Closed-Form Solution for System Availability Distribution

Highly reliable systems with long mission time, that can tolerate no down time, have motivated the study of system reliability. The emergence of fault-tolerant computing systems, where small down times may be tolerable, and preventive and corrective maintenance permitted, motivates a revisit to measures like mean availability. Vendors of computer systems are being required to specify the level of availability that will be met by their systems over a finite time interval, and pay a penalty for non-compliance. Since no closed-form solution has been reported in the literature, numerical approaches have often been used to compute systems availability over a finite time, even for simple Markov models. We report a Laplace transform solution for the distribution of availability over a finite interval, for a semi-Markov model. The transform of the distribution is analytically inverted to obtain a closed-form solution for the corresponding Markov model.     

Effect of Maintenance on the Dependability and Performance of Mulitprocessor Systems

This paper presents analytic models for dependability and performance evaluation of multiprocessor systems with both on-line and off-line maintenance. Markov models are developed to compute the system reliability and performance availability incorporating the reliability of the maintenance processor. The maintenance processor failure is considered separately in order to emphasize its effect on system performance and dependability. The reliability of the maintenance processor can not be ignored for degradable multiprocessors. Probabilistic models are presented to compute the system downtime and the service cost for three off-line maintenance policies: scheduled maintenance (SM), unscheduled maintenance (UM), and scheduled & unscheduled maintenance (SUM). The SUM policy, that combines both SM and UM, can be used to give a compromise between cost and downtime.     

Reliability Modeling Using SHARPE

Combinatorial models such as fault trees and reliability block diagrams are efficient for model specification and often efficient in their evaluation. But it is difficult, if not impossible, to allow for dependencies (such as repair dependency and near-coincident-fault type dependency), transient and intermittent faults, standby systems with warm spares, and so on. Markov models can capture such important system behavior, but the size of a Markov model can grow exponentially with the number of components in this system. This paper presents an approach for avoiding the large state space problem. The approach uses a hierarchical modeling technique for analyzing complex reliability models. It allows the flexibility of Markov models where necessary and retains the efficiency of combinatorial solution where possible. Based on this approach a computer program called SHARPE (Symbolic Hierarchical Automated Reliability and Performance Evaluator) has been written. The hierarchical modeling technique provides a very flexible mechanism for using decomposition and aggregation to model large systems; it allows for both combinatorial and Markov or semi-Markov submodels, and can analyze each model to produce a distribution function. The choice of the number of levels of models and the model types at each level is left up to the modeler. Component distribution functions can be any exponential polynomial whose range is between zero and one. Examples show how combinations of models can be used to evaluate the reliability and availability of large systems using SHARPE.     

Decomposition in Reliability Analysis of Fault-Tolerant Systems

Two important problems which arise in modeling fault-tolerant systems with ultra-high reliability requirements are discussed. 1) Any analytic model of such a system has a large number of states, making the solution computationally intractable. This leads to the need for decomposition techniques. 2) The common assumption of exponential holding times in the states is intolerable while modeling such systems. Approaches to solving this problem are reviewed. A major notion described in the attempt to deal with reliability models with a large number of states is that of behavioral decomposition followed by aggregation. Models of the fault-handling processes are either semi-Markov or simulative in nature, thus removing the usual restrictions of exponential holding times within the coverage model. The aggregate fault-occurrence model is a non-homogeneous Markov chain, thus allowing the times to failure to possess Weibull-like distributions. There are several potential sources of error in this approach to reliability modeling. The decomposition/aggregation process involves the error in estimating the transition parameters. The numerical integration involves discretization and round-off errors. Analysis of these errors and questions of sensitivity of the output (R(t)) to the inputs (failure rates and recovery model parameters) and to the initial system state acquire extreme importance when dealing with ultra-high reliability requirements.     

RAM analysis of a navigational system by Markov technique: A case study

An analysis is made for a typical repairable Dual-VHF Omni Range (Dual-VOR), for prediction of its reliability, availability and maintainability by adopting Markov technique. Dual-VOR is a ground based azimuth navigational system for aircraft. The failure rates of various functional blocks for the Dual-VOR have been derived from the MIL-HDBK. The repair rates have been based on maintenance practices and repair policies adopted for the system. Markov modelling for reliability involves solving a set of differential equations. Markov modelling for availability involves solving a set of linear simultaneous equations. A simple technique which involves only the availability model has been adopted for determining the mean time to repair (MTTR). RAM analysis of “hot-standby” configuration of Dual-VOR has been discussed.     

An approximation analysis for the availability of a parallel redundant system with general distributions

This paper presents an approximation method for deriving the availability of a parallel redundant system with general distributions. The system discussed is composed of two identical units. A single service facility is available for the performance of preventive maintenance(PM) and repair. The failure times, repair times and PM times are assumed to be arbitrarily distributed. The presented method formulates the problem of the availability analysis of a parallel redundant system as a semi-Markov process which represents the state transitions of one specified unit in the system. This method derives the availability easily and accurately. Further, when all the distributions are exponential, the availability obtained by this method is exact.     

Semi-Markov models with an application to power-plant reliabilityanalysis

Systems with, (1) a finite number of states, and (2) random holding times in each state, are often modeled using semi-Markov processes. For general holding-time distributions, closed formulas for transition probabilities and average availability are usually not available. Recursion procedures are derived to approximate these quantities for arbitrarily distributed holding-times; these recursion procedures are then used to fit the semi-Markov model with Weibull distributed holding-times to actual power-plant operating data. The results are compared to the more familiar Markov models; the semi-Markov model using Weibull holding-times fits the data remarkably well. In particular comparing the transition probabilities shows that the probability of the system being in the state of refitting converges more quickly to its limiting value as compared to convergence in the Markov model. This could be because the distribution of the holding-times in this state is rather unlike the exponential distribution. The more flexible semi-Markov model with Weibull holding-times describes more accurately the operating characteristics of power-plants, and produces a better fit to the actual operating data     

Cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair

This paper deals with the cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair where each unit of the system has two modes, normal and failed. It is assumed that the failure, repair, replacement and inspection times are stochastically independent random variables each having an arbitrary distribution. The cold standby unit replaces the failed operative unit after a random amount of time. An inspection is required to decide whether it needs type I (minor repair) or type 2 (major repair). In this system the repairman is not always available with the system, but is called whenever the operative unit fails. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Pointwise availability, steady-state availability, busy period by a server and the expected cost per unit time of the system are obtained. Certain important results have been derived as particular cases.     

Failure correlation in software reliability models

Perhaps the most stringent restriction in most software reliability models is the assumption of statistical independence among successive software failures. The authors research was motivated by the fact that although there are practical situations in which this assumption could be easily violated, much of the published literature on software reliability modeling does not seriously address this issue. The research work in this paper is devoted to developing the software reliability modeling framework that can consider the phenomena of failure correlation and to study its effects on the software reliability measures. The important property of the developed Markov renewal modeling approach is its flexibility. It allows construction of the software reliability model in both discrete time and continuous time, and (depending on the goals) to base the analysis either on Markov chain theory or on renewal process theory. Thus, their modeling approach is an important step toward more consistent and realistic modeling of software reliability. It can be related to existing software reliability growth models. Many input-domain and time-domain models can be derived as special cases under the assumption of failure s-independence. This paper aims at showing that the classical software reliability theory can be extended to consider a sequence of possibly s-dependent software runs, viz, failure correlation. It does not deal with inference nor with predictions, per se. For the model to be fully specified and applied to estimations and predictions in real software development projects, we need to address many research issues, e.g., the detailed assumptions about the nature of the overall reliability growth, way modeling-parameters change as a result of the fault-removal attempts     

Reliability and availability analysis of transit systems

This paper presents three newly developed Markov models representing on-surface transit systems. Transit system reliability, steady-state availability, mean time to failure (MTTF) and variance of time to failure formulas are developed. Selective plots are shown for each model. These plots clearly exhibit the impact of various parameters on transit system reliability, steady-state availability, and MTTF.     

Time-varying failure rates in the availability and reliabilityanalysis of repairable systems

This paper combines time varying failure rates and Markov chain analysis to obtain a hybrid reliability and availability analysis. However, combining these techniques can, depending on the size of the system, result in solutions of the Markov chain differential matrix equations that are intractable. This paper identifies solutions that are tractable, These form the analytical baseline for the reliability and availability analysis of systems with time varying failure rates. Tractable solutions were found for the 1-component 2-state and the 2-component 4-state configurations. Time varying failure rates were characterized by a general polynomial expression. Constant, linear, and Weibull failure rate functions are special cases of this polynomial. The general polynomial failure rate provides flexibility in modeling the time varying failure rates that occur in practice     

Reliability analysis of a three-unit system in a changing environment

Consider a parallel redundant repairable system consisting of three identical units and one repair facility, which operates in a changing environment subject to a Markov process with two states. Using Markov renewal processes we obtain the system availability, failure frequency and reliability function.     

Reliability and availability analysis of on surface transit systems

This paper presents four Markov models pertaining to repairable and non-repairable on surface transit systems. The expressions for state probabilities, system reliability, mean time to failure (MTTF) and steady state availability are developed. System reliability, MTTF and steady state availability plots for various assumed values of system parameters are shown.     

Availability modeling

Availability, which for years has been a critical design parameter in the defense and aerospace industry, has now become a very important design parameter in the commercial world. System designers must evaluate the availability implications of various architectures along with cost, performance, and other measures. Availability models provide a way to predict and compare the availability of system architectures. Design engineers can use these models for performing front-end design tradeoffs, evaluating product modifications, and selecting maintenance strategies. This article describes some common methods used to evaluate the availability of various architectures, particularly Markov models. It illustrates the methods with a simple computer architecture evaluation and shows how single points of failure tend to drive system availability. Markov models are a very flexible and powerful tool for calculating system availability. They can be reduced to linear equations and solved with most commercial spreadsheets. As availability becomes more important in the commercial marketplace, Markov models are becoming a standard design engineering technique     

Maintenance strategy optimization using a continuous-state partially observable semi-Markov decision process

Due to the limitation of current condition monitoring technologies, the estimates of asset health states may contain some uncertainties. A maintenance strategy ignoring this uncertainty of asset health state can cause additional costs or downtime. The partially observable Markov decision process (POMDP) is a commonly used approach to derive optimal maintenance strategies when asset health inspections are imperfect. However, existing applications of the POMDP to maintenance decision-making largely adopt the discrete time and state assumptions. The discrete-time assumption requires the health state transitions and maintenance activities only happen at discrete epochs, which cannot model the failure time accurately and is not cost-effective. The discrete health state assumption, on the other hand, may not be elaborate enough to improve the effectiveness of maintenance. To address these limitations, this paper proposes a continuous state partially observable semi-Markov decision process (POSMDP). An algorithm that combines the Monte Carlo-based density projection method and the policy iteration is developed to solve the POSMDP. Different types of maintenance activities (i.e., inspections, replacement, and imperfect maintenance) are considered in this paper. The next maintenance action and the corresponding waiting durations are optimized jointly to minimize the long-run expected cost per unit time and availability. The result of simulation studies shows that the proposed maintenance optimization approach is more cost-effective than maintenance strategies derived by another two approximate methods, when regular inspection intervals are adopted. The simulation study also shows that the maintenance cost can be further reduced by developing maintenance strategies with state-dependent maintenance intervals using the POSMDP. In addition, during the simulation studies the proposed POSMDP shows the ability to adopt a cost-effective strategy structure when multiple types of maintenance activities are involved.     

Reliability Analysis of Systems Comprised of Units with Arbitrary Repair-Time Distributions

This work demonstrates the feasibility of reliability modeling of systems with repair capability using a semi-Markov process. A two-unit system with exponential failure times but general repair times is studied. Formulas for state-transition probabilities, waiting-time distribution functions, and mean time in each state are developed. These quantities are expressed in terms of the Laplace transform of repair time distribution functions. Once these quantities are known, mean time to system failure and system availability, as well as other system parameters, can be found using matrix manipulations. In addition, time-dependent results may be obtained. A numerical example varying the parameter in a repair-time law is presented. The formulas developed can be extended to larger systems with repair capability for only one unit at a time and exponential failure times.     

Inspection policy for two-unit parallel redundant system

This paper deals with a single-server two-unit parallel redundant system with non-negligible inspection time. We shall assume that a failure of a unit or the system failure is detected by inspection only. We consider two inspection policies and under each inspection policy the stationary availability is derived by applying Piecewise Markov Process. Optimum inspection schedule is discussed to maximize the stationary availability of the system. A numerical example is presented.     

Systems with two dual failure modes — A survey

A survey of the research done on reliability of systems exhibiting two dual modes of failure, which are often called open and short, is given. Particular attention is paid to books and papers published after the 1977 review by Dhillon [32]. The present survey discusses the different terminology and approaches that appear in the literature and covers results concerning mainly the following topics: the optimal structure of the system, the optimal allocation of units, Markov and semi-Markov models of systems with repairable units and investigation of the structures of particular importance — series-parallel, k-out-of-n:G and logic circuits.     

Reliability of k-out-of-n:G systems with imperfect fault-coverage

k-out-of-n:G systems are modeled to determine their reliability and availability. Markov models are obtained to examine the fault-tolerant operation of the system. From the Markov chains, reliability and availability measures are found as state probabilities. Recursive expressions for mean time-between-failures and mean time-to-failure are obtained for repairable systems, considering perfect and imperfect fault-coverage     

Two models for two-dissimilar-unit cold standby redundant system with partial failure and two types of repairs

This paper deals with the availability function and the mean time to the first failure for two models of a cold standby redundant system with two different types of repair. Each model consists of two dissimilar units. In the first model, the operative unit has two modes of operation, normal mode and partial failure mode. The standby unit has one mode of operation, normal mode. In the second model, each unit has three modes of operation, normal mode, failure mode and total failure mode. Both models are analyzed by the semi-Markov process technique, assuming that the failure time and repair time distributions are general and arbitrary. Some reliability measures of interest to system designers as well as operations managers have been obtained. Moreover, we give computer programs for calculating the MTSF for each model (see Appendix).