On selecting the most reliable components

Dylan Shi

doi:10.1155/s1173912698000078

Abstract

Consider a series system consisting of n components of k types. Whenever a unit fails, it is replaced immediately by a new one to keep the system working. Under the assumption that all the life lengths of the components are independent and exponentially distributed and that the replacement policies depend only on the present state of the system at each failure, the system may be represented by a birth and death process. The existence of the optimum replacement policies are discussed and the ε-optimal policies axe derived. If the past experience of the system can also be utilized, the process is not a Markov process. The optimum Bayesian policies are derived and the properties of the resulting process axe studied. Also, the stochastic processes are simulated and the probability of absorption, the mean time to absorption and the average proportion of the retrograde motion are approximated.

Highlights

Consider a series system consisting of n components of k types
Since the rank order of these k failure rates is assumed to be completely unknown, the natural problem arises of how to select the most reliable components associated with the smallest failure rate
The decision problem could be classiiied in the areas of multiple comparisons, ranking and selection and reliability theory, but especially in the area of ranking and selection

Summary

Introduction

Suppose we have k types of components. Assume that all the life lengths of the components are independent and exponentially distributed with unknown constant failure rates ),..-, ). Instead of observing the k parallel running, independent experiments, the k types of components are examined in a series operating system where it is emphasized that at each failure, an immediate replacement is made in some optimum way to keep the system running Under this model Miescke (1990) has derived some natural ’look-ahead-one-failure’ Bayes replacement policies to maximize, at each failure, the expected waiting time for the failure in the system, or the posterior probability that the waiting time for the failure exceeds a given value. Miescke and Shi (1995) have studied the process for k = 2 and, under certain conditions, derived Markov replacement policies in some optimum way to ensure that, with maximum absorption probability, the system will be composed of only the components of the better type, possessing the smaller failure rate. The absorbing property of the non-Markov processes will be studied and the results of the computer simulation of the processes will be given at the end

Model Assumptions

The existence of optimum replacement policies

SHI aiO where 6 n-1 aln j

The e-optimal policies

Policies based on both the present and the past information

The likelihood

Bayes replacement policy

An Example

The absorbing property of the process

Simulation of the process

The retrograde motions