Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

Boyan Dimitrov,Vladimir Rykov,Tatiana Milovanova

doi:10.3390/math8030459

Abstract

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements.

Highlights

In 1967, Marshall and Olkin proposed a bivariate distribution, called (MO), with dependent components, defined via three independent Poisson processes that represent three types of shocks. Two of these act individually on each component and the third one acts simultaneously on both components. This model possesses the what is known as a bivariate lack of memory property—BLMP
The corresponding joint distributions encapsulate “aging.” In 2014 the model was extended to the multidimensional case by Lin and Li [4]
In 2015 Pinto and Kolev [5] introduced the extended BLMP model assuming the dependence between individual shocks, but keeping the third one independent of the previous two

Summary

Introduction

In 1967, Marshall and Olkin proposed a bivariate distribution, called (MO), with dependent components, defined via three independent Poisson processes that represent three types of shocks. Their motivation is that the individual shocks might be dependent if the items share a common environment Almost all of those investigations were focused on some generalizations of the bivariate or multivariate distributions, while lacking, respectively, the memory property and studies of their additional properties. In this paper the stationary probabilities for such a system with a MO renewable failure model between its components are derived and investigated using common means of Markov chains. For this reason the assessments of their time dependent probabilities are not required. We conclude with a wish list for further possible research on similar reliability maintenance models, which could be based on the already existing models of extended MO distribution

The Problem Setting and Notations

Life Cycle and System Life Time

Number of Passages between the States During a Cycle

Sojourn Times during a Life Cycle

Stationary Probabilities

Conclusions