Markov model solution techniques

Markov models are a reliability and safety modeling technique that uses state diagrams. These diagrams have only two simple s)rmbols (see Figure 5-17) a circle representing a working or a failed system state and a transition arc representing a movement between states caused by a failure or a repair. Solution techniques for Markov models can directly calculate many different metrics compared to other reliability and safety evaluation techniques (Ref. 9). 

Most analysts doing safety instrumented system modeling use either fault trees or Markov models. Both methods provide a clear way to express the reality of multiple failure modes. Both methods, however, require careful modeling and appropriate solution techniques. Realistic levels of detad... 

This definition is used to obtain numerical results in several of the system modeling techniques. In a discrete time Markov model using numerical solution techniques, a direct average of the time dependent numerical values will provide the most accurate answer. When analytical equations for PFD are obtained using a fault tree, the above equation can be used to... 

The steady-state solution technique is useful for many situations. However, it is not appropriate for situations where the probability of moving from state to state is not constant (a non-homogeneous Markov model). It is also not appropriate for absorbing Markov models. This solution technique is not appropriate for safety instrumented functions where many failures are not detected until a periodic inspection and repair is performed. In the case of failures detected by a non-constant inspection and test process, the probability of repair is not constant. It is zero for most time periods. Do not use steady-state techniques to model repair processes with inspection and test. 

For discrete time, discrete state Markov models numerical solutions for probability of being in any state can be obtained by simple matrix multiplication. This technique can be used to solve many realistic models, regular or absorbing. The technique may be even used on certain non-homogeneous models to include deterministic events as well as probabilistic events. 

One of the attractive features of Markov models is that numerical results can be obtained without an extensive knowledge of probabihty theory and techniques. The intuitive ideas of probabihty discussed in the previous section are sufficient for the construction of a Markov model. Once the model is constructed, it can be computed by differential equations. The solution does not require any additional knowledge of probabihty. 

Different techniques are suitable for different tasks. For example, BD focuses on molecules and particles in solution where the solvent is implicitly lumped into a friction force. On the other hand, DSMC and LB are typically applied to various fluid-related problems. MD is the only fundamental, first principles tool where the equations of motion are solved using as input an interparticle potential. MC methods map the system description into a stochastic Markov-based framework. MD and MC are often thought of as molecular modeling tools, whereas the rest are mesoscopic tools (lattice MC is also a mesoscopic tool). 

---