On the Consecutive k-of-n System.
Abstract
The consecutive k-of-n system is considered in which there are n components linearly ordered. Each component either functions or fails and the system is said to be failed if any k consecutive components are failed. Let r(p) = r(p sub 1, ..., p sub n) denote the probability that the system does not fail given that the components are independent, component i functions with probability p sub i, i = l, ..., n. The function r(p) is called the reliability function. The above system is studied both when the components are linearly ordered and also when they are arranged in a circular order. In Section 2, the case is considered where all p sub i are identical and present a recursion for obtaining the reliability of a consecutive k-of-n in terms of the reliability of a consecutive k - 1 on n system. This yields simple explicit formulas when k is small and differs from the recursion obtained. In Section 3, we show how upper and lower bounds on r(p) can be simply obtained. In Section 4, we consider a dynamic version in which each component independently functions for random time having distribution F. We show that when F is increasing failure rate (IFR), then system lifetime is also IFR only in the circular case when k = 2. In Section 5, we consider a sequential optimization model in the linear k = 2 case. In this model, components are put in place one at a time with complete knowledge as to whether the previous component has worked or not. We show that the optimal policy is such that whenever a success occurs we follow it with the worst of the remaining components and whenever a failure occurs we follow it with the best of the remainder.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1980
- Accession Number
- ADA096951
Entities
People
- Cyrus Derman
- Gerald J. Lieberman
- Sheldon M. Ross
Organizations
- Stanford University