Generalized q‐Weibull model and the bathtub curve

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 backgrou...

[1]  S. Kannan,et al.  A diagnostic approach to Weibull‐Weibull stress‐strength model and its generalization , 2011 .

[2]  S. Picoli,et al.  An improved description of the dielectric breakdown in oxides based on a generalized Weibull distribution , 2006 .

[3]  Joanne S. Utley,et al.  Using reliability tools in service operations , 2011 .

[4]  Edilson Machado de Assis,et al.  Reliability Modeling of a Natural Gas Recovery Plant Using q-Weibull Distribution , 2009 .

[5]  R. S. Mendes,et al.  q-exponential, Weibull, and q-Weibull distributions: an empirical analysis , 2003, cond-mat/0301552.

[6]  Hoang Pham,et al.  On Recent Generalizations of the Weibull Distribution , 2007, IEEE Transactions on Reliability.

[7]  Per Bak,et al.  How Nature Works , 1996 .

[8]  T. Yamano Some properties of q-logarithm and q-exponential functions in Tsallis statistics , 2002 .

[9]  Constantino Tsallis Nonextensive Statistical Mechanics, Anomalous Diffusion and Central Limit Theorems , 2004 .

[10]  M. Xie,et al.  Exponential approximation for maintained Weibull distributed component , 2000 .

[11]  C. Tsallis,et al.  Nonextensive foundation of Lévy distributions. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Samuel Kotz,et al.  q exponential is a Burr distribution , 2006 .

[13]  C. Tsallis Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World , 2009 .

[14]  Horst Rinne,et al.  The Weibull Distribution: A Handbook , 2008 .

[15]  Kiyoyuki Kaito,et al.  Random proportional Weibull hazard model for large‐scale information systems , 2011 .

[16]  Constantino Tsallis,et al.  On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics , 2008 .

[17]  I. W. Burr Cumulative Frequency Functions , 1942 .

[18]  C. Tsallis,et al.  Escort mean values and the characterization of power-law-decaying probability densities , 2008, 0802.1698.

[19]  C. Tsallis,et al.  Statistical-mechanical foundation of the ubiquity of Lévy distributions in Nature. , 1995, Physical review letters.

[20]  W. Weibull A Statistical Distribution Function of Wide Applicability , 1951 .

[21]  C. Tsallis Possible generalization of Boltzmann-Gibbs statistics , 1988 .