Repairable models with operating and repair times governed by phase type distributions

Neuts, Marcel F.; Pérez‐Ocón, Rafael; Castro, Inma T.

doi:10.1239/aap/1013540174

Cited by 51 publications

(13 citation statements)

References 4 publications

(1 reference statement)

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“…The use of mathematical maintenance models have been emphasized over the past few decades, and much research on the analysis and control of such models has appeared in the literature. For example, Neuts et al . considered the analysis of failing systems governed by phase‐type distributions.…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation for a Hidden Semi‐Markov Model with Multivariate Observations

Jiang

Kim

Makiš

2012

Quality & Reliability Eng

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In this paper, a parameter estimation procedure for a condition‐based maintenance model under partial observations is presented. The deterioration process of the partially observable system is modeled as a continuous‐time homogeneous semi‐Markov process. The system can be in a healthy or unhealthy operational state, or in a failure state, and the sojourn time in the operational state follows a phase‐type distribution. Only the failure state is observable, whereas operational states are unobservable. Vector observations that are stochastically related to the system state are collected at equidistant sampling times. The objective is to determine maximum likelihood estimates of the model parameters using the Expectation–Maximization (EM) algorithm. We derive explicit formulae for both the pseudo likelihood function and the parameter updates in each iteration of the EM algorithm. A numerical example is developed to illustrate the efficiency of the estimation procedure. Copyright © 2012 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation for a Hidden Semi‐Markov Model with Multivariate Observations

Jiang

Kim

Makiš

2012

Quality & Reliability Eng

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“…We extend the paper of Van der Duyn Schouten and Wartenhorst (1994) introducing the Markovian degrading in the units by means of phase-type distributions, and the successive degrading states have not to be successively occupied. On the other hand, we extend the paper of Neuts et al (2000), considering a multiple component system with different lifetimes and repair ones under transient regime.…”

Section: Introductionmentioning

confidence: 94%

“…, N k − 1, indicate that a repair is completed. For unit k, the transitions among its macro-states are given by: For more details see Neuts et al (2000). Note that for the conservativity of the generator of unit k, denoted by Q k , the diagonal blocks B k i,i , i = 0, .…”

Section: Infinitesimal Generatormentioning

confidence: 99%

“…Performance measures for unit 1. occurrence of external failures, υ k 2 (t) the rate of occurrence of wear-out failures and υ k 3 (t) the total rate of occurrence of failures. These quantities have been calculated in Neuts et al (2000).…”

Section: General Modelmentioning

confidence: 99%

“…More recently, Neuts (2000), studied a unit system in stationary regime; Pérez-Ocón and Montoro-Cazorla (2004) a unit system in transient regime; and Pérez-Ocón and Ruiz-Castro (2004) a two-system in stationary regime. On the other side, Van der Duyn Schouten and Wartenhorst (1994) studied a two-system with Markovian degrading.…”

Section: Introductionmentioning

confidence: 99%

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Transient Analysis of a Multi-Component System Modeled by a General Markov Process

Pérez‐Ocón

Montoro-Cazorla

Castro

2006

Asia Pac. J. Oper. Res.

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An M-unit system in dynamic environment with operational and repair times following phase-type distributions and incorporating geometrical processes is considered. A general Markov process with vectorial states is the appropriate structure for modeling this system. A transient analysis is performed for this complex system and the transition probabilities are calculated. Some performance measures of general interest in the study of systems are obtained using an algorithmic approach, and applied to G-out-of-M systems. A numerical example is presented and the transient performance measures are calculated and compared with the stationary ones. This paper extends previous reliability systems, that can be considered as particular cases of this one. Throughout the paper, the mathematical expressions are given by algorithmic methods, that emphasized the utility of phase-type distributions in the analysis of lifetime data. 311 Asia Pac. J. Oper. Res. 2006.23:311-327. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/18/15. For personal use only. Cui et al. (2004) and references in it. This paper, we present studies a general multicomponent system, and it is different from the previous ones in several ways: (1) A system with multiple components phase-type distributed is studied; (2) the transition probability functions for the system are expressed in terms of the correponding functions of the components, and they are explicitly calculated in the applications; (3) we use the general methodology for studying a G-out-of-M system; (4) the operational time and performance measures for the G-out-of-M system are determined; (5) a parallel system with two components illustrates the algorithmic procedures; and (6) a numerical example is also performed for illustrating the computational implementation of the expressions obtained throughout the paper. Asia Pac. J. Oper. Res. 2006.23:311-327. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/18/15. For personal use only.

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Phase‐Type (PH) Distributions

Pérez‐Ocón

2011

Wiley Encyclopedia of Operations Research and Management Science

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The phase‐type (PH) distributions were introduced by Neuts (1981). They are playing an important role in modeling random phenomena due to their interesting properties, and it has a widespread use in applied probability. A continuous PH distribution is the distribution of the absorption time in a Markov chain in continuous time with a finite‐state space and one absorbent state. The cumulative distribution function (CDF) has exponential form, and it is deduced from the underlying Markov process. The class of PH distributions has important closure properties which make it very versatile in applications; it is closed for finite convolution, finite mixture, formation of coherent systems, and order statistics. On the other hand, it has interesting properties of approximation. This class is dense, in the sense of distributions, in the class of distribution functions defined on the positive real half line, so it is possible to approach general distributions by elements of this class. The use of PH distributions in stochastic modeling leads to results that can be presented in algorithmic form and the results are susceptible to computational implementation. The PH distributions together with the Markovian arrival processes are the basic elements in what is known as matrix‐analytic methods.

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Repairable models with operating and repair times governed by phase type distributions

Cited by 51 publications

References 4 publications

Maximum Likelihood Estimation for a Hidden Semi‐Markov Model with Multivariate Observations

Maximum Likelihood Estimation for a Hidden Semi‐Markov Model with Multivariate Observations

Transient Analysis of a Multi-Component System Modeled by a General Markov Process

Phase‐Type (PH) Distributions

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