Andrew L. Reibman

Numerical transient analysis of Markov models

Abstract We consider the numerical evaluation of Markov model transient behavior. Our research is motivated primarily by computer system dependability modeling. Other application areas include finitecapacity queueing models, closed queueing networks and inventory models. We focus our attention on the general problem of finding the state probability vector of a large, continuous-time, discrete-state Markov chain. Two computational approaches are examined in detail: uniformization and numerical linear multistep methods for ordinary differential equation solution. In general, uniformization provides greater accuracy but deals poorly with stiffness. A special stable ordinary differential equation solver deals well with stiffness, but it provides increased accuracy only at much greater cost. Examples are presented to illustrate the behavior of the techniques discussed as a function of model size, model stiffness, increased accuracy requirements and mission time.

Markov and Markov reward model transient analysis: An overview of numerical approaches

Abstract The advent of fault-tolerant, distributed systems has led to increased interest in analytic techniques for the prediction of reliability, availability, and combined performance and reliability measures. Markov and Markov reward models are common tools for fault-tolerant system reliability prediction. In this paper, we first derive instantaneous and cumulative measures of Markov and Markov reward model behavior. We then compare the complexity of several competing algorithms for the computation of these measures. Better approaches for Markov model solution should lead to more effective techniques for fault-tolerant system modeling.

Sensitivity analysis of reliability and performability measures for multiprocessor systems

Traditional evaluation techniques for multiprocessor systems use Markov chains and Markov reward models to compute measures such as mean time to failure, reliability, performance, and performability. In this paper, we discuss the extension of Markov models to include parametric sensitivity analysis. Using such analysis, we can guide system optimization, identify parts of a system model sensitive to error, and find system reliability and performability bottlenecks. As an example we consider three models of a 16 processor. 16 memory system. A network provides communication between the processors and the memories. Two crossbar-network models and the Omega network are considered. For these models, we examine the sensitivity of the mean time to failure, unreliability, and performability to changes in component failure rates. We use the sensitivities to identify bottlenecks in the three system models.

Transient analysis of acyclic markov chains

Abstract Continuous-time Markov chains are commonly used insystem reliability modeling. In this paper, we discuss a method for automatically deriving transient solutions that are symbolic in t for acyclic Markov chains. Our method also includes parametric sensitivity analysis of the transient solution and several cumulative measures associated with Markov chain behavior. We include three examples, one to show the use of our method in evaluating approximate solution techniques, one showing parametric sensitivity analysis of a large Markov model, and one demonstrating the computation of cumulative measures for an acyclic Markov reward process.

Analysis of Stiff Markov Chains

Continuous-time Markov chains (CTMC) are widely used mathematical models. Reliability models, queueing networks, and inventory models all require transient solutions of CTMC. The cost of CTMC transient solution increases with size, stiffness, and mission time. To eliminate stiffness and reduce the cost of solution, approximation techniques have been proposed. In this paper, we describe a software package for the specification and solution of stiff CTMC. As an interface, we use a language for the description of Markov chains. The language also provides facilities for controlling the solution procedure. Both exact and approximate solution techniques are provided. To conclude the paper, we use several examples to show the use of our specification language and the utility of our approximation technique. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Markov Reliability Models for Digital Flight Control Systems

The reliability of digital flight control systems can often be accurately predicted! using Markov chain models. We begin our discussion of flight control system reliability models with definitions of key terms. We then construct a single-fault one-processor model based on the results of fault-injecti on experiments. To illustrate more complex system models, we consider several models of a triple modular redundant system. Once we have constructed some representative Markov reliability models, we discuss numerical techniques for their solution. The cost of numerical solution depends on a models 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, we extend our discussion to phased-mission models. We classify phased-mission reliability models based on the state restoration behavior that occurs between mission phases: As an economical approach for the solution of such models, we introduce the mean failure rate solution method. We use a numerical example to show the influence of fault-model parameters and interphase behavior on system unreliability.