Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

Antonio Di Crescenzo,Suchandan Kayal,Patrizia Di Gironimo

doi:10.3390/math8081203

Antonio Di Crescenzo, Suchandan Kayal + Show 1 more

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https://doi.org/10.3390/math8081203

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Abstract

A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a fixed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at x ∈ ( u , t ) , which is finalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the firing activity of a stochastic neuronal model is provided.

Highlights

The problem of the reliability and survival analysis of a system has been widely studied in recent years
Differently from the previous investigation, which was centered on the residual lifetime, in this case we focus on the past lifetime of the system
In order to illustrate the effect of the replacement in a context of interest in theoretical neurobiology, in this paper we investigate an application to a solvable stochastic neuronal model

Summary

Introduction

The problem of the reliability and survival analysis of a system has been widely studied in recent years. The reliability of a system is analyzed on the ground of partially available information that is concerning the status of the system or its components at certain fixed instants. In these instances, it is necessary to study the reliability measures of interest under the condition of truncated or doubly truncated random variables. It is necessary to study the reliability measures of interest under the condition of truncated or doubly truncated random variables In this contribution, we refer to a previous investigation centered on a stochastic model dealing with the replacement of items occurring at deterministic arbitrary instants (see Di Crescenzo and Di Gironimo [3]). Due to the nature of the treated model, specific attention is given to the Mathematics 2020, 8, 1203; doi:10.3390/math8081203 www.mdpi.com/journal/mathematics

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