Markov modelling technique

IEC 61165 and ISA TR 84.00.02-4 illustrate the Markov modelling technique for calculating the probabilities of failures for safety instrumented functions designed in accordance with lEG 64511-1 ANSI/ISA-84.00.01-2004 Part 1 (IEC 61511-1 Modi and this standard. 

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The following is provided as an example to illustrate the various steps performed to develop a SIS arohiteoture, whioh satisfies the requirements of lEC 61511-1 ANSI/ISA-84.00.01-2004 Part 1 (lEC 61511-1 ModT SIS engineering follows guidelines and praotices and uses standardized equipment as outlined below. 

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Markov modeling is a technique for calculating system reliability as exponential transitions between various states of operability, much like atomic transitions. In addition to the use of constant transition rates, the model depends only on the initial and final states (no memory). 

Finally, using a linear phonemic representation has the benefit that it makes automatic database labelling considerably easier. In Chapter 17 we shall consider the issue of labelling a speech database with phoneme units, both by hand and by computer. As we shall see, it is easier to perform this labelling if we assume a linear pronunciation model, as this works well with automatic techniques such as hidden Markov models. 

While many possible approaches to statistical synthesis are possible, most work has focused on using hidden Markov models (HMMs). This along with the unit selection techniques of the next chapter are termed third generation techniques. This chapter gives a full introduction to these and explains how they can be used in synthesis. In addition we also show how these can be used to automatically label speech databases, which finds use in many areas of speech technology, including unit selection synthesis. Finally, we introduce some other statistical synthesis techniques. 

Several different modeling techniques have been developed. In this chapter block diagrams (also called network modeling), fault trees, and Markov models are presented in a simple introductory way. More advanced and realistic modeling techniques are covered in later chapters. 

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). 

Markov models are generally considered more flexible than other methods. On a single drawing, a Markov model can show the entire operation of a fault tolerant control system including multiple failure modes. Different repair rates can be modeled for different failure situations. If the model is created completely, it will show full system success states. It will also show degraded states where the system is still operating successfully but is vulnerable to further failures. The modeling technique provides clear ways to express failure sequences and can be used to model time dependent probabilities. 

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 calculations may be done with simplified equations, fault trees, Markov models or other techniques depending on the complexity of the model and the demand mode of operation. 

The PFDavg may be calculated using one of several different techniques as described in Chapter 5. The most common approaches involve simplified equations, fault trees or Markov models. The resultant PFDavg value must be compared with a table from ANSl/ISA-84.00.01-2004 (lEC 61511 Mod)... 

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. 

The technique can be extended to calculate the MTTF for any given failure state. If the metric of interest was mean time to dangerous failure in a multi-failure state Markov model then the model could be modified and the MTTF to a particular state could be calculated (see Ref. 3). 

In many ways modeling the repair process is difficult because the repair process is quite different from the failure process. Random failures are due to a stochastic process and most of our modeling techniques were created for these stochastic processes. Certain aspects of the repair process are deterministic. Other aspects of the repair process are stochastic. Fortunately, we can approximate the repair process more accurately with Markov models than most other techniques. 

This situation can be modeled using probability combination techniques or with Markov models. Consider the multiple failure state Markov model of Figure G-7. This model shows a simplified loo2 system without common cause or diagnostics. (For a complete model of a loo2 system, see Appendix F.)... 

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