Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model

Antonio Di Crescenzo,Patrizia Di Gironimo

doi:10.3390/math6100204

Abstract

We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order to assess conditions leading to an improvement of the system. Attention is also given to the relative ratio of improvement, which is proposed as a suitable index finalized to measure the goodness of the replacement procedure. Finally, we provide various results on the dynamic differential entropy of the lifetime of the improved system.

Highlights

In reliability theory, various stochastic models have been proposed in the past in order to describe replacement policies of system components
This measure has some analogies with the entropy of discrete random variables, even though the differential entropy lacks a number of properties that the Shannon discrete entropy possesses
In the context of lifetimes truncated over intervals of the form [0, t] or (t, ∞), specific forms of the differential entropy have been investigated in the recent decades

Summary

Introduction

Various stochastic models have been proposed in the past in order to describe replacement policies of system components. A classical model in this area is the relevation transform, which describes the overall lifetime of a component which is replaced at its (random) failure time by another component of the same age, whose lifetime distribution is possibly different. In order to describe the total lifetime of an item given that it is less than an independent random inspection time Such transforms deserve a large interest since they can be employed in restoration models of failed units, or in the determination of optimal redundancy policy in coherent systems. In both cases, the above models involve a random replacement (or inspection) time. We assume that an item having random lifetime X is planned to be replaced at time t by another item having the same age but possibly different random lifetime Y

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Conclusion