## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 1, Number 2 (1991), 207-218.

### Stochastic Order for Inspection and Repair Policies

Philip J. Boland, Emad El-Neweihi, and Frank Proschan

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 1, Number 2 (1991), 207-218.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005934

**Digital Object Identifier**

doi:10.1214/aoap/1177005934

**Mathematical Reviews number (MathSciNet)**

MR1102317

**Zentralblatt MATH identifier**

0737.62089

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62N05: Reliability and life testing [See also 90B25]

Secondary: 62N10 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 90B35: Scheduling theory, deterministic [See also 68M20]

**Keywords**

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

#### Citation

Boland, Philip J.; El-Neweihi, Emad; Proschan, Frank. Stochastic Order for Inspection and Repair Policies. Ann. Appl. Probab. 1 (1991), no. 2, 207--218. doi:10.1214/aoap/1177005934. https://projecteuclid.org/euclid.aoap/1177005934