Innovative Applications of O.R.
Optimal allocation policy of one redundancy in a n-component series system

https://doi.org/10.1016/j.ejor.2016.07.055Get rights and content

Highlights

  • We study redundancy allocation in a n-component series system.

  • We use the tool of stochastic orderings.

  • Two optimal allocation principles are proposed.

  • Examples are provided to illustrate the results.

Abstract

It is of great importance to optimize system performance by allocating redundancies in a coherent system in reliability engineering and system security. In this paper, we focus on the problem of how to optimally allocate one active [standby] redundancy in a n-component series system in the sense of stochastic ordering. For the active case, it is showed that allocating the redundancy to the relatively weaker component leads to longer system’s lifetime in the likelihood ratio and reversed hazard rate orders, respectively. For the standby case, we show that the redundancy should be allocated to the weakest component of the series system in the likelihood ratio order. Based on these results, two optimal allocation policies are proposed. Also, some numerical examples are presented to explicate the theoretic results established here.

Introduction

The reliability level of a coherent system can be enhanced by adding redundancies to the original system, ant this topic is of great interest and importance in reliability engineering and system security. In real world, two most commonly used types of redundancies are active (or hot) redundancy and standby (or cold) redundancy. In the active case, available spares are used in parallel with the original components of the system and function simultaneously with those components; in the standby case, spares are attached to the components of the system in such a way that each spare functions only after the failure of its corresponding original component. For these two kinds of allocation ways, one is effectively able to measure the performances through stochastic comparisons between the lifetimes of the resulting systems in the sense of various stochastic orders. Many researchers have paid their attentions on this topic in the past decades; see, for example, Boland, El-Neweihi, Proschan, 1988, Boland, El-Neweihi, Proschan, 1992, Shaked and Shanthikumar (1992), Singh and Misra (1994), Singh and Singh (1997), Valdés, Zequeira, 2003, Valdés, Zequeira, 2004, Valdés, Zequeira, 2006, da Costa Bueno (2005), da Costa Bueno and do Carmo (2007), Li and Hu (2008), Hu and Wang (2009), Valdés, Arango, and Zequeira (2010), Brito, Zequeira, and Valdés (2011), Misra, Dhariyal, and Gupta (2009), Misra, Misra, Dhariyal, 2011a, Misra, Misra, Dhariyal, 2011b, Li, Yan, and Hu (2011), Zhao, Chan, and Ng (2012), Belzunce, Martinez-Puertas, and Ruiz (2013), Zhao, Zhang, and Li (2015), and Caserta and Voss (2015) and the references therein.

We first recall some pertinent definitions of stochastic orders that will be used in the sequel. Throughout this paper, the term increasing is used for monotone non-decreasing and decreasing is used for monotone non-increasing. Let X and Y be two random variables with common support +=[0,), density functions fX and fY, distribution functions FX and FY, respectively. Then, F¯X=1FX and F¯Y=1FY are the survival functions of X and Y, respectively. Denote by hX=fX/F¯X and hY=fY/F¯Y the hazard rate functions of X and Y, and rX=fX/FX and rY=fY/FY the reversed hazard rate functions of X and Y, respectively. X is said to be smaller than Y in the usual stochastic order (denoted by XstY) if F¯X(x)F¯Y(x) for all x+; X is said to be smaller than Y in the hazard rate order (denoted by XhrY) if F¯Y(x)/F¯X(x) is increasing in x+, or hX(x) ≥ hY(x) for all x+; X is said to be smaller than Y in the reversed hazard rate order (denoted by XrhY) if FY(x)/FX(x) is increasing in x+, or rX(x) ≤ rY(x) for all x+; X is said to be smaller than Y in the likelihood ratio order (denoted by XlrY) if fY(x)/fX(x) is increasing in x+. For a comprehensive discussion on various stochastic orders, one may refer to Shaked and Shanthikumar (2007).

Let independent random variables X1, X2 and X be the lifetimes of the components C1, C2 and redundancy R, respectively. Assume that S is a two-component series system with its components C1 and C2, and a natural question is how to allocate the redundancy R so that the resulting system performs better. In active redundancy case, we want to compare U1=((X1,X),X2)andU2=(X1,(X2,X)),where the symbols ``∧” and ``∨” mean min and max, respectively. In standby redundancy case, we want to compare the lifetimes W1=(X1+X,X2)andW2=(X1,X2+X).In the literature, there has been some work treating this topic. For instance, in the active case, Boland et al. (1992) proved in their seminal work that X1stX2U1stU2.Singh and Misra (1994) showed that, if X1, X2 and X in (1) have exponential lifetimes with hazard rates λ1, λ2 and λ, then λ1{λ2,λ}U1hrU2.Recently, Zhao et al. (2012) strengthened (2) as λ1{λ2,λ}U1lrU2.You and Li (2014) generalized the result in (3) to the proportional hazard rates (PHR) model.

In the standby case, Boland et al. (1992) showed that X1hrX2W1stW2,which was improved by Zhao et al. (2012) under the exponential setup, i.e., if X1, X2 and X have exponential distributions with parameters λ1, λ2 and λ, then λ1λ2W1lrW2.However, the result in (5) does not hold for the PHR models as pointed out through a counterexample in You and Li (2014).

It should be noticed that all the results in (1)–(5) mentioned above are limited in the case of two-component series system. Inspired by this, in this paper, we will further pursue the problem of optimal allocation of one active [standby] redundancy in a series system consisting of more than two components. It should be also mentioned here that Boland et al. (1992), Valdés and Zequeira (2006), Misra, Misra, and Dhariyal (2011b) presented several results for n-component series and parallel systems, but here our research methods seem to be quite different from them.

Let X1,X2,,Xn and X be independent exponential random variables with respective parameters λ1,λ2,,λn and λ representing the lifetimes of the components C1,C2,,Cn and redundancy, respectively. In the active case, we would like to stochastically compare Ui=(X1,,Xi1,(Xi,X),Xi+1,,Xn)and Uj=(X1,,Xj1,(Xj,X),Xj+1,,Xn),for 1 ≤ ijn and n ≥ 2. In the standby case, we would like to stochastically compare Wi=(X1,,Xi1,Xi+X,Xi+1,,Xn)and Wj=(X1,,Xj1,Xj+X,Xj+1,,Xn).In this regard, we prove for the active redundancy that λi{λj,λ}UilrUjand λλiλjUirhUj.Combining (6) with (7), an equivalent characterization can be derived as λiλjUirhUj.Also, all the results in (6)–(8) can be extended to the PHR models. For the standby redundancy, we show that λiλjWilrWj.These new results in (6)–(9) generalize and strengthen those in (1)–(5) established earlier in the literature. Most importantly, the results established here can help provide a criterion on how to optimally allocate one redundancy in a n-component series system for the design engineer.

The remainder of the paper is organized as follows. The main results are presented in Section 2, in which Section 2.1 puts forward the optimal strategy for allocating an active redundancy in a n-component series system, and the optimal allocation policy for a standby redundancy in a n-component series system is provided in Section 2.2. Some remarks can be found in Section 3 to conclude this paper. For ease of presentation, the detailed proofs of the theorems are deferred in the appendix.

Section snippets

Active case

As mentioned earlier, Zhao et al. (2012) improved the results in (1) and (2) established by Boland et al. (1992), Singh and Misra (1994) under the exponential framework in the likelihood ratio order. However, all these existing results are based on the framework of the two-component series system. Motivated by this, here we aim to present some general results in a n-component series system instead.

Theorem 2.1

LetX1,X2,,Xn and X be independent exponential components having respective hazard ratesλ1,λ2,,λn

Concluding remarks

Let X1,X2,,Xn and X be independent exponential random variables with respective parameters λ1,λ2,,λn and λ representing the lifetimes of the components C1,C2,,Cn and redundancy, respectively. In the case of active redundancy, upon using the likelihood ratio and reversed hazard rate orders, we carry out the stochastic comparison on Ui=(X1,,Xi1,(Xi,X),Xi+1,,Xn)and Uj=(X1,,Xj1,(Xj,X),Xj+1,,Xn),where 1 ≤ i, jn and n ≥ 2. In the case of standby redundancy, we carry out the stochastic

Acknowledgments

The authors thank the editor and anonymous referees for the insight comments and suggestions, which resulted in an improvement in the presentation of this manuscript. This work was supported by National Natural Science Foundation of China (11422109), Natural Science Foundation of Jiangsu Province (BK20141145), Qing Lan Project and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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