APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind. 2009; 25:323–337 Published online 3 December 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.750 Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems Jorge Navarro 1, ∗, † , Antonio Guillam´ on 2 and Mar´ ıa del Carmen Ruiz 2 1 Facultad de Matem´ aticas, Universidad de Murcia, 30100 Murcia, Spain 2 Universidad Polit´ ecnica de Cartagena, Av. Dr. Fleming, 30202 Cartagena (Murcia), Spain SUMMARY In this paper, we obtain and discuss some general properties of hazard rate (HR) functions constructed via generalized mixtures of two members. These results are applied to determine the shape of generalized mixtures of an increasing hazard rate (IHR) model and an exponential model. In addition, we note that these kind of generalized mixtures can be used to construct bathtub-shaped HR models. As examples, we study in detail two cases: when the IHR model chosen is a linear HR function and when the IHR model is the extended exponential-geometric distribution. Finally, we apply the results and show the utility of generalized mixtures in determining the shape of the HR function of different systems, such as mixed systems or consecutive k -out-of-n systems. Copyright 2008 John Wiley & Sons, Ltd. Received 5 May 2008; Revised 3 October 2008; Accepted 3 October 2008 KEY WORDS: hazard rate; failure rate; bathtub shaped; generalized mixtures; coherent systems; mixed systems 1. INTRODUCTION Stochastic models with bathtub-shaped hazard rate (BHR) functions are common models in practice in reliability and survival studies. However, it is not always straightforward to construct a sufficiently flexible BHR model from standard models such as exponential or Weibull models. In the last two decades several techniques have been developed to obtain BHR models (see the survey in [1]). These techniques include mixtures, extensions of known models (e.g. Weibull or gamma), piecewise distributions, stochastic processes, etc. For example, several authors consider the introduction of new parameters in classical models such as Weibull [2–6]. Thus, one can choose a model ∗ Correspondence to: Jorge Navarro, Facultad de Matem´ aticas, Universidad de Murcia, 30100 Murcia, Spain. † E-mail: jorgenav@um.es Contract/grant sponsor: Ministerio de Educaci´ on y Ciencia; contract/grant number: MTM2006-12834 Copyright 2008 John Wiley & Sons, Ltd.