# Markov Chains and Dependability Theory

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Product Highlights Covers fundamental and applied results of Markov chain analysis for the evaluation of dependability metrics, for graduate students and researchers. About This Item We aim to show you accurate product information. Manufacturers, suppliers and others provide what you see here, and we have not verified it.

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Dependability metrics are omnipresent in every engineering field, from simple ones through to more complex measures combining performance and dependability aspects of systems. This book presents the mathematical basis of the analysis of these metrics in the most used framework, Markov models, describing both basic results and specialized techniques.

The authors first present both discrete and continuous time Markov chains before focusing on dependability measures, which necessitate the study of Markov chains on a subset of states representing different user satisfaction levels for the modelled system. Topics covered include Markovian state lumping, analysis of sojourns on subset of states of Markov chains, analysis of most dependability metrics, fundamentals of performability analysis, and bounding and simulation techniques designed to evaluate dependability measures.

Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. Email or Customer ID. Forgot password? Old Password. New Password. Password Changed Successfully Your password has been changed. The precise determination of the factors that influence this degradation is a complex issue.

### Description

In the engineering field, many problems are related to natural processes and phenomena that are inherently random, and some variables related to them may not be considered as constant in time. Balaji Rao and Anoop 1 reported that the corrosion state of a reinforcing bar in a specified bridge girder, exposed to given nominal environmental conditions, is a random variable. The corrosion state of the reinforcing bar also varies along the length of the bridge. Thus, the corrosion state of the reinforcing bar in a bridge has to be modeled as a random process. Therefore, many decisions made during the planning and project phases of engineering processes are invariably accomplished under uncertain conditions 2.

## Markov Chains and reliability analysis for reinforced concrete structure service life

In some cases, the use of stochastic processes, such as the Reliability Theory of Markov Chains, may be advantageous to account for the variability in the main parameters that have a significant influence on the degradation process. Reliability theory is one of the first stochastic methods to be used for probabilistic predictions of the service life SL Appendix 1 of reinforced concrete structures RC , whose probability reliability is related to the perfect operation of a certain component during a specific period of time in its normal conditions of use 3.

Other probabilistic approaches are currently employed for studies of engineering systems, such as Fuzzy Logic, Neural Networks and Markov Chains 1, In this study, due to the short bibliography about the theme, the basic principles of Markov Chains are discussed. Higher considerations about the reliability theory and its applications may be found in the studies of Stewart and Melchers 8 and Ang and Tang 2.

The Markov Chain is a special case of a stochastic process of discreet parameters whose development may be conducted through a series of transitions between the states of a system. According to Ang and Tang 2 , take a system with m possible states, called 1, 2, The probability of the system is in a future state depending exclusively on the present state of the system, thus creating a Markov Chain, whose conditional probability is given by Equation 1.

This type of process is also known as memoryless process because the past is "forgotten"; only the present time is actually taken into consideration.

## Semi-Markov Processes: Applications in System Reliability and Maintenance

For a Markov Chain of discreet parameters, the probability that the system takes the state j at the time t n once it was in the state i at the given time t m is called the transition probability p ij , and it is represented by Equation 2. Because m represents the number of possible states for the system over time, the transition probabilities between the states of the system may be represented by a matrix m x m called the transition probability matrix P Equation 3. Because the system states are mutually exclusive and collectively exhaustive after each transition, the values that compose the matrix must be non-negative and lie between 0 and 1, and the addition of the entrance of each line must equal 1.

The probabilities of the initial state of a system P 0 may be represented by a line matrix Equation 4 , also called the initial condition vector in which p i 0 is the probability of the system in the initial state i. After a transition, the probability of the system in state j is given by Equation 5, which is represented in matrix denomination by Equation 6. Therefore, the probability of the system being in state j after n transitions is given by Equation 7.

For engineering applications, the Markov Chains models are based on cumulative probabilities of the degradation of a determined system or component of the system. These probabilities, in general, are obtained by means of visual inspections, degradation models or experts' knowledge and reliability theory 9, Attention must be given to the limitations related to the determination of the initial probabilities of the degradation system, which are usually made by visual inspections of a structure and taken as constant for the system From information derived from experts, Balaji Rao et al. In the methodology used by the researchers, the experts evaluated the damage state of the structure, which is expressed in terms of probability of the structure to achieve a given level of degradation.

An example problem of a reinforced concrete bridge girder is presented, which illustrates the usefulness of the proposed methodology in facilitating decision making regarding repair. The project Lifecon 5 employed Markov Chains for SL studies, emphasizing the benefits of this application when combined with degradation models.

Some authors 7,9 also used Markov Chains for durability studies and SL prediction in RC, proving them advantageous for engineering studies. Markov Chains may be used at any time of an SL of a structure, considering the possible actions of maintenance and repair and the costs inherent to them. Moreover, one of the most important properties of this method is that it provides a system that is mathematically linear, enabling linear optimization. Before these facts, and as a result of field observation, for the initiation period of corrosion due to degradation by chlorides, reliability analysis and Markov Chains are jointly employed for the purposes of the SL prediction of RC following the Brazilian normative specifications.

The decision to employ Markov Chains to predict the SL of RC, together with the reliability theory, aims to consider the uncertainties of the degradation process until the structures reach the durability limit state DLS. It can be said that the period of corrosion initiation corresponds to the time t that the chloride ions take to reach the steel bars. Through natural aging tests Figure 1 , data about chloride penetration into concrete over time were obtained.

## Markov Chains and Dependability Theory (E-Book, PDF)

The solution of Fick's Second Law was used for the adjustment of the penetration profiles to determine the values of the diffusion coefficient D and the superficial concentration C s. The cover thicknesses related to the environmental classification adopted by national Brazilian standards 12 were used.

For each variable of study, the uncertainties of the degradation process, as shown in Table 1 , were included. The coefficient of variation CV and the probability density function p. Considering a system with two possible states depending on the C cr value, the initial probabilities were determined by reliability analysis. The experimental data of a six-month exposure to chlorides were considered in Markov as p ij in the P , also called the probability matrix of system degradation.

The p ij values of the system were estimated using Markov Chains of discreet processes, by means of simulation, thus determining the DLS for the studied concretes. After seven days of humid curing and 21 days in a laboratory environment, the beams were submerged in a NaCl solution with a concentration of 3. A cylindrical concrete specimen for compression strength testing at 28 days was also manufactured. The chloride amount was determined by titulometry based on powdery samples collected at depths of 5, 10, 15, 20 and 25 mm in relation to the surface Figure 2c.

With these data, it was possible to make penetration profiles of the chloride. Through adjustments using the least square method Equation 8 , the values of C s and D for the analyzed concretes were calculated. In this study, concrete was considered a homogenous and isotropic material. The only transportation mechanism is the diffusion, and it is a one-dimensional analysis. The possible local variations in the surfaces of the beams were not considered, assuming that penetration is uniform and that the concrete structure was well executed.

For simulation purposes, it was assumed that two structures with coating thicknesses x corresponding to 30 mm and 20 mm would be constructed in the same degradation environment sea environment of moderate aggression. It is assumed that a structure standing between and meters away from the sea is found in an environment of moderate aggressiveness.