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May, 1991 Stochastic Order for Inspection and Repair Policies
Philip J. Boland, Emad El-Neweihi, Frank Proschan
Ann. Appl. Probab. 1(2): 207-218 (May, 1991). DOI: 10.1214/aoap/1177005934

Abstract

Inspection and repair policies for $(n - r + 1)$-out-of-$n$ systems are compared stochastically with respect to two partial orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations of $\{1, 2, \ldots, n\}$. The partial ordering $\geq^{b_1}$ is finer than the partial ordering $\geq^{b_2}$. A given permutation $\pi$ of $\{1,2, \ldots, n\}$ determines the order in which components are visited and inspected. We assume that the reliability of the $i$th independent component is given by $P_i$, where $P_1 \leq P_2 \leq \cdots \leq P_n$. If $\pi$ and $\pi'$ are two permutations such that $\pi \geq^{b_1} \pi'$, then we show that the number of inspections necessary to achieve minimal or complete repair is stochastically smaller with $\pi$ than with $\pi'$. We also consider three policies for minimal repair when the components are each made up of $t$ "parts" assembled in parallel. It is shown that if $\pi \geq^{b_2} \pi'$, then the number of repairs necessary under $\pi$ is stochastically smaller than the number necessary under $\pi'$, but that in general this is not true for the finer ordering $\geq^{b_1}$. The results enable one to make interesting comparisons between various inspection and repair policies, as well as to understand better the relationship between the orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations.

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Philip J. Boland. Emad El-Neweihi. Frank Proschan. "Stochastic Order for Inspection and Repair Policies." Ann. Appl. Probab. 1 (2) 207 - 218, May, 1991. https://doi.org/10.1214/aoap/1177005934

Information

Published: May, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0737.62089
MathSciNet: MR1102317
Digital Object Identifier: 10.1214/aoap/1177005934

Subjects:
Primary: 62N05
Secondary: 60G40 , 62N10 , 90B35

Keywords: $(n - r + 1)$-out-of-$n$ systems , complete repair , Inspection policy , minimal repair , partial orderings , permutation , stochastic order

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.1 • No. 2 • May, 1991
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