Abstract
In many applications, the failure rate function may present a bathtub shape curve. In this paper, an expectation maximization algorithm is proposed to construct a suitable continuous-time Markov chain which models the failure time data by the first time reaching the absorbing state. Assume that a system is described by methods of supplementary variables, the device of stage, and so on. Given a data set, the maximum likelihood estimators of the initial distribution and the infinitesimal transition rates of the Markov chain can be obtained by our novel algorithm. Suppose that there aremtransient states in the system and that there arenfailure time data. The devised algorithm only needs to compute the exponential ofm×mupper triangular matrices forO(nm2)times in each iteration. Finally, the algorithm is applied to two real data sets, which indicates the practicality and efficiency of our algorithm.
Highlights
Among many quantitative analysis methods of system reliability, the state space representation has often been employed by reliability engineers, for example, 1–5
Inspired by the theory of continuous-time Markov chains observed in continuous-time channels see 19, 20, we propose a new expectation maximization EM algorithm for data of TTFs in this paper
Based on the theories of continuous-time Markov chains and EM algorithms, this paper deals with the problem of parameters estimation for extended Markov-models
Summary
Among many quantitative analysis methods of system reliability, the state space representation has often been employed by reliability engineers, for example, 1–5. The state space representation has been recognized by industrial IEC61511 standards 6 This method assumes that the structure of the system is modelled with a continuous-time Markov chain CTMC. In the state space representation, nodes represent states of the system and arcs represent the transitions between nodes This method is well adapted to study the reliability of various systems and allows an exact analysis of their probability of failure. In the extended Markov-model, an operation state is divided into substates with different levels of failure rates, which result in a nonconstant failure rate of the operation state That is, in these models, the bug of failure rate functions is fixed within the framework of the state space representation see 12, 13 and references therein for some applications.
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