Generalized q‐Weibull model and the bathtub curve

Abstract

Purpose

The purpose of this paper is to analyze mathematical aspects of the q‐Weibull model and explore the influence of the parameter q.

Design/methodology/approach

The paper uses analytical developments with graph illustrations and an application to a practical example.

Findings

The q‐Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented.

Practical implications

The generalized q‐Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U‐shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q‐Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long‐range interactions and/or long‐term memory. This theoretical background may help the understanding of the underlying mechanisms for failure events in engineering problems.

Originality/value

Q‐Weibull model has already been introduced in the literature, but it was not realized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings a mapping of the parameters, showing the range of the parameters that should be used for each type of curve.

Citation

Assis, E.M., Borges, E.P. and Vieira de Melo, S.A.B. (2013), "Generalized q‐Weibull model and the bathtub curve", International Journal of Quality & Reliability Management, Vol. 30 No. 7, pp. 720-736. https://doi.org/10.1108/IJQRM-Oct-2011-0143