Markov Method

This is a widely used method to perform reliability analysis of engineering systems and is named after the Russian mathematician, Andrei A. Markov (1856-1922). The method is commonly used to model repairable systems with constant failure and repair rates. The following assumptions are associated with the method [11]  

The application of the Markov method is demonstrated by solving the following example. 

Assume that a transportation system can either be in an operating or a failed state. 

Transportation system operating normally f t5 Transportation system failed 

At = probability of transportatiori system failure in finite time interval At Pj,At = probability of transportation system repair in finite time interval At RJit + At) = probability of the transportation system being in operating state 0 at time (t + At) 

Assume that a system used in the oil and gas industry can either be in an operating state or a failed state. The constant failure rate and the 

X is the probability of the oil and gas industry system failure in finite time interval At. 

The following assumptions are associated with this method [9]  

State space diagram representing the nursing professionals. 

By solving Equations (4.7) and (4.8) using Laplace transforms, we obtain 

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In Section 2.5.4, we found the availability of a repairable emergency generator (EG) by Markov methods. If a plant requires that two identical, independent EGs must both work for time T for success. What is the probability of this Assume the failure rates are A., A., and the repair rates are p, p-,. 

Farrell, J. A., S. Pang, and W. Li. Plume mapping via hidden Markov methods. Ieee Trans. Syst., Man, and Cybernetics—Part B Cybernetics 33(6), 850-863 (2003). 

Given the transition rates ayt standard Markov methods may be used to find the steady state and time dependent probabilities at component level, see Rausand Hoyland (2004). 

By applying the dimension reduction procedure indicated above we are able to limit the number of states to a manageable size. At the system level we then obtain the steady state probabilities and hence system regularity by standard Markov methods. The model has been applied in relation to oil production where reservoir characteristics have to be included in the model. For example for a gas injection system a degraded performance (e.g., 66.67% injection capacity) may have no immediate effect on oil production. This is due to the fact that produced gas is sent to flare for a shorter period of time without affecting the reservoir performance. Due to environmental restriction imlimited flaring is not acceptable, hence after a period of some hours production has to be reduced... 

The authors have so far not succeeded to identify practically used systems where not even custom formulas can be developed. Such systems can of cause be constructed by negating the assmnptions of lEC 61508, part 6. B.3.1, but that does not make these systems common in industry . E.g. a multi-channel system with complex, but specified interaction would be a candidate for system simulation. If in addition exponential failure rates are not appropriate, even Markov Methods would be difficult to use. Such systems do actually exist in industrial apphcations, but normally the information about the interaction between the channels cannot be specified well enough to make dedicated calculation worthwhile and one is often left with 8-model estimations to tackle the common cause issues. 

The results which are obtained by using the three methods are are smnmarized in Table 4. As seen in this Table the /8-factor model and the PDS method provides a result which corresponds to SIL 2, while the Markov method gives a result which satisfies SIL 1 (see Table 1 for definition). 

Markov methods are also applicable for general MooN methods, and is in fact mathematically more accurate since none of these make the conservative assumptions when deriving the formulas for the 8-factor model and the PDS method (Rausand Hoyland 2004). 

AU of them require simplifying assumptions about time to failure behavior of the system components. Moreover, Markov method analyses the system by identifying all the different states in which the system can reside and is able to produce accurate system reUabUity measures by assigning rates of transition between these states. However, the Markov method has its own drawbacks in its appUcation for a relatively large system to establish the state transition model is an intractable task. 

Today, various mathematics and probability concepts are being used to study various types of safety-related problems. For example, probability distributions are used to represent times to human error in performing various types of time-continuous tasks in the area of safety [3-7]. In addition, the Markov method is used to conduct human performance reliability analysis in regard to engineering systems safety [7-9]. 

Assume that the nursing professionals at a health care facility make errors at a constant rate X. A state space diagram, shown in Figure 4.7, exhibits this scenario. The numerals in the figure denote system states. With the aid of the Markov method, develop expressions for calculating the nursing professionals ... 

Using the Markov method, we write down the following equations for states 1 and 2, respectively, shown in Figure 4.7 [1,9] ... 

By using the Markov method presented in Chapter 4, we write down the following set of equations for the Figure 11.2 diagram [2,12] ... 

Other classical methods to predict the fatigue behavior in complex or random load patterns involve the so-called rainflow and Markov methods (28). These methods utilize any form of a cumulative damage theory to predict the lifetime in random loading situations. 

Zajac, M. Kierzkowski, A. (2012) Uncertainty assessment in semi Markov methods for Wdbull functions distributions. Advances in Safety, Reliability and Risk Management—Proceedings of the European Safety and Reliability Conference, ESREL2011 1161 1166. 

The state-transition model can be analyzed using a number of approaches as a Markov chains, using semi-Markov processes or using Monte Carlo simulation (Fishman 1996). The applicability of each method depends on the assumptions that can be made regarding faults occurrence and a repair time. In case of the Markov approach, it is necessary to assume that both the faults and renewals occur with constant intensities (i.e. exponential distribution). Also the large number of states makes Markov or semi-Markov method more difficult to use. Presented in the previous section reliability model includes random values with exponential, truncated normal and discrete distributions as well as some periodic relations (staff working time), so it is hard to be solved by analytical methods. 

Fault tree analysis (FTA), failure modes and effect analysis (FMEA), and the Markov method are the examples of methods that can be used in both safety and reliability fields. The FTA method was developed in the early 1960s for analyzing the safety of rocket launch control systems, and FMEA was developed in the early 1950s for analyzing the reliability of engineering systems. 

The Markov method is named after the Russian mathematician Andrei A. Markov (1856-1922) and is a highly mathematical approach that is frequently used for performing various types of safety and reliability analyses in engineering systems. This chapter presents a number of methods and approaches extracted from the published literature, considered useful to perform safety and reliability analyses in the oil and gas industry. 

Chapter 3 presents introductory aspects of safety and reliability. Chapter 4 presents a number of methods considered useful for performing safety and reliability analyses in the oil and gas industry. These methods are root cause analysis, hazard and operability analysis, technique of operations review, interface safety analysis, preliminary hazard analysis, job safety analysis, failure modes and effect analysis, fault tree analysis, and the Markov method. 

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