A study of a two-phase inspection policy for a preparedness system with a defective state and heterogeneous lifetime

This paper considers an inspection policy for a single component protection or preparedness system, in which the component arises from a heterogeneous population. At any point in time, the system may be in one of three states, good, defective or failed. The system is only required in an emergency, and in order to ensure high availability of the system on-demand, the system undergoes a sequence of inspections. Inspection determines the system state, so that if a transition from the good state occurs between inspections it is not revealed until subsequent inspection. When a defect or failure is revealed, the component is replaced. At the final inspection the component is replaced. We suppose that a component may be either weak or strong, so that the time in the good state has a distribution that is a mixture. In these circumstances, the efficacy of a two-phase inspection policy, with an anticipated high inspection frequency in early life and low inspection frequency in later life, is considered using availability and cost criteria. The policy is investigated in the context of a valve in a natural gas supply network. If the lifetime distributions in the mixture are quite distinct, then cost savings of the order of 5% can be achieved by using the two-phase policy in place of the simpler single phase policy. Furthermore, only if the mean time in the defective state is small or the required availability is very high does the two-phase policy tend to mimic a burn-in policy.     

Production policies under deteriorating process conditions

This paper examines a single-stage production system that manufactures multiple products under deteriorating equipment conditions. The machine condition worsens with production, and improves with maintenance. The condition of the process can be in any one of several discrete states, and transitions from state to state follow a semi-Markov process. In many production environments, the quality or yield of output depends heavily on the condition of the production process. The problem considers the trade-offs between manufacturing products that have a higher profit, a longer processing time, and therefore, a higher deterioration probability versus products that have a smaller profit, shorter processing time with a lower process deterioration probability. The firm needs to determine the optimal production choice in each state in a way that maximizes the long-run expected average reward per unit time.

The paper makes three sets of contributions. First, it introduces the concept of critical ratios for the firm's manufacturing decision at each state regarding whether to switch from one product to another. Second, through the use of critical ratios, the main result shows that the optimal production choice for each state can be determined independently of the actions taken in other states, despite the complex interconnections between the production decisions and state transitions. Third, the paper provides generalizations that illustrate the depth, scope and richness of the proposed solution technique by extending the model in the number of machine states, to settings where maintenance is performed in intermediate states, and to settings where transition probabilities are influenced by both mean and variance of processing times.     

Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations

A new approach for an efficient numerical implementation of the path integral (PI) method based on non-Gaussian transition probability density function (PDF) and the Gauss-Legendre integration scheme is developed. This modified PI method is used to solve the Fokker-Planck (FP) equation and to study the nature of the stochastic and chaotic response of the nonlinear systems. The steady state PDF, periodicity, jump phenomenon, noise induced changes in joint PDF of the states are studied by the modified PI method. A computationally efficient higher order, finite difference (FD) technique is derived for the solution of higher-dimensional FP equation. A two degree of freedom nonlinear system having Coulomb damping with a variable friction coefficient subjected to Gaussian white noise excitation is considered as an example which can represent a bladed disk assembly of turbo-machinery blades. Effects of normal force and viscous damping on the mean square response are investigated.     

Accounting for randomness in measurement and sampling in studying cancer cell population dynamics

Knowing the expected temporal evolution of the proportion of different cell types in sample tissues gives an indication about the progression of the disease and its possible response to drugs. Such systems have been modelled using Markov processes. We here consider an experimentally realistic scenario in which transition probabilities are estimated from noisy cell population size measurements. Using aggregated data of FACS measurements, we develop MMSE and ML estimators and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes. Our numerical results show that the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach for noisy measurements. This provides support for our argument that for the analysis of FACS data one should consider the observed state as a random variable. The second problem we address is about the consequences of estimating the probability of a cell being in a particular state from measurements of small population of cells. We show how the uncertainty arising from small sample sizes can be captured by a distribution for the state probability.Inspec keywords: cancer, tumours, cellular biophysics, biomedical measurement, Gaussian distribution, maximum likelihood estimation, mean square error methods, hidden Markov models, fluorescence, random processes, convergence of numerical methodsOther keywords: cancer cell population dynamics, malignant tumours, tissue samples, normal tissue cells, disease, drugs, Markov process, cell population size measurement, hidden Markov model, noisy measurement, state transition probability, fluorescence‐activated cell sorting measurement, minimum mean square error estimator, maximum likelihood estimator, transition probability matrix, noise distributions, Gaussian distributions, MMSE, convergence mechanism, standard deterministic approach, stochastic phenomena, random variable     

Performability indicators for the traffic analysis of wide area networks

In this paper, a repairable circular consecutive-k-out-of-n:F system with one repairman is studied. It is assumed that the working time and the repair time of each component are both exponentially distributed and every component after repair is ‘as good as new’. Each component is classified as either a key component or an ordinary component. Key components have priority in repair when failed. By using the definition of generalized transition probability, the state transition probabilities of the system are derived. Important reliability indices are evaluated for an example.     

Optimal periodic replacement for a deteriorating production system with inspection and minimal repair

Here we discuss an inspection policy model for a deteriorating production system with minimal repair. A minimal repair is resorted to as and when the system is found to be in a failed state during an inspection unless it is apre-set overhaul/replacement time in which case the system is overhauled or replaced. Using a dynamic programming formulation, and assuming that the cost of minimal repair is a non-decreasing function of age, we arrive at the optimal inspection time that maximizes the profit per unit time for a given overhaul/replacement time. The procedure is then extended to determine the optimal periodic overhaul/replacement time and the corresponding optimal number of inspections and their schedule.     

Risk-based reconfiguration of safety monitoring system using dynamic Bayesian network

To prevent an abnormal event from leading to an accident, the role of its safety monitoring system is very important. The safety monitoring system detects symptoms of an abnormal event to mitigate its effect at its early stage. As the operation time passes by, the sensor reliability decreases, which implies that the decision criteria of the safety monitoring system should be modified depending on the sensor reliability as well as the system reliability. This paper presents a framework for the decision criteria (or diagnosis logic) of the safety monitoring system. The logic can be dynamically modified based on sensor output data monitored at regular intervals to minimize the expected loss caused by two types of safety monitoring system failure events: failed-dangerous (FD) and failed-safe (FS). The former corresponds to no response under an abnormal system condition, while the latter implies a spurious activation under a normal system condition. Dynamic Bayesian network theory can be applied to modeling the entire system behavior composed of the system and its safety monitoring system. Using the estimated state probabilities, the optimal decision criterion is given to obtain the optimal diagnosis logic. An illustrative example of a three-sensor system shows the merits and characteristics of the proposed method, where the reasonable interpretation of sensor data can be obtained.     

State discrimination of two pure states with a fixed rate of inconclusive answer

We investigate the optimum strategy for state discrimination of two pure states when the probability of inconclusive answers is fixed. By varying a given rate of inconclusive probabilities, the strategy optimally interpolates between Unambiguous and Minimum-Error discrimination, the two standard approaches to quantum state discrimination. We derive the explicit expressions of all the probabilities using an ancillary system and introducing unitary transformations which act on the input states and produce the output measured on the ancilla. By exploring physically accessible transformation acting on the input states, we present an optical scheme to implement the state discrimination.     

基于可靠性的最优检测策略研究

本文研究带有多维诊断参数的两部件系统,其中一个部件具有正常和故障两种状态,另一个部件具有正常、异常和故障三种状态。当系统工作时,为了判断后一个部件是正常还是异常,每隔一段随机时间对系统检测一次,直到系统故障或检测结果为该部件异常为止。利用概率分析、补充变量、Laplace变换和最优化方法,我们得到了系统的可靠性指标,并以此为基础导出了诊断参数的最优临界值和最优检测周期。     

Reliability analysis for a circular consecutive-2-out-of-n:F repairable system with priority in repair

A circular consecutive-2-out-of-n:F repairable system with one repairman is studied in this paper. When there are more than one failed component, priorities are assigned to the failed components. Both the working time and the repair time of each component is assumed to be exponentially distributed. Every component after repair is as good as new. By using the definition of generalized transition probability and the concept of critical component, we derive the state transition probability matrix of the system. Methodologies are then presented for the derivation of system reliability indexes such as availability, rate of occurrence of failure, mean time between failures, reliability, and mean time to first failure.     

Quantification of sequential failure logic for fault tree analysis

Fault tree analysis (FTA) is generally accepted as an efficient method for analyzing system failures. It is well known that a fault tree (FT) is equivalent to a minimal cut set fault tree with all minimal cut-AND structures. The minimal cut-AND structure is an AND conjunction of an output and all inputs that compose a minimal cut set. For the structure, the failed state of the output becomes true when all failed states of inputs exist simultaneously. There are cases where the output of the minimal cut-AND structure depends not only on all failed states of inputs but also on the sequence of occurrences of those failures. This sequential failure logic (SFL) is equivalently expressed with Priority-AND gates in FTA, where inputs to the gates have constant failure and repair rates. A probabilistic model for analysis of SFL was proposed and equations with multiple integration for arbitrary number of inputs were derived from the model. However, it is usually difficult to solve the multiple integration when the number of inputs exceeds a certain range. This paper presents analytical solutions of the probability that the output is in a failed state at time t and the statistically expected number of failures of the output per unit time at time t for the special case where inputs are characterized by common failure and repair rates. In addition, the analysis of FT involving SFL is demonstrated by means of software Mathematica.     

A general formulation for transition probabilities of Markov model and the application to fracture of composite materials

A generalized method for formulating unit-jump Markov chain model is presented. This method uses geometric series to determine the transition probabilities instead of using a moment generation function. Recursive equation for the transition probabilities has been derived in terms of the mean values. Present method is also compared by other method for robustness. Application of the unit-jump Markov chain model for characterizing fracture process of composites is described. By introducing a proper scaling factor, the entire probability history of the crack propagation at any loading states can be evaluated by the present model.     

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.     

Quantitative reliability analysis of different design alternatives for steer-by-wire system

For a fault-tolerant SBW (Steer-By-Wire) system, two structural designs with and without a backup mechanical coupling are proposed and assessed by a unified approach from a reliability point of view. An operation procedure to cope with partial system degradations prior to a complete loss of steering is presented for each design. The procedure can be represented by a transition diagram that, in turn, is converted into a layered Markov state transition diagram by introduction of augmented states. Reliabilities of the two designs are quantitatively compared by numerical integration of the Markov diagrams based on realistic failure rate data; system degradation probabilities are obtained as well as the complete steering loss probability. A power source is identified as a key component for the SBW to function without a mechanical backup.     

Online short-term reliability evaluation using a fast sorting technique

To consider time varying system operating conditions and operating reserve units, the online short-term reliability of a power system is investigated. Real-time system reliability based on the online operation data provided by the SCADA/EMS/WAMS is assessed using time- dependent state probabilities of components. Considering both speed and accuracy requirements, a fast sorting technique (FST) is proposed to quickly select the required number of system states in descending order probability. The number of computations and comparisons used in the evaluation is minimum. The relative accuracy of a given reliability index is denned as the stopping rule for the evaluation. As only a small number of system states are required to achieve the high accuracy of the short-term reliability indices when using the FST, the evaluation can be performed in an online environment. The proposed evaluation technique is illustrated by the application to the IEEE-RTS and China Southern Power Grid.     

A delay-time model with safety constraint

We consider the basic delay-time model in which a system has three states, the perfect functioning state, a defective state and the failure state. The system is deteriorating and to reduce the number of failures, preventive replacements are carried out when the system is in the defective state. The time in the defective state is referred to as the delay time. Inspections are required to check whether the system is in the defective state. System failures are safety critical and to control the risk, management considers two types of safety constraints: (i) the probability of at least one failure in the interval [0,A] should not exceed a fixed probability ω₁ and (ii) the fraction of time the system is in the defective state should not exceed a fixed limit ω₂. The problem is to determine optimal inspection intervals T, minimizing the expected discounted costs under the safety constraints. Conditions are established for when the safety constraints affect the optimal inspection time and causes increased costs.     

Communication theoretic image restoration for binary-valued imagery

We present a new image-restoration algorithm for binary-valued imagery. A trellis-based search method is described that exploits the finite alphabet of the target imagery. This algorithm seeks the maximum-likelihood solution to the image-restoration problem and is motivated by the Viterbi algorithm for traditional binary data detection in the presence of intersymbol interference and noise. We describe a blockwise method to restore two-dimensional imagery on a row-by-row basis and in which a priori knowledge of image pixel correlation structure can be included through a modification to the trellis transition probabilities. The performance of the new Viterbi-based algorithm is shown to be superior to Wiener filtering in terms of both bit error rate and visual quality. Algorithmic choices related to trellis state configuration, complexity reduction, and transition probability selection are investigated, and various trade-offs are discussed.     

Transient rate matrix determination for the complex system

This paper presents new techniques dealing with the reliability of complex systems. A power station reliability assessment model is used to illustrate the approach. Before calculating the state transient probabilities for the overall electrical power plant, the failure and repair rates of each generating unit are needed. These two coefficients can form a transition rate matrix, which represents each generator. To form the overall system a special technique is required. In the present study the direct‐differentiation and the Kronecker algebra methods are used to form the overall transition rate matrix, which represents the whole system. Forming the overall system is the first part (major part) of the current study. The second part is to calculate the state transient probabilities, following the Adams method. Two electric power plants from the State of Bahrain are analysed as examples. Copyright © 2001 John Wiley & Sons, Ltd.     

Reliability estimation and prediction of multi-state components and coherent systems

This paper discusses the multi-state coherent system composed of multi-state components. First, using the min cut sets or min path sets, we present our simulation algorithm, instead of the general structure function, to calculate the probability that the system is in a specified state. Second, we check the components per period, e.g. one check per year, to obtain the state sequences of all components. When the state sequences are Markovian chains, we can predict the reliability of the components in several periods, such as the probability that the components are in specified states. Also, we give two methods to compute the system reliability in a number of periods: one employs the states of the components in these periods, which can be predicted by the state transition probability matrixes of the components; the other uses the state transition probability matrix of the system obtained by the simulated states of the components.