Markov Reliability Models Research Articles

Markov reliability models of series systems with redundancy and repair facilities

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Statistical Taylor series expansion: An approach for epistemic uncertainty propagation in Markov reliability models

In this paper we develop a new Taylor series expansion method for computing model output metrics under epistemic uncertainty in the model input parameters. Specifically, we compute the expected value and the variance of the stationary distribution associated with Markov reliability models. In the multi-parameter case, our approach allows to analyze the impact of correlation between the uncertainty on the individual parameters the model output metric. In addition, we also approximate true risk by using the Chebyshev’ inequality. Numerical results are presented and compared to the corresponding Monte Carlo simulations ones.

Characterization Studies of Dual-Input Single-Output Converters for PV Applications

ABSTRACTIntegrating renewable energy source with storage devices and the grid depends heavily upon the power electronic converters. Multi-input converters play an essential role in nanogrid and microgrid. In this paper, two novel cost-effective configurations of dual input single output (DISO) converters are designed for the low voltage PV applications in a standalone microgrid with the objective to improve both reliability and efficiency. The performance characterization models of these converters viz. reliability and sensitivity models are developed to validate the robustness and effectiveness when there is a parametric variation or under the influence of various stresses. Markov reliability models are developed to estimate the mean time to system failure, using the Military Handbook for Reliability Prediction of Electronic Equipment (MIL-HDBK-217F).

Analysis of stationary availability factor of two-level backbone computer networks with arbitrary topology

This scientific paper deals with the two-level backbone computer networks with arbitrary topology. A specialized method, offered by the author for calculation of the stationary availability factor of the two-level backbone computer networks, based on the Markov reliability models for the set of the independent repairable elements with the given failure and repair rates and the methods of the discrete mathematics, is also discussed. A specialized algorithm, offered by the author for analysis of the network connectivity, taking into account different kinds of the network equipment failures, is also observed. Finally, this paper presents an example of calculation of the stationary availability factor for the backbone computer network with the given topology.

Design, development, and reliability assessment of dual output converters for SPV based DC nanogrid

A DC nanogrid for residential and commercial purposes supplies both AC and DC output voltages at different utilization levels to meet the load requirements. In this paper, the author(s) have developed a Solar Photovoltaic based DC nanogrid using dual output converter configurations which aims to improve both reliability and efficiency. The converter configurations are developed and analyzed for different levels of DC and AC output voltages in a nanogrid. Further, the performance characterization models of these converters such as sensitivity and reliability models are developed to test the robustness and effects when there is parametric variation. Markov reliability models are developed to estimate the mean time to system failure, as assessed in the Military Handbook for Reliability Prediction of Electronic Equipment (MIL-HDBK-217F). Also, the developed converter configurations are investigated in detail using MATLAB along with Simulink toolbox. Finally, the converter configurations are experimentally validated using a 100 W prototype, built, and tested in the laboratory for practical applications. The prototype model is a basic building block for further study and practical implementation for Power System designers and is useful in the areas where there is no grid. Also, the developed dual output-based system has improved energy security and reliability.

A Set-Theoretic Method for Parametric Uncertainty Analysis in Markov Reliability and Reward Models

This paper proposes a set-theoretic method to capture the effect of parametric uncertainty in reliability and performability indices obtained from Markov reliability and reward models. We assume that model parameters, i.e., component failure and repair rates, are not perfectly known, except for upper and lower bounds obtained from engineering judgment or field data. Thus, the values that these parameters can take are constrained to lie within a set. In our method, we first construct a minimum volume ellipsoid that upper bounds this set, and hence contains all possible values that the parameters can take. This ellipsoid is then propagated via set operations through a second-order Taylor series expansion of the Markov chain stationary distribution, resulting in a set that provides approximate bounds on reliability and performability indices of interest. Case studies pertaining to a two-component shared load system with common-cause failures, and preventative maintenance of an electric-power distribution transformer, are presented.

A Parametric Uncertainty Analysis Method for Markov Reliability and Reward Models

A common concern with Markov reliability and reward models is that model parameters, i.e., component failure and repair rates, are seldom perfectly known. This paper proposes a numerical method based on the Taylor series expansion of the underlying Markov chain stationary distribution (associated to the reliability and reward models) to propagate parametric uncertainty to reliability and performability indices of interest. The Taylor series coefficients are expressed in closed form as functions of the Markov chain generator-matrix group inverse. Then, to compute the probability density functions of the reliability and performability indices, random variable transformations are applied to the polynomial approximations that result from the Taylor series expansion. Additionally, closed-form expressions that approximate the expectation and variance of the indices are also derived. A significant advantage of the proposed framework is that only the parametrized Markov chain generator matrix is required as an input, i.e., closed-form expressions for the reliability and performability indices as a function of the model parameters are not needed. Several case studies illustrate the accuracy of the proposed method in approximating distributions of reliability and performability indices. Additionally, analysis of a large model demonstrates lower execution times compared to Monte Carlo simulations.

A Unified Approach to Reliability Assessment of Multiphase DC–DC Converters in Photovoltaic Energy Conversion Systems

A systematic framework for reliability assessment and fault-tolerant design of multiphase dc-dc converters deployed in photovoltaic applications is presented. System-level steady-state models allow a detailed specification of component failure rates, and in turn establish the effects of ambient conditions and converter design on reliability. Markov reliability models are derived to estimate the mean time to system failure. Case studies applied to two- and three-phase, 250-W converters demonstrate that topological redundancy does not necessarily translate to improved reliability for all choices of switching frequency and capacitance. Capacitor voltage rating is found to be the dominant factor that affects system reliability.

From differential to difference importance measures for Markov reliability models

This paper presents the development of the differential importance measures (DIM), proposed recently for the use in risk-informed decision-making, in the context of Markov reliability models. The proposed DIM are essentially based on directional derivatives. They can be used to quantify the relative contribution of a component (or a group of components, a state or a group of states) of the system on the total variation of system performance provoked by the changes in system parameters values. The estimation of DIM at steady state using only a single sample path of a Markov process is also investigated. A numerical example of a dynamic system is finally introduced to illustrate the use of DIM, as well as the advantages of proposed evaluation approaches.

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.

Joint interval reliability for Markov systems with an application in transmission line reliability

We consider Markov reliability models whose finite state space is partitioned into the set of up states U and the set of down states D . Given a collection of k disjoint time intervals I ℓ = [ t ℓ , t ℓ + x ℓ ] , ℓ = 1 , … , k , the joint interval reliability is defined as the probability of the system being in U for all time instances in I 1 ∪ ⋯ ∪ I k . A closed form expression is derived here for the joint interval reliability for this class of models. The result is applied to power transmission lines in a two-state fluctuating environment. We use the L inux versions of the free packages M axima and S cilab in our implementation for symbolic and numerical work, respectively.

Approximate sensitivity analysis for acyclic Markov reliability models

Acyclic Markov chains are frequently used for reliability analysis of nonmaintained mission-critical computer-based systems. Since traditional sensitivity (or importance) analysis using Markov chains can be computationally expensive, an approximate approach is presented which is easy to compute and which performs quite well in test cases. This approach is presented in terms of a Markov chain which is used for solving a dynamic fault-tree, but the approach applies to any acyclic Markov reliability model.

Computationally efficient and numerically stable reliability bounds for repairable fault-tolerant systems

The transient analysis of large continuous time Markov reliability models of repairable fault-tolerant systems is computationally expensive due to model stiffness. We develop and analyze a method to compute bounds for a measure defined on a particular, but quite wide class of continuous time Markov models, encompassing both exact and bounding continuous time Markov reliability models of fault-tolerant systems. The method is numerically stable and computes the bounds with well-controlled and specifiable-in-advance error. Computational effort can be traded off with bounds accuracy. For a class of continuous time Markov models, class C, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, the method can yield reasonably tight bounds at a very small computational cost. The method builds upon a recently proposed numerical method for the transient analysis of continuous time Markov models called regenerative randomization.

Splitting-based importance-sampling algorithm for fast simulation of Markov reliability models with general repair-policies

Markov chains with small transition probabilities occur while modeling the reliability of systems where the individual components are highly reliable and quickly repairable. Complex inter-component dependencies can exist and the state space involved can be huge, making these models analytically and numerically intractable. Naive simulation is also difficult because the event of interest (system failure) is rare, so that a prohibitively large amount of computation is needed to obtain samples of these events. An earlier paper (Juneja et al., 2001) proposed an importance sampling scheme that provides large efficiency increases over naive simulation for a very general class of models including reliability models with general repair policies such as deferred and group repairs. However, there is a statistical penalty associated with this scheme when the corresponding Markov chain has high probability cycles as may be the case with reliability models with general repair policies. This paper develops a splitting-based importance-sampling technique that avoids this statistical penalty by splitting paths at high probability cycles and thus achieves bounded relative-error in a stronger sense than in previous attempts.

Neural network realization of Markov reliability and fault-tolerance models

A feed-forward recursive neural network consisting of 2n neurons forming 2 layers, n neurons in each layer, is set to represent a discrete-time n-state Markov model of a fault-tolerant hardware. A quadratic energy function for the neural net is presented and the appropriate update equations for the weights are derived using the least mean square gradient-descent technique.

Markov reliability models for digital flight control systems

The reliability of digital flight control systems can often be accurately predicted using Markov chain models. The cost of numerical solution depends on a model's size and stiffness. Acyclic Markov models, a useful special case, are particularly amenable to efficient numerical solution. Even in the general case, instantaneous coverage approximation allows the reduction of some cyclic models to more readily solvable acyclic models. After considering the solution of single-phase models, the discussion is extended to phased-mission models. Phased-mission reliability models are classified based on the state restoration behavior that occurs between mission phases. As an economical approach for the solution of such models, the mean failure rate solution method is introduced. A numerical example is used to show the influence of fault-model parameters and interphase behavior on system unreliability.

An abstract language for specifying Markov reliability models : Ricky W. Butler. IEEE Trans. Reliab.R-35 (5), 595 (1986)

An abstract language for specifying Markov reliability models : Ricky W. Butler. IEEE Trans. Reliab.R-35 (5), 595 (1986)

Markov reliability models of fault-tolerant distributed computing systems

A hierachical view of fault-tolerant distributed computers is presented, viewing a distributed computing system as composed of interconnected, interacting, functional modules. Each module, modeled by a directed-state graph, is governed by internal random failure events and counteracting recovery processes, and also by coupling of external random events from other modules. It is shown that, under certain assumptions, the system is governed by a multidimensional Markov process, with non-Markov module processes as components. Mathematical properties of this model are formally analyzed. Performance measures are found from the steady-state distribution and visitation rate of each system and module state. A numerical example is presented exemplifying its practical application. The results are shown to fit very well the actual statistical data collected on an AT&T Bell Laboratories Electronic Switching System.

The robustness of markov reliability models

AbstractMarkov models are an established part of current systems reliability and availability analysis. They are extensively used in various applications, including, in particular, electrical power supply systems. One of their advantages is that they considerably simplify availability evaluation so that the availability of very large and complex systems can be computed. It is generally assumed, with some justification, that the results obtained from such Markov reliability models are relatively robust.It has, however, been known for some time, that practical time to failure distributions are frequently non‐exponential, particular attention being given in much reliability work to the Weibull family. Morover, recently additional doubt has been case on the validity of the Markov approach, both because of the work of Professor Kline and others on the non‐exponentiality of practical repair time distribution, and because of the advantages to be obtained in terms of modelling visibility of the alternative simulation approach.In this paper we employ results on the ability of the k‐out‐of‐n systems to span the coherent set to investigate the robustness of Markov reliability models based upon a simulation investigation of coherent systems of up to 10 identical components. We treat the case where adequate repair facilities are available for all components. The effects upon the conventional transient and steady‐state measures of Weibull departures from exponentiality are considered. In general, the Markov models are found to be relatively robust, with alterations to failure distributions being more important than those to repair distributions, and decreasing hazard rates more critical than increasing hazard rates. Of the measures studied, the mean time to failure is most sensitive to variations in distributional shape.

Procedure for weighting failure rates in Markov reliability models for automation systems: P. Fries. Siemens Res. Devel. Rep.9 (5), 305 (1980)

Procedure for weighting failure rates in Markov reliability models for automation systems: P. Fries. Siemens Res. Devel. Rep.9 (5), 305 (1980)