Aggregated Markov-based reliability analysis of multi-state systems under combined dynamic environments

Reliability of a system may differ greatly when operating under different environments. However, the existing works have either neglected the environment factor in system reliability analysis or considered this factor for binary systems or systems subject to a single environment (parameter). In this paper, we make contributions by modeling a multi-state system operating under hybrid dynamic environments affected by multiple environmental parameters. Different Markov chains with finite states are used to represent the random system behavior and dynamic environments, leading to an aggregated Markov process that models the overall system behavior. An effective approach based on state partitions and aggregations is suggested for assessing the system reliability indexes, including reliability, availability, multi-point availability, and environment-based reliability. A high-pressure homogenizer system is analyzed to demonstrate the proposed model and show the comparison of the reliability of system under fixed and dynamic environment.     

A semi-Markovian model allowing for inhomogenities with respect to process time

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q_ij(t), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ_ij( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N² integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.     

Reliability assessment of reliquefaction systems on LNG carriers

The paper gives an introduction to reliability assessment of reliquefaction systems for boil-off gas (BOG) on LNG carriers with focus on redundancy optimization and maintenance strategies. The reliability modeling is based on a time-dependent Markov approach. Four different system options are studied, with varying degree of redundancy. Failures in the reliquefaction system may require flaring of the BOG, and the associated cost is compared with the cost of introducing redundancy and the cost of onboard maintenance. A model for maintenance optimization is developed and exemplified on a main unit of the reliquefaction system. Reliability and maintenance cost data for reliquefaction systems on LNG ships are very scarce. The input data have been collected from the best available data sources and adjusted by expert judgement. A tailor-made computer program has been developed and may be supplied to interested readers.     

An optimized technique for reliability analysis of safety‐critical systems: A case study of nuclear power plant

Stochastic models are extensively used in quantifying the reliability of safety critical systems. These models use the state‐space model for reliability quantification. Markov chain is comprehensively used in describing a sequence of possible events of any system in which the probability of each event depends only on the state attained in the previous event. Markov chains are convenient to model the software system of the SCS with the help of Petri Nets, a directed bipartite graph widely used for the verification and validation of real‐time systems. However, the stochastic model suffers from the state‐space explosion problem. In this paper, we proposed a technique for reliability analysis of safety critical systems, excavating into the coherent optimization of Markov chain. The approach has been validated on 17 safety critical systems of nuclear power plants.     

Stochastic demand patterns for Markov service facilities with neutral and active periods

In an earlier paper, a closed form expression was obtained for the joint interval reliability of a Markov system with a partitioned state space S=U∪D, i.e. for the probability that the system will reside in the set of up states U throughout the union of some specific disjoint time intervals I_?=[θ_?,θ_?+ζ_?],?=1,…,k. The deterministic time intervals I_? formed a demand pattern specifying the desired active periods.In the present paper, we admit stochastic demand patterns by assuming that the lengths of the active periods, ζ_?, as well as the lengths of the neutral periods, θ_?-(θ_?-1+ζ_?-1), are random. We explore two mechanisms for modelling random demand: (1) by alternating renewal processes; (2) by sojourn times of some continuous time Markov chain with a partitioned state space. The first construction results in an expression in terms of a revised version of the moment generating functions of the sojourns of the alternating renewal process. The second construction involves the probability that a Markov chain follows certain patterns of visits to some groups of states and yields an expression using Kronecker matrix operations.The model of a small computer system is analysed to exemplify the ideas.     

Condition-based fault tree analysis (CBFTA): A new method for improved fault tree analysis (FTA), reliability and safety calculations

Condition-based maintenance methods have changed systems reliability in general and individual systems in particular. Yet, this change does not affect system reliability analysis. System fault tree analysis (FTA) is performed during the design phase. It uses components failure rates derived from available sources as handbooks, etc. Condition-based fault tree analysis (CBFTA) starts with the known FTA. Condition monitoring (CM) methods applied to systems (e.g. vibration analysis, oil analysis, electric current analysis, bearing CM, electric motor CM, and so forth) are used to determine updated failure rate values of sensitive components. The CBFTA method accepts updated failure rates and applies them to the FTA. The CBFTA recalculates periodically the top event (TE) failure rate (λ_TE) thus determining the probability of system failure and the probability of successful system operation—i.e. the system's reliability.FTA is a tool for enhancing system reliability during the design stages. But, it has disadvantages, mainly it does not relate to a specific system undergoing maintenance.CBFTA is tool for updating reliability values of a specific system and for calculating the residual life according to the system's monitored conditions. Using CBFTA, the original FTA is ameliorated to a practical tool for use during the system's field life phase, not just during system design phase.This paper describes the CBFTA method and its advantages are demonstrated by an example.     

Markov models for evaluating risk-informed in-service inspection strategies for nuclear power plant piping systems

As part of an EPRI sponsored research project to develop technology for risk informed in-service inspection evaluations, new methods and databases were developed to predict piping system reliability. The methods include a Markov modeling technique for predicting the influence of alternative inspection strategies on piping system reliability, and Bayes' uncertainty analysis methods for quantifying uncertainties in piping system reliability parameters. This article describes these methods and associated databases needed for their quantification with particular emphasis on the application of the Markov piping reliability model. Insights are developed regarding reliability metrics that should be used in Probabilistic Risk Assessments for estimating time dependent frequencies of loss of coolant accidents and internal flooding events. The methodology for developing estimates of all the input parameters of the piping reliability models is described including the quantitative treatment of uncertainties in risk informed applications. Examples are presented to demonstrate the practical aspects of applying the Markov model and developing the inputs needed for its quantification.     

Overshoot in biological systems modelled by Markov chains: a non‐equilibrium dynamic phenomenon

A number of biological systems can be modelled by Markov chains. Recently, there has been an increasing concern about when biological systems modelled by Markov chains will perform a dynamic phenomenon called overshoot. In this study, the authors found that the steady‐state behaviour of the system will have a great effect on the occurrence of overshoot. They showed that overshoot in general cannot occur in systems that will finally approach an equilibrium steady state. They further classified overshoot into two types, named as simple overshoot and oscillating overshoot. They showed that except for extreme cases, oscillating overshoot will occur if the system is far from equilibrium. All these results clearly show that overshoot is a non‐equilibrium dynamic phenomenon with energy consumption. In addition, the main result in this study is validated with real experimental data.Inspec keywords: Markov processes, physiological modelsOther keywords: biological systems, Markov chains, nonequilibrium dynamic phenomenon, overshoot, steady‐state behaviour, oscillating overshoot, simple overshoot     

A discrete-time Bayesian network reliability modeling and analysis framework

Dependability tools are becoming an indispensable tool for modeling and analyzing (critical) systems. However the growing complexity of such systems calls for increasing sophistication of these tools. Dependability tools need to not only capture the complex dynamic behavior of the system components, but they must be also easy to use, intuitive, and computationally efficient. In general, current tools have a number of shortcomings including lack of modeling power, incapacity to efficiently handle general component failure distributions, and ineffectiveness in solving large models that exhibit complex dependencies between their components. We propose a novel reliability modeling and analysis framework based on the Bayesian network (BN) formalism. The overall approach is to investigate timed Bayesian networks and to find a suitable reliability framework for dynamic systems. We have applied our methodology to two example systems and preliminary results are promising. We have defined a discrete-time BN reliability formalism and demonstrated its capabilities from a modeling and analysis point of view. This research shows that a BN based reliability formalism is a powerful potential solution to modeling and analyzing various kinds of system components behaviors and interactions. Moreover, being based on the BN formalism, the framework is easy to use and intuitive for non-experts, and provides a basis for more advanced and useful analyses such as system diagnosis.     

Beyond its cost, the value of maintenance: An analytical framework for capturing its net present value

Maintenance planning and activities have grown dramatically in importance across many industries and are increasingly recognized as drivers of competitiveness if managed appropriately. Correlated with this observation is the proliferation of maintenance optimization techniques in the technical literature. But while all these models deal with the cost of maintenance (as an objective function or a constraint), only a handful addresses the notion of value of maintenance, and seldom in an analytical or quantitative way.In this paper, we propose that maintenance has intrinsic value and argue that existing cost-centric models ignore an important dimension of maintenance, namely its value, and in so doing, they can lead to sub-optimal maintenance strategies. We develop a framework for capturing and quantifying the value of maintenance activities. Our framework is based on four key components. First, we consider systems that deteriorate stochastically and exhibit multi-state failures, and model their state evolution using Markov chains and directed graphs. Second, we consider that the system provides a flow of service per unit time. This flow in turn is “priced” and a discounted cash flow is calculated resulting in a present value (PV) for each branch of the graph—or “value trajectory” of the system. Third as the system ages or deteriorates, it migrates towards lower PV branches of the graph, or lower value trajectories. Fourth, we conceptualize maintenance as an operator (in a mathematical sense) that raises the system to a higher PV branch in the graph. We refer to the value of maintenance as the incremental PV between the pre- and post-maintenance branches of the graphs minus the cost of maintenance. The framework presented here offers rich possibilities for future work in benchmarking existing maintenance strategies based on their value implications, and in deriving new maintenance strategies that are “value-optimized.”     

A multi-state Markov model for a short-term reliability analysis of a power generating unit

This paper presents a multi-state Markov model for a coal power generating unit. The paper proposes a technique for the estimation of transition intensities (rates) between the various generating capacity levels of the unit based on field observation. The technique can be applied to such units where output generating capacity is uniformly distributed. In order to estimate the transition intensities a special Markov chain embedded in the observed capacity process was defined. By using this technique, all transition intensities can be estimated from the observed realization of the unit generating capacity stochastic process. The proposed multi-state Markov model was used to calculate important reliability indices such as the Forced Outage Rate (FOR), the Expected Energy Not Supplied (EENS) to consumers, etc. These indices were found for short-time periods (about 100 h). It was shown that these indices are sensibly different from those calculated for a long-term range. Such Markov models could be very useful for power system security analysis and short-term operating decisions.     

Reliability analysis of standby systems with multi‐state elements subject to constant transition rates

Standby redundancy has been extensively applied to critical engineering systems to enhance system reliability. Researches on reliability evaluation for standby systems focus more on systems with binary‐state elements. However, multi‐state elements with different performances have played a significant role in engineering systems. This paper presents an approach for reliability analysis of standby systems composed of multi‐state elements with constant state transition rates and absorbing failure states. The approach allows modelling different standby systems beyond cold, warm and hot ones by taking into account differences in possible maintenance of elements in standby and operation modes and dependence of elements' operational behavior on their initial state at the time of activation. An iterative algorithm for reliability evaluation based on element state probabilities is suggested. Illustrating examples of evaluating reliability of different types of homogeneous and heterogeneous standby systems are demonstrated.     

Spin Fluctuations in A-like Superfluid Phase of 3He in Aerogel

The frequency spectrum of the spin fluctuations δ S(t) superimposed on the coherently precessing spin modes in the A-like superfluid phase in aerogel is analysed. It is shown that the low amplitude spin fluctuations could be most easily observed in the case of an uni-axially deformed aerogel. It is demonstrated, in particular, that for axially stretched (radially squeezed) aerogel described by the U(1)LIM model the fourth order harmonic in δ S _z (t) is erased, in contrast with what is expected for the long range Leggett orbital configuration in the ³He-A (ABM state).     

Reliability updating with equality information

In many instances, information on engineering systems can be obtained through measurements, monitoring or direct observations of system performances and can be used to update the system reliability estimate. In structural reliability analysis, such information is expressed either by inequalities (e.g. for the observation that no defect is present) or by equalities (e.g. for quantitative measurements of system characteristics). When information Z is of the equality type, the a priori probability of Z is zero and most structural reliability methods (SRM) are not directly applicable to the computation of the updated reliability. Hitherto, the computation of the reliability of engineering systems conditional on equality information was performed through first- and second-order approximations. In this paper, it is shown how equality information can be transformed into inequality information, which enables reliability updating by solving a standard structural system reliability problem. This approach enables the use of any SRM, including those based on simulation, for reliability updating with equality information. It is demonstrated on three numerical examples, including an application to fatigue reliability.     

Modeling safety instrumented systems with MooN voting architectures addressing system reconfiguration for testing

This paper addresses the modeling of probability of dangerous failure on demand and spurious trip rate of safety instrumented systems that include MooN voting redundancies in their architecture. MooN systems are a special case of k-out-of-n systems. The first part of the article is devoted to the development of a time-dependent probability of dangerous failure on demand model with capability of handling MooN systems. The model is able to model explicitly common cause failure and diagnostic coverage, as well as different test frequencies and strategies. It includes quantification of both detected and undetected failures, and puts emphasis on the quantification of common cause failure to the system probability of dangerous failure on demand as an additional component. In order to be able to accommodate changes in testing strategies, special treatment is devoted to the analysis of system reconfiguration (including common cause failure) during test of one of its components, what is then included in the model. Another model for spurious trip rate is also analyzed and extended under the same methodology in order to empower it with similar capabilities. These two models are powerful enough, but at the same time simple, to be suitable for handling of dependability measures in multi-objective optimization of both system design and test strategies for safety instrumented systems. The level of modeling detail considered permits compliance with the requirements of the standard IEC 61508. The two models are applied to brief case studies to demonstrate their effectiveness. The results obtained demonstrated that the first model is adequate to quantify time-dependent PFD of MooN systems during different system states (i.e. full operation, test and repair) and different MooN configurations, which values are averaged to obtain the PFD_avg. Also, it was demonstrated that the second model is adequate to quantify STR including spurious trips induced by internal component failure and by test itself. Both models were tested for different architectures with 1≤N≤5 and 2≤M≤5 subject to uniform staggered test. The results obtained also showed the effects that modifying M and N has on both PFD_avg and STR, and also demonstrated the conflicting nature of these two measures with respect to one another.     

Construction of event-tree/fault-tree models from a Markov approach to dynamic system reliability

While the event-tree (ET)/fault-tree (FT) methodology is the most popular approach to probability risk assessment (PRA), concerns have been raised in the literature regarding its potential limitations in the reliability modeling of dynamic systems. Markov reliability models have the ability to capture the statistical dependencies between failure events that can arise in complex dynamic systems. A methodology is presented that combines Markov modeling with the cell-to-cell mapping technique (CCMT) to construct dynamic ETs/FTs and addresses the concerns with the traditional ET/FT methodology. The approach is demonstrated using a simple water level control system. It is also shown how the generated ETs/FTs can be incorporated into an existing PRA so that only the (sub)systems requiring dynamic methods need to be analyzed using this approach while still leveraging the static model of the rest of the system.     

基于HMM的多态系统状态识别模型研究

在分析装备三级工作模式基础上,建立了多态系统的可靠性向量模型,基于隐马尔可夫模型(HMM)原理分析了多态系统的状态转移过程,并建立了多状态系统HMM模型,在此基础上利用MATLAB对系统隐藏状态转移和观察状态之间的关系进行了仿真.仿真结果表明,在一定观察序列的情况下,该模型可以实现较高的状态识别率和状态预测精度,为科学合理地诊断多状态系统的潜在故障提供了技术支持,具有重要的理论意义和应用价值.     

Availability optimisation for stochastic degrading systems under imperfect preventive maintenance

This paper deals with imperfect preventive maintenance (PM) optimisation problem. The system to be maintained is typically a production system assumed to be continuously monitored and subject to stochastic degradation. To assess such degradation, the proposed maintenance model takes into account both corrective maintenance (CM) and PM. The system undergoes PM whenever its reliability reaches an appropriate value, while CM is performed at system failure. After a given number of maintenance actions, the system is preventively replaced by a new one. Both CM as well as PM are considered imperfect, i.e. they bring the system to an operating state which lies between two extreme states, namely the as bad as old state and as good as new state. The imperfect effect of CM and PM is modelled on the basis of the hybrid hazard rate model. The objective of the proposed PM optimisation model consists on finding the optimal reliability threshold together with the optimal number of PM actions to maximise the average availability of the system. A mathematical model is then proposed. To solve this problem an algorithm is provided. A numerical example is presented to illustrate the proposed maintenance optimisation model.     

Optimal joint maintenance and operation policies to maximise overall systems effectiveness

In this paper, the problem of determining the optimal maintenance and operation policies for a multi-state, multi-stage machine maintenance problem is considered. This problem has been formulated in the literature as a Partially Observed Markov Decision Process (POMDP). A new formulation that explicitly ties maintenance, operation, and quality within the POMDP framework is provided. The new formulation maximises Overall Systems Effectiveness for an n-state system with multiple speed and maintenance actions. The model provides, for each time epoch, a set of optimal maintenance and production-rate actions. The decision-maker (controller) can select the optimal policy depending on the system state occupancy vector (belief state). A realistic numerical model is presented to demonstrate the model utility.