Markovian Modelling

This chapter describes the Markov graph approach. It is based on a graphic description of the states and transitions of the modelled systems and represents an underlying random process where all the events are modelled by exponential distributions (i.e. constant transition rates). It is equivalent to a set of homogeneous differential equations with constant coefficients and its future, from a given instant, depends only on its state at this given instant (memoryless property). Beyond the classical numerical calculation, a powerful algorithm based on the series expansion of the transition matrix is proposed. This allows to calculate at the same time both the state probabilities and the accumulated sojourn times in the states. In conjunction with the critical states this, in turn, allows to calculate all the various probabilistic parameters of interest (availability, reliability, failure frequency, failure intensity, failure rate, Vesely failure rate, MUT, MDT MTTF or MTBF). Approximations are provided in the case where the failures of all the components are quickly detected and repaired. Multistate (e.g. production systems) and multiphase (e.g. periodically tested systems) modelling are described as well as sequence modelling and common cause failures (beta-factor, shock and semi-catastrophic models) modelling. Exercises are provided in Chap. 34 .