Allocation of two redundancies in two-component series systems

Abstract

The allocation of redundancies in a system to optimize the reliability of system performance is an interesting problem in reliability engineering and system security. In this article, we focus on the optimal allocation of two exponentially distributed active (standby) redundancies in a two‐component series system using the tool of stochastic ordering. For the case of active redundancy, stochastic comparisons are carried out in terms of the likelihood ratio and reversed hazard rate orders. For the case of standby redundancy, a likelihood ratio ordering result is developed. The results established here generalize and strengthen corresponding results in the recent literature. In addition, several numerical examples are used to explicate the results. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013

Allocations of cold standbys to series and parallel systems with dependent components

“…Barlow and Proschan, [2]). The past several decades have witnessed comprehensive developments on investigating optimal allocation policies of active redundancies for coherent systems (especially k-out-of-n systems) consisting of independent components; see for example Boland et al [5,6], Singh and Misra [31], Valdés and Zequeira [33], Valdés et al [32], Brito et al [7], Misra et al [25], Hazra and Nanda [18], Zhao et al [42,43,44,45], Da and Ding [12], Ding et al [13], and Zhang [38]. On the other hand, some research work has appeared on redundancies allocation for coherent systems with dependent components.…”

Optimal redundancy allocations for series systems under hierarchical dependence structures

“…(i) The first scenario is to use only one of the two redundancies: either 𝑅 1 with 𝐴 1 , or 𝑅 2 with 𝐴 2 , namely, 𝑈 1 = ∧{∨{𝑋 1 , 𝑌 1 }, 𝑋 2 , 𝑋 3 , … , 𝑋 𝑛 } and 𝑈 2 = ∧{𝑋 1 , ∨{𝑋 2 , 𝑌 2 }, 𝑋 3 , … , 𝑋 𝑛 }, where "∧" means minima and "∨" means maxima. To determine the best allocation policy, one has to make stochastic comparisons on 𝑈 1 and 𝑈 2 , which have been considered by Romera et al, 9 Valdés et al, 2 Misra et al, 10 and Zhao et al 11 by means of various stochastic orderings when all the components and redundancies are independent. The results in these works indicate that the worse component should be allocated by a better redundancy to enhance the system reliability.…”

“… (i)The first scenario is to use only one of the two redundancies: either $$R_1$$ with $$A_1$$, or $$R_2$$ with $$A_2$$, namely, $$$$\begin{equation*} U_{1}=\wedge {\left\lbrace \vee \lbrace X_{1},Y_{1}\rbrace,X_{2},X_{3},\ldots,X_{n}\right\rbrace} \quad \mbox{and}\quad U_{2}=\wedge {\left\lbrace X_{1},\vee \lbrace X_{2},Y_{2}\rbrace,X_{3},\ldots,X_{n}\right\rbrace}, \end{equation*}$$$$where “$$\wedge$$” means minima and “$$\vee$$” means maxima. To determine the best allocation policy, one has to make stochastic comparisons on $$U_{1}$$ and $$U_2$$, which have been considered by Romera et al., 9 Valdés et al., 2 Misra et al., 10 and Zhao et al 11 . by means of various stochastic orderings when all the components and redundancies are independent.…”

Orderings of coherent systems with randomized dependent components

“…In the theory of coherent systems, it is important to study the performance of a system composed by different kinds of units, and to define the optimal allocation of these units in the system. Some results on this topic are given in da Costa Bueno [5], da Costa Bueno and do Carmo [6], Li and Ding [15], Brito et al [4], Misra et al [18], Navarro and Rychlik [23], Zhang [27], Eryilmaz [10], Kuo and Zhu [13], Zhao et al [28,29], Belzunce et al [3], Levitin et al [14] and Hazra and Nanda [11]. In this context, a translation of the Parrondo's paradox was proposed in Di Crescenzo [7].…”