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# k-out-of-n Systems

1. Deborah K. Shepherd

Published Online: 15 MAR 2008

DOI: 10.1002/9780470061572.eqr342

## Encyclopedia of Statistics in Quality and Reliability

#### How to Cite

Shepherd, D. K. 2008. k-out-of-n Systems. Encyclopedia of Statistics in Quality and Reliability. II.

#### Author Information

1. Louisiana State University in Shreveport, Shreveport, LA, USA

#### Publication History

1. Published Online: 15 MAR 2008

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### Abstract

A k-out-of-n system is a system with n components which functions or works if at least k out of the n components work. Parallel and series systems are special cases of a k-out-of-n system. There are many different arrangements of the components in a k-out-of-n system and many assumptions that can be made about the lifetimes of the components. In this chapter, the lifetimes of the components in the system are assumed to be independent and identically distributed. The reliability function of the system is found by calculating the probability that at least k components out of the n components work. The mean time to failure of the system is also found for the case of components with exponential distributions of lifetimes. In addition, the reliability function for a linear and circular consecutive k-out-of-n system is given.

### Keywords:

• k-out-of-n system;
• reliability;
• exponential distribution;
• consecutive k-out-of-n system;
• circular k-out-of-n system;
• linear k-out-of-n system;
• binomial distribution