Bounding steady-state availability models with group repair and phase type repair distributions

We propose an algorithm to obtain bounds for the steady-state availability using Markov models in which only a small portion of the state space is generated. The algorithm is applicable to models with group repair and phase type repair distributions and involves the solution of only four linear systems of the size of the generated state space, independently on the number of “return” states. Numerical examples are presented to illustrate the algorithm and compare it with a previous bounding algorithm.Postprint (published version

Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations

Computing the stationary distributions of a continuous-time Markov chain involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a manageable size. In this review, we first give a comprehensive theoretical account of the stationary distributions and their relation to the long-term behaviour of the Markov chain, which is readily accessible to non-experts and free of irreducibility assumptions made in standard texts. We then review truncation-based approximation schemes paying particular attention to their convergence and to the errors they introduce, and we illustrate their performance with an example of a stochastic reaction network of relevance in biology and chemistry. We conclude by elaborating on computational trade-offs associated with error control and some open questions

Comparing Markov Chains: Aggregation and Precedence Relations Applied to Sets of States, with Applications to Assemble-to-Order Systems

International audienceSolving Markov chains is, in general, difficult if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and provide bounds for a reward function of interest. We present a new bounding method for Markov chains inspired by Markov reward theory: Our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications of the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for an initial chain by analyzing a much smaller chain. We illustrate our method by using it to prove monotonicity results and bounds for assemble-to-order systems

Bound computation of dependability and performance measures

Theme 1 - Reseaux et systemes. Projet ModelSIGLEAvailable from INIST (FR), Document Supply Serviceunder shelf-number : 22588, issue : a.1997 n.1093 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Bound Computation of Dependability and Performance Measures

When evaluating quantitative measures of complex systems using Markov models, a major drawback is the size of the generated state space, due to the intrinsic combinatorics associated with these models. If the number of states is excessively large, the model may be simply untractable. Observe also that in some cases, infinite models are appropriate, in which case, except for particular structures, the numerical procedures are not applicable. Recently, methods have been proposed to deal with this problem. The idea is to derive bounds of the measures of interest, by replacing a large part of the state space by some compact information. This has been done mainly for asymptotic dependability measures, and the proposed methods are valid under specific and restrictive conditions. In this paper, we extend these techniques to more general cases. In particular, we show that our approach can also give tight bounds of performance measures. We also show how to handle, in some cases, infinite models. We illustrate the new method with some analytically untractable open queueing networks, as well as with dependability models that cannot be analyzed by previous proposed techniques