,
∂N = (2x1 − x21 )e−x1 (1 − e−x2 )2 + 2x22 e−(x1 +x2 ) (1 − e−x1 ), ∂ x1 ∂D = g (x1 )e−x1 (1 − e−x2 )2 + 2x2 e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 ), ∂ x1
J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
where g (x) = 1 − x − e−x + 2xe−x . So,
We have,
∂N ∂D D−N = x21 d1 (x1 )e−2x1 (1 − e−x2 )4 ∂ x1 ∂ x1 + x1 x2 d2 (x1 )e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 )3 + x22 d3 (x1 )e−(x1 +x2 ) (1 − e−x1 )2 (1 − e−x2 )2 ,
η′ (x) = (1 − α) + α e−x − (1 + α x)e−x = (1 − α) − (1 − α)e−x − α xe−x = (1 − α){1 − (1 + x)e−x } + (1 − 2α)xe−x .
where d1 (x) = 1 − e−x − xe−x , d2 (x) = 2 − x − 2e−x − xe−x , and d3 (x) = −1 + x + e−x . It is easy to show that for x ≥ 0, d1 (x) ≥ 0, and d3 (x) ≥ 0. Notice d2 (x) = d1 (x) − d3 (x), then, we have,
∂N ∂D D−N ∂ x1 ∂ x1 = e−x1 (1 − e−x2 )2 x21 d1 (x1 )e−x1 (1 − e−x2 )2
A =
+ x1 x2 d2 (x1 )e−x2 (1 − e−x1 )(1 − e−x2 ) + x22 d3 (x1 )e−x2 (1 − e−x1 )2 = e−x1 (1 − e−x2 )2 d1 (x1 )(1 − e−x2 )x1 x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ) + d3 (x1 )(1 − e−x1 )x2 e−x2 x2 (1 − e−x1 ) − x1 (1 − e−x2 ) .
197
So, for α ≤ 1/2, η′ (x) ≥ 0. The conclusion η(x) ≥ 0 comes out with the condition η(0) = 0. Thus, ξ ′ (x) ≥ 0, and ξ (x) ≥ ξ (x1 ) > 0 for x ≥ x1 . Therefore, A + α B ≥ 0, which, in turn, means the function Ψ (x1 , x2 ) is increasing along the direction v = (1, α) for 0 ≤ α ≤ 1/2. For a point (γ1 , γ2 ) in the triangle region, pick a point (γ1′ , γ2′ ) ¯ λ) ¯ , such that, on the line segment connected points (λ1 , λ2 ) and (λ, the slope of points (γ1 , γ2 ) and (γ1′ , γ2′ ) is 1/2. Since (λ1 , λ2 ) ≻ (γ1′ , γ2′ ), Dykstra’s theorem ensures T (λ1 , λ2 ) ≥lr T (γ1′ , γ2′ ). Because the function Ψ (x1 , x2 ) is increasing along the direction v = (1, 21 ), we know
ϕt′ (γ1′ , γ2′ ; t ) ϕt′ (γ1 , γ2 ; t ) − ≥ 0, ϕ(γ1′ , γ2′ ; t ) ϕ(γ1 , γ2 ; t ) so, T (γ1′ , γ2′ ) ≥lr T (γ1 , γ2 ). Thus, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). This completes the proof of theorem.
By symmetry,
∂N ∂D D−N ∂ x2 ∂ x2 = e−x2 (1 − e−x1 )2 d1 (x2 )(1 − e−x1 )x2 x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ) + d3 (x2 )(1 − e−x2 )x1 e−x1 x1 (1 − e−x2 ) − x2 (1 − e−x1 ) .
B =
So,
▽v Ψ = A + α B = p(x1 , x2 ) e−x1 (1 − e−x2 )3 x1 d1 (x1 ) + α e−x2 (1 − e−x1 )3 x2 d1 (x2 ) + q(x1 , x2 ) (1 − e−x2 )x2 d3 (x1 ) − α(1 − e−x1 )x1 d3 (x2 ) , where p(x1 , x2 ) = x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ), and q(x1 , x2 ) = e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 ) × [x2 (1 − e−x1 ) − x1 (1 − e−x2 )].
4. Computer simulations We can easily confirm that the conclusion T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) will not hold for all (γ1 , γ2 ) ∈ Ωλ1 ,λ2 . To see the reason, consider the situation when γ1 = λ1 and γ2 = ∞. In this case, the likelihood ratio for the two systems is
ψ(t ) = 1 +
λ2 −λ2 t λ2 −(λ2 −λ1 )t e − 1+ e , λ1 λ1
this function is clearly not monotonous for all t > 0. In this section, we check where T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds when (γ1 , γ2 ) ∈ Ωλ1 ,λ2 − Θλ1 ,λ2 . Since theoretical investigation for such problem is not so tractable, we thus resort to computer simulations. Set (λ1 , λ2 ) = (1, 2). We checked the points in the triangle region enclosed by lines x = 1, y = 4, and y = x with grid length 0.1. For a point (γ1 , γ2 ), if T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ), we dot the point. If T (λ1 , λ2 )
By Lemma 3.2, x2 (1 − e−x1 ) − x1 (1 − e−x2 ) x2 x1
=
1 − e −x 2
−
1 − e −x 1
(1 − e−x1 )(1 − e−x2 ) ≥ 0,
thus, q(x1 , x2 ) ≥ 0. Denote r (x1 , x2 ) = (1 − e−x2 )x2 d3 (x1 ) − α(1 − e−x1 )x1 d3 (x2 ). Regard it as a function of x2 , then r (x1 , x2 ) becomes the function
ξ (x) = d3 (x1 )x(1 − e−x ) − α(1 − e−x1 )x1 d3 (x),
x ≥ x1 .
We have,
ξ (x1 ) = d3 (x1 )x1 (1 − e−x1 ) − α(1 − e−x1 )x1 d3 (x1 ) = (1 − α)d3 (x1 )x1 (1 − e−x1 ) > 0, and ξ ′ (x) = {d3 (x1 ) − α x1 (1 − e−x1 )}(1 − e−x ) + d3 (x1 )xe−x . We want to show, for 0 ≤ α ≤ 1/2, d3 (x1 ) − α x1 (1 − e−x1 ) ≥ 0. To this end, consider the function
η(x) = d3 (x) − α x(1 − e−x ) = (1 − α)x − 1 + (1 + α x)e−x .
From the picture, we can see that T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds almost for all points in Ωλ1 ,λ2 , excepting some points on the line
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J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
γ1 = λ1 . As we have just explained, when γ1 = λ1 , there may have no likelihood ratio order between T (λ1 , λ2 ) and T (γ1 , γ2 ) when γ2 is sufficiently large. As the picture shows, T (1, 2) ≥lr T (1, γ2 ), when 2 ≤ γ2 ≤ 2.8. For γ2 > 2.8, there is no likelihood ratio order between T (1, 2) and T (1, γ2 ). A clear boundary between the regions of having or not having likelihood ratio order in general, however, is not so easy to describe. 5. Discussions Stochastic comparison of systems is an important issue in reliability analysis. However, theoretical study for the issue can be elusive even for the simple parallel systems with exponential components. Misra and Misra [9] showed that when λ1 ≤ γ1 , and λ1 + λ2 ≤ γ1 + γ2 , or, the point (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). This result can be interpreted as: in reversed hazard ordering, the performance of a parallel system depends on its best component and its average quality of the components. In this paper, we confirm, by computer simulations, that the above conclusion is almost true in terms of likelihood ratio ordering. By theoretical investigation, we reveal a subregion of Ωλ1 ,λ2 , in which T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds. However, the likelihood ratio ordering in whole region of Ωλ1 ,λ2 has not been well depicted. This merits further investigation. Acknowledgments The authors are very grateful to the referees for their valuable suggestions and comments.
References [1] P.J. Boland, E. EL-Neweihi, F. Proschan, Applications of hazard rate ordering in reliability and order statistics, J. Appl. Probab. 31 (1994) 180–192. [2] G. Da, W. Ding, X. Li, On hazard rate ordering of parallel systems with two independent components, J. Statist. Plann. Inference 140 (2010) 2148–2154. [3] R. Dykstra, S.C. Kochar, J. Rojo, Stochastic comparisons of parallel systems of heterogeneous exponential components, J. Statist. Plann. Inference 65 (1997) 203–211. [4] S. Joo, J. Mi, Some properties of hazard rate functions of systems with two components, J. Statist. Plann. Inference 140 (2010) 444–453. [5] B. Khaledi, S.C. Kochar, Some new results on stochastic comparisons of parallel systems, J. Appl. Probab. 37 (2000) 283–291. [6] B. Khaledi, S.C. Kochar, Stochastic orderings among order statistics and sample spacings, in: J.C. Misra (Ed.), Uncertainty and Optimality–Probability, Statistics and Operations Research, World Scientific Publishers, Singapore, 2000, pp. 167–203. [7] S.C. Kochar, J. Rojo, Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions, J. Multivariate Anal. 59 (1996) 272–281. [8] S.C. Kochar, M. Xu, Stochastic comparisons of parallel systems when components have proportional hazard rates, Probab. Engrg. Inform. Sci. 21 (2007) 597–609. [9] N. Misra, A.K. Misra, On comparison of reversed hazard rates of two parallel systems comprising of independent gamma components, Statist. Probab. Lett. 83 (2013) 1567–1570. [10] A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, John Wiley & Sons, New York, 2002. [11] P. Pledger, F. Proschan, Comparisons of order statistics and of spacings from heterogeneous distributions, in: J.S. Rustagi (Ed.), Optimizing Methods in Statistics, Academic Press, New York, 1971, pp. 89–113. [12] M. Shaked, J.G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. [13] R. Yan, G. Da, P. Zhao, Further results for parallel systems with two heterogeneous exponential components, Statistics 47 (2013) 1128–1140. [14] P. Zhao, N. Balakrishnan, Some characterization results for parallel systems with two heterogeneous exponential components, Statistics 45 (2011) 593–604.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On likelihood ratio ordering of parallel system with two exponential components Jiantian Wang a,∗ , Henry Laniado b a
Department of Mathematics, Kean University, Union, NJ, 07083, USA
b
Departamento de Estadística, Universidad Carlos III de Madrid, 28911, Leganés, Spain
article
info
Article history: Received 8 August 2014 Received in revised form 31 January 2015 Accepted 31 January 2015 Available online 7 February 2015
abstract This paper considers stochastic comparison for parallel systems with two exponential components. For a given such system, we identify a region, such that, if the hazard rate pair of another parallel system lies in that region, then there exists likelihood ratio ordering between the two systems. The new results presented in this paper extend most existing ones in the literature. Published by Elsevier B.V.
Keywords: Parallel system Allocation policy Stochastic order Likelihood rate order
1. Introduction Parallel systems are commonly used to improve the reliability of a device. For economical reason, parallel systems with only two components are quite popular. To compare the lifetimes of two such systems is fundamental in engineering reliability theory. For systems made up of components with general distributed lifetimes, the stochastic comparison for those systems is quite elusive, since the distribution theory becomes quite complicated. For this reason, in this paper, we confine ourselves to the parallel systems with components whose lifetimes follow exponential distributions. Pledger and Proschan [11] were the first ones to compare stochastically for such systems. Since then, many researchers have worked in this field, including Kochar and Rojo [7], Dykstra et al. [3], Khaledi and Kochar [5,6], Kochar and Xu [8], Joo and Mi [4], Da et al. [2], and Zhao and Balakrishnan [14]. Let T (λ1 , . . . , λn ) be the lifetime of a parallel system with n exponential components whose hazard rates are λ1 , . . . , λn , respectively. For technical reason, we just focus on the case of n = 2 in the present paper. For the systems T (λ1 , λ2 ) and T (γ1 , γ2 ), by symmetry, we assume λ1 ≤ λ2 and γ1 ≤ γ2 . Denote by ≻ as majorization order, ≥st , ≥hr , ≥rh , and ≥lr as the usual stochastic
∗
Corresponding author. E-mail address: [email protected] (J. Wang).
http://dx.doi.org/10.1016/j.orl.2015.01.012 0167-6377/Published by Elsevier B.V.
order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order, respectively. The formal definitions of these orders will be given in the next section. So far, several stochastic comparison results for the lifetimes T (λ1 , λ2 ) and T (γ1 , γ2 ) have been established. For instance, Pledger and Proschan [11] showed that if (λ1 , λ2 ) ≻ (γ1 , γ2 ), T (λ1 , λ2 ) ≥st T (γ1 , γ2 ). Boland et al. [1] strengthened it as T (λ1 , λ2 ) ≥hr T (γ1 , γ2 ) and T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). Dykstra et al. [3] further enhanced it as T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). Joo and Mi [4] revealed that when λ1 ≤ γ1 ≤ γ2 ≤ λ2 and λ1 + λ2 ≤ γ1 + γ2 , T (λ1 , λ2 ) ≥hr T (γ1 , γ2 ). Zhao and Balakrishnan [14] improved this result from hazard rate order to likelihood ratio order. Yan et al. [13] showed, when λ1 ≤ γ1 ≤ λ2 ≤ γ2 and λ1 + γ2 ≤ γ1 + λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). Misra and Misra [9] proved that, when (λ1 , λ2 ) weakly majorizes (γ1 , γ2 ), then T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). For a given point (λ1 , λ2 ), denote Ωλ1 ,λ2 as the region such that, if (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , (γ1 , γ2 ) is weakly majorized by (λ1 , λ2 ). The result of Misra and Misra [9] thus can be stated as: when (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). A natural question now is: can we extend the result of Misra and Misra [9] to likelihood ratio order? Or, in what a subregion of Ωλ1 ,λ2 that T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds? This paper intends to answer this question. In this paper, we reveal a subregion of Ωλ1 ,λ2 , denote it as Θλ1 ,λ2 , such that, for any (γ1 , γ2 ) ∈ Θλ1 ,λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). The picture below ¯ = (λ1 + λ2 )/2. shows the regions Ωλ1 ,λ2 and Θλ1 ,λ2 , where λ
196
J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
Lemma 3.2. For x > 0, the function f (x) = 1−xe−x is increasing and convex. The following lemma is a classical result of Dykstra et al. [3]: Lemma 3.3 (Theorem 3.1, Dykstra et al. [3]). If (λ1 , λ2 ) ≻ (γ1 , γ2 ),
The region Ωλ1 ,λ2 is formed by the inequalities y ≥ x ≥ λ1 , and x + y ≥ λ1 + λ2 , and the region Θλ1 ,λ2 is formed by the ¯ , or, x + y ≥ λ1 + λ2 , and y ≤ λ2 + inequalities y ≥ x ≥ λ 1 ( x − λ ) . Notice (λ , λ 1 1 2 ) ≻ (γ1 , γ2 ) is equivalent to that the 2 point (γ1 , γ2 ) is on the line segment connecting the point (λ1 , λ2 ) ¯ λ) ¯ . Thus, Dykstra’s result can be stated as: for any point and (λ, (γ1 , γ2 ) on that line segment, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). The result of Zhao and Balakrishnan [14] can be stated as: for any point (γ1 , γ2 ) ¯ λ) ¯ , and in the triangle formed by the three points (λ1 , λ2 ), (λ, (λ2 , λ2 ), T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ), and the result of Yan et al. [13] can be stated as: for any point (γ1 , γ2 ) lies in the triangle with vertexes (λ1 , λ2 ), (λ2 , λ2 ), and (λ2 , 2λ2 −λ1 ), T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). As we can see, the result established in this paper extends the results of Dykstra et al. [3], Zhao and Balakrishnan [14], improves part of the result of Da et al. [2] from hazard ratio ordering to likelihood ratio ordering, and overlaps that of Yan et al. [13]. 2. Definitions and notations
Proof of Theorem 3.1. At first, by using Theorem 1. C. 33 in ¯ T (λ1 , λ2 ) Shaked and Shanthikumar [12], we have, when γ1 ≥ λ, ≥lr T (γ1 , γ2 ). Now we want to show T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) for all point (γ1 , γ2 ) in the triangle region formed by the lines x + y = λ1 + λ2 , y = x, and y = λ2 + 21 (x − λ1 ). By the result of Misra and Misra [9], T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ) on that region. From Theorem 1.C.4(b) of Shaked and Shanthikumar [12], it is enough to show that the ratio of the reversed hazard rate functions is increasing for t > 0. Denote the reversed hazard rate function of T (λ1 , λ2 ) as r(λ1 ,λ2 ) (t ). We have,
λ1 e−λ1 t (1 − e−λ2 t ) + λ2 e−λ2 t (1 − e−λ1 t ) (1 − e−λ1 t )(1 − e−λ2 t ) −λ1 t λ2 e−λ2 t λ1 e + . = 1 − e−λ1 t 1 − e−λ2 t
r(λ1 ,λ2 ) (t ) =
sgn
For our convenience, we denote A = B if the signs of A and B are the same. We have,
ψ(t ) =
To state the results, we introduce some notations and concepts first. Let X be a nonnegative continuous random variable with distribution function FX (t ), survival function F¯X (t ) = 1 − FX (t ), and density function fX (t ). The hazard function and the reversed hazard function of X are defined as λX = fX /F¯X and rX = fX /FX , respectively. For two nonnegative continuous random variables X and Y , X is said to be smaller than Y in the usual stochastic order (denoted by X ≤st Y ), if F¯X (t ) ≤ F¯Y (t ). X is said to be smaller than Y in hazard rate order (denoted by X ≤hr Y ), if λX (t ) ≥ λY (t ). X is said to be smaller than Y in reversed hazard rate order (denoted by X ≤rh Y ), if rX (t ) ≤ rY (t ). X is said to be smaller than Y in likelihood ratio order (denoted by X ≤lr Y ), if the ratio fY (t )/fX (t ) is increasing in t. It is well known that likelihood ratio order implies both hazard rate order and reversed hazard rate order, while these two orders imply usual stochastic order. Given two vectors a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ), let a(1) ≤ a(2) ≤ · · · ≤ a(n) and b(1) ≤ b(2) ≤ · · · ≤ b(n) be the increasing arrangements of the components of the two vectors, then the vector a is said to majorize the vector b (denoted by a ≻ b) n n k k if and only if, i=1 a(i) = i=1 b(i) , and i=1 a(i) ≤ i=1 b(i) , for k = 1, . . . , n − 1. If for k = 1, . . . , n, i=1 a(i) ≤ i=1 b(i) , then the vector a is said to weakly majorize the vector b (denoted by
k
k
w
a ≻ b). For some extensive and comprehensive discussions on the theory of these orders and their applications, one can see Müller and Stoyan [10], or, Shaked and Shanthikumar [12]. 3. Main result and proof We state the main result as the following theorem. Theorem 3.1. For any (γ1 , γ2 ) ∈ Θλ1 ,λ2 , T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). To prove the theorem, the following lemmas will be used. The proof of the first lemma is easy and thus is omitted.
then, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ).
r(λ1 ,λ2 ) (t ) r(γ1 ,γ2 ) (t )
λ1 e−λ1 t
=
1−e−λ1 t γ1 e−γ1 t 1−e−γ1 t
+ +
λ2 e−λ2 t 1−e−λ2 t γ2 e−γ2 t 1−e−γ2 t
def
=
ϕ(λ1 , λ2 ; t ) , ϕ(γ1 , γ2 ; t )
so, sgn
ψ ′ (t ) = ϕt′ (λ1 , λ2 ; t )ϕ(γ1 , γ2 ; t ) − ϕ(λ1 , λ2 ; t )ϕt′ (γ1 , γ2 ; t ) ′ ϕ ′ (γ1 , γ2 ; t ) sgn ϕt (λ1 , λ2 ; t ) = − t , ϕ(λ1 , λ2 ; t ) ϕ(γ1 , γ2 ; t ) where
ϕt′ (λ1 , λ2 ; t ) ϕ(λ1 , λ2 ; t ) =−
λ1
e−λ1 t (1
λ21 e−λ1 t (1 − e−λ2 t )2 + λ22 e−λ2 t (1 − e−λ1 t )2 , − e−λ1 t )(1 − e−λ2 t )2 + λ2 e−λ2 t (1 − e−λ2 t )(1 − e−λ1 t )2
and similarly to ϕt′ (γ1 , γ2 ; t )/ϕ(γ1 , γ2 ; t ). Consider the function
Ψ (x1 , x2 )
=
x21 e−x1 (1 − e−x2 )2 + x22 e−x2 (1 − e−x1 )2 x1 e−x1 (1 − e−x1 )(1 − e−x2 )2 + x2 e−x2 (1 − e−x2 )(1 − e−x1 )2 0 < x1 ≤ x2 .
,
To show ψ ′ (t ) ≥ 0 for all (γ1 , γ2 ) in that triangle region, we just need to verify that the function Ψ (x1 , x2 ) is increasing along the direction v = (1, α) with 0 ≤ α ≤ 1/2. Denote the numerator part of Ψ (x1 , x2 ) as N, and the denominator part as D. We have,
∂Ψ (1, α) ∂ x1 ∂ x2 ∂N ∂D ∂N ∂D sgn = D−N , D−N (1, α). ∂ x1 ∂ x1 ∂ x2 ∂ x2
▽v Ψ =
∂Ψ
,
Some calculations lead,
∂N = (2x1 − x21 )e−x1 (1 − e−x2 )2 + 2x22 e−(x1 +x2 ) (1 − e−x1 ), ∂ x1 ∂D = g (x1 )e−x1 (1 − e−x2 )2 + 2x2 e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 ), ∂ x1
J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
where g (x) = 1 − x − e−x + 2xe−x . So,
We have,
∂N ∂D D−N = x21 d1 (x1 )e−2x1 (1 − e−x2 )4 ∂ x1 ∂ x1 + x1 x2 d2 (x1 )e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 )3 + x22 d3 (x1 )e−(x1 +x2 ) (1 − e−x1 )2 (1 − e−x2 )2 ,
η′ (x) = (1 − α) + α e−x − (1 + α x)e−x = (1 − α) − (1 − α)e−x − α xe−x = (1 − α){1 − (1 + x)e−x } + (1 − 2α)xe−x .
where d1 (x) = 1 − e−x − xe−x , d2 (x) = 2 − x − 2e−x − xe−x , and d3 (x) = −1 + x + e−x . It is easy to show that for x ≥ 0, d1 (x) ≥ 0, and d3 (x) ≥ 0. Notice d2 (x) = d1 (x) − d3 (x), then, we have,
∂N ∂D D−N ∂ x1 ∂ x1 = e−x1 (1 − e−x2 )2 x21 d1 (x1 )e−x1 (1 − e−x2 )2
A =
+ x1 x2 d2 (x1 )e−x2 (1 − e−x1 )(1 − e−x2 ) + x22 d3 (x1 )e−x2 (1 − e−x1 )2 = e−x1 (1 − e−x2 )2 d1 (x1 )(1 − e−x2 )x1 x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ) + d3 (x1 )(1 − e−x1 )x2 e−x2 x2 (1 − e−x1 ) − x1 (1 − e−x2 ) .
197
So, for α ≤ 1/2, η′ (x) ≥ 0. The conclusion η(x) ≥ 0 comes out with the condition η(0) = 0. Thus, ξ ′ (x) ≥ 0, and ξ (x) ≥ ξ (x1 ) > 0 for x ≥ x1 . Therefore, A + α B ≥ 0, which, in turn, means the function Ψ (x1 , x2 ) is increasing along the direction v = (1, α) for 0 ≤ α ≤ 1/2. For a point (γ1 , γ2 ) in the triangle region, pick a point (γ1′ , γ2′ ) ¯ λ) ¯ , such that, on the line segment connected points (λ1 , λ2 ) and (λ, the slope of points (γ1 , γ2 ) and (γ1′ , γ2′ ) is 1/2. Since (λ1 , λ2 ) ≻ (γ1′ , γ2′ ), Dykstra’s theorem ensures T (λ1 , λ2 ) ≥lr T (γ1′ , γ2′ ). Because the function Ψ (x1 , x2 ) is increasing along the direction v = (1, 21 ), we know
ϕt′ (γ1′ , γ2′ ; t ) ϕt′ (γ1 , γ2 ; t ) − ≥ 0, ϕ(γ1′ , γ2′ ; t ) ϕ(γ1 , γ2 ; t ) so, T (γ1′ , γ2′ ) ≥lr T (γ1 , γ2 ). Thus, T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ). This completes the proof of theorem.
By symmetry,
∂N ∂D D−N ∂ x2 ∂ x2 = e−x2 (1 − e−x1 )2 d1 (x2 )(1 − e−x1 )x2 x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ) + d3 (x2 )(1 − e−x2 )x1 e−x1 x1 (1 − e−x2 ) − x2 (1 − e−x1 ) .
B =
So,
▽v Ψ = A + α B = p(x1 , x2 ) e−x1 (1 − e−x2 )3 x1 d1 (x1 ) + α e−x2 (1 − e−x1 )3 x2 d1 (x2 ) + q(x1 , x2 ) (1 − e−x2 )x2 d3 (x1 ) − α(1 − e−x1 )x1 d3 (x2 ) , where p(x1 , x2 ) = x1 e−x1 (1 − e−x2 ) + x2 e−x2 (1 − e−x1 ), and q(x1 , x2 ) = e−(x1 +x2 ) (1 − e−x1 )(1 − e−x2 ) × [x2 (1 − e−x1 ) − x1 (1 − e−x2 )].
4. Computer simulations We can easily confirm that the conclusion T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) will not hold for all (γ1 , γ2 ) ∈ Ωλ1 ,λ2 . To see the reason, consider the situation when γ1 = λ1 and γ2 = ∞. In this case, the likelihood ratio for the two systems is
ψ(t ) = 1 +
λ2 −λ2 t λ2 −(λ2 −λ1 )t e − 1+ e , λ1 λ1
this function is clearly not monotonous for all t > 0. In this section, we check where T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds when (γ1 , γ2 ) ∈ Ωλ1 ,λ2 − Θλ1 ,λ2 . Since theoretical investigation for such problem is not so tractable, we thus resort to computer simulations. Set (λ1 , λ2 ) = (1, 2). We checked the points in the triangle region enclosed by lines x = 1, y = 4, and y = x with grid length 0.1. For a point (γ1 , γ2 ), if T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ), we dot the point. If T (λ1 , λ2 )
By Lemma 3.2, x2 (1 − e−x1 ) − x1 (1 − e−x2 ) x2 x1
=
1 − e −x 2
−
1 − e −x 1
(1 − e−x1 )(1 − e−x2 ) ≥ 0,
thus, q(x1 , x2 ) ≥ 0. Denote r (x1 , x2 ) = (1 − e−x2 )x2 d3 (x1 ) − α(1 − e−x1 )x1 d3 (x2 ). Regard it as a function of x2 , then r (x1 , x2 ) becomes the function
ξ (x) = d3 (x1 )x(1 − e−x ) − α(1 − e−x1 )x1 d3 (x),
x ≥ x1 .
We have,
ξ (x1 ) = d3 (x1 )x1 (1 − e−x1 ) − α(1 − e−x1 )x1 d3 (x1 ) = (1 − α)d3 (x1 )x1 (1 − e−x1 ) > 0, and ξ ′ (x) = {d3 (x1 ) − α x1 (1 − e−x1 )}(1 − e−x ) + d3 (x1 )xe−x . We want to show, for 0 ≤ α ≤ 1/2, d3 (x1 ) − α x1 (1 − e−x1 ) ≥ 0. To this end, consider the function
η(x) = d3 (x) − α x(1 − e−x ) = (1 − α)x − 1 + (1 + α x)e−x .
From the picture, we can see that T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds almost for all points in Ωλ1 ,λ2 , excepting some points on the line
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J. Wang, H. Laniado / Operations Research Letters 43 (2015) 195–198
γ1 = λ1 . As we have just explained, when γ1 = λ1 , there may have no likelihood ratio order between T (λ1 , λ2 ) and T (γ1 , γ2 ) when γ2 is sufficiently large. As the picture shows, T (1, 2) ≥lr T (1, γ2 ), when 2 ≤ γ2 ≤ 2.8. For γ2 > 2.8, there is no likelihood ratio order between T (1, 2) and T (1, γ2 ). A clear boundary between the regions of having or not having likelihood ratio order in general, however, is not so easy to describe. 5. Discussions Stochastic comparison of systems is an important issue in reliability analysis. However, theoretical study for the issue can be elusive even for the simple parallel systems with exponential components. Misra and Misra [9] showed that when λ1 ≤ γ1 , and λ1 + λ2 ≤ γ1 + γ2 , or, the point (γ1 , γ2 ) ∈ Ωλ1 ,λ2 , T (λ1 , λ2 ) ≥rh T (γ1 , γ2 ). This result can be interpreted as: in reversed hazard ordering, the performance of a parallel system depends on its best component and its average quality of the components. In this paper, we confirm, by computer simulations, that the above conclusion is almost true in terms of likelihood ratio ordering. By theoretical investigation, we reveal a subregion of Ωλ1 ,λ2 , in which T (λ1 , λ2 ) ≥lr T (γ1 , γ2 ) holds. However, the likelihood ratio ordering in whole region of Ωλ1 ,λ2 has not been well depicted. This merits further investigation. Acknowledgments The authors are very grateful to the referees for their valuable suggestions and comments.
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