On Extreme Residual Lives after the Failure of the System

Abstract

The concept of residual lifetime has attracted considerable research interest in reliability theory. It is useful for evaluating the dynamic behavior of a system. In this paper, we study the extreme residual lives, that is, the minimum and maximum residual lives of the remaining components after the failure of the system. The system is assumed to have an arbitrary structure. We obtain signature-based distributional and ordering results for the extreme residual lives.

1. Introduction

A considerable attention has been given to the concept of residual lifetime in reliability and survival analysis. There are various works in the literature not only on the residual lifetime of systems but also on their components. See, for example Asadi and Bayramoglu [1], Navarro et al. [2], Bairamov and Arnold [3], Asadi and Goliforushani [4], Sadegh [5], and Eryilmaz [6].

In this paper, we study the extreme residual lifetimes of the remaining components after the complete failure of the system. These extremes might be useful to determine if the remaining components should be used after repair in the renewed system. For example, a decision maker may suggest the reuse of the remaining components if the minimum residual live is above a given threshold, or the refuse of them if the maximum residual live is below a given threshold. We aim to study the marginal and joint distributions of extreme residual lives for an arbitrary coherent structure. Our results are based on system signature, so we can easily evaluate these random variables for a given structure with a known signature.

Consider a coherent system with the lifetime 𝑇=𝜙(𝑋₁,…,𝑋_𝑛), where 𝑋₁,…,𝑋_𝑛 are independent and identically distributed (iid) random variables representing the lifetimes of components, and have common absolutely continuous distribution 𝐹 and density 𝑓. It is well known that the survival function of 𝑇 can be written as 𝑃{𝑇>𝑡}=^𝑛_𝑖=1𝑝_𝑖𝑃𝑋_𝑖∶𝑛>𝑡, (1.1) where 𝑋_𝑖∶𝑛 is the 𝑖th smallest lifetime among 𝑋₁,…,𝑋_𝑛 and 𝑝_𝑖=𝑃{𝑇=𝑋_𝑖∶𝑛},𝑖=1,…,𝑛. In words, 𝑝_𝑖 is the proportion of permutations among the 𝑛! equally likely permutations of 𝑋₁,…,𝑋_𝑛 that result in a minimal cut set failure when 𝑖 components break down. More explicitly, 𝑝_𝑖=#oforderingsforwhichthe𝑖thfailurecausessystemfailure,𝑛! (1.2)𝑖=1,…,𝑛. The vector 𝐩=(𝑝₁,…,𝑝_𝑛) defines a discrete probability distribution and is called system signature. The computation of 𝐩 is a well-defined combinatorial problem. The 𝑖th element of 𝐩 can be computed from 𝑝_𝑖=𝑎_{𝑛−𝑖+1}(𝑛)−𝑎_𝑛−𝑖(𝑛), (1.3) where 𝑎_𝑖(𝑛)=𝑟_𝑖(𝑛)(^𝑛_𝑖), (1.4)𝑖=1,…,𝑛, and 𝑟_𝑖(𝑛) is the number of path sets of the structure including 𝑖 working components (see, e.g., Boland [7]).

The concept of system signature is a useful tool for the evaluation of reliability characteristics and ordering properties of coherent systems. An extensive review of the signature of coherent system and its applications can be found in Samaniego [8].

Let 𝑁 be a random variable which represents the number of surviving components at the time when the system with lifetime 𝑇 fails. It is known that 𝑃{𝑁=𝑛−𝑖}=𝑃𝑇=𝑋_𝑖∶𝑛=𝑝_𝑖, (1.5) for 𝑖=1,…,𝑛−1 (Eryilmaz [9]). Let 𝑋^′_𝑖 denote the residual lifetime of a surviving component after the failure of the system, that is 𝑋=^′_𝑖^st𝑋_𝑖−𝑇∣𝑋_𝑖>𝑇, (1.6) for 𝑖=1,…,𝑁.

Bairamov and Arnold [3] studied the joint distribution of the residual lifetimes of the remaining (surviving) components after the 𝑘th failure in the system, that is when the system has 𝑘-out-of-𝑛:F structure. In this case 𝑃{𝑇=𝑋_𝑘∶𝑛}=1, and hence the remaining number of components is 𝑛−𝑘, that is, 𝑃{𝑁=𝑛−𝑘}=1. They have shown that the joint survival function of the residual life lengths is 𝑃𝑋^′₁>𝑥₁,…,𝑋^′_𝑛−𝑘>𝑥_𝑛−𝑘=^∞₀^𝑛−𝑘_𝑗=1𝐹𝑥_𝑗+𝑡𝐹(𝑡)𝑑𝐹_𝑘∶𝑛(𝑡), (1.7) where 𝑑𝐹_𝑘∶𝑛(𝑡)=𝑛!()!()!𝐹𝑘−1𝑛−𝑘^𝑘−1(𝑡)(1−𝐹(𝑡))^𝑛−𝑘𝑑𝐹(𝑡). (1.8)

In the present paper, we study the residual lifetimes of the remaining components, in particular, minimum and maximum residual lives for an arbitrary coherent structure. The paper is organized as follows. In Section 2, we obtain mixture representations for the marginal and joint distributions of minimum and maximum residual lives. Section 3 contains stochastic ordering results on the minimum and maximum residual lives of two systems having different structures. Finally in Section 4, we present illustrative examples.

2. Extreme Residual Lives

Obviously, for an arbitrary coherent structure (different from 𝑘-out-of-𝑛:F structure), the number of surviving components is a random variable. Thus, in general, it should be taken into account that we have random number of surviving components on hand after the failure of the system. This case is illustrated in the following example.

Example 2.1. Consider the system which functions if and only if at least two consecutive of 𝑛=6 components function. That is, the system has consecutive 2-out-of-6:G structure. The lifetime of this system is given by the following: 𝑇=maxmin𝑋₁,𝑋₂,min𝑋₂,𝑋₃,min𝑋₃,𝑋₄,min𝑋₄,𝑋₅,min𝑋₅,𝑋₆. (2.1) For a consecutive 𝑘-out-of-𝑛:G system, it is known that 𝑟_𝑖(𝑛)=𝑛𝑖−^{min([𝑖/𝑘],𝑛−𝑖+1)}_𝑗=0(−1)^𝑗𝑗𝑛−𝑖+1𝑛−𝑗𝑘𝑛−𝑖, (2.2) where [𝑥] denotes the integer part of 𝑥 (see, e.g., Salehi et al. [10]). Using (2.2) for 𝑛=6 and 𝑘=2 in (1.3), the signature of consecutive 2-out-of-6:G system is found to be the following: 3𝑝=0,0,,715,515.15,0 (2.3) Therefore, at the end of the lifetime of the system we may have 1, 2, or 3 surviving components with respective probabilities as follows: 𝑃{}=5𝑁=1{}=715,𝑃𝑁=2{}=315,𝑃𝑁=3,15 (2.4) and hence the expected number of surviving components at the time when the system fails is 𝐸(𝑁)≅2.

In Table 1, we compute 𝐸(𝑁) for consecutive 2-out-of-𝑛:G system whose lifetime is defined by the following:𝑇=max_{1≤𝑖≤𝑛−1}min𝑋_𝑖,𝑋_𝑖+1. (2.5) In view of Table 1, we observe that 𝐸(𝑁) depends on 𝑛, and there is no limiting point for 𝐸(𝑁) when 𝑛 tends to infinity.

Table 1 

𝐸(𝑁) for consecutive 2-out-of-𝑛:G systems.

The minimum and maximum residual lifetimes after the failure of the system are defined respectively as follows: 𝑋^′_1∶𝑁=min𝑋^′₁,…,𝑋^′_𝑁,𝑋^′_𝑁∶𝑁=max𝑋^′₁,…,𝑋^′_𝑁, (2.6) where 𝑋^′₁,…,𝑋^′_𝑁 represent the residual lifetimes of the remaining components. Obviously, the problem is ridiculous for a system satisfying 𝑃{𝑁=0}=𝑃{𝑇=𝑋_𝑛∶𝑛}=𝑝_𝑛>0. Thus we consider the systems satisfying 𝑃{𝑁>0}=1 or equivalently 𝑝_𝑛=0.

In the following Theorem, we obtain mixture representations for the distributions of the random variables 𝑋^′_1∶𝑁 and 𝑋^′_𝑁∶𝑁.

Theorem 2.2. Let 𝑇=𝜙(𝑋₁,…,𝑋_𝑛) denote the lifetime of a coherent system with iid components and signature 𝐩=(𝑝₁,…,𝑝_𝑛) satisfying 𝑝_𝑛=0. Then 𝑃𝑋^′_1∶𝑁≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖},𝑃𝑋≤𝑥_{′𝑁∶𝑁}≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑥, (2.7) where 𝑋^′_{1∶𝑛−𝑖} and 𝑋^′_{𝑛−𝑖∶𝑛−𝑖} are the minimum and maximum order statistics corresponding to exchangeable random variables 𝑋^′₁,…,𝑋^′_𝑛−𝑖.

Proof . For a coherent system with lifetime 𝑇=𝜙(𝑋₁,…,𝑋_𝑛), via the order statistic equivalent of 𝑇, one obtains 𝑃𝑋^′₁>𝑥₁,…,𝑋^′_𝑁>𝑥_𝑁=^𝑛−1_𝑖=1𝑃𝑋^′₁>𝑥₁,…,𝑋^′_𝑛−𝑖>𝑥_𝑛−𝑖∣𝑇=𝑋_𝑖∶𝑛𝑃𝑇=𝑋_𝑖∶𝑛. (2.8)
The conditional probability in the last equation is actually the probability given in (1.7) by replacing 𝑘 by 𝑖. Thus we have 𝑃𝑋^′₁>𝑥₁,…,𝑋^′_𝑁>𝑥_𝑁=^𝑛−1_𝑖=1𝑝_𝑖^∞₀^𝑛−𝑖_𝑗=1𝐹𝑥_𝑗+𝑡𝐹(𝑡)𝑑𝐹_𝑖∶𝑛(𝑡). (2.9)
Similarly, we can obtain 𝑃𝑋^′₁≤𝑥₁,…,𝑋^′_𝑁≤𝑥_𝑁=^𝑛−1_𝑖=1𝑝_𝑖^∞₀^𝑛−𝑖_𝑗=1𝐹𝑥_𝑗+𝑡−𝐹(𝑡)𝐹(𝑡)𝑑𝐹_𝑖∶𝑛(𝑡). (2.10)
The results of the theorem follow immediately because 𝑃𝑋^′_1∶𝑁≤𝑥=1−𝑃𝑋^′₁>𝑥,…,𝑋^′_𝑁>𝑥=1−^𝑛−1_𝑖=1𝑝_𝑖^∞₀𝐹(𝑥+𝑡)𝐹(𝑡)^𝑛−𝑖𝑑𝐹_𝑖∶𝑛(𝑡)=1−^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖},𝑃𝑋>𝑥_{′𝑁∶𝑁}≤𝑥=𝑃𝑋^′₁≤𝑥,…,𝑋^′_𝑁=≤𝑥_{𝑛−1𝑖=1}𝑝_𝑖^∞₀𝐹(𝑥+𝑡)−𝐹(𝑡)𝐹(𝑡)^𝑛−𝑖𝑑𝐹_𝑖∶𝑛(𝑡)=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑥. (2.11)

Under the assumption that the component lifetime distribution is exponential, the distributions of the random variables 𝑋^′_1∶𝑁 and 𝑋^′_𝑁∶𝑁 can be written as mixtures of the distributions of order statistics corresponding to 𝑋₁,…,𝑋_𝑛. This is due to the independence of residual lifetimes and the preservation of the original lifetime distribution of a component in the case of exponential distribution. That is, the residual lifetime distribution of a component is same as the original lifetime distribution of a component. The results are presented in the following Corollary.

Corollary 2.3. Let 𝑋₁,…,𝑋_𝑛 be iid lifetime random variables with common cdf 𝐹(𝑡)=1−exp(−𝜆𝑡),𝑡≥0. Then 𝑃𝑋^′_1∶𝑁≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋_{1∶𝑛−𝑖},𝑃𝑋≤𝑥_{′𝑁∶𝑁}≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋_{𝑛−𝑖∶𝑛−𝑖}≤𝑥. (2.12)

In the following, we present the joint distribution of the random variables 𝑋^′_1∶𝑁 and 𝑋^′_𝑁∶𝑁.

Theorem 2.4. Let 𝑇=𝜙(𝑋₁,…,𝑋_𝑛) denote the lifetime of a coherent system with iid components and signature 𝐩=(𝑝₁,…,𝑝_𝑛) satisfying 𝑝_𝑛=0. Then 𝑃𝑋^′_1∶𝑁≤𝑥,𝑋^′_𝑁∶𝑁≤𝑦=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}≤𝑥,𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑦, (2.13) for 𝑥<𝑦.

Proof. For 𝑥<𝑦, it is obvious that 𝑃𝑋^′_1∶𝑁≤𝑥,𝑋^′_𝑁∶𝑁≤𝑦=𝑃𝑋^′_𝑁∶𝑁≤𝑦−𝑃𝑋^′_1∶𝑁>𝑥,𝑋^′_𝑁∶𝑁≤𝑦. (2.14)
By conditioning on 𝑁, 𝑃𝑋^′_1∶𝑁>𝑥,𝑋^′_𝑁∶𝑁=≤𝑦_{𝑛−1𝑖=1}𝑃𝑥<𝑋^′₁≤𝑦,…,𝑥<𝑋^′_𝑛−𝑖≤𝑦∣𝑇=𝑋_𝑖∶𝑛𝑃𝑇=𝑋_𝑖∶𝑛. (2.15)
It is easy to see that 𝑃𝑥<𝑋^′₁≤𝑦,…,𝑥<𝑋^′_𝑛−𝑖≤𝑦∣𝑇=𝑋_𝑖∶𝑛=^∞₀𝐹(𝑦+𝑡)−𝐹(𝑥+𝑡)𝐹(𝑡)^𝑛−𝑖𝑑𝐹_𝑖∶𝑛(𝑡)=𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥,𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑦. (2.16)
Therefore, 𝑃𝑋^′_1∶𝑁≤𝑥,𝑋^′_𝑁∶𝑁=≤𝑦_{𝑛−1𝑖=1}𝑝_𝑖𝑃𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑦−^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥,𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑦, (2.17) which completes the proof.

Corollary 2.5. Let 𝑋₁,…,𝑋_𝑛 be iid lifetime random variables with common cdf 𝐹(𝑡)=1−exp(−𝜆𝑡),𝑡≥0. Then 𝑃𝑋^′_1∶𝑁≤𝑥,𝑋^′_𝑁∶𝑁≤𝑦=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋_{1∶𝑛−𝑖}≤𝑥,𝑋_{𝑛−𝑖∶𝑛−𝑖}≤𝑦, (2.18) for 𝑥<𝑦.

3. Stochastic Ordering Results

Let us consider two systems with different structures having lifetimes 𝑇₁=𝜙₁(𝑋₁,…,𝑋_𝑛), and 𝑇₂=𝜙₂(𝑋₁,…,𝑋_𝑛). The respective signatures of the systems are defined by the following: 𝑝_𝑖=𝑃𝑇₁=𝑋_𝑖∶𝑛=𝑃𝑁₁,𝑞=𝑛−𝑖_𝑖=𝑃𝑇₂=𝑋_𝑖∶𝑛=𝑃𝑁₂=𝑛−𝑖, (3.1) for 𝑖=1,…,𝑛−1.

The residual lifetimes of the remaining components in systems 𝜙₁ and 𝜙₂ are defined respectively, as follows 𝑋^′_𝑖=𝑋_𝑖−𝑇₁∣𝑋_𝑖>𝑇₁,𝑖=1,…,𝑁₁,𝑌^′_𝑖=𝑋_𝑖−𝑇₂∣𝑋_𝑖>𝑇₂,𝑖=1,…,𝑁₂. (3.2) Although 𝑋_𝑖s and 𝑌_𝑖s have different joint distributions, given {𝑇₁=𝑋_𝑖∶𝑛} and {𝑇₂=𝑋_𝑖∶𝑛} they have the same joint distributions. If 𝑋^′_1∶𝑁1 (𝑌^′_1∶𝑁2) and 𝑋^′_{𝑁1∶𝑁1} (𝑌^′_{𝑁2∶𝑁2}) denote, respectively, the minimum and maximum residual lives corresponding to the system 𝜙₁ (𝜙₂), then using Theorem 2.2 we have 𝑃𝑋^′_1∶𝑁1≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖},𝑃𝑌≤𝑥_′1∶𝑁2≤𝑥=^𝑛−1_𝑖=1𝑞_𝑖𝑃𝑌^′_{1∶𝑛−𝑖}≤𝑥, (3.3) where 𝑋^′_{1∶𝑛−𝑖} and 𝑌^′_{1∶𝑛−𝑖} are the minimum residual lives after the failure of the 𝑖th component, that is after time 𝑋_𝑖∶𝑛. Since the systems have the common components 𝑋^′𝑑_{1∶𝑛−𝑖}=𝑌^′_{1∶𝑛−𝑖}. Thus 𝑃{𝑌^′_1∶𝑁2≤𝑥} can be equivalently written as follows 𝑃𝑌^′_1∶𝑁2≤𝑥=^𝑛−1_𝑖=1𝑞_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}≤𝑥. (3.4) That is, the distributions of 𝑋^′_1∶𝑁1 and 𝑌^′_1∶𝑁2 differ from each other through systems' structures and this is taken into account by the coefficients of 𝑃{𝑋^′_{1∶𝑛−𝑖}≤𝑥}.

For two discrete distributions 𝐩=(𝑝₁,…,𝑝_𝑛) and 𝐪=(𝑞₁,…,𝑞_𝑛), let ≤_st,≤_hr, and ≤_rh represent, respectively, the usual stochastic order, hazard rate order, and reversed hazard rate order. Then(a)𝐩≤_st𝐪 if ∑^𝑛_𝑗=𝑖𝑝_𝑗≤∑^𝑛_𝑗=𝑖𝑞_𝑗 for all 𝑖=1,2,…,𝑛.(b)𝐩≤_hr𝐪 if ∑^𝑛_𝑗=𝑖𝑝_𝑗/∑^𝑛_𝑗=𝑖𝑞_𝑗 is decreasing in 𝑖.(c)𝐩≤_rh𝐪 if ∑^𝑖_𝑗=1𝑝_𝑗/∑^𝑖_𝑗=1𝑞_𝑗 is decreasing in 𝑖.

Theorem 3.1. If 𝐩≤_st𝐪, then 𝑋^′_1∶𝑁1≤_st𝑌^′_1∶𝑁2 and 𝑋^′_{𝑁1∶𝑁1}≥_st𝑌^′_{𝑁2∶𝑁2}.

Proof . Because 𝑋^′_{1∶𝑛−𝑖}≤_st𝑋^′_{1∶𝑛−𝑖−1}, the function 𝑃{𝑋^′_{1∶𝑛−𝑖}>𝑥} is nondecreasing in 𝑖 for all 𝑥. Thus if 𝐩≤_st𝐪, then 𝑃𝑋^′_1∶𝑁1>𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}≤>𝑥_{𝑛−1𝑖=1}𝑞_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥=𝑃𝑌^′_1∶𝑁2>𝑥, (3.5) which implies 𝑋^′_1∶𝑁1≤_st𝑌^′_1∶𝑁2. On the other hand, 𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≥_st𝑋^′_{𝑛−𝑖−1∶𝑛−𝑖−1} and hence 𝑃{𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑥} is nondecreasing in 𝑖 for all 𝑥. Thus if 𝐩≤_st𝐪, then 𝑃𝑋^′_{𝑁1∶𝑁1}≤𝑥=^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤≤𝑥_{𝑛−1𝑖=1}𝑞_𝑖𝑃𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≤𝑥=𝑃𝑌^′_{𝑁2∶𝑁2}≤𝑥, (3.6) which implies 𝑋^′_{𝑁1∶𝑁1}≥_st𝑌^′_{𝑁2∶𝑁2}.

Lemma 3.2 (see [11]). Let 𝛼 and 𝛽 be two real valued functions such that 𝛽 is nonnegative and 𝛼/𝛽 and 𝛽 are nondecreasing. If 𝑋_𝑖 has distribution 𝐹_𝑖,𝑖=1,2 and 𝑋₁≤_hr𝑋₂, then ∫^∞_−∞𝛼(𝑥)𝑑𝐹₁(𝑥)∫^∞_−∞𝛽(𝑥)𝑑𝐹₁(𝑥)≤∫^∞_−∞𝛼(𝑥)𝑑𝐹₂(𝑥)∫^∞_−∞𝛽(𝑥)𝑑𝐹₂(𝑥). (3.7)

Lemma 3.3 (see [12]). Let 𝛼 and 𝛽 be two real valued functions such that 𝛼 is nonnegative and 𝛽/𝛼 and 𝛼 are nonincreasing. If 𝑋_𝑖 has distribution 𝐹_𝑖,𝑖=1,2 and 𝑋₁≤_rh𝑋₂, then ∫^∞_−∞𝛽(𝑥)𝑑𝐹₁(𝑥)∫^∞_−∞𝛼(𝑥)𝑑𝐹₁(𝑥)≥∫^∞_−∞𝛽(𝑥)𝑑𝐹₂(𝑥)∫^∞_−∞𝛼(𝑥)𝑑𝐹₂(𝑥). (3.8)

Theorem 3.4. (a)If 𝐩≤_hr𝐪, and 𝑋^′_{1∶𝑛−𝑖}≤_hr𝑋^′_{1∶𝑛−𝑖−1} for 𝑖=1,…,𝑛−2, then 𝑋^′_1∶𝑁1≤_hr𝑌^′_1∶𝑁2.(b)If 𝐩≤_rh𝐪, and 𝑋^′_{𝑛−𝑖∶𝑛−𝑖}≥_hr𝑋^′_{𝑛−𝑖−1∶𝑛−𝑖−1} for 𝑖=1,…,𝑛−2, then 𝑋^′_{𝑁1∶𝑁1}≥_hr𝑌^′_{𝑁2∶𝑁2}.

Proof. Because 𝑋^′_{1∶𝑛−𝑖}≤_ℎ𝑟𝑋^′_{1∶𝑛−𝑖−1}, 𝑃{𝑋^′_{1∶𝑛−𝑖−1}>𝑥}/𝑃{𝑋^′_{1∶𝑛−𝑖}>𝑥} is nondecreasing in 𝑥. That is, for 𝑥₁≤𝑥₂, 𝑃𝑋^′_{1∶𝑛−𝑖−1}>𝑥₂𝑃𝑋^′_{1∶𝑛−𝑖−1}>𝑥₁≥𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₂𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₁. (3.9) which implies that 𝑃{𝑋^′_{1∶𝑛−𝑖}>𝑥₂}/𝑃{𝑋^′_{1∶𝑛−𝑖}>𝑥₁} is nondecreasing in 𝑖. Applying Lemma 3.2, and using 𝐩≤_hr𝐪 we obtain ∑^𝑛−1_𝑖=1𝑞_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₂∑^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₂≥∑^𝑛−1_𝑖=1𝑞_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₁∑^𝑛−1_𝑖=1𝑝_𝑖𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥₁. (3.10) for 𝑥₁≤𝑥₂, which implies that 𝑋^′_1∶𝑁1≤_hr𝑌^′_1∶𝑁2. The proof of part (b) is based on similar arguments and the usage of Lemma 3.3.

As stated before, if the original lifetime distribution is exponential, then the residual lifetimes are iid. For a sequence of iid random variables 𝑋₁,…,𝑋_𝑛 we have 𝑋_1∶𝑚+1≤_hr𝑋_1∶𝑚, and 𝑋_𝑚∶𝑚≤_hr𝑋_{𝑚+1∶𝑚+1},𝑚=1,2,.... Thus, the problem of comparing extreme residual lives reduces to the comparison of systems' signatures for the exponential lifetime distribution and we obtain the following Corollary.

Corollary 3.5. Let 𝑋₁,…,𝑋_𝑛 be iid lifetime random variables with common cdf 𝐹(𝑡)=1−exp(−𝜆𝑡),𝑡≥0. Then(a)If 𝐩≤_hr𝐪, then 𝑋^′_1∶𝑁1≤_hr𝑌^′_1∶𝑁2.(b)If 𝐩≤_rh𝐪, then 𝑋^′_{𝑁1∶𝑁1}≥_hr𝑌^′_{𝑁2∶𝑁2}.

4. Illustrative Examples

Consider the systems with structure functions 𝜙₁𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅=maxmin𝑥₁,𝑥₂,𝑥₃,min𝑥₁,𝑥₄,𝑥₅,𝜙₂𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅=maxmin𝑥₁,𝑥₂,𝑥₃,min𝑥₁,𝑥₂,𝑥₄,min𝑥₁,𝑥₃,𝑥₅. (4.1) The respective signatures of these systems are 𝐩=(1/5,3/5,1/5,0,0), and 𝐪=(1/5,1/2,3/10,0,0) (see Navarro and Rubio [13]). Thus we have 𝑃𝑁₁=1=25,𝑃𝑁₁=3=35,𝑃𝑁₁=1=45,𝑃𝑁₂=3=2𝑁10,𝑃₂=1=32,𝑃𝑁₂=1=45. (4.2)

Because 𝐩≤_st𝐪, from Theorem 3.1, we have 𝑋^′_1∶𝑁1≤_st𝑌^′_1∶𝑁2 and 𝑋^′_{𝑁1∶𝑁1}≥_st𝑌^′_{𝑁2∶𝑁2}. Suppose that the common lifetime distribution of components is Lomax with cdf 𝐹(𝑡)=1−(1+𝑡)^−𝛼,𝑡>0,𝛼>0. Note that for 𝛼≤1, the expected value of this distribution is infinite and hence we consider the case 𝛼>1. In Table 2, we compute the expected values of the extreme residual lives for the two systems for different values of the parameter 𝛼.

Table 2 

Expected values of the extreme residual lives.

For 𝛼=2, from the Figure 1, we observe that 𝑟_𝑛()=𝑃𝑋𝑖,𝑥^{′1∶𝑛−𝑖−1}>𝑥𝑃𝑋^′_{1∶𝑛−𝑖}>𝑥 (4.3) is nondecreasing in 𝑥 for 𝑖=1,2,3. That is, 𝑋^′_1∶4≤_hr𝑋^′_1∶3,𝑋^′_1∶3≤_hr𝑋^′_1∶2,𝑋^′_1∶2≤_hr𝑋^′_1∶1. Because 𝐩≤_hr𝐪, by Theorem 3.4 we conclude that 𝑋^′_1∶𝑁1≤_hr𝑌^′_1∶𝑁2.

Figure 1 

Plot of the ratio 𝑟𝑛(𝑖,𝑥) for 𝑖 = 1, 2, 3 from the above to below.

Acknowledgment

The authors wish to thank three anonymous referees for their useful comments and suggestions which improved the presentation of the paper.