On Extreme Residual Lives after the Failure of the System

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.


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 X 1 , . . ., X n , where X 1 , . . ., X n are independent and identically distributed iid random variables representing the lifetimes of 2 Mathematical Problems in Engineering 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 where X i:n is the ith smallest lifetime among X 1 , . . ., 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 X 1 , . . ., X n that result in a minimal cut set failure when i components break down.More explicitly, # of orderings for which the ith failure causes system failure n! , 1.2 i 1, . . ., n.The vector p p 1 , . . ., 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 where . ., 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 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 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 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.

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: 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: , 0 .Therefore, at the end of the lifetime of the system we may have 1, 2, or 3 surviving components with respective probabilities as follows:

2.3
and hence the expected number of surviving components at the time when the system fails is In Table 1, we compute E N for consecutive 2-out-of-n:G system whose lifetime is defined by the following: 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.The minimum and maximum residual lifetimes after the failure of the system are defined respectively as follows: 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 X 1 , . . ., X n denote the lifetime of a coherent system with iid components and signature p p 1 , . . ., p n satisfying p n 0. Then 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 X 1 , . . ., X n , via the order statistic equivalent of T , one obtains

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 2.9 Similarly, we can obtain The results of the theorem follow immediately because

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 X 1 , . . ., 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.

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 X 1 , . . ., X n denote the lifetime of a coherent system with iid components and signature p p 1 , . . ., p n satisfying p n 0. Then for x < y.
Proof.For x < y, it is obvious that

2.14
By conditioning on N,

2.15
It is easy to see that

2.16
Therefore, 2.17 which completes the proof.

Stochastic Ordering Results
Let us consider two systems with different structures having lifetimes T 1 φ 1 X 1 , . . ., X n , and T 2 φ 2 X 1 , . . ., X n .The respective signatures of the systems are defined by the following: The residual lifetimes of the remaining components in systems φ 1 and φ 2 are defined respectively, as follows

3.2
Although X i s and Y i s have different joint distributions, given {T 1 X i:n } and {T 2 X i:n } they have the same joint distributions.If X 1:N 1 Y 1:N 2 and X N 1 :N 1 Y N 2 :N 2 denote, respectively, the minimum and maximum residual lives corresponding to the system φ 1 φ 2 , then using Theorem 2.2 we have 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 1:n−i d Y 1:n−i .Thus P {Y 1:N 2 ≤ x} can be equivalently written as follows That is, the distributions of X 1:N 1 and Y 1:N 2 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 p 1 , . . ., p n and q q 1 , . . ., q n , let ≤ st , ≤ hr , and ≤ rh represent, respectively, the usual stochastic order, hazard rate order, and reversed hazard rate order.Then a p≤ st q if n j i p j ≤ n j i q j for all i 1, 2, . . ., n. b p≤ hr q if n j i p j / n j i q j is decreasing in i.
c p≤ rh q if i j 1 p j / i j 1 q j is decreasing in i.
On the other hand, X n−i:n−i ≥ st X 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≤ st q, then 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.

3.8
Theorem 3.4.(a) If p≤ hr q, and X 1:n−i ≤ hr X 1:n−i−1 for i 1, . . ., n − 2, then X 1: 3.9 which implies that P {X 1:n−i > x 2 }/P {X 1:n−i > x 1 } is nondecreasing in i. Applying Lemma 3.2, and using p ≤ hr q we obtain 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 X 1 , . . ., X n we have X 1:m 1 ≤ hr X 1:m , and X m:m ≤ hr X 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 X 1 , . . ., X n be iid lifetime random variables with common cdf F t

Illustrative Examples
Consider the systems with structure functions

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   Because p≤ st q, from Theorem 3.1, we have X 1:N 1 ≤ st Y 1:N 2 and X N 1 :N 1 ≥ st Y N 2 :N 2 .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.

XFigure 1 :
Figure 1: Plot of the ratio r n i,x for i 1, 2, 3 from the above to below.

Table 1 :
E N for consecutive 2-out-of-n:G systems.
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

Table 2 :
Expected values of the extreme residual lives.