An extension of the Freund's bivariate distribution to model cascading failures

ABSTRACT

The notion of cascading failures is a common phenomenon we observe around us. Here the initial failure alters the structure function of the system, which leads to subsequent failures within a short period of time referred to as threshold time. The concept of cascading failures within the framework of reliability theory and the Freund bivariate exponential distribution to model cascading failures has been studied by few authors. The Freund bivariate exponential distribution allows modelling a parallel redundant system with two components. In this system, the lifetimes of the two components behave as if they are independent, until one of the components fail, after which the remaining component suffers an increased/decreased stress. In this article, we further generalize this model to accommodate cascading failures. Various properties of the model are investigated and statistical inference procedures are developed using L-moments and method of moments. A practical application of this model is illustrated using data from www.espncricinfo.com. Also well analysed Diabetic Retinopathy Study (DRS) data is further analysed using this model and our findings are presented.

Keywords:

• Cascading failures
• Cricket data
• Freund's bivariate exponential model
• L-moments
• Proportional hazard model

MATHEMATICS SUBJECT CLASSIFICATION:

• 62N05
• 60E99

Acknowledgments

The first author wish to express her sincere gratitude to Prof. Debasis Kundu, Department of Mathematics and Statistics, IIT Kanpur, India for the initial discussions. The authors also wish to thank the anonymous referee and associate editor for their comments that brought further clarity to the article.