A bivariate two-state Markov modulated Poisson process for failure modeling

• Published in: Reliability engineering & system safety Vol. 208; p. 107318
• Main Authors: Yera, Yoel G., Lillo, Rosa E., Nielsen, Bo F., Ramírez-Cobo, Pepa, Ruggeri, Fabrizio
• Format: Journal Article
• Language: English
• Published: Barking Elsevier Ltd 01.04.2021; Elsevier BV
• Subjects: ABC; Algorithms; Bayesian analysis; Bivariate analysis; Bivariate process; Datasets; Failure; Failure times; Identifiability; Markov modulated Poisson process (MMPP); Moments matching method; Poisson density functions; Probability distribution functions; Public transportation; Reliability engineering; Train reliability data; Two dimensional models; Identifiability; ABC; Bivariate process; Markov modulated Poisson process (MMPP); Moments matching method; Train reliability data
• ISSN: 0951-8320; 1879-0836
• DOI: 10.1016/j.ress.2020.107318

Motivated by a real failure dataset in a two-dimensional context, this paper presents an extension of the Markov modulated Poisson process (MMPP) to two dimensions. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-failure times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence between the two sequences of inter-failure times, while at the same time preserves the MMPP properties, marginally. The generalization is based on the Marshall–Olkin exponential distribution. Inference is undertaken for the new model through a method combining a matching moments approach with an Approximate Bayesian Computation (ABC) algorithm. The performance of the method is shown on simulated and real datasets representing times and distances covered between consecutive failures in a public transport company. For the real dataset, some quantities of importance associated with the reliability of the system are estimated as the probabilities and expected number of failures at different times and distances covered by trains until the occurrence of a failure. •A bivariate Markov modulated Poisson process is introduced.•A matrix formulation of the model is proposed that allows attacking the reliability of the process in a feasible way.•A sequential fitting approach via a moments matching combined with an ABC algorithm is provided.•The suitability of the results is illustrated through the analysis of a real dataset from the reliability context.