#### 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 , where are independent and identically distributed (iid) random variables representing the lifetimes of components, and have common absolutely continuous distribution and density . It is well known that the survival function of can be written as where is the th smallest lifetime among and . In words, is the proportion of permutations among the equally likely permutations of that result in a minimal cut set failure when components break down. More explicitly, . The vector defines a discrete probability distribution and is called system signature. The computation of is a well-defined combinatorial problem. The th element of can be computed from where , and is the number of path sets of the structure including 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 be a random variable which represents the number of surviving components at the time when the system with lifetime fails. It is known that for (Eryilmaz [9]). Let denote the residual lifetime of a surviving component after the failure of the system, that is for .

Bairamov and Arnold [3] studied the joint distribution of the residual lifetimes of the remaining (surviving) components after the th failure in the system, that is when the system has -out-of-:F structure. In this case , and hence the remaining number of components is , that is, . They have shown that the joint survival function of the residual life lengths is where

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 -out-of-: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 components function. That is, the system has consecutive -out-of-:G structure. The lifetime of this system is given by the following:
For a consecutive -out-of-:G system, it is known that
where denotes the integer part of (see, e.g., Salehi et al. [10]). Using (2.2) for and in (1.3), the signature of consecutive -out-of-:G system is found to be the following:
Therefore, at the end of the lifetime of the system we may have , , or surviving components with respective probabilities as follows:
and hence the expected number of surviving components at the time when the system fails is .

In Table 1, we compute for consecutive -out-of-:G system whose lifetime is defined by the following: In view of Table 1, we observe that depends on , and there is no limiting point for when tends to infinity.

The minimum and maximum residual lifetimes after the failure of the system are defined respectively as follows: where represent the residual lifetimes of the remaining components. Obviously, the problem is ridiculous for a system satisfying . Thus we consider the systems satisfying or equivalently .

In the following Theorem, we obtain mixture representations for the distributions of the random variables and .

Theorem 2.2. *Let denote the lifetime of a coherent system with iid components and signature satisfying . Then
**
where and are the minimum and maximum order statistics corresponding to exchangeable random variables .*

*Proof . *For a coherent system with lifetime , via the order statistic equivalent of , one obtains

The conditional probability in the last equation is actually the probability given in (1.7) by replacing by . Thus we have

Similarly, we can obtain

The results of the theorem follow immediately because

Under the assumption that the component lifetime distribution is exponential, the distributions of the random variables and can be written as mixtures of the distributions of order statistics corresponding to . 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 be iid lifetime random variables with common cdf . Then
*

In the following, we present the joint distribution of the random variables and .

Theorem 2.4. * Let denote the lifetime of a coherent system with iid components and signature satisfying . Then
**
for .*

*Proof. * For , it is obvious that

By conditioning on ,

It is easy to see that

Therefore,
which completes the proof.

Corollary 2.5. * Let be iid lifetime random variables with common cdf . Then
**
for .*

#### 3. Stochastic Ordering Results

Let us consider two systems with different structures having lifetimes , and . The respective signatures of the systems are defined by the following: for .

The residual lifetimes of the remaining components in systems and are defined respectively, as follows Although s and s have different joint distributions, given and they have the same joint distributions. If () and () denote, respectively, the minimum and maximum residual lives corresponding to the system (), then using Theorem 2.2 we have where and are the minimum residual lives after the failure of the th component, that is after time . Since the systems have the common components . Thus can be equivalently written as follows That is, the distributions of and differ from each other through systems' structures and this is taken into account by the coefficients of .

For two discrete distributions and , let , and represent, respectively, the usual stochastic order, hazard rate order, and reversed hazard rate order. Then(a) if for all .(b) if is decreasing in .(c) if is decreasing in .

Theorem 3.1. *If , then and .*

*Proof . *Because , the function is nondecreasing in for all . Thus if , then
which implies . On the other hand, and hence is nondecreasing in for all . Thus if , then
which implies .

Lemma 3.2 (see [11]). *Let and be two real valued functions such that is nonnegative and and are nondecreasing. If has distribution and , then
*

Lemma 3.3 (see [12]). *Let and be two real valued functions such that is nonnegative and and are nonincreasing. If has distribution and , then
*

Theorem 3.4. (a)*If , and for , then .*(b)*If , and for , then .*

*Proof. * Because , is nondecreasing in . That is, for ,
which implies that is nondecreasing in . Applying Lemma 3.2, and using we obtain
for , which implies that . 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 we have , and .. 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 be iid lifetime random variables with common cdf . Then*(a)*If , then .*(b)*If , then .*

#### 4. Illustrative Examples

Consider the systems with structure functions The respective signatures of these systems are , and (see Navarro and Rubio [13]). Thus we have

Because , from Theorem 3.1, we have and . Suppose that the common lifetime distribution of components is Lomax with cdf . Note that for , the expected value of this distribution is infinite and hence we consider the case . In Table 2, we compute the expected values of the extreme residual lives for the two systems for different values of the parameter .

For , from the Figure 1, we observe that is nondecreasing in for . That is, . Because , by Theorem 3.4 we conclude that .

#### Acknowledgment

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