Reversed Failure Rate Research Articles

ACTIVE REDUNDANCY ALLOCATIONS IN SERIES SYSTEMS

We consider the problem of optimally allocating one/two active spares in series systems. Lifetimes of different configurations are compared with respect to the reversed failure rate order, the failure rate order, the increasing concave order, and the stochastic precedence order.

OPTIMAL BACKWARD AND BACKUP POLICIES IN RELIABILITY THEORY

We are sometimes faced with the problem of searching for the actual failure time of a unit when it has failed until time t0. This is called a backward time problem of how much time we go back to search for the failure time from time t0, and is solved by using the reversed failure rate. This paper proposes an extended model of multiple backward trials in which we find the actual failure time from time t0. Another problem is that when an operating unit fails, we have to go back to the newest checking time and reconstruct it as recovery techniques, which is called a backup policy. This paper takes up three backup policies where the unit is checked at periodic, successive and random working times, and discusses analytically their optimal policies. Finally, it would be important to keep a record of operational behavior of a unit when its failure is detected, which is called traceability. This paper compares two models with traceability and without traceability, and proposes three traceability models where the unit operates for a finite interval, and is checked at periodic and successive times. Optimal policies for each model are discussed by using backup techniques.

Optimal Allocation of Active Spares in Series Systems and Comparison of Component and System Redundancies

We consider the problem of allocating k active spares to n components of a series system in order to optimize its lifetime. Under the hypotheses that lifetimes of n components are identically distributed with distribution function F(⋅), lifetimes of k spares are identically distributed with distribution function G(⋅), lifetimes of components and spares are independently distributed, and that ln(G(x))/ln(F(x)) is increasing in x, we show that the strategy of balanced allocation of spares optimizes the failure rate function of the system. Furthermore, under the hypotheses that lifetimes of n components are stochastically ordered, lifetimes of k spares are identically distributed, and that lifetimes of components and spares are independently distributed, we show that the strategy of balanced allocation of spares is superior to the strategy of allocating a larger number of components to stronger components. For coherent systems consisting of n identical components with n identical redundant (spare) components, we compare strategies of component and system redundancies under the criteria of reversed failure rate and likelihood ratio orderings. When spares and original components do not necessarily match in their life distributions, we provide a sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under reversed failure rate ordering. As a consequence, we show that, for r-out-of-n systems, the strategy of component redundancy is superior to the strategy of system redundancy under the criterion of reversed failure rate ordering. When spares and original components match in their life distributions, we provide a necessary and sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under the likelihood ratio ordering. As a consequence, we show that, for r-out-of-n systems, with spares and original components matching in their life distributions, the strategy of component redundancy is superior to the strategy of system redundancy under the likelihood ratio ordering.

Optimal Allocation of Active Spares in Series Systems and Comparison of Component and System Redundancies

We consider the problem of allocating k active spares to n components of a series system in order to optimize its lifetime. Under the hypotheses that lifetimes of n components are identically distributed with distribution function F(⋅), lifetimes of k spares are identically distributed with distribution function G(⋅), lifetimes of components and spares are independently distributed, and that ln(G(x))/ln(F(x)) is increasing in x, we show that the strategy of balanced allocation of spares optimizes the failure rate function of the system. Furthermore, under the hypotheses that lifetimes of n components are stochastically ordered, lifetimes of k spares are identically distributed, and that lifetimes of components and spares are independently distributed, we show that the strategy of balanced allocation of spares is superior to the strategy of allocating a larger number of components to stronger components. For coherent systems consisting of n identical components with n identical redundant (spare) components, we compare strategies of component and system redundancies under the criteria of reversed failure rate and likelihood ratio orderings. When spares and original components do not necessarily match in their life distributions, we provide a sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under reversed failure rate ordering. As a consequence, we show that, for r-out-of-n systems, the strategy of component redundancy is superior to the strategy of system redundancy under the criterion of reversed failure rate ordering. When spares and original components match in their life distributions, we provide a necessary and sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under the likelihood ratio ordering. As a consequence, we show that, for r-out-of-n systems, with spares and original components matching in their life distributions, the strategy of component redundancy is superior to the strategy of system redundancy under the likelihood ratio ordering.

Optimum problems in backward times of reliability models

This paper considers the problem of searching for the actual time of failure for a system when the only information that is available is that it is has failed by a time t. This situation can be analyzed by using the concept of the reversed failure rate. This paper considers optimization problems that can be solved by using the reversed failure rate. When a unit is detected to have failed at time t, we discuss an optimum backward time from time t to search for its failure time which minimizes the expected cost. The recovery of a database system and of reweighing products using a scale are taken to be two typical applications of the backward time concept. Two models are proposed for appropriate maintenance actions for these situations when a unit is detected to have failed at time t.

Characterizations from Relationships Between Failure Rate Functions and Conditional Moments

We obtain general results on characterization of distributions from relationships between failure rate functions and conditional moments. Specifically, we study characterizations from linear relationships between the usual failure rate function and the left truncated conditional moment m h (x) = E(h(X) | X > x), extending some characterizations of mixtures given recently. Moreover, we also study characterizations from relationships between the generalized failure rate functions and the doubly truncated conditional moment function m h (x, y) = E(h(X)|x < X < y). Our results extend some particular characterizations based on mean residual life and failure rate. As a consequence, we obtain characterizations based on reversed failure rate and reversed mean residual life (expected inactivity time). Finally, we apply our theoretical results to obtain some new characterizations for usual models.

ON PARTIAL ORDERINGS BETWEEN COHERENT SYSTEMS WITH DIFFERENT STRUCTURES

In this article, we give some results on the preservation of orderings between the components under the formation of coherent systems with different structures. We consider the stochastic, failure rate, reversed failure rate, and likelihood ratio orderings to compare two coherent structures formed from a set of components and from two sets of components, in both cases, when components are independent and either identically distributed or not necessarily identically distributed. As a consequence, we get new stochastic comparisons of systems with different k-out-of-n structures, and some results of partial orderings given by other authors may be also obtained from ours.