Models Of Fault-tolerant Systems Research Articles

Математическое моделирование отказоустойчивости информационных систем

The article deals with the problem of reliability of information systems, in order to solve this problem, a method for assessing the fault tolerance of data transmission systems using the apparatus of mathematical modeling of faulttolerant systems with three levels of operation was introduced. Also the mathematical model in the form of a graph of states and the probability of failure with a system of differential equations for its description was formulated. The solution of mathematical model allows to obtain the probability of failure at different levels of functioning of information systems in order to detect and find faults and equipment failures.

Reliability Bounds for Fault-Tolerant Systems with Deferred Repair using Bounding Split Regenerative Randomization

A numerically stable method is developed which computes seemingly tight bounds at a small computational cost relative to the model size, when that model size is large, for the unreliability and bounds for the unreliability using, respectively, exact and bounding failure/repair continuous-time Markov chain models of fault-tolerant systems with exponential failure and repair time distributions, in which repair is deferred until some condition on the collection of failed components is satisfied, and, then, proceeds until reaching the state without failed components, with failure rates much smaller than repair rates and not too different output rates from states with deferred repair.

An Efficient and Numerically Stable Method for Computing Bounds for the Interval Availability Distribution

This paper is concerned with the computation of the interval availability (proportion of time in a time interval in which the system is up) distribution of a fault-tolerant system modeled by a finite (homogeneous) continuous-time Markov chain (CTMC). General-purpose methods for performing that computation tend to be very expensive when the CTMC and the time interval are large. Based on a previously available method (regenerative transformation) for computing the interval availability complementary distribution, we develop a method called bounding regenerative transformation for the computation of bounds for that measure. Similar to regenerative transformation, bounding regenerative transformation requires the selection of a regenerative state. The method is targeted at a certain class of models, including both exact and bounding failure/repair models of fault-tolerant systems with increasing structure function, with exponential failure and repair time distributions and repair in every state with failed components having failure rates much smaller than repair rates (F/R models), with a “natural” selection for the regenerative state. The method is numerically stable and computes the bounds with well-controlled error. For models in the targeted class and the natural selection for the regenerative state, computational cost should be traded off with bounds tightness through a control parameter. For large models in the class, the version of the method that should have the smallest computational cost should have small computational cost relative to the model size if the value above which the interval availability has to be guaranteed to be is close to 1. In addition, under additional conditions satisfied by F/R models, the bounds obtained with the natural selection for the regenerative state by the version that should have the smallest computational cost seem to be tight for all time intervals or not small time intervals, depending on whether the initial probability distribution of the CTMC is concentrated in the regenerative state or not. CORRECTED VERSION OF RECORD, SEE LAST PAGE OF ARTICLE

Transient analysis of Markov models of fault‐tolerant systems with deferred repair using split regenerative randomization

AbstractThe (standard) randomization method is an attractive alternative for the transient analysis of continuous time Markov models. The main advantages of the method are numerical stability, well‐controlled computation error, and ability to specify the computation error in advance. However, the fact that the method can be computationally very expensive limits its applicability. In this paper, we develop a new method called split regenerative randomization, which, having the same good properties as standard randomization, can be significantly more efficient. The method covers reliability‐like models with a particular but quite general structure and requires the selection of a subset of states and a regenerative state satisfying some conditions. For a class of continuous time Markov models, model class C2, including typical failure/repair reliability‐like models with exponential failure and repair time distributions and deferred repair, natural selections are available for both the subset of states and the regenerative state and, for those natural selections, theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. Those results can be used to anticipate when the method can be expected to be competitive. We illustrate the application of the method using a large class C2 model and show that for models in that class the method can indeed be significantly more efficient than previously available randomization‐based methods. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Two methods for computing bounds for the distribution of cumulative reward for large Markov models

Degradable fault-tolerant systems can be evaluated using rewarded continuous-time Markov chain (CTMC) models. In that context, a useful measure to consider is the distribution of the cumulative reward over a time interval [ 0 , t ] . All currently available numerical methods for computing that measure tend to be very expensive when the product of the maximum output rate of the CTMC model and t is large and, in that case, their application is limited to CTMC models of moderate size. In this paper, we develop two methods for computing bounds for the cumulative reward distribution of CTMC models with reward rates associated with states: BT/RT (Bounding Transformation/Regenerative Transformation) and BT/BRT (Bounding Transformation/Bounding Regenerative Transformation). The methods require the selection of a regenerative state, are numerically stable and compute the bounds with well-controlled error. For a class of rewarded CTMC models, class C 1 ‴ , and a particular, natural selection for the regenerative state the BT/BRT method allows us to trade off bound tightness with computational cost and will provide bounds at a moderate computational cost in many cases of interest. For a class of models, class C 1 ″ , slightly wider than class C 1 ‴ , and a particular, natural selection for the regenerative state, the BT/RT method will yield tighter bounds at a higher computational cost. Under additional conditions, the bounds obtained using the less expensive version of BT/BRT and BT/RT seem to be tight for any value of t or not small values of t , depending on the initial probability distribution of the model. Class C 1 ″ and class C 1 ‴ models with these additional conditions include both exact and bounding typical failure/repair performability models of fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components and a reward rate structure which is a non-increasing function of the collection of failed components. We illustrate both the applicability and the performance of the methods using a large CTMC performability example of a fault-tolerant multiprocessor system.

A Generalized Method for the Transient Analysis of Markov Models of Fault-Tolerant Systems with Deferred Repair

Abstract Randomization is an attractive alternative for the transient analysis of continuous time Markov models. The main advantages of the method are numerical stability, well-controlled computation error, and ability to specify the computation error in advance. However, the fact that the method can be computationally expensive limits its applicability. Recently, a variant of the (standard) randomization method, called split regenerative randomization has been proposed for the efficient analysis of reliability-like models of fault-tolerant systems with deferred repair. In this article, we generalize that method so that it covers more general reward measures: the expected transient reward rate and the expected averaged reward rate. The generalized method has the same good properties as the standard randomization method and, for large models and large values of the time t at which the measure has to be computed, can be significantly less expensive. The method requires the selection of a subset of states an...

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.

Almost Sure Fault Tolerance in Random Graphs

We analyze the PMC [10] model for fault tolerant systems by means of random directed graphs. Previous work has shown that the minimum in-degree of the testing digraph had to exceed the expected number of faulty units (vertices). We show, for much sparser digraphs than those described above, that the asymptotic probability of correct diagnosis of a faulty system tends to 1. Specifically we show that if our directed graph has n vertices, arc probability $p = {{(c\log n)} / n}$ and vertex failure probability $q < \frac{1}{2}$ then a simple algorithm will correctly diagnose the system “almost always” when ${{c > 1} / {(1 - q)}}$. However, when ${{c < 1} / {(1 - q)}}$ no algorithm can correctly diagnose the system. We present a similar analysis for a family of directed graphs which are not random. We conclude with analogous results for undirected graphs.

Computer-aided reliability analysis of fault-tolerant systems

We present an overview of the major problems inherent in reliability modeling of fault-tolerant systems. The problems faced while modeling such systems include the need to consider a very large state space, non-exponential distributions, error analysis, the need to perform a combined evaluation of performance and reliability, and the need to include the details of fault/error handling behavior. Some of the proposed solutions are discussed and current tools (HARP, SAVE, DEEP and SHARPE) to facilitate evaluation of such systems are described. References are provided to many of the important techniques utilized in reliability, availability, and performance modeling of such systems.

Analysis of Performability for Stochastic Models of Fault-Tolerant Systems

Performability, a composite measure for the performance and reliability, may be interpreted as the probability density function of the aggregate reward obtained from a system during its mission time. For large mission times we show that known limit theorems lead to an asymptotic normal distribution for the aggregate reward. For finite mission times and Markovian models we obtain the expressions for all moments of performability and give recursions to compute coefficients involved in the expressions. We illustrate the use of the results through an example of a multiple processor computer system.

An Abstract Language for Specifying Markov Reliability Models

In principle, Markov models can be used to describe the reliability of virtually any fault-tolerant system. However, the process of delineating all of the states and transitions in a model of a complex system can be devastatingly tedious and error-prone. This paper presents a new approach to this problem by using an abstract model-definition language. The language essentially defines a set of rules which are used to generate the Markov model automatically. These rules correspond to the basic concepts used to create models of fault-tolerant systems. A small number of statements in the language can be used to describe a very large model. A variation in the system (such as in the number of initial spares) can be accomplished by changing only one line in the model definition, although such a change represents a large increase in the size of the Markov model. This high-level language is described in a non-formal manner and illustrated by several examples. A computer program has been developed which translates the abstract language described in this paper into the input language for the SURE (Semi-Markov Unreliability Range Evaluator) program. The program has been named ASSIST (Abstract Semi-Markov Specification Interface to the SURE Tool). It is written in Pascal and runs on a VAX 11/750 in the NASA AIRLAB Facility.