Markov analysis provides a means of analysing the reliability and availability of systems whose components exhibit strong dependencies. Other systems analysis methods (such as the Kinetic Tree Theory method employed in fault tree analyses) generally assume component independence that may lead to optimistic predictions for the system availability and reliability parameters. Some typical dependencies that can be handled using Markov models are:
- Components in cold or warm standby
- Common maintenance personnel
- Common spares with a limited on-site stock
The major drawback of Markov methods is that Markov diagrams for large systems are generally exceedingly large and complicated and difficult to construct. However, Markov models may be used to analyse smaller systems with strong dependencies requiring accurate evaluation. Other analysis techniques, such as fault tree analysis, may be used to evaluate large systems using simpler probabilistic calculation techniques. Large systems which exhibit strong component dependencies in isolated and critical parts of the system may be analysed using a combination of Markov analysis and simpler quantitative models.
The state transition diagram identifies all the discrete states of the system and the possible transitions between those states. In a Markov process the transition frequencies between states depends only on the current state probabilities and the constant transition rates between states. In this way the Markov model does not need to know about the history of how the state probabilities have evolved in time in order to calculate future state probabilities. Although a true Markovian process would only consider constant transition rates, computer programs such as FaultTree+ and MKV allow time-varying transition rates to be defined. These time-varying rates must be defined with respect to absolute time or phase time (the time elapsed since the beginning of the current phase).
As the size of the Markov diagram increases the task of evaluating the expressions for time-dependent unavailability by hand becomes impractical. Computerised numerical methods may be employed, however, to provide a fast solution to large and complicated Markov systems. In addition these numerical methods may be extended to allow the modelling of phased behaviour and time-dependent transition rates.
For more information on Markov Analysis and its integration with other reliability methods visit Isograph’s web site at www.isograph.com.
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